diff --git a/.gitattributes b/.gitattributes index 9fea288..0abcd53 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1,3 +1 @@ -expreduce/resources.go filter=lfs diff=lfs merge=lfs -text *.m linguist-language=Mathematica -expreduce/rubi_snapshot/rubi_resources.go filter=lfs diff=lfs merge=lfs -text diff --git a/expreduce/builtin.go b/expreduce/builtin.go index adac752..b32affa 100644 --- a/expreduce/builtin.go +++ b/expreduce/builtin.go @@ -1,4 +1,4 @@ -//go:generate go-bindata -pkg expreduce -o resources.go resources/... +//go:generate go-bindata -pkg expreduce -nocompress -nometadata -o resources.go resources/... package expreduce diff --git a/expreduce/builtin_rubi.go b/expreduce/builtin_rubi.go index 8286a39..7d078fb 100644 --- a/expreduce/builtin_rubi.go +++ b/expreduce/builtin_rubi.go @@ -1,5 +1,5 @@ //go:generate go run ../utils/gensnapshots/gensnapshots.go -rubi_snapshot_location=./rubi_snapshot/rubi_snapshot.expred -//go:generate go-bindata -pkg rubi_snapshot -o rubi_snapshot/rubi_resources.go -nocompress rubi_snapshot/rubi_snapshot.expred +//go:generate go-bindata -pkg rubi_snapshot -o rubi_snapshot/rubi_resources.go -nocompress -nometadata rubi_snapshot/rubi_snapshot.expred package expreduce diff --git a/expreduce/resources.go b/expreduce/resources.go index 9b8dbee..7e87fd0 100644 --- a/expreduce/resources.go +++ b/expreduce/resources.go @@ -1,3 +1,56073 @@ -version https://git-lfs.github.com/spec/v1 -oid sha256:28775e4c399c2a6f25749208d51100398e3cb6050e67894e828eba5c01a66e7d -size 1347198 +// Code generated for package expreduce by go-bindata DO NOT EDIT. (@generated) +// sources: +// resources/arithmetic.m +// resources/atoms.m +// resources/boolean.m +// resources/calculus.m +// resources/combinatorics.m +// resources/comparison.m +// resources/equationdata.m +// resources/expression.m +// resources/flowcontrol.m +// resources/functional.m +// resources/init.m +// resources/list.m +// resources/manip.m +// resources/matrix.m +// resources/numbertheory.m +// resources/pattern.m +// resources/plot.m +// resources/power.m +// resources/random.m +// resources/replacement.m +// resources/rubi/1.1.1 Linear binomial products.m +// resources/rubi/1.1.3 General binomial products.m +// resources/rubi/1.1.4 Improper binomial products.m +// resources/rubi/1.2.1 Quadratic trinomial products.m +// resources/rubi/1.2.2 Quartic trinomial products.m +// resources/rubi/1.2.3 General trinomial products.m +// resources/rubi/1.2.4 Improper trinomial products.m +// resources/rubi/1.3 Miscellaneous algebraic functions.m +// resources/rubi/2 Exponentials.m +// resources/rubi/3 Logarithms.m +// resources/rubi/4.1 Sine.m +// resources/rubi/4.2 Tangent.m +// resources/rubi/4.3 Secant.m +// resources/rubi/4.4 Miscellaneous trig functions.m +// resources/rubi/5 Inverse trig functions.m +// resources/rubi/6 Hyperbolic functions.m +// resources/rubi/7 Inverse hyperbolic functions.m +// resources/rubi/8 Special functions.m +// resources/rubi/9.1 Integrand simplification rules.m +// resources/rubi/9.2 Derivative integration rules.m +// resources/rubi/9.3 Piecewise linear functions.m +// resources/rubi/9.4 Miscellaneous integration rules.m +// resources/rubi/Integration Utility Functions.m +// resources/rubi/MakeRubiMxFile.m +// resources/rubi/README +// resources/rubi/Rubi.m +// resources/rubi/Rubi4.12.nb +// resources/rubi/ShowStep Routines.m +// resources/rubi.m +// resources/rubi_loader.m +// resources/simplify.m +// resources/solve.m +// resources/sort.m +// resources/specialsyms.m +// resources/stats.m +// resources/string.m +// resources/system.m +// resources/tests.m +// resources/time.m +// resources/trig.m +package expreduce + +import ( + "fmt" + "io/ioutil" + "os" + "path/filepath" + "strings" + "time" +) +type asset struct { + bytes []byte + info os.FileInfo +} + +type bindataFileInfo struct { + name string + size int64 + mode os.FileMode + modTime time.Time +} + +// Name return file name +func (fi bindataFileInfo) Name() string { + return fi.name +} + +// Size return file size +func (fi bindataFileInfo) Size() int64 { + return fi.size +} + +// Mode return file mode +func (fi bindataFileInfo) Mode() os.FileMode { + return fi.mode +} + +// Mode return file modify time +func (fi bindataFileInfo) ModTime() time.Time { + return fi.modTime +} + +// IsDir return file whether a directory +func (fi bindataFileInfo) IsDir() bool { + return fi.mode&os.ModeDir != 0 +} + +// Sys return file is sys mode +func (fi bindataFileInfo) Sys() interface{} { + return nil +} + +var _resourcesArithmeticM = []byte(`Plus::usage = "`+"`"+`(e1 + e2 + ...)`+"`"+` computes the sum of all expressions in the function."; +(*{"Verbatim[Plus][beg___, Optional[c1_?NumberQ]*a_, Optional[c2_?NumberQ]*a_, end___]", "beg+(c1+c2)*a+end"},*) + (*The world is not ready for this madness.*) +(*{"Verbatim[Plus][beg___, Verbatim[Times][Optional[c1_?NumberQ],a__], Verbatim[Times][Optional[c2_?NumberQ],a__], end___]", "beg+(c1+c2)*a+end"},*) +Attributes[Plus] = {Flat, Listable, NumericFunction, OneIdentity, Orderless, Protected}; +Tests`+"`"+`Plus = { + ESimpleExamples[ + ESameTest[2, 1 + 1], + EComment["If Reals are present, other Integers are demoted to Reals:"], + ESameTest[0., (5.2 - .2) - 5], + EComment["Plus automatically combines like terms:"], + ESameTest[a+6*b^2, a + b^2 + 5*b^2], + ESameTest[((5 * c^a) + (3 * d)), (a+b)-(a+b)+c-c+2*c^a+2*d+5*d+d-5*d+3*c^a], + ESameTest[-3 a b c d e f, 4*a*b*c*d*e*f + -7*a*b*c*d*e*f] + ], ETests[ + (*Test automatic expansion*) + EStringTest["(a + b)", "1*(a + b)"], + EStringTest["(1.*(a + b))", "1.*(a + b)"], + EStringTest["(2.*(a + b))", "2.*(a + b)"], + EStringTest["(a + b)", "(a + b)/1"], + EStringTest["(1.*(a + b))", "(a + b)/1."], + EStringTest["(2*(a + b))", "2*(a + b)"], + EStringTest["(a*(b + c))", "a*(b + c)"], + EStringTest["(-a - b)", "-1*(a + b)"], + EStringTest["(-a - b)", "-(a + b)"], + EStringTest["((-1.)*(a + b))", "-1.*(a + b)"], + EStringTest["(-a - b)", "(a + b)/-1"], + EStringTest["((-1.)*(a + b))", "(a + b)/-1."], + + (*Test that we do not delete all the addends*) + ESameTest[0., (5.2 - .2) - 5], + ESameTest[0, 0 + 0], + + (*Test empty Plus expressions*) + ESameTest[0, Plus[]], + + (*Test proper accumulation of Rationals*) + EStringTest["(47/6 + sym)", "Rational[5, 2] + Rational[7, 3] + 3 + sym"], + EStringTest["(17/6 + sym)", "Rational[5, -2] + Rational[7, 3] + 3 + sym"], + EStringTest["(-19/6 + sym)", "Rational[5, -2] + Rational[7, 3] - 3 + sym"], + EStringTest["(-47/6 + sym)", "Rational[5, -2] + Rational[-7, 3] - 3 + sym"], + + (*Test combining monomials of degree 1*) + ESameTest[a+7*b, a + 2*b + 5*b], + + (*Test a more general version*) + ESameTest[a+7*b, a + 2*b + 5*b], + ESameTest[a+7*b^2, a + 2*b^2 + 5*b^2], + ESameTest[a+3*b^2, a - 2*b^2 + 5*b^2], + + (*Test using terms without a coefficient*) + ESameTest[a+6*b^2, a + b^2 + 5*b^2], + + (*Test additive identity*) + ESameTest[a, a+0], + ESameTest[a+b, (a+b)+0], + + (*Test additive inverse*) + ESameTest[0, a-a], + ESameTest[0, -a + a], + ESameTest[0, (a+b)-(a+b)], + ESameTest[0, -(a+b)+(a+b)], + ESameTest[0, (a+b)-(a+b)], + ESameTest[0, -(a+b)+(a+b)], + + (*Test basic simplifications*) + ESameTest[d, (a+b)-(a+b)+c-c+d], + ESameTest[((5 * c^a) + (3 * d)), (a+b)-(a+b)+c-c+2*c^a+2*d+5*d+d-5*d+3*c^a], + ESameTest[87.5 + 3 * x, ((x + 80. + 3. + x) + 2. + x + 2.5)], + ESameTest[87.5 + (7. * x), ((x + 80. + 3. + x) + 2. + (x * 2. * 2.) + (0. * 3. * x) + x + 2.5)], + ESameTest[50*a, a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a], + + (*More complicated term combining*) + ESameTest[-3 * m - 10 * n, -9 * n - n - 3 * m], + ESameTest[7*a * b - 2*a * c, 3*a*b - 2*a*c + 4*a*b], + ESameTest[-3*a - 2*b + 3*a*b, 2*a - 4*b + 3*a*b - 5*a + 2*b], + ESameTest[7*x - 11*y + x*y, 8*x - 9*y - 3*x*y - 2*y - x + 4*x*y], + ESameTest[-3*a*b*c*d*e*f, 4*a*b*c*d*e*f + -7*a*b*c*d*e*f], + ESameTest[-3*a*b*c*d*e*f, a*b*c*4*d*e*f + -a*b*c*d*e*f*7], + ESameTest[-3*a*b*c*d*e*f, a*b*2*c*2*d*e*f + -a*b*c*d*e*f*7], + ESameTest[2 r + 2 t, 2 r - 3 s - t + 3 t + 3 s], + ESameTest[3 (x - 2 y) - 4 x y + 2 (-1 + x y), 2 (x*y - 1) + 3 (x - 2 y) - 4 x*y], + ESameTest[-4 s + 4 r s - 3 (1 + r s), 4 r*s - 2 s - 3 (r*s + 1) - 2 s], + ESameTest[7 y - z + 3 y z, 8 y - 2 z - (y - z) + 3 y*z] + ] +}; + +Sum::usage = "`+"`"+`Sum[expr, n]`+"`"+` returns the sum of `+"`"+`n`+"`"+` copies of `+"`"+`expr`+"`"+`. + +`+"`"+`Sum[expr, {sym, n}]`+"`"+` returns the sum of `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = 1 to `+"`"+`n`+"`"+`. + +`+"`"+`Sum[expr, {sym, m, n}]`+"`"+` returns the sum of `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`."; +Sum[i_Symbol, {i_Symbol, 0, n_Integer}] := 1/2*n*(1 + n); +Sum[i_Symbol, {i_Symbol, 1, n_Integer}] := 1/2*n*(1 + n); +Sum[i_Symbol, {i_Symbol, end_}] := 1/2*end*(1 + end); +Sum[i_Symbol, {i_Symbol, 0, n_Symbol}] := 1/2*n*(1 + n); +Sum[i_Symbol, {i_Symbol, 1, n_Symbol}] := 1/2*n*(1 + n); +Attributes[Sum] = {HoldAll, ReadProtected, Protected}; +Tests`+"`"+`Sum = { + ESimpleExamples[ + ESameTest[45, Sum[i, {i, 5, 10}]], + ESameTest[55, Sum[i, {i, 1, 10}]], + ESameTest[55, Sum[i, {i, 0, 10}]], + ESameTest[450015000, Sum[i, {i, 1, 30000}]], + ESameTest[450015000, Sum[i, {i, 0, 30000}]], + ESameTest[1/2*n*(1 + n), Sum[i, {i, 0, n}]], + ESameTest[1/2*n*(1 + n), Sum[i, {i, 1, n}]], + ESameTest[30, Sum[a + b, {a, 0, 2}, {b, 0, 3}]], + ESameTest[b+c+d+e, Sum[a, {a, {b, c, d, e}}]], + ESameTest[b g + c g + d g + e g + b h + c h + d h + e h, Sum[a*f, {a, {b, c, d, e}}, {f, {g, h}}]], + ESameTest[25 n (1 + 50 n), Sum[i, {i, n*50}]], + ] +}; + +Times::usage = "`+"`"+`(e1 * e2 * ...)`+"`"+` computes the product of all expressions in the function."; +(* This is likely the most compute-intensive rule in the system. Modify with +care. *) +Verbatim[Times][beg___, a_^Optional[m_], a_^Optional[n_], end___] := beg*a^(m+n)*end /; (!NumberQ[a] || !NumberQ[m] || !NumberQ[n]); +(*Verbatim[Times][a_Integer, mid___, b_Integer^n_, end___] := mid*-(b^(n+1))*end /; (a == -b);*) +a_Integer^c_Rational*b_Integer^c_Rational*rest___ := (a*b)^c*rest; +(*Verbatim[Times][beg___, a_^Optional[m_], a_^Optional[n_], end___] := beg*a^(m+n)*end;*) +(* This qualifier is so that the simplification for 3^(4/3) -> 3*3^(1/3) does not produce an infinite evaluation loop *) +wouldntBeLessThanNegOne[x_] := ((x - 1) < -1) =!= True; +Verbatim[Times][Rational[1, a_Integer], inner___, a_Integer^n_?wouldntBeLessThanNegOne, end___] := inner*a^(n-1)*end; +Times[den_Integer^-1, num_Integer, rest___] := Rational[num,den] * rest; +Times[ComplexInfinity, rest___] := ComplexInfinity; +a_Integer?Negative^b_Rational*c_Integer^d_Rational*rest___ := (-1)^b*rest /; (a == -c && b == -d); +Verbatim[Times][beg___, a_Integer^m_Rational, a_Integer^n_Rational, end___] := beg*a^(m+n)*end; +Times[c : (Rational[_Integer, d_Integer] | + Complex[_Integer, Rational[_Integer, d_Integer]]), + Power[a_Integer, Rational[1, r_Integer]], rest___] := + Times[c*a, a^(1/r - 1), rest] /; (Mod[d, a] === 0 && a > 1) +Sin[x_]*Cos[x_]^(-1)*rest___ := Tan[x]*rest; +Cos[x_]*Sin[x_]^(-1)*rest___ := Cot[x]*rest; +Attributes[Times] = {Flat, Listable, NumericFunction, OneIdentity, Orderless, Protected}; +Tests`+"`"+`Times = { + ESimpleExamples[ + EComment["Simplification rules apply automatically:"], + ESameTest[3/2, (3 + (x^2 * 0)) * 2^-1], + ESameTest[a^(2+c), a^2*a^c], + ESameTest[a/(b*c*d), a/b/c/d] + ], EFurtherExamples[ + EComment["Rational numbers are suppported (explicit rational declaration added for clarity):"], + ESameTest[-2/3, Rational[1, -2]*Rational[-2, 3]*-2], + EComment["The product of nothing is defined to be one:"], + ESameTest[1, Times[]] + ], ETests[ + (*Test that we do not delete all the multiplicands*) + ESameTest[1, 1*1], + ESameTest[1, 5*1/5*1], + + (*Test empty Times expressions*) + ESameTest[1, Times[]], + + (*Test fraction simplification*) + ESameTest[25, 50/2], + ESameTest[50, 100/2], + ESameTest[50, 1/2*100], + ESameTest[5/4, 1/2*5/2], + ESameTest[1/4, 1/2*1/2], + ESameTest[a/(b*c*d), a/b/c/d], + + (*Test Rational detection*) + EStringTest["10", "40/2^2"], + EStringTest["10", "40/4"], + ESameTest[40/3, 40/3], + ESameTest[20/3, 40/6], + EStringTest["10", "1/4*40"], + EStringTest["10", "1/(2^2)*40"], + + (*Test proper accumulation of Rationals*) + EStringTest["(2*sym)", "sym*Rational[1,2]*Rational[2,3]*6"], + ESameTest[-2/3, Rational[1, -2]*Rational[-2, 3]*-2], + EStringTest["Rational", "Rational[1, -2]*Rational[-2, 3]*-2 // Head"], + + (*Test multiplicative identity*) + EStringTest["5", "5*1"], + EStringTest["a", "1*a"], + EStringTest["(1.*a)", "1.*a"], + + (*Test multiplicative inverse*) + EStringTest["1", "8*1/8"], + EStringTest["1", "a*1/a"], + EStringTest["1", "1/a*a"], + + (*Test multiplicative property of zero*) + ESameTest[3/2, (3 + (x^2 * 0)) * 2^-1], + + (*Simplifications with Power*) + ESameTest[a^(2+c), a^2*a^c], + ESameTest[a^(2-c), a^2/a^c], + ESameTest[m^2, m*m], + ESameTest[1, m/m], + ESameTest[1, m^2/m^2], + ESameTest[Times[Power[2,Rational[-1,2]],a], (1/2)*a*2^(1/2)], + + (*Conversion of exact numeric functions to reals*) + ESameTest[True, MatchQ[Sqrt[2*Pi]*.1, _Real]], + + ESameTest[(-1)^(1/3) a b c, (-2)^(1/3)*2^(-1/3)*a*b*c], + ESameTest[1/12, (1/12)*2^(-2/3)*2^(2/3)], + ESameTest[I/(4 Sqrt[3]), (0+I/12) Sqrt[3]], + (* 1/12*Sqrt[3]===1/12*3^(1/2)===12^-1*3^(1/2)===(2*2*3)^-1*3^(1/2)===(2*2)^-1*3^-1*3^(1/2)===(2*2)^-1*3^(1/2-1)===1/4*3^(-1/2)===1/(4 Sqrt[3]) *) + ESameTest[1/(4 Sqrt[3]), 1/12*Sqrt[3]], + ESameTest[-(I/(4 Sqrt[3])), (0-I/4) 1/Sqrt[3]], + ESameTest[-(1/(4 Sqrt[3])), Times[-1/12,Sqrt[3]]], + ESameTest[-(5/(4 Sqrt[3])), Times[-5/12,Sqrt[3]]], + ESameTest[(3+I/4)/Sqrt[3], Times[1+1/12 I,Sqrt[3]]], + ESameTest[(I Sqrt[3])/4, Times[3/12 I,Sqrt[3]]], + ESameTest[-(I/(2 Sqrt[3])), Times[-2/12 I,Sqrt[3]]], + ESameTest[-(I/(2 Sqrt[3])), Times[2/-12 I,Sqrt[3]]], + ESameTest[(3+(5 I)/4)/Sqrt[3], Times[1+5/12 I,Sqrt[3]]], + ESameTest[I/(2 Sqrt[3] a^2), (0+1/6*I)*3^(1/2)*a^(-2)], + (* Test wouldntBeLessThanNegOne. *) + ESameTest[(1/3)*3^(-1/2), (1/3)*3^(-1/2)], + + ESameTest[Sqrt[2/\[Pi]], Sqrt[2]*Sqrt[1/Pi]], + ESameTest[Sqrt[3/(2 \[Pi])], Sqrt[3/2]*Sqrt[1/Pi]], + ], EKnownFailures[ + ESameTest[-2^(1/3), (-2)*2^(-2/3)], + ESameTest[-2^(1+a), (-2)*2^(a)], + ] +}; + +Product::usage = "`+"`"+`Product[expr, n]`+"`"+` returns the product of `+"`"+`n`+"`"+` copies of `+"`"+`expr`+"`"+`. + +`+"`"+`Product[expr, {sym, n}]`+"`"+` returns the product of `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = 1 to `+"`"+`n`+"`"+`. + +`+"`"+`Product[expr, {sym, m, n}]`+"`"+` returns the product of `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`."; +Attributes[Product] = {HoldAll, ReadProtected, Protected}; +Tests`+"`"+`Product = { + ESimpleExamples[ + ESameTest[120, Product[a, {a, 1, 5}]], + ESameTest[f[1] * f[2] * f[3] * f[4] * f[5], Product[f[a], {a, 1, 5}]], + ESameTest[576, Product[a^2, {a, 4}]], + ESameTest[1440, Product[a + b, {a, 1, 2}, {b, 1, 3}]] + ] +}; + +Abs::usage = "`+"`"+`Abs[expr]`+"`"+` calculates the absolute value of `+"`"+`expr`+"`"+`."; +Abs[a_?NumberQ] := If[a<0,-a,a]; +Abs[Infinity] := Infinity; +Abs[ComplexInfinity] := Infinity; +Abs[-a_] := Abs[a]; +Abs[a_?((!FreeQ[#,I|_Complex])&)] := Sqrt[Total[ReIm[a]^2]] ;/ (FreeQ[ReIm[a], Re] && FreeQ[ReIm[a], Im]); +Attributes[Abs] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`Abs = { + ESimpleExamples[ + ESameTest[5, Abs[-5]], + ESameTest[5, Abs[5]], + EComment["Absolute values of unspecified inputs will be left unevaluated:"], + ESameTest[Abs[a], Abs[a]], + EComment["But sometimes simplifications can occur:"], + ESameTest[Abs[Sin[x]], Abs[-Sin[x]]] + ], ETests[ + ESameTest[True, Abs[-5.2] > 0], + ESameTest[0, Abs[0]], + ESameTest[Abs[x^a], Abs[-x^a]], + ESameTest[Abs[x^(a + b)], Abs[-x^(a + b)]], + ESameTest[1/2, Abs[1/2 E^(\[ImaginaryJ]*\[Pi]/4)]], + ] +}; + +Divide::usage = "`+"`"+`Divide[a, b]`+"`"+` computes `+"`"+`a/b`+"`"+`."; +Divide[a_,b_] := a/b; +Attributes[Divide] = {Listable, NumericFunction, Protected}; +Tests`+"`"+`Divide = { + ESimpleExamples[ + ESameTest[2, Divide[10, 5]] + ] +}; + +Increment::usage = "`+"`"+`Increment[a]`+"`"+` adds 1 to `+"`"+`a`+"`"+` and returns the original value."; +Increment[a_] := (a = a + 1; a - 1); +Attributes[Increment] = {HoldFirst, Protected, ReadProtected}; +Tests`+"`"+`Increment = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[3, toModify++], + ESameTest[4, toModify] + ] +}; + +Decrement::usage = "`+"`"+`Decrement[a]`+"`"+` subtracts 1 from `+"`"+`a`+"`"+` and returns the original value."; +Decrement[a_] := (a = a - 1; a + 1); +Attributes[Decrement] = {HoldFirst, Protected, ReadProtected}; +Tests`+"`"+`Decrement = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[3, toModify--], + ESameTest[2, toModify] + ] +}; + +PreIncrement::usage = "`+"`"+`PreIncrement[a]`+"`"+` adds 1 to `+"`"+`a`+"`"+` and returns the new value."; +PreIncrement[a_] := (a = a + 1); +Attributes[PreIncrement] = {HoldFirst, Protected, ReadProtected}; +Tests`+"`"+`PreIncrement = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[4, ++toModify], + ESameTest[4, toModify] + ] +}; + +PreDecrement::usage = "`+"`"+`PreDecrement[a]`+"`"+` subtracts 1 from `+"`"+`a`+"`"+` and returns the new value."; +PreDecrement[a_] := (a = a - 1); +Attributes[PreDecrement] = {HoldFirst, Protected, ReadProtected}; +Tests`+"`"+`PreDecrement = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[2, --toModify], + ESameTest[2, toModify] + ] +}; + +AddTo::usage = "`+"`"+`AddTo[a, b]`+"`"+` adds `+"`"+`b`+"`"+` to `+"`"+`a`+"`"+` and returns the new value."; +AddTo[a_, b_] := (a = a + b); +Attributes[AddTo] = {HoldFirst, Protected}; +Tests`+"`"+`AddTo = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[5, toModify += 2], + ESameTest[5, toModify] + ] +}; + +SubtractFrom::usage = "`+"`"+`SubtractFrom[a, b]`+"`"+` subtracts `+"`"+`b`+"`"+` from `+"`"+`a`+"`"+` and returns the new value."; +SubtractFrom[a_, b_] := (a = a - b); +Attributes[SubtractFrom] = {HoldFirst, Protected}; +Tests`+"`"+`SubtractFrom = { + ESimpleExamples[ + ESameTest[3, toModify = 3], + ESameTest[1, toModify -= 2], + ESameTest[1, toModify] + ] +}; +`) + +func resourcesArithmeticMBytes() ([]byte, error) { + return _resourcesArithmeticM, nil +} + +func resourcesArithmeticM() (*asset, error) { + bytes, err := resourcesArithmeticMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/arithmetic.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesAtomsM = []byte(`Rational::usage = "`+"`"+`Rational`+"`"+` is the head for the atomic rational type."; +Attributes[Rational] = {Protected}; +Tests`+"`"+`Rational = { + ESimpleExamples[ + EComment["Rationals are created from `+"`"+`Times`+"`"+` when a rational form is encountered:"], + ESameTest[Rational, Times[5, 6^-1] // Head], + EComment["Which is equivalent to typing them in directly:"], + ESameTest[Rational, 5/6 // Head], + EComment["Or being even more explicit:"], + ESameTest[Rational, Rational[5, 6] // Head], + EComment["Rationals simplify on evaluation:"], + ESameTest[5/3, Rational[10, 6]], + EComment["Which might include evaluating to an Integer:"], + ESameTest[Integer, Rational[-100, 10] // Head], + EComment["Rationals of non-Integer types are not allowed:"], + EStringTest["Rational[0, n]", "Rational[0, n]"] + ], EFurtherExamples[ + EComment["Undefined rationals are handled accordingly:"], + EStringTest["Indeterminate", "Rational[0, 0]"], + EStringTest["ComplexInfinity", "Rational[1, 0]"], + EComment["Rational numbers have some special handling for pattern matching:"], + ESameTest[2/3, test = Rational[2, 3]], + ESameTest[True, MatchQ[test, 2/3]], + ESameTest[True, MatchQ[test, Rational[a_Integer, b_Integer]]], + ESameTest[{2, 3}, 2/3 /. Rational[a_Integer, b_Integer] -> {a, b}], + ESameTest[2/3, 2/3 /. a_Integer/b_Integer -> {a, b}] + ], ETests[ + ESameTest[10/7, Rational[10, 7]], + EStringTest["Rational[x, 10]", "Rational[x, 10]"], + EStringTest["10", "Rational[100, 10]"], + ESameTest[-10, Rational[-100, 10]], + EStringTest["10", "Rational[-100, -10]"], + ESameTest[-5/3, Rational[-10, 6]], + ESameTest[5/3, Rational[-10, -6]], + EStringTest["0", "Rational[0, 5]"], + EStringTest["Rational[0, n]", "Rational[0, n]"], + EStringTest["ComplexInfinity", "Rational[-1, 0]"], + EStringTest["ComplexInfinity", "Rational[-1, -0]"], + EStringTest["Indeterminate", "Rational[-0, -0]"], + EStringTest["Indeterminate", "Rational[-0, 0]"], + ESameTest[buzz[bar], foo[bar, 1/2] /. foo[base_, 1/2] -> buzz[base]], + ESameTest[buzz[bar], foo[bar, 1/2] /. foo[base_, Rational[1, 2]] -> buzz[base]], + ESameTest[buzz[bar], foo[bar, Rational[1, 2]] /. foo[base_, 1/2] -> buzz[base]], + ESameTest[buzz[bar], foo[bar, Rational[1, 2]] /. foo[base_, Rational[1, 2]] -> buzz[base]], + ESameTest[True, MatchQ[1/2, Rational[1, 2]]], + ESameTest[True, MatchQ[Rational[1, 2], 1/2]], + ESameTest[False, Hold[Rational[1, 2]] === Hold[1/2]] + ] +}; + +Complex::usage = "`+"`"+`Complex`+"`"+` is the head for the atomic rational type."; +(a : (_Integer|_Real|_Rational)) * Complex[real_, im_] * rest___ := Complex[a * real, a * im] * rest; +(a : (_Integer|_Real|_Rational)) + Complex[real_, im_] + rest___ := Complex[a + real, im] + rest; +Complex[x_,y_] + Complex[u_,v_] + rest___ := Complex[x+u,y+v] + rest; +Complex[x_,y_] * Complex[u_,v_] * rest___ := Complex[x*u-y*v,x*v+y*u] * rest; +Attributes[Complex] = {Protected}; +Tests`+"`"+`Complex = { + ESimpleExamples[ + ESameTest[Complex[-16, 28], (4 + 8I)(2 + 3I)] + ] +}; + +String::usage = "`+"`"+`String`+"`"+` is the head for the atomic string type."; +Attributes[String] = {Protected}; +Tests`+"`"+`String = { + ESimpleExamples[ + ESameTest["Hello", "Hello"], + ESameTest[True, "Hello" == "Hello"], + ESameTest[False, "Hello" == "Hello world"], + ESameTest[String, Head["Hello"]] + ] +}; + +Real::usage = "`+"`"+`Real`+"`"+` is the head for the atomic floating point type."; +Attributes[Real] = {Protected}; +Tests`+"`"+`Real = { + ESimpleExamples[ + ESameTest[Real, Head[1.53]], + EComment["One can force Real interperetation on an Integer by appending a decimal point:"], + ESameTest[Real, Head[1.]], + EComment["Real numbers are backed by arbitrary-precision floating points:"], + EStringTest["10.", "10.^5000 / 10.^4999"] + ], EFurtherExamples[ + ESameTest[True, MatchQ[1.53, _Real]] + ] +}; + +Integer::usage = "`+"`"+`Integer`+"`"+` is the head for the atomic integer type."; +Attributes[Integer] = {Protected}; +Tests`+"`"+`Integer = { + ESimpleExamples[ + ESameTest[Integer, Head[153]], + EComment["Integer numbers are backed by arbitrary-precision data structures:"], + ESameTest[815915283247897734345611269596115894272000000000, Factorial[40]] + ], EFurtherExamples[ + ESameTest[True, MatchQ[153, _Integer]] + ] +}; + +IntegerQ::usage = "`+"`"+`IntegerQ[e]`+"`"+` returns True if `+"`"+`e`+"`"+` is an Integer, False otherwise."; +IntegerQ[e_] := Head[e] === Integer; +Attributes[IntegerQ] = {Protected}; +Tests`+"`"+`IntegerQ = { + ESimpleExamples[ + ESameTest[False, IntegerQ[a]], + ESameTest[True, IntegerQ[1]], + ESameTest[False, IntegerQ[2.]] + ] +}; + +realNumberQ[x_Integer] := True; +realNumberQ[x_Real] := True; +realNumberQ[x_Rational] := True; +realNumberQ[(b_Integer?Positive)^Rational[-1, n_Integer?Positive]] := True; +realNumberQ[(b_Integer?Positive)^Rational[1, 2]] := True; +realNumberQ[x_] := Which[ + x === Pi, True, + True, False +]; + +Im::usage = "`+"`"+`Im[e]`+"`"+` finds the imaginary part of `+"`"+`e`+"`"+`."; +Im[x_?realNumberQ] := 0; +Im[a_Integer * x_Integer?Positive^y_Rational] := 0; +Im[Complex[_,im_]] := im; +Im[x_?realNumberQ + rest__] := Im[Plus[rest]]; +Im[x_?realNumberQ * rest__] := x * Im[Times[rest]]; +Im[I * rest__] := Re[Times[rest]]; +Im[Complex[a_, b_] * rest__] := Im[a rest] + Re[b rest]; +Im[Complex[a_, b_] * c__ + d__] := Im[a c + d] + Re[b c]; +Im[E^(x_?NumericQ)] := E^Re[x] Sin[Im[x]]; +Attributes[Im] = {Listable, NumericFunction, Protected}; +Tests`+"`"+`Im = { + ESimpleExamples[ + ESameTest[0, Im[1]], + ESameTest[0, Im[0.5]], + ESameTest[0, Im[2/3]], + ESameTest[1, Im[2 + I]], + ESameTest[1/(2 Sqrt[2]), Im[1/2 E^(I*\[Pi]/4)]], + ESameTest[-1/Sqrt[2], Im[(-I)/Sqrt[2]]], + ESameTest[Re[a*b], Im[a*I*b]], + ESameTest[-(1/(2 Sqrt[2])), Im[-(I/Sqrt[2])+1/2 E^((I \[Pi])/4)]], + ], EKnownFailures[ + ESameTest[Im[a c]+Re[b c], Im[a c + I b c]], + ] +}; + +Re::usage = "`+"`"+`Re[e]`+"`"+` finds the real part of `+"`"+`e`+"`"+`."; +Re[x_?realNumberQ] := x; +Re[Complex[re_,_]] := re; +Re[Complex[0, 1] + rest__] := Re[Plus[rest]]; +Re[x_?realNumberQ + rest__] := x + Re[Plus[rest]]; +Re[x_?realNumberQ * rest__] := x * Re[Times[rest]]; +Re[Complex[0, 1] * rest__] := -Im[Times[rest]]; +Re[E^(x_?NumericQ)] := E^Re[x] Cos[Im[x]]; +Attributes[Re] = {Listable, NumericFunction, Protected}; +Tests`+"`"+`Re = { + ESimpleExamples[ + ESameTest[1, Re[1]], + ESameTest[0.5, Re[0.5]], + ESameTest[2/3, Re[2/3]], + ESameTest[2, Re[2 + I]], + ESameTest[1/(2 Sqrt[2]), Re[1/2 E^(I*\[Pi]/4)]], + ESameTest[1 + Re[foo], Re[foo+1]], + ESameTest[1 + Re[foo+bar], Re[foo+1+bar]], + ESameTest[Re[foo], Re[foo+I]], + ESameTest[Re[foo+bar], Re[foo+I+bar]], + ESameTest[Re[a]/2, Re[a/2]], + ] +}; + +ReIm[x_] := {Re[x], Im[x]}; +`) + +func resourcesAtomsMBytes() ([]byte, error) { + return _resourcesAtomsM, nil +} + +func resourcesAtomsM() (*asset, error) { + bytes, err := resourcesAtomsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/atoms.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesBooleanM = []byte(`And::usage = "`+"`"+`e1 && e2 && ...`+"`"+` returns `+"`"+`True`+"`"+` if all expressions evaluate to `+"`"+`True`+"`"+`."; +Attributes[And] = {Flat, HoldAll, OneIdentity, Protected}; +Tests`+"`"+`And = { + ESimpleExamples[ + ESameTest[False, True && False], + ESameTest[True, True && True && True] + ], ETests[ + ESameTest[True, And[]], + ESameTest[1, 1 && True && True], + ESameTest[False, True && False], + ESameTest[False, False && True], + ESameTest[True, True && True], + ESameTest[False, False && 1], + ESameTest[False, 1 && False], + ESameTest[1 && 1, 1 && 1], + ESameTest[1 && 1 && kfdkkfd, 1 && 1 && kfdkkfd], + ESameTest[1 && 1 && kfdkkfd, 1 && 1 && True && kfdkkfd], + ESameTest[False, 1 && 1 && True && False && kfdkkfd] + ] +}; + +Or::usage = "`+"`"+`e1 || e2 || ...`+"`"+` returns `+"`"+`True`+"`"+` if any expressions evaluate to `+"`"+`True`+"`"+`."; +Attributes[Or] = {Flat, HoldAll, OneIdentity, Protected}; +Tests`+"`"+`Or = { + ESimpleExamples[ + ESameTest[True, True || False], + ESameTest[False, False || False || False] + ], ETests[ + ESameTest[a || b, a || b], + ESameTest[True, a || True || b], + ESameTest[True, a || True || False], + ESameTest[a || b, a || b || False], + ESameTest[a || b, a || b || False || False], + ESameTest[a || b, a || False || b || False || False], + ESameTest[True, True || False], + ESameTest[False, False || False], + ESameTest[False, Or[False]], + ESameTest[False, Or[]] + ] +}; + +Not::usage = "`+"`"+`!e`+"`"+` returns `+"`"+`True`+"`"+` if `+"`"+`e`+"`"+` is `+"`"+`False`+"`"+` and `+"`"+`False`+"`"+` if `+"`"+`e`+"`"+` is `+"`"+`True`+"`"+`."; +!!e_ := e; +Attributes[Not] = {Protected}; +Tests`+"`"+`Not = { + ESimpleExamples[ + ESameTest[False, !True], + ESameTest[True, !False], + ESameTest[!a, !a], + ESameTest[a, !!a] + ] +}; + +TrueQ::usage = "`+"`"+`TrueQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is True, False otherwise."; +Attributes[TrueQ] = {Protected}; +Tests`+"`"+`TrueQ = { + ESimpleExamples[ + ESameTest[True, TrueQ[True]], + ESameTest[False, TrueQ[False]], + ESameTest[False, TrueQ[1]] + ] +}; + +BooleanQ::usage = "`+"`"+`BooleanQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is True or False, False otherwise."; +Attributes[BooleanQ] = {Protected}; +Tests`+"`"+`BooleanQ = { + ESimpleExamples[ + ESameTest[True, BooleanQ[True]], + ESameTest[True, BooleanQ[False]], + ESameTest[False, BooleanQ[1]] + ] +}; + +AllTrue::usage = "`+"`"+`AllTrue[list, condition]`+"`"+` returns True if all parts of `+"`"+`list`+"`"+` satisfy `+"`"+`condition`+"`"+`."; +AllTrue[_[elems___], cond_] := And @@ (cond /@ {elems}); +Attributes[AllTrue] = {Protected}; +Tests`+"`"+`AllTrue = { + ESimpleExamples[ + ESameTest[False, AllTrue[{1, a}, NumberQ]], + ESameTest[True, AllTrue[{1, 2}, NumberQ]] + ] +}; + +Boole::usage = "`+"`"+`Boole[e]`+"`"+` returns 0 if `+"`"+`e`+"`"+` is False and 1 if `+"`"+`e`+"`"+` is True."; +Boole[True] := 1; +Boole[False] := 0; +Attributes[Boole] = {Listable, Protected}; +Tests`+"`"+`Boole = { + ESimpleExamples[ + ESameTest[1, Boole[True]], + ESameTest[0, Boole[False]] + ], ETests[ + ESameTest[Boole[1], Boole[1]], + ESameTest[Boole[a], Boole[a]], + ESameTest[Boole[False,False], Boole[False, False]] + ] +}; +`) + +func resourcesBooleanMBytes() ([]byte, error) { + return _resourcesBooleanM, nil +} + +func resourcesBooleanM() (*asset, error) { + bytes, err := resourcesBooleanMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/boolean.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesCalculusM = []byte(`D::usage = "`+"`"+`D[f, x]`+"`"+` finds the partial derivative of `+"`"+`f`+"`"+` with respect to `+"`"+`x`+"`"+`."; +D[Indeterminate, x_] := Indeterminate; +D[x_,x_] := 1; +D[a_,x_] := 0 /; (FreeQ[a,x] && a =!= Indeterminate); +D[a_+b_,x_] := D[a,x]+D[b,x]; +D[a_ b_,x_] := D[a,x]*b + a*D[b,x]; +(*Chain rule*) +(*Prime notation would be nicer here*) +D[f_[g_[x_Symbol]], x_Symbol] := (Evaluate[D[f[#], #]] &)[g[x]]*D[g[x],x]; +(*The times operator is needed here. Whitespace precedence is messed up*) +D[a_^(b_), x_] := a^b*(D[b,x] Log[a]+D[a,x]/a*b); +D[Abs[a_], x_] := If[FreeQ[a, x], 0, Derivative[1][Abs][x]]; +D[Log[a_], x_] := D[a, x]/a; +D[Sin[a_], x_] := D[a,x] Cos[a]; +D[Cos[a_], x_] := -D[a,x] Sin[a]; +D[Exp[x_Symbol], x_Symbol] := Exp[x]; +D[l_List, x_] := Table[D[l[[idx]], x], {idx, Length[l]}]; +Attributes[D] = {ReadProtected, Protected}; +Tests`+"`"+`D = { + ESimpleExamples[ + ESameTest[Sqrt[x] + x^(3/2), D[2/3*x^(3/2) + 2/5*x^(5/2), x]], + ESameTest[2/x, D[Log[5 x^2], x]], + ESameTest[-(Sin[Log[x]]/x), D[Cos[Log[x]], x]] + ], ETests[ + ESameTest[1, D[x,x]], + ESameTest[1, D[foo,foo]], + ESameTest[0, D[foo,bar]], + ESameTest[2, D[bar+foo+bar,bar]], + ESameTest[2x, D[x^2,x]], + ESameTest[2x+3x^2, D[x^2+x^3,x]], + ESameTest[-4x^3, D[-x^4,x]], + ESameTest[-n x^(-1 - n) + n x^(-1 + n), D[x^n+x^(-n),x]], + ESameTest[4 x (1 + x^2), D[(x^2 + 1)^2, x]], + ESameTest[((1 + x + (1/6 * x^3) + (1/2 * x^2))), D[1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4, x]], + ESameTest[-10*Power[x,-3] - 7*Power[x,-2], D[1 + 7/x + 5/(x^2), x]], + ESameTest[-2 Sin[2 x], D[Cos[2 x], x]], + ESameTest[Cos[x]/x - Sin[x]*Power[x,-2], D[(Sin[x]*x^-1), x]], + ESameTest[-((2 Cos[x])*Power[x,-2]) + (2 Sin[x])*Power[x,-3] - Sin[x]/x, D[D[(Sin[x]*x^-1), x], x]], + ESameTest[-((2 Cos[x])*Power[x,-2]) + (2 Sin[x])*Power[x,-3] - Sin[x]/x, D[D[(Sin[x]*x^-1+Sin[y]), x], x]], + ESameTest[-Cos[Cos[x]] Sin[x], D[Sin[Cos[x]],x]], + ESameTest[Cos[Log[x]]/x, D[Sin[Log[x]],x]], + ESameTest[-(Sin[Log[x]]/x), D[Cos[Log[x]],x]], + ESameTest[1-(1+Cot[x]) Sin[x+Log[Sin[x]]], D[Cos[Log[Sin[x]]+x]+x,x]], + ESameTest[{a,b}, D[{a*x, b*x}, x]], + ] +}; + +Grad::usage = "`+"`"+`Grad[e, {var1, var2, ...}]`+"`"+` finds the gradient of `+"`"+`e`+"`"+` with respect to the named variables."; +Grad[e_,vars_List]:=Table[D[e,vars[[idx]]],{idx,Length[vars]}]; +Attributes[Grad] = {ReadProtected, Protected}; +Tests`+"`"+`Grad = { + ESimpleExamples[ + ESameTest[{-Sin[x+2 y],-2 Sin[x+2 y]}, Grad[Cos[x+2y],{x,y}]], + ] +}; + +findSubscripts[expr_] := Module[{subscripts = {}}, + Map[ + (If[MatchQ[#, Subscript[_, _]], + AppendTo[subscripts, #]; + ]; #) &, expr, {0, Infinity}]; + subscripts // DeleteDuplicates + ]; +genSubscriptReplacements[expr_] := + Module[{subscripts, uniques, n, fwd, bwd, tmpi}, + subscripts = findSubscripts[expr]; + n = Length[subscripts]; + uniques = Table[Unique[], {tmpi, 1, n}]; + fwd = Table[subscripts[[i]] -> uniques[[i]], {i, n}]; + bwd = Table[uniques[[i]] -> subscripts[[i]], {i, n}]; + {fwd, bwd} + ]; + +Integrate::usage = "`+"`"+`Integrate[f, x]`+"`"+` finds the indefinite integral of `+"`"+`f`+"`"+` with respect to `+"`"+`x`+"`"+`. + +!!! warning \"Under development\" + This function is under development, and as such will be incomplete and inaccurate."; +Integrate[a_,{x_Symbol,start_,end_}] := + (ReplaceAll[Integrate[a, x],x->end] - ReplaceAll[Integrate[a, x],x->start]) // Simplify; +Integrate[a_,x_Symbol] := Module[{cleanedA, replaceRules}, + If[!MemberQ[$ContextPath, "Rubi`+"`"+`"], + Print["Loading Rubi rules for integration. This happens once. Preload on startup with -preloadrubi."]; + LoadRubiBundledSnapshot[] + ]; + replaceRules = genSubscriptReplacements[a]; + cleanedA = a /. replaceRules[[1]]; + (Rubi`+"`"+`Int[cleanedA, x] /. replaceRules[[2]]) // Simplify +]; +Attributes[Integrate] = {ReadProtected, Protected}; +Tests`+"`"+`Integrate = { + ESimpleExamples[ + ESameTest[Null, LoadRubiBundledSnapshot[]], + ESameTest[2 x + (3 x^(5/3))/5 + (3 x^2)/2, Integrate[x^(2/3) + 3 x + 2, x]], + ESameTest[-((3 x^2)/4) + (1/2) (x^2) Log[x] - Sin[x], Integrate[Integrate[Sin[x] + Log[x], x], x]], + ESameTest[1/3, Integrate[x^2, {x, 0, 1}]], + ESameTest[True, (Integrate[x^2, {x, 0.5, 1.}] - 0.29166667) < .00001], + (*ESameTest[-(E^(3 x)/9)+1/3 E^(3 x) x, Integrate[E^(3*x)*x, x]//Expand],*) + ESameTest[E^(3*x)/3, Integrate[E^(3*x), x]], + ESameTest[x^(1 + a + b)/(1 + a + b), Integrate[x^(a + b), x]], + ESameTest[n^3/3, Integrate[x^2, {x, 0, n}]], + ESameTest[x^3/3, Integrate[x^2, {x, 0, x}]] + ], ETests[ + (*Test some trig definitions*) + ESameTest[Tan[x], Integrate[Sec[x]^2,x]], + ESameTest[-Cot[x], Integrate[Csc[x]^2,x]], + ESameTest[Sec[x], Integrate[Sec[x]Tan[x],x]], + ESameTest[-Csc[x], Integrate[Csc[x]Cot[x],x]], + ESameTest[ArcSin[x], Integrate[1/Sqrt[1-x^2],x]], + ESameTest[ArcTan[x], Integrate[1/(1+x^2),x]] + ], EKnownFailures[ + ESameTest[Log[x] - 1/2 Log[1 + 2 x^2], Integrate[1/(2 x^3 + x), x]] + ] +}; +`) + +func resourcesCalculusMBytes() ([]byte, error) { + return _resourcesCalculusM, nil +} + +func resourcesCalculusM() (*asset, error) { + bytes, err := resourcesCalculusMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/calculus.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesCombinatoricsM = []byte(`IntegerPartitions::usage = "`+"`"+`IntegerPartitions[n]`+"`"+` lists the possible ways to partition `+"`"+`n`+"`"+` into smaller integers. + +`+"`"+`IntegerPartitions[n, k]`+"`"+` lists the possible ways to partition `+"`"+`n`+"`"+` into smaller integers, using up to `+"`"+`k`+"`"+` elements."; +Attributes[IntegerPartitions] = {Protected}; +Tests`+"`"+`IntegerPartitions = { + ESimpleExamples[ + EComment["Find the partitions of 4:"], + ESameTest[{{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}, IntegerPartitions[4]], + EComment["Find the partitions of 10, using a maximum of k = 2 integers:"], + ESameTest[{{10}, {9, 1}, {8, 2}, {7, 3}, {6, 4}, {5, 5}}, IntegerPartitions[10, 2]] + ], EFurtherExamples[ + EComment["The partitions of zero is a nested empty List:"], + ESameTest[{{}}, IntegerPartitions[0]] + ], ETests[ + ESameTest[{{1}}, IntegerPartitions[1]], + ESameTest[{}, IntegerPartitions[-1]], + ESameTest[{}, IntegerPartitions[-5]], + ESameTest[IntegerPartitions[.5], IntegerPartitions[.5]], + ESameTest[{{10}}, IntegerPartitions[10, 1]], + ESameTest[{}, IntegerPartitions[10, 0]] + ] +}; + +Permutations::usage = "`+"`"+`Permutations[list]`+"`"+` lists the possible permutations for a given list."; +Attributes[Permutations] = {Protected}; +Tests`+"`"+`Permutations = { + ESimpleExamples[ + EComment["Find the permutations of `+"`"+`{1, 2, 3}`+"`"+`:"], + ESameTest[{{1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}}, Permutations[Range[3]]], + EComment["`+"`"+`Permutations`+"`"+` ignores duplicates:"], + ESameTest[{{1, 2, 2}, {2, 1, 2}, {2, 2, 1}}, Permutations[{1, 2, 2}]] + ] +}; + +Multinomial::usage = "`+"`"+`Multinomial[n1, n2, ...]`+"`"+` gives the multinomial coefficient for the given term."; +Multinomial[seq___] := Factorial[Apply[Plus, {seq}]] / Apply[Times, Map[Factorial, {seq}]]; +Attributes[Multinomial] = {Listable, NumericFunction, Orderless, ReadProtected, Protected}; +Tests`+"`"+`Multinomial = { + ESimpleExamples[ + EComment["Find the multinomial coefficient for the 1, 3, 1 term:"], + ESameTest[20, Multinomial[1, 3, 1]], + EComment["`+"`"+`Multinomial`+"`"+` handles symbolic arguments:"], + ESameTest[Factorial[k+2] / Factorial[k], Multinomial[1,k,1]] + ] +}; + +Factorial::usage = "`+"`"+`n!`+"`"+` returns the factorial of `+"`"+`n`+"`"+`."; +Attributes[Factorial] = {Listable, NumericFunction, ReadProtected, Protected}; +Tests`+"`"+`Factorial = { + ESimpleExamples[ + ESameTest[2432902008176640000, 20!], + ESameTest[120, Factorial[5]] + ], EFurtherExamples[ + ESameTest[1, Factorial[0]], + ESameTest[ComplexInfinity, Factorial[-1]] + ], ETests[ + ESameTest[1, Factorial[1]], + ESameTest[1, Factorial[0]], + ESameTest[1, Factorial[-0]], + ESameTest[ComplexInfinity, Factorial[-10]], + ESameTest[120, Factorial[5]], + ESameTest[Indeterminate, 0 * Infinity], + ESameTest[Indeterminate, 0 * ComplexInfinity] + ] +}; +`) + +func resourcesCombinatoricsMBytes() ([]byte, error) { + return _resourcesCombinatoricsM, nil +} + +func resourcesCombinatoricsM() (*asset, error) { + bytes, err := resourcesCombinatoricsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/combinatorics.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesComparisonM = []byte(`NumericQ::usage = "`+"`"+`NumericQ[expr]`+"`"+` returns `+"`"+`True`+"`"+` if `+"`"+`expr`+"`"+` is a numeric quantity, `+"`"+`False`+"`"+` otherwise."; +NumericQ[e_] := If[NumberQ[N[e]], True, False]; +Attributes[NumericQ] = {Protected}; +Tests`+"`"+`NumericQ = { + ESimpleExamples[ + ESameTest[True, NumericQ[5]], + ESameTest[False, NumericQ[a]], + ESameTest[False, NumericQ[Sin[a]]], + ESameTest[True, NumericQ[Sin[2]]] + ], ETests[ + ESameTest[True, NumericQ[Cos[2]]], + ESameTest[False, NumericQ[Sqrt[a]]], + ESameTest[False, NumericQ[Sqrt[Sin[2]]*Sqrt[Sin[x]]]], + ESameTest[True, NumericQ[Sqrt[2]]] + ] +}; +Equal::usage = "`+"`"+`lhs == rhs`+"`"+` evaluates to True or False if equality or inequality is known."; +(*TODO(corywalker): Ideally this should be handled in the core. Also should + support arbitrary number of arguments.*) +Equal[a_?NumericQ, b_?NumericQ] := (a//N) == (b//N); +Attributes[Equal] = {Protected}; +Tests`+"`"+`Equal = { + ESimpleExamples[ + EComment["Expressions known to be equal will evaluate to True:"], + EStringTest["True", "9*x==x*9"], + EComment["Sometimes expressions may or may not be equal, or Expreduce does not know how to test for equality. In these cases, the statement will remain unevaluated:"], + EStringTest["(9*x == 10*x)", "9*x==x*10"], + EComment["Equal considers Integers and Reals that are close enough to be equal:"], + EStringTest["5", "tmp=5"], + EStringTest["True", "tmp==5"], + EStringTest["True", "tmp==5."], + EStringTest["True", "tmp==5.00000"], + EComment["Equal can test for Rational equality:"], + EStringTest["False", "4/3==3/2"], + EStringTest["True", "4/3==8/6"] + ], EFurtherExamples[ + EStringTest["True", "If[xx == 2, yy, zz] == If[xx == 2, yy, zz]"], + EComment["Equal does not match patterns:"], + ESameTest[{1, 2, 3} == _List, {1, 2, 3} == _List], + EComment["This functionality is reserved for MatchQ:"], + ESameTest[True, MatchQ[{1, 2, 3}, _List]] + ], ETests[ + EStringTest["5", "tmp=5"], + EStringTest["True", "tmp==5"], + EStringTest["True", "5==tmp"], + EStringTest["False", "tmp==6"], + EStringTest["False", "6==tmp"], + EStringTest["(a == b)", "a==b"], + EStringTest["True", "a==a"], + EStringTest["(a == 2)", "a==2"], + EStringTest["(2 == a)", "2==a"], + EStringTest["(2 == a + b)", "2==a+b"], + EStringTest["(2. == a)", "2.==a"], + EStringTest["(2^k == a)", "2^k==a"], + EStringTest["(2^k == 2^a)", "2^k==2^a"], + EStringTest["(2^k == 2 + k)", "2^k==k+2"], + EStringTest["(k == 2*k)", "k==2*k"], + EStringTest["(2*k == k)", "2*k==k"], + EStringTest["True", "1+1==2"], + EStringTest["(y == b + m*x)", "y==m*x+b"], + EStringTest["True", "1==1."], + EStringTest["True", "1.==1"], + EStringTest["True", "(x==2)==(x==2)"], + EStringTest["True", "(x==2.)==(x==2)"], + EStringTest["True", "(x===2.)==(x===2)"], + EStringTest["(If[xx == 3, yy, zz] == If[xx == 2, yy, zz])", "If[xx == 3, yy, zz] == If[xx == 2, yy, zz]"], + EStringTest["True", "(1 == 2) == (2 == 3)"], + EStringTest["False", "(1 == 2) == (2 == 2)"], + ESameTest[True, foo[x == 2, y, x] == foo[x == 2, y, x]], + ESameTest[True, foo[x == 2, y, x] == foo[x == 2., y, x]], + ESameTest[foo[x == 2, y, x] == foo[x == 2., y, y], foo[x == 2, y, x] == foo[x == 2., y, y]], + ESameTest[foo[x == 2, y, x] == bar[x == 2, y, x], foo[x == 2, y, x] == bar[x == 2, y, x]], + EStringTest["(foo[x, y, z] == foo[x, y])", "foo[x, y, z] == foo[x, y]"], + EStringTest["(foo[x, y, z] == foo[x, y, 1])", "foo[x, y, z] == foo[x, y, 1]"], + ESameTest[True, foo[x, y, 1] == foo[x, y, 1]], + ESameTest[True, foo[x, y, 1.] == foo[x, y, 1]], + ESameTest[True, Equal[test]], + ESameTest[True, Equal[]], + ESameTest[False, (-1)^(1/6)==-I], + ESameTest[True, (-1)^(1/6)==(-1)^(1/6)//N], + ESameTest[True, (2^(-1/2)*E^((-1/2)*x^2)*Pi^(-1/2)/.x->(-Sqrt[2] Sqrt[Log[5 Sqrt[2/\[Pi]]]]//N))==.1], + ESameTest[True, 1.0000000000005==1.00000000000051], + ESameTest[False, 1.000000000005==1.0000000000051], + ESameTest[True, 100.00000000005==100.000000000051], + ESameTest[True, 1000000000000.5==1000000000000.51], + ] +}; + +Unequal::usage = "`+"`"+`lhs != rhs`+"`"+` evaluates to True if inequality is known or False if equality is known."; +Attributes[Unequal] = {Protected}; +Tests`+"`"+`Unequal = { + ESimpleExamples[ + EComment["Expressions known to be unequal will evaluate to True:"], + EStringTest["True", "9 != 8"], + EComment["Sometimes expressions may or may not be unequal, or Expreduce does not know how to test for inequality. In these cases, the statement will remain unevaluated:"], + EStringTest["((9*x) != (10*x))", "9*x != x*10"], + EComment["Unequal considers Integers and Reals that are close enough to be equal:"], + EStringTest["5", "tmp=5"], + EStringTest["False", "tmp != 5"], + EStringTest["False", "tmp != 5."], + EStringTest["False", "tmp != 5.00000"], + EComment["Unequal can test for Rational inequality:"], + EStringTest["True", "4/3 != 3/2"], + EStringTest["False", "4/3 != 8/6"] + ] +}; + +SameQ::usage = "`+"`"+`lhs === rhs`+"`"+` evaluates to True if `+"`"+`lhs`+"`"+` and `+"`"+`rhs`+"`"+` are identical after evaluation, False otherwise."; +Attributes[SameQ] = {Protected}; +Tests`+"`"+`SameQ = { + ESimpleExamples[ + EStringTest["True", "a===a"], + EStringTest["True", "5 === 5"], + EComment["Unlike Equal, SameQ does not forgive differences between Integers and Reals:"], + EStringTest["False", "5 === 5."], + EComment["SameQ considers the arguments of all expressions and subexpressions:"], + ESameTest[True, foo[x == 2, y, x] === foo[x == 2, y, x]], + ESameTest[False, foo[x == 2, y, x] === foo[x == 2., y, x]] + ], EFurtherExamples[ + EComment["SameQ does not match patterns:"], + ESameTest[False, {1, 2, 3} === _List], + EComment["This functionality is reserved for MatchQ:"], + ESameTest[True, MatchQ[{1, 2, 3}, _List]] + ], ETests[ + EStringTest["5", "tmp=5"], + EStringTest["False", "a===b"], + EStringTest["True", "tmp===5"], + EStringTest["False", "tmp===5."], + EStringTest["True", "1+1===2"], + EStringTest["False", "y===m*x+b"], + EStringTest["False", "1===1."], + EStringTest["False", "1.===1"], + EStringTest["True", "(x===2.)===(x===2)"], + EStringTest["False", "(x==2.)===(x==2)"], + EStringTest["True", "If[xx == 2, yy, zz] === If[xx == 2, yy, zz]"], + EStringTest["False", "If[xx == 2, yy, zz] === If[xx == 2., yy, zz]"], + EStringTest["False", "If[xx == 3, yy, zz] === If[xx == 2, yy, zz]"], + EStringTest["False", "(x == y) === (y == x)"], + EStringTest["True", "(x == y) === (x == y)"], + ESameTest[False, foo[x == 2, y, x] === foo[x == 2., y, y]], + ESameTest[False, foo[x == 2, y, x] === bar[x == 2, y, x]], + ESameTest[False, foo[x, y, z] === foo[x, y]], + ESameTest[False, foo[x, y, z] === foo[x, y, 1]], + ESameTest[True, foo[x, y, 1] === foo[x, y, 1]], + ESameTest[False, foo[x, y, 1.] === foo[x, y, 1]], + ESameTest[True, SameQ[test]], + ESameTest[True, SameQ[]] + ] +}; + +UnsameQ::usage = "`+"`"+`lhs =!= rhs`+"`"+` evaluates to False if `+"`"+`lhs`+"`"+` and `+"`"+`rhs`+"`"+` are identical after evaluation, True otherwise."; +Attributes[UnsameQ] = {Protected}; +Tests`+"`"+`UnsameQ = { + ESimpleExamples[ + EStringTest["False", "a=!=a"], + EStringTest["False", "5 =!= 5"], + EStringTest["True", "a=!=b"] + ], ETests[ + EStringTest["False", "a=!=b=!=a"], + EStringTest["True", "UnsameQ[]"] + ] +}; + +AtomQ::usage = "`+"`"+`AtomQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is an atomic type, and False if `+"`"+`expr`+"`"+` is a full expression."; +Attributes[AtomQ] = {Protected}; +Tests`+"`"+`AtomQ = { + ESimpleExamples[ + ESameTest[True, AtomQ["hello"]], + ESameTest[True, AtomQ[5/3]], + ESameTest[True, AtomQ[hello]], + ESameTest[False, AtomQ[a/b]], + ESameTest[False, AtomQ[bar[x]]] + ] +}; + +NumberQ::usage = "`+"`"+`NumberQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is numeric, otherwise False."; +Attributes[NumberQ] = {Protected}; +Tests`+"`"+`NumberQ = { + ESimpleExamples[ + ESameTest[True, NumberQ[2]], + ESameTest[True, NumberQ[2.2]], + ESameTest[True, NumberQ[Rational[5, 2]]], + ESameTest[False, NumberQ[Infinity]], + ESameTest[False, NumberQ[Sqrt[2]]], + ESameTest[False, NumberQ[randomvar]], + ESameTest[False, NumberQ["hello"]] + ] +}; + +Less::usage = "`+"`"+`a < b`+"`"+` returns True if `+"`"+`a`+"`"+` is less than `+"`"+`b`+"`"+`."; +Attributes[Less] = {Protected}; +Tests`+"`"+`Less = { + ESimpleExamples[ + ESameTest[a < b, a < b], + ESameTest[True, 1 < 2], + ESameTest[True, 3 < 5.5], + ESameTest[False, 5.5 < 3], + ESameTest[False, 3 < 3] + ] +}; + +Greater::usage = "`+"`"+`a > b`+"`"+` returns True if `+"`"+`a`+"`"+` is greater than `+"`"+`b`+"`"+`."; +Attributes[Greater] = {Protected}; +Tests`+"`"+`Greater = { + ESimpleExamples[ + ESameTest[a > b, a > b], + ESameTest[False, 1 > 2], + ESameTest[False, 3 > 5.5], + ESameTest[True, 5.5 > 3], + ESameTest[False, 3 > 3] + ] +}; + +LessEqual::usage = "`+"`"+`a <= b`+"`"+` returns True if `+"`"+`a`+"`"+` is less than or equal to `+"`"+`b`+"`"+`."; +-Infinity <= (_Integer | _Real | _Rational) := True; +Infinity <= (_Integer | _Real | _Rational) := False; +(_Integer | _Real | _Rational) <= -Infinity := False; +(_Integer | _Real | _Rational) <= Infinity := True; +Attributes[LessEqual] = {Protected}; +Tests`+"`"+`LessEqual = { + ESimpleExamples[ + ESameTest[a <= b, a <= b], + ESameTest[True, 1 <= 2], + ESameTest[True, 3 <= 5.5], + ESameTest[False, 5.5 <= 3], + ESameTest[True, 3 <= 3] + ] +}; + +GreaterEqual::usage = "`+"`"+`a >= b`+"`"+` returns True if `+"`"+`a`+"`"+` is greater than or equal to `+"`"+`b`+"`"+`."; +Attributes[GreaterEqual] = {Protected}; +Tests`+"`"+`GreaterEqual = { + ESimpleExamples[ + ESameTest[a >= b, a >= b], + ESameTest[False, 1 >= 2], + ESameTest[False, 3 >= 5.5], + ESameTest[True, 5.5 >= 3], + ESameTest[True, 3 >= 3] + ] +}; + +Positive::usage = "`+"`"+`Positive[x]`+"`"+` returns `+"`"+`True`+"`"+` if `+"`"+`x`+"`"+` is positive."; +Positive[x_?NumberQ] := x > 0; +Attributes[Positive] = {Listable, Protected}; +Tests`+"`"+`Positive = { + ESimpleExamples[ + ESameTest[{True, False, False, Positive[a]}, Map[Positive, {1, 0, -1, a}]] + ] +}; + +Negative::usage = "`+"`"+`Negative[x]`+"`"+` returns `+"`"+`True`+"`"+` if `+"`"+`x`+"`"+` is positive."; +Negative[x_?NumberQ] := x < 0; +Attributes[Negative] = {Listable, Protected}; +Tests`+"`"+`Negative = { + ESimpleExamples[ + ESameTest[{False, False, True, Negative[a]}, Map[Negative, {1, 0, -1, a}]] + ] +}; + +Max::usage = "`+"`"+`Max[e1, e2, ...]`+"`"+` the maximum of the expressions."; +Attributes[Max] = {Flat, NumericFunction, OneIdentity, Orderless, Protected}; +Tests`+"`"+`Max = { + ESimpleExamples[ + ESameTest[3, Max[1,2,3]], + ESameTest[Max[3,a], Max[1,a,3]] + ], ETests[ + ESameTest[Max[3,a,b], Max[b,1,a,3]], + ESameTest[Max[3.,a,b], Max[b,1,a,3,3.]], + ESameTest[Max[3.1,a,b], Max[b,1,a,3,3.,3.1]], + ESameTest[Max[99/2,a,b], Max[b,1,a,3,3.,3.1 ,Rational[99,2]]], + ESameTest[-Infinity, Max[]], + ESameTest[Max[99/2,a,b], Max[{b,1,a},3,3.,3.1 ,Rational[99,2]]], + ESameTest[Max[99/2,foo[b,1,a]], Max[foo[b,1,a],3,3.,3.1 ,Rational[99,2]]] + ], EKnownDangerous[ + ESameTest[Max[a,b,c,d], Max[{c,d},{b,a}]], + ESameTest[Max[a,b,c,d], Max[{c,{d}},{b,a}]] + ] +}; + +Min::usage = "`+"`"+`Min[e1, e2, ...]`+"`"+` the maximum of the expressions."; +Attributes[Min] = {Flat, NumericFunction, OneIdentity, Orderless, Protected}; +Tests`+"`"+`Min = { + ESimpleExamples[ + ESameTest[1, Min[1,2,3]], + ESameTest[Min[1,a], Min[1,a,3]] + ], ETests[ + ESameTest[Min[1,a,b], Min[b,1,a,3]], + ESameTest[Min[1,a,b], Min[b,1,a,3,3.]], + ESameTest[Min[1,a,b], Min[b,1,a,3,3.,3.1]], + ESameTest[Min[1,a,b], Min[b,1,a,3,3.,3.1 ,Rational[99,2]]], + ESameTest[Infinity, Min[]], + ], EKnownDangerous[ + ESameTest[Min[a,b,c,d], Min[{c,d},{b,a}]], + ESameTest[Min[a,b,c,d], Min[{c,{d}},{b,a}]] + ] +}; + +PossibleZeroQ::usage = "`+"`"+`PossibleZeroQ[e]`+"`"+` returns True if `+"`"+`e`+"`"+` is most likely equivalent to zero."; +Attributes[PossibleZeroQ] = {Listable, Protected}; +PossibleZeroQ[e_] := e === 0 || e === 0.; +Tests`+"`"+`PossibleZeroQ = { + ESimpleExamples[ + ESameTest[True, PossibleZeroQ[a-a]], + ESameTest[False, PossibleZeroQ[a-b]] + ] +}; + +MinMax::usage = "`+"`"+`MinMax[l]`+"`"+` returns `+"`"+`{Min[l], Max[l]}`+"`"+`."; +Attributes[MinMax] = {Protected}; +MinMax[l_List] := {Min[l], Max[l]}; +Tests`+"`"+`MinMax = { + ESimpleExamples[ + ESameTest[{1, 5}, MinMax[Range[5]]] + ] +}; + +Element::usage = "`+"`"+`Element[i, s]`+"`"+` checks if `+"`"+`i`+"`"+` is an element of `+"`"+`s`+"`"+`."; +Attributes[Element] = {Protected}; +Element[i_Integer, Integers] := True; +Tests`+"`"+`Element = { + ESimpleExamples[ + ESameTest[True, Element[-1, Integers]] + ] +}; + +Attributes[Inequality] = {Protected}; +Tests`+"`"+`Inequality = { + ESimpleExamples[ + ESameTest[True, Inequality[-Pi,Less,0,LessEqual,Pi]], + ESameTest[0<=a, Inequality[-Pi,Less,0,LessEqual,a]], + ], ETests[ + ESameTest[Inequality[c, Less, 0, Less, a], Inequality[c,Less,0,Less,a]], + ESameTest[c<0, Inequality[c,Less,0]], + ESameTest[Inequality[c,Less], Inequality[c,Less]], + ESameTest[True, Inequality[c]], + ESameTest[Inequality[], Inequality[]], + ESameTest[True, Inequality[False]], + ESameTest[Inequality[-1, Less], Inequality[-1,Less]], + ESameTest[Inequality[-1, Less, a, Less, 0], Inequality[-1,Less,a,Less,0,Less,1]], + ESameTest[Inequality[0, Less, a, Less, 1], Inequality[-1,Less,0,Less,a,Less,1]], + ESameTest[False, Inequality[-1,Less,a,Less,-2]], + ESameTest[0>=a, Inequality[-Pi,Less,0,GreaterEqual,a]], + ESameTest[0a&&a<0, Inequality[0,Greater,a,Less,0]], + ESameTest[Inequality[0, Less, a, Less, 1], Inequality[0,Less,a,Less,1]], + ESameTest[0=e>f, Inequality[a,Less,b,LessEqual,c,Equal,d,GreaterEqual,e,Greater,f]], + ESameTest[a>b>=c==d&&d<=eb>=c==d>=e&&e(-b-Sqrt[b^2-4 a c])/(2 a)},{x->(-b+Sqrt[b^2-4 a c])/(2 a)}}; +solveCubic[d_,c_,b_,a_,x_] := {{x->-(c/(3 d))-(2^(1/3) (-c^2+3 b d))/(3 d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))+(-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)/(3 2^(1/3) d)},{x->-(c/(3 d))+((1+I Sqrt[3]) (-c^2+3 b d))/(3 2^(2/3) d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))-(1/(6 2^(1/3) d))(1-I Sqrt[3]) (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)},{x->-(c/(3 d))+((1-I Sqrt[3]) (-c^2+3 b d))/(3 2^(2/3) d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))-(1/(6 2^(1/3) d))(1+I Sqrt[3]) (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)}}; +solveQuartic[e_,d_,c_,b_,a_,x_] := {{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))}}/;FreeQ[{a,b,c,d,e},x]; +`) + +func resourcesEquationdataMBytes() ([]byte, error) { + return _resourcesEquationdataM, nil +} + +func resourcesEquationdataM() (*asset, error) { + bytes, err := resourcesEquationdataMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/equationdata.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesExpressionM = []byte(`Head::usage = "`+"`"+`Head[expr]`+"`"+` returns the head of the expression."; +Attributes[Head] = {Protected}; +Tests`+"`"+`Head = { + ESimpleExamples[ + ESameTest[f, Head[f[x]]], + ESameTest[Symbol, Head[x]], + ESameTest[List, Head[{x}]], + ESameTest[Plus, Head[a + b]], + ESameTest[Integer, Head[1]], + ESameTest[Real, Head[1.]], + ESameTest[Rational, Head[2/7]], + ESameTest[Rational, Head[1/7]], + ESameTest[String, Head["1"]], + ESameTest[Plus, Head[Head[(a + b)[x]]]] + ] +}; + +Depth::usage = "`+"`"+`Depth[expr]`+"`"+` returns the depth of `+"`"+`expr`+"`"+`."; +Attributes[Depth] = {Protected}; +Tests`+"`"+`Depth = { + ESimpleExamples[ + ESameTest[1, Depth[foo]], + ESameTest[2, Depth[{foo}]], + ESameTest[2, Depth[bar[foo, bar]]], + ESameTest[3, Depth[foo[foo[]]]], + ESameTest[1, Depth[3]], + ESameTest[1, Depth[3.5]], + ESameTest[1, Depth[3/5]], + ESameTest[2, Depth[foo[{{{}}}][]]] + ] +}; + +Length::usage = "`+"`"+`Length[expr]`+"`"+` returns the length of `+"`"+`expr`+"`"+`."; +Attributes[Length] = {Protected}; +Tests`+"`"+`Length = { + ESimpleExamples[ + ESameTest[4, Length[{1,2,3,4}]], + ESameTest[0, Length[{}]], + ESameTest[1, Length[{5}]] + ], EFurtherExamples[ + EComment["`+"`"+`expr`+"`"+` need not have a `+"`"+`List`+"`"+` head:"], + ESameTest[2, Length[foo[1, 2]]], + EComment["The length of an atomic expression is zero:"], + ESameTest[0, Length[a]], + ESameTest[0, Length[2.5]], + ESameTest[0, Length["hello"]] + ] +}; + +Sequence::usage = "`+"`"+`Sequence[e1, e2, ...]`+"`"+` holds a list of expressions to be automatically inserted into another function."; +Attributes[Sequence] = {Protected}; +Tests`+"`"+`Sequence = { + ESimpleExamples[ + EComment["Sequence arguments are automatically inserted into the parent functions:"], + ESameTest[foo[a, 2, 3], foo[a, Sequence[2, 3]]], + EComment["Outside of the context of functions, Sequence objects do not merge:"], + ESameTest[Sequence[2, 3], Sequence[2, 3]], + ESameTest[14, Sequence[2, 3] + Sequence[5, 4]], + ESameTest[120, Sequence[2, 3]*Sequence[5, 4]] + ], EFurtherExamples[ + EComment["Empty `+"`"+`Sequence[]`+"`"+` objects effectively disappear:"], + ESameTest[foo[], foo[Sequence[]]] + ], ETests[ + ESameTest[Sequence[2], Sequence[2]], + ESameTest[Sequence[2, 3], Sequence[2, 3]], + ESameTest[foo[2, 3], foo[Sequence[2, 3]]], + ESameTest[foo[2], foo[Sequence[2]]], + ESameTest[foo[14], foo[Sequence[2, 3] + Sequence[5, 4]]], + ESameTest[foo[2, 3, 5, 4], foo[Sequence[2, 3], Sequence[5, 4]]], + ESameTest[False, Sequence[2, 3] == Sequence[2, 3]], + ESameTest[True, Sequence[2, 2] == Sequence[2]], + ESameTest[False, Sequence[2, 3] === Sequence[2, 3]], + ESameTest[True, Sequence[2, 2] === Sequence[2]] + ] +}; + +Evaluate::usage = "`+"`"+`Evaluate[expr]`+"`"+` evaluates to an evaluated form of `+"`"+`expr`+"`"+`, even when under hold conditions."; +Attributes[Evaluate] = {Protected}; +Tests`+"`"+`Evaluate = { + ESimpleExamples[ + EStringTest["Hold[4, 2 + 1]", "Hold[Evaluate[1 + 3], 2 + 1]"], + EStringTest["Hold[foo[Evaluate[1 + 1]]]", "Hold[foo[Evaluate[1 + 1]]]"], + EStringTest["Hold[4, 7, 2 + 1]", "Hold[Evaluate[1 + 3, 5 + 2], 2 + 1]"], + EStringTest["Hold[1 + 3, 5 + 2, 2 + 1]", "Hold[Sequence[1 + 3, 5 + 2], 2 + 1]"] + ] +}; + +Hold::usage = "`+"`"+`Hold[expr]`+"`"+` prevents automatic evaluation of `+"`"+`expr`+"`"+`."; +Attributes[Hold] = {HoldAll, Protected}; +Tests`+"`"+`Hold = { + ESimpleExamples[ + EStringTest["Hold[5^3]", "Hold[Power[5, 3]]"], + EStringTest["Hold[5.^3.]", "Hold[Power[5., 3.]]"] + ] +}; + +HoldForm::usage = "`+"`"+`HoldForm[expr]`+"`"+` prevents automatic evaluation of `+"`"+`expr`+"`"+`. Prints as `+"`"+`expr`+"`"+`."; +Attributes[HoldForm] = {HoldAll, Protected}; +Tests`+"`"+`HoldForm = { + ESimpleExamples[ + EStringTest["(5^3)", "HoldForm[Power[5, 3]]"], + EStringTest["(5.^3.)", "HoldForm[Power[5., 3.]]"] + ] +}; + +Flatten::usage = "`+"`"+`Flatten[list]`+"`"+` flattens out lists in `+"`"+`list`+"`"+`."; +Attributes[Flatten] = {Protected}; +Tests`+"`"+`Flatten = { + ESimpleExamples[ + ESameTest[Flatten[1], Flatten[1]], + EComment["Input must be nonatomic:"], + ESameTest[{1}, Flatten[{1}]], + ESameTest[{1}, Flatten[{{{{1}}}}]], + ESameTest[{1, 2, 3}, Flatten[{{{{1}, 2}}, 3}]], + ESameTest[{1, 2, 3, 4}, Flatten[{{{{1}, 2}}, 3, 4}]], + ESameTest[{-1, 1, 2, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}]], + EComment["A level of zero means no change:"], + ESameTest[{-1, {{{1}, 2}}, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}, 0]], + ESameTest[{-1, {{1}, 2}, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}, 1]], + ESameTest[{-1, {1}, 2, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}, 2]], + ESameTest[{-1, 1, 2, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}, 3]], + ESameTest[{-1, 1, 2, 3, 4}, Flatten[{-1, {{{1}, 2}}, 3, 4}, 4]], + ESameTest[Flatten[{-1, {{{1}, 2}}, 3, 4}, a], Flatten[{-1, {{{1}, 2}}, 3, 4}, a]], + ESameTest[{-1, foo[{{1}, 2}], 3, 4}, Flatten[{-1, {foo[{{1}, 2}]}, 3, 4}, 999]], + ESameTest[{-1, foo[{{1}, 2}], 3, 4}, Flatten[{-1, {foo[{{1}, 2}]}, 3, 4}, 999]], + ESameTest[{-1, 1[{{1}, 2}], 3, 4}, Flatten[{-1, {1[{{1}, 2}]}, 3, 4}, 999]] + ] +}; + +LeafCount::usage = "`+"`"+`LeafCount[e]`+"`"+` returns the count of leaves in `+"`"+`e`+"`"+`."; +Attributes[LeafCount] = {Protected}; +Tests`+"`"+`LeafCount = { + ESimpleExamples[ + ESameTest[3, LeafCount[a+b]], + ESameTest[8, LeafCount[a^2 + b^(c!)]], + ESameTest[1, LeafCount[a]] + ] +}; + +Unevaluated::usage = "`+"`"+`Unevaluated[e]`+"`"+` do not evaluate `+"`"+`e`+"`"+` in an expression, but treat as `+"`"+`e`+"`"+`."; +Attributes[Unevaluated] = {HoldAllComplete, Protected}; +Tests`+"`"+`Unevaluated = { + ESimpleExamples[ + ESameTest[{1,2}, List@@Unevaluated[1+2]], + ] +}; + +HoldComplete::usage = "`+"`"+`HoldComplete[e1, e2, ...]`+"`"+` holds evaluation of its arguments, even evaluation that would take place under `+"`"+`Hold`+"`"+`."; +Attributes[HoldComplete] = {HoldAllComplete, Protected}; +Tests`+"`"+`HoldComplete = { + ESimpleExamples[ + ESameTest[HoldComplete[Evaluate[a+a],2+2,Sequence[a,b]], HoldComplete[Evaluate[a+a],2+2,Sequence[a,b]]], + ESameTest[3, HoldComplete[Evaluate[a + a], 2 + 2, Sequence[a, b]] // Length], + ] +}; +`) + +func resourcesExpressionMBytes() ([]byte, error) { + return _resourcesExpressionM, nil +} + +func resourcesExpressionM() (*asset, error) { + bytes, err := resourcesExpressionMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/expression.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesFlowcontrolM = []byte(`If::usage = "`+"`"+`If[cond, iftrue, iffalse]`+"`"+` returns `+"`"+`iftrue`+"`"+` if `+"`"+`cond`+"`"+` is True, and `+"`"+`iffalse`+"`"+` if `+"`"+`cond`+"`"+` is False."; +Attributes[If] = {HoldRest, Protected}; +Tests`+"`"+`If = { + ESimpleExamples[ + EStringTest["9", "x=9"], + EStringTest["18", "If[x+3==12, x*2, x+3]"], + EStringTest["12", "If[x+3==11, x*2, x+3]"] + ], EFurtherExamples[ + EComment["Undefined conditions leave the statement unevaluated."], + EStringTest["If[undefined, a, b]", "If[undefined, a, b]"] + ], ETests[ + EStringTest["True", "t=True"], + EStringTest["True", "t"], + EStringTest["False", "f=False"], + EStringTest["False", "f"], + EStringTest["True", "If[t, True, False]"], + EStringTest["False", "If[f, True, False]"], + EStringTest["False", "If[t, False, True]"], + EStringTest["True", "If[f, False, True]"], + ESameTest[itsfalse, If[1 == 2, itstrue, itsfalse]], + ESameTest[itsfalse, If[1 == 2, itstrue, itsfalse] /. (2 -> 1)], + ESameTest[itstrue, If[1 == k, itstrue, itsfalse] /. (k -> 1)], + ESameTest[If[1 == k, itstrue, itsfalse], If[1 == k, itstrue, itsfalse]], + ESameTest[a, If[True, a]], + ESameTest[Null, If[False, a]] + ] +}; + +While::usage = "`+"`"+`While[cond, body]`+"`"+` evaluates `+"`"+`cond`+"`"+`, and if it returns True, evaluates `+"`"+`body`+"`"+`. This happens repeatedly."; +Attributes[While] = {HoldAll, Protected}; +Tests`+"`"+`While = { + ESimpleExamples[ + ESameTest[1, a=1], + ESameTest[Null, While[a != 5, a = a + 1]], + ESameTest[5, a] + ] +}; + +CompoundExpression::usage = "`+"`"+`CompoundExpression[e1, e2, ...]`+"`"+` evaluates each expression in order and returns the result of the last one."; +Attributes[CompoundExpression] = {HoldAll, ReadProtected, Protected}; +Tests`+"`"+`CompoundExpression = { + ESimpleExamples[ + EComment["The result of the first expression is not included in the output, but the result of the second is:"], + ESameTest[3, a = 5; a - 2], + EComment["Including a trailing semicolon causes the expression to return `+"`"+`Null`+"`"+`:"], + ESameTest[Null, a = 5; a - 2;] + ] +}; + +Return::usage = "`+"`"+`Return[x]`+"`"+` returns `+"`"+`x`+"`"+` immediately."; +Attributes[Return] = {Protected}; +Tests`+"`"+`Return = { + ESimpleExamples[ + ESameTest[x, myreturnfunc:=(Return[x];hello);myreturnfunc], + ESameTest[3, ret[x_]:=(Return[x];hello);ret[3]], + ESameTest[3, myfoo:=(i=1;While[i<5,If[i===3,Return[i]];i=i+1]);myfoo], + ESameTest[Return[3], Return[3]], + ESameTest[Null, retother:=(Return[];hello);retother] + ] +}; + +Which::usage = "`+"`"+`Which[cond, res, cond, res, ...]`+"`"+` tries each `+"`"+`cond`+"`"+` in sequence and returns the corresponding result if True."; +Attributes[Which] = {HoldAll, Protected}; +Tests`+"`"+`Which = { + ESimpleExamples[ + ESameTest[b, Which[1>2, a, 1<2, b]], + ESameTest[Null, Which[2>2, a, 2<2, b]] + ], ETests[ + ESameTest[Which[True, a, b], Which[True, a, b]], + ESameTest[Null, Which[False,a,False,b]] + ] +}; + +Switch::usage = "`+"`"+`Switch[e, case1, val1, case2, val2, ...]`+"`"+` attempts to match `+"`"+`e`+"`"+` with the cases in order. If a match is found, returns the corresponding value."; +Attributes[Switch] = {HoldRest, Protected, ReadProtected}; +Tests`+"`"+`Switch = { + ESimpleExamples[ + ESameTest[b, Switch[z,_,b,z,c]], + ESameTest[k, Switch[z,k_Symbol,k]], + ESameTest[Switch[z,1], Switch[z,1]], + ESameTest[Switch[z,d,b,l,c], Switch[z,d,b,l,c]] + ], ETests[ + ESameTest[Switch[], Switch[]], + ESameTest[Switch[z], Switch[z]] + ] +}; + +With::usage = "`+"`"+`With[{s1=v1, s2=v2, ...}, body]`+"`"+` locally replaces the specified symbols in body with their respective values."; +Attributes[With] = {HoldAll, Protected}; +Tests`+"`"+`With = { + ESimpleExamples[ + ESameTest[{2, 6}, With[{x=2},{x,3*x}]], + ESameTest[{2, 9}, With[{x:=2,y:=3},{x,3*y}]] + ] +}; + +Do::usage = "`+"`"+`Do[expr, n]`+"`"+` evaluates `+"`"+`expr`+"`"+` `+"`"+`n`+"`"+` times. + +`+"`"+`Table[expr, {sym, n}]`+"`"+` evaluates `+"`"+`expr`+"`"+` with `+"`"+`sym`+"`"+` = 1 to `+"`"+`n`+"`"+`. + +`+"`"+`Table[expr, {sym, m, n}]`+"`"+` evaluates `+"`"+`expr`+"`"+` with `+"`"+`sym`+"`"+` = `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`."; +Attributes[Do] = {HoldAll, Protected}; +Tests`+"`"+`Do = { + ESimpleExamples[ + ESameTest[7, Catch[Do[If[a > 6, Throw[a]], {a, 10}]; False]] + ] +}; + +For::usage = "`+"`"+`For[beg, cond, incr, expr]`+"`"+` runs a for loop."; +Attributes[For] = {HoldAll, Protected}; +For[beg_, cond_, incr_, expr_] := (beg; While[cond, expr; incr]); +Tests`+"`"+`For = { + ESimpleExamples[ + ESameTest[11, For[n = 1, n < 1000, n++, If[PrimeQ[n] && (n > 7), Return[]]]; n] + ] +}; +`) + +func resourcesFlowcontrolMBytes() ([]byte, error) { + return _resourcesFlowcontrolM, nil +} + +func resourcesFlowcontrolM() (*asset, error) { + bytes, err := resourcesFlowcontrolMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/flowcontrol.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesFunctionalM = []byte(`Function::usage = "`+"`"+`Function[inner]`+"`"+` defines a pure function where `+"`"+`inner`+"`"+` is evaluated with `+"`"+`Slot`+"`"+` parameters. + +`+"`"+`Function[x, inner]`+"`"+` defines a pure function where `+"`"+`inner`+"`"+` is evaluated a single parameter `+"`"+`x`+"`"+`."; +Attributes[Function] = {HoldAll, Protected}; +Tests`+"`"+`Function = { + ESimpleExamples[ + ESameTest[1 + x, Function[1 + #][x]], + ESameTest[1 + x + 2y, Function[1 + # + 2#2][x, y]], + ESameTest[a^2, Function[x, x^2][a]], + ESameTest[a^2, Function[x, x^2][a, b]], + ESameTest[x^2, Function[x, x^2][x]], + ESameTest[4, Function[x, x^2][-2]] + ] +}; + +Slot::usage = "`+"`"+`#`+"`"+` serves as a pure function's first parameter. + +`+"`"+`#n`+"`"+` serves as a pure function's `+"`"+`n`+"`"+`'th parameter."; +Attributes[Slot] = {NHoldAll, Protected}; +Tests`+"`"+`Slot = { + ESimpleExamples[ + ESameTest[1 + x, Function[1 + #][x]], + ESameTest[1 + x + 2y, Function[1 + # + 2#2][x, y]], + ESameTest[True, # === Slot[1]], + ESameTest[True, #2 === Slot[2]], + ESameTest[2a + 4b, (4 # + (2 # &)[a] &)[b]] + ], ETests[ + ESameTest[foo[test, k], (foo[test, #] &) &[j][k]] + ] +}; + +Apply::usage = "`+"`"+`Apply[f, e]`+"`"+` (`+"`"+`f@@e`+"`"+`) replaces the head of expression `+"`"+`e`+"`"+` with `+"`"+`f`+"`"+`."; +Attributes[Apply] = {Protected}; +Tests`+"`"+`Apply = { + ESimpleExamples[ + ESameTest[bar[syma, symb], Apply[bar, foo[syma, symb]]], + ESameTest[bar[syma, symb], bar@@foo[syma, symb]], + ESameTest[{syma, symb}, List@@(syma + symb)], + EComment["`+"`"+`Apply`+"`"+` is useful in performing aggregations on `+"`"+`List`+"`"+`s:"], + ESameTest[12, Times @@ {2, 6}], + ESameTest[a b, Times @@ {a, b}] + ], EFurtherExamples[ + EComment["`+"`"+`Apply`+"`"+` has no effect on atoms:"], + ESameTest[1, foo @@ 1], + ESameTest[bar, foo @@ bar] + ], ETests[ + ESameTest[foo[a,b,c], Apply[foo, {a,b,c}]], + ESameTest[foo[bar, buzz], Apply[foo, {bar, buzz}]], + ESameTest[foo[bar, buzz], foo @@ {bar, buzz}], + ESameTest[foo[1, 2], foo @@ {1, 2}], + ESameTest[f[{{a}}], Apply[f,{{{a}}}]], + ESameTest[{{f[a]}}, Apply[f,{{{a}}},{2}]], + ESameTest[{{{a}}}, Apply[f,{{{a}}},{3}]], + ESameTest[{f[a],f[b]}, f@@@{{a},foo[b]}], + ESameTest[f[{a},foo[b]], Apply[f,{{a},foo[b]},{0}]], + ESameTest[f[f[a],f[b]], Apply[f,{{a},foo[b]},{0,1}]], + ESameTest[f[f[a,{c}],f[b]], Apply[f,{{a,{c}},foo[b]},{0,1}]], + ESameTest[f[f[a,f[c,{d}]],f[b]], Apply[f,{{a,{c,{d}}},foo[b]},{0,2}]], + ESameTest[{f[a,f[c,f[d]]],f[b]}, Apply[f,{{a,{c,{d}}},foo[b]},Infinity]], + ESameTest[{f[a,f[c]],f[b]}, Apply[f,{{a,{c}},foo[b]},2]] + ] +}; + +Map::usage = "`+"`"+`Map[f, expr]`+"`"+` returns a new expression with the same head as `+"`"+`expr`+"`"+`, but with `+"`"+`f`+"`"+` mapped to each of the arguments. +Map[f, expr, levelspec] maps f to all subexpressions that match the level specification levelspec."; +Attributes[Map] = {Protected}; +Tests`+"`"+`Map = { + ESimpleExamples[ + ESameTest[{foo[a], foo[b], foo[c]}, Map[foo, {a, b, c}]], + ESameTest[{foo[a], foo[b], foo[c]}, foo /@ {a, b, c}], + ESameTest[{2, 4, 9}, Times /@ {2, 4, 9}], + ESameTest[{foo[{a, b}], foo[c]}, Map[foo, {{a, b}, c}]], + ESameTest[Map[foo], Map[foo]], + ESameTest[foo, Map[foo, foo]], + ESameTest[Map[foo, foo, foo], Map[foo, foo, foo]], + EComment["Pure functions are useful with `+"`"+`Map`+"`"+`:"], + ESameTest[{4,16}, Function[x, x^2] /@ {2,4}], + ESameTest[{4,16}, Function[#^2] /@ {2,4}], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], {-Infinity, Infinity}], + f[a[f[b[f[c], f[d], f[e]]], f[l[f[g[f[h], f[j]]], f[k]]]]] + ], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], Infinity], + a[f[b[f[c], f[d], f[e]]], f[l[f[g[f[h], f[j]]], f[k]]]] + ], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], {-2, Infinity}], + a[f[b[f[c], f[d], f[e]]], l[f[g[f[h], f[j]]], f[k]]] + ], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], {2, -2}], + a[b[c, d, e], l[f[g[h, j]], k]] + ], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], 2], + a[f[b[f[c], f[d], f[e]]], f[l[f[g[h, j]], f[k]]]] + ], + ESameTest[ + Map[f, a[b[c, d, e], l[g[h, j], k]], -2], + a[f[b[c, d, e]], f[l[f[g[h, j]], k]]] + ] + ] +}; + +MapIndexed::usage = "`+"`"+`MapIndexed[f, expr]`+"`"+` returns a new expression with the same head as `+"`"+`expr`+"`"+`, but with `+"`"+`f`+"`"+` mapped to each of the arguments. +Additionally, MapIdnexed supplies the part specification of the subexpression as the second argument of f. +Map[f, expr, levelspec] maps f to all subexpressions that match the level specification levelspec, also supplying the part specification for +each subexpression as the second argument to f."; +Attributes[MapIndexed] = {Protected} +Tests`+"`"+`MapIndexed = { + ESimpleExamples[ + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], {-Infinity, Infinity}], + f[a[f[b[f[c, {1, 1}], f[d, {1, 2}], f[e, {1, 3}]], {1}], f[l[f[g[f[h, {2, 1, 1}], f[j, {2, 1, 2}]], {2, 1}], f[k, {2, 2}]], {2}]], {}] + ], + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], Infinity], + a[f[b[f[c, {1, 1}], f[d, {1, 2}], f[e, {1, 3}]], {1}], f[l[f[g[f[h, {2, 1, 1}], f[j, {2, 1, 2}]], {2, 1}], f[k, {2, 2}]], {2}]] + ], + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], {-2, Infinity}], + a[f[b[f[c, {1, 1}], f[d, {1, 2}], f[e, {1, 3}]], {1}], l[f[g[f[h, {2, 1, 1}], f[j, {2, 1, 2}]], {2, 1}], f[k, {2, 2}]]] + ], + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], {2, -2}], + a[b[c, d, e], l[f[g[h, j], {2, 1}], k]] + ], + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], 2], + a[f[b[f[c, {1, 1}], f[d, {1, 2}], f[e, {1, 3}]], {1}], f[l[f[g[h, j], {2, 1}], f[k, {2, 2}]], {2}]] + ], + ESameTest[ + MapIndexed[f, a[b[c, d, e], l[g[h, j], k]], -2], + a[f[b[c, d, e], {1}], f[l[f[g[h, j], {2, 1}], k], {2}]] + ] + ] +}; + +FoldList::usage = "`+"`"+`FoldList[f, x, {a, b}] returns {x, f[x, a], f[f[x, a], b]}" +FoldList[f_, expr_] := FoldList[f, First[expr], Rest[expr]] +(* FoldList[f_][expr__] := FoldList[f, expr] When subvalues are allowed *) +Attributes[FoldList] = {Protected}; +Tests`+"`"+`FoldList = { + ESimpleExamples[ + ESameTest[{1, f[1, 2], f[f[1, 2], 3]}, FoldList[f, 1, {2, 3}]], + ESameTest[{1, f[1, 2], f[f[1, 2], 3]}, FoldList[f, {1, 2, 3}]], + (* ESameTest[{1, f[1, 2], f[f[1, 2], 3]}, FoldList[f][{1, 2, 3}]], *) + ESameTest[h[e1, f[e1, e2], f[f[e1, e2], e3], f[f[f[e1, e2], e3], e4]], FoldList[f, e1, h[e2, e3, e4]]], + ESameTest[{h}, FoldList[f, h, {}]] + ] +} + +Fold::usage = "`+"`"+`Fold[f, x, {a, b}]`+"`"+` returns `+"`"+`f[f[x, a], b]`+"`"+`, and this nesting continues for lists of arbitrary length. `+"`"+`Fold[f, {a, b, c}]`+"`"+` returns `+"`"+`Fold[f, a, {b, c}]`+"`"+`. `+"`"+`Fold[f]`+"`"+` is an operator form that can be applied to expressions such as `+"`"+`{a, b, c}`+"`"+`." +Fold[f_, x_, expr_] := Last[FoldList[f, x, expr]] +Fold[f_, expr_] := Last[FoldList[f, First[expr], Rest[expr]]] +(* Fold[f_][expr__] := Last[FoldList[f, expr]] When subvalues are allowed *) +Attributes[Fold] = {Protected}; +Tests`+"`"+`Fold = { + ESimpleExamples[ + ESameTest[f[f[1, 2], 3], Fold[f, 1, {2, 3}]], + ESameTest[f[f[1, 2], 3], Fold[f, {1, 2, 3}]], + (* ESameTest[f[f[1, 2], 3], Fold[f][{1, 2, 3}]], *) + ESameTest[f[f[f[e1, e2], e3], e4], Fold[f, e1, h[e2, e3, e4]]], + ESameTest[h, Fold[f, h, {}]] + ] +} + +NestList::usage = "`+"`"+`NestList[f, expr, n]`+"`"+` returns `+"`"+`f`+"`"+` wrapped around `+"`"+`expr`+"`"+` first once, then twice, and so on up to `+"`"+`n`+"`"+` times." +Attributes[NestList] = {Protected} +Tests`+"`"+`NestList = { + ESimpleExamples[ + ESameTest[{x, f[x], f[f[x]], f[f[f[x]]]}, NestList[f, x, 3]], + ESameTest[{{1, 2, 3}, {1, 4, 9}, {1, 16, 81}, {1, 256, 6561}}, NestList[#^2 &, {1, 2, 3}, 3]] + ] +} + +Nest::usage = "`+"`"+`Nest[f, expr, n]`+"`"+` returns `+"`"+`f`+"`"+` wrapped around `+"`"+`expr`+"`"+` `+"`"+`n`+"`"+` times." +Nest[f_, expr_, n_] := Last[NestList[f, expr, n]] +Attributes[Nest] = {Protected} +Tests`+"`"+`Nest = { + ESimpleExamples[ + ESameTest[f[f[f[x]]], Nest[f, x, 3]], + ESameTest[{1, 256, 6561}, Nest[#^2 &, {1, 2, 3}, 3]] + ] +} + +NestWhileList::usage = "`+"`"+`NestWhileList[f, expr, test, m, max, n]`+"`"+` applies `+"`"+`f`+"`"+` to `+"`"+`expr`+"`"+` until `+"`"+`test`+"`"+` does not return `+"`"+`True`+"`"+`. +It returns a list of all intermediate results. `+"`"+`test`+"`"+` is a function that takes as its argument the last `+"`"+`m`+"`"+` results. +`+"`"+`max`+"`"+` denotes the maximum number of applications of `+"`"+`f`+"`"+` and `+"`"+`n`+"`"+` denotes that `+"`"+`f`+"`"+` should be applied another `+"`"+`n`+"`"+` times after +`+"`"+`test`+"`"+` has terminated the recursion." +Attributes[NestWhileList] = {Protected} +Tests`+"`"+`NestWhileList = { + ESimpleExamples[ + ESameTest[7, Length@NestWhileList[(# + 3/#)/2 &, 1.0, UnsameQ[#1, #2] &, 2]], + ESameTest[{2, 4, 16, 256}, NestWhileList[#^2 &, 2, # < 256 &]], + ESameTest[{1, 2, 3, 4, 5, 6, 7}, NestWhileList[#+1 &, 1, # + #4 < 10 &, 4]], + ESameTest[{1, 2, 3, 4, 5}, NestWhileList[#+1 &, 1, True &, 1, 4]], + ESameTest[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, NestWhileList[#+1 &, 1, True &, 1, 4, 5]] + ] +} + +NestWhile::usage = "`+"`"+`NestWhile[f, expr, test, m, max, n]`+"`"+` applies `+"`"+`f`+"`"+` to `+"`"+`expr`+"`"+` until `+"`"+`test`+"`"+` does not return `+"`"+`True`+"`"+`. +`+"`"+`test`+"`"+` is a function that takes as its argument the last `+"`"+`m`+"`"+` results. `+"`"+`max`+"`"+` denotes the maximum number of applications +of `+"`"+`f`+"`"+` and `+"`"+`n`+"`"+` denotes that `+"`"+`f`+"`"+` should be applied another `+"`"+`n`+"`"+` times after `+"`"+`test`+"`"+` has terminated the recursion." +Attributes[NestWhile] = {Protected} +NestWhile[args__] := Last[NestWhileList[args]] +Tests`+"`"+`NestWhile = { + ESimpleExamples[ + ESameTest[256, NestWhile[#^2 &, 2, # < 256 &]], + ESameTest[7, NestWhile[#+1 &, 1, # + #4 < 10 &, 4]], + ESameTest[5, NestWhile[#+1 &, 1, True &, 1, 4]], + ESameTest[10, NestWhile[#+1 &, 1, True &, 1, 4, 5]] + ] +} + +FixedPointList::usage = "`+"`"+`FixedPointList[f, expr]`+"`"+` applies `+"`"+`f`+"`"+` to `+"`"+`expr`+"`"+` until `+"`"+`UnsameQ`+"`"+` applied to the two most recent results +returns False. It returns a list of all intermediate results." +FixedPointList[f_, expr_] := NestWhileList[f, expr, UnsameQ, 2] +Tests`+"`"+`FixedPointList = { + ESimpleExamples[ + ESameTest[7, Length@FixedPointList[(# + 3/#)/2 &, 1.0]], + ESameTest[{x^3, 3 x^2, 6 x, 6, 0, 0}, FixedPointList[D[#, x] &, x^3]] + ] +} + +FixedPoint::usage = "`+"`"+`FixedPoint[f, expr]`+"`"+` applies `+"`"+`f`+"`"+` to `+"`"+`expr`+"`"+` until `+"`"+`UnsameQ`+"`"+` applied to the two most recent results +returns False." +(*FixedPoint[f_, expr_] := Module[{currVal=expr, nextVal=f[expr]},*) + (*While[UnsameQ[currVal, nextVal],*) + (*currVal = nextVal;*) + (*nextVal = f[currVal];*) + (*];*) + (*nextVal]*) +Tests`+"`"+`FixedPoint = { + ESimpleExamples[ + ESameTest[0, FixedPoint[D[#, x] &, x^3]] + ] +} + +Array::usage = "`+"`"+`Array[f, n]`+"`"+` creates a list of `+"`"+`f[i]`+"`"+`, with `+"`"+`i`+"`"+` = 1 to `+"`"+`n`+"`"+`."; +Attributes[Array] = {Protected}; +Tests`+"`"+`Array = { + ESimpleExamples[ + ESameTest[{f[1], f[2], f[3]}, Array[f, 3]], + ESameTest[Null, mytest[x_] := 5], + ESameTest[{5, 5, 5}, Array[mytest, 3]], + ESameTest[{(a + b)[1], (a + b)[2], (a + b)[3]}, Array[a + b, 3]], + ESameTest[Array[a, a], Array[a, a]] + ] +}; + +Identity::usage = "`+"`"+`Identity[expr_]`+"`"+` returns `+"`"+`expr`+"`"+`."; +Identity[expr_] := expr; +Attributes[Identity] = {Protected}; +Tests`+"`"+`Identity = { + ESimpleExamples[ + ESameTest[5, Identity[5]], + ESameTest[a, Identity[Identity[a]]] + ] +}; +`) + +func resourcesFunctionalMBytes() ([]byte, error) { + return _resourcesFunctionalM, nil +} + +func resourcesFunctionalM() (*asset, error) { + bytes, err := resourcesFunctionalMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/functional.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesInitM = []byte(`(* This file is executed upon initialization. *) +SeedRandom[UnixTime[]]; +$Version = "Expreduce"; +$ModuleNumber = 1; +$Path = {"."}; +$Line = 1; +`) + +func resourcesInitMBytes() ([]byte, error) { + return _resourcesInitM, nil +} + +func resourcesInitM() (*asset, error) { + bytes, err := resourcesInitMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/init.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesListM = []byte(`List::usage = "`+"`"+`{e1, e2, ...}`+"`"+` groups expressions together."; +Attributes[List] = {Locked, Protected}; +Tests`+"`"+`List = { + ESimpleExamples[ + ESameTest[{1, 2, a}, List[1,2,a]] + ] +}; + +Total::usage = "`+"`"+`Total[list]`+"`"+` sums all the values in `+"`"+`list`+"`"+`."; +Total[l__List] := Apply[Plus, l]; +Attributes[Total] = {Protected}; +Tests`+"`"+`Total = { + ESimpleExamples[ + ESameTest[10, Total[{1,2,3,4}]] + ], EFurtherExamples[ + EComment["The total of an empty list is zero:"], + ESameTest[0, Total[{}]] + ] +}; + +Mean::usage = "`+"`"+`Mean[list]`+"`"+` calculates the statistical mean of `+"`"+`list`+"`"+`."; +Mean[l__List] := Total[l]/Length[l]; +Attributes[Mean] = {Protected}; +Tests`+"`"+`Mean = { + ESimpleExamples[ + ESameTest[11/2, Mean[{5,6}]] + ] +}; + +Table::usage = "`+"`"+`Table[expr, n]`+"`"+` returns a list with `+"`"+`n`+"`"+` copies of `+"`"+`expr`+"`"+`. + +`+"`"+`Table[expr, {sym, n}]`+"`"+` returns a list with `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = 1 to `+"`"+`n`+"`"+`. + +`+"`"+`Table[expr, {sym, m, n}]`+"`"+` returns a list with `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`."; +(*Use a UniqueDefined`+"`"+` prefix, or else Table[i, 5] will return*) +(*incorrect results.*) +Table[a_, b_Integer] := Table[a, {UniqueDefined`+"`"+`i, 1, b}]; +Attributes[Table] = {HoldAll, Protected}; +Tests`+"`"+`Table = { + ESimpleExamples[ + ESameTest[{a, a, a, a, a}, Table[a, 5]], + ESameTest[{5, 6, 7, 8, 9, 10}, Table[i, {i, 5, 10}]], + EComment["Create a list of the first 10 squares:"], + ESameTest[{1, 4, 9, 16, 25, 36, 49, 64, 81, 100}, Table[n^2, {n, 1, 10}]], + EComment["Iteration definitions do not have side effects:"], + EStringTest["i", "i"], + ESameTest[22, listTableTestState`+"`"+`i = 22], + ESameTest[{5, 6, 7, 8, 9, 10}, Table[i, {i, 5, 10}]], + EStringTest["22", "i"] + ], EFurtherExamples[ + ESameTest[{0,1,2}, Table[x[99], {x[_], 0, 2}]] + ], ETests[ + EComment["Test proper evaluation of the iterspec."], + ESameTest[Null, testn := 5;], + ESameTest[{1, 2, 3, 4, 5}, Table[i, {i, testn}]] + ] +}; + +ParallelTable::usage = "`+"`"+`ParallelTable[expr, n]`+"`"+` returns a list with `+"`"+`n`+"`"+` copies of `+"`"+`expr`+"`"+`, evaluated in parallel. + +`+"`"+`ParallelTable[expr, {sym, n}]`+"`"+` returns a list with `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = 1 to `+"`"+`n`+"`"+`, evaluated in parallel. + +`+"`"+`ParallelTable[expr, {sym, m, n}]`+"`"+` returns a list with `+"`"+`expr`+"`"+` evaluated with `+"`"+`sym`+"`"+` = `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`, evaluated in parallel."; +ParallelTable[a_, b_Integer] := ParallelTable[a, {UniqueDefined`+"`"+`i, 1, b}]; +Attributes[ParallelTable] = {HoldAll, Protected}; +Tests`+"`"+`ParallelTable = { + ESimpleExamples[ + ESameTest[{5, 6, 7, 8, 9, 10}, ParallelTable[i, {i, 5, 10}]], + ] +}; + +MemberQ::usage = "`+"`"+`MemberQ[expr, pat]`+"`"+` returns True if any of the elements in `+"`"+`expr`+"`"+` match `+"`"+`pat`+"`"+`, otherwise returns False."; +Attributes[MemberQ] = {Protected}; +Tests`+"`"+`MemberQ = { + ESimpleExamples[ + ESameTest[False, MemberQ[{1, 2, 3}, 0]], + ESameTest[True, MemberQ[{1, 2, 3}, 1]], + ESameTest[False, MemberQ[{1, 2, 3}, {1}]], + EComment["`+"`"+`MemberQ`+"`"+` works with patterns:"], + ESameTest[True, MemberQ[{1, 2, 3}, _Integer]], + ESameTest[True, MemberQ[{1, 2, 3}, _]], + ESameTest[False, MemberQ[{1, 2, 3}, _Real]], + ESameTest[True, MemberQ[{1, 2, 3}, testmatch_Integer]], + EStringTest["testmatch", "testmatch"], + ESameTest[False, MemberQ[a, a]], + ESameTest[False, MemberQ[a, _]], + (*More tests to be used in OrderlessIsMatchQ*) + ESameTest[False, MemberQ[{a, b}, c]], + ESameTest[True, MemberQ[{a, b}, a]] + ], EFurtherExamples[ + EComment["`+"`"+`MemberQ`+"`"+` works with BlankSequences:"], + ESameTest[True, MemberQ[{a, b}, ___]], + ESameTest[True, MemberQ[{a, b}, __]], + ESameTest[False, MemberQ[{a, b}, __Integer]], + ESameTest[False, MemberQ[{a, b}, ___Integer]], + ESameTest[True, MemberQ[{a, b}, ___Symbol]], + ESameTest[True, MemberQ[{a, b}, __Symbol]], + ESameTest[True, MemberQ[{a, b, 1}, __Symbol]], + ESameTest[True, MemberQ[{a, b, 1}, __Integer]], + EComment["`+"`"+`expr`+"`"+` need not be a List:"], + ESameTest[True, MemberQ[bar[a, b, c], a]], + ESameTest[False, MemberQ[bar[a, b, c], bar]] + ] +}; + +Cases::usage = "`+"`"+`Cases[expr, pat]`+"`"+` returns a new `+"`"+`List`+"`"+` of all elements in `+"`"+`expr`+"`"+` that match `+"`"+`pat`+"`"+`."; +Attributes[Cases] = {Protected}; +Tests`+"`"+`Cases = { + ESimpleExamples[ + ESameTest[{5, 2, 3.5, x, y, 4}, Cases[{5, 2, 3.5, x, y, 4}, _]], + ESameTest[{5,2,4}, Cases[{5, 2, 3.5, x, y, 4}, _Integer]], + ESameTest[{3.5}, Cases[{5, 2, 3.5, x, y, 4}, _Real]], + ESameTest[{2,c}, Cases[{b^2,1,a^c},_^e_->e]] + ], EFurtherExamples[ + EComment["`+"`"+`expr`+"`"+` need not be a list:"], + ESameTest[{a}, Cases[bar[a, b, c], a]] + ] +}; + +DeleteCases::usage = "`+"`"+`DeleteCases[expr, pat]`+"`"+` returns a new expression of all elements in `+"`"+`expr`+"`"+` that do not match `+"`"+`pat`+"`"+`."; +Attributes[DeleteCases] = {Protected}; +Tests`+"`"+`DeleteCases = { + ESimpleExamples[ + ESameTest[{3.5,x,y}, DeleteCases[{5,2,3.5,x,y,4},_Integer]], + ESameTest[{5,2,x,y,4}, DeleteCases[{5,2,3.5,x,y,4},_Real]], + ESameTest[x+y, DeleteCases[3.5+x+y,_Real]] + ] +}; + +Union::usage = "`+"`"+`Union[expr1, expr2, ...]`+"`"+` returns a sorted union of the items in the expressions."; +Attributes[Union] = {Protected}; +Tests`+"`"+`Union = { + ESimpleExamples[ + ESameTest[{a,b}, Union[{b,a,a,b,a}]], + ESameTest[{a,b,y,z}, Union[{b,a,a,b,a},{y,z}]], + ESameTest[foo[a,b,y,z], Union[foo[b,a,a,b,a],foo[y,z]]] + ], ETests[ + ESameTest[Union[foo[b,a,a,b,a],{y,z}], Union[foo[b,a,a,b,a],{y,z}]], + ESameTest[{}, Union[]], + ESameTest[Union[{b,a,a,b,a},z], Union[{b,a,a,b,a},z]], + ESameTest[{a}, Union[{a}]], + ESameTest[{List}, Union[{List}]] + ] +}; + +Complement::usage = "`+"`"+`Complement[expr1, expr2, ...]`+"`"+` returns a sorted union of the items in the expressions."; +Attributes[Complement] = {Protected}; +Tests`+"`"+`Complement = { + ESimpleExamples[ + ESameTest[{b}, Complement[{a, b}, {a}]], + ESameTest[{c}, Complement[{a, b, c}, {a}, {b}]] + ], ETests[ + ESameTest[foo[], Complement[foo[a], foo[a]]], + ESameTest[Complement[], Complement[]], + ESameTest[{}, Complement[{}]], + ESameTest[Complement[{}, foo[a]], Complement[{}, foo[a]]], + ESameTest[{b, c}, Complement[{a, b, c}, {a}, {_}]], + ESameTest[{b, c}, Complement[{a, b, c, _}, {a}, {_}]], + ESameTest[{b, c}, Complement[{a, b, c, _}, {a}, {_}, {}]], + ESameTest[{b, c}, Complement[{a, c, b, _}, {a}, {_}, {}]], + ESameTest[{b, c}, Complement[{a, c, c, b, _}, {a}, {_}, {}]] + ] +}; + +Intersection::usage = "`+"`"+`Intersection[expr1, expr2, ...]`+"`"+` returns a sorted intersection of the items in the expressions."; +Attributes[Intersection] = {Flat, OneIdentity, Protected, ReadProtected}; +Tests`+"`"+`Complement = { + ESimpleExamples[ + ESameTest[{a}, Intersection[{a, b}, {a}]], + ESameTest[{}, Intersection[{a, b, c}, {a}, {b}]] + ], ETests[ + ESameTest[foo[a], Intersection[foo[a], foo[a]]], + ] +}; + +Range::usage = "`+"`"+`Range[n]`+"`"+` returns a `+"`"+`List`+"`"+` of integers from 1 to `+"`"+`n`+"`"+`. + +`+"`"+`Range[m, n]`+"`"+` returns a `+"`"+`List`+"`"+` of integers from `+"`"+`m`+"`"+` to `+"`"+`n`+"`"+`."; +Attributes[Range] = {Listable, Protected}; +Tests`+"`"+`Range = { + ESimpleExamples[ + ESameTest[{1, 2, 3}, Range[3]], + ESameTest[{2, 3, 4, 5}, Range[2, 5]], + ESameTest[{}, Range[2, -5]] + ] +}; + +Part::usage = "`+"`"+`expr[[i]]`+"`"+` or `+"`"+`Part[expr, i]`+"`"+` returns the `+"`"+`i`+"`"+`th element of `+"`"+`expr`+"`"+`."; +Attributes[Part] = {NHoldRest, ReadProtected, Protected}; +Tests`+"`"+`Part = { + ESimpleExamples[ + EComment["Return the second item in a list:"], + ESameTest[b, {a, b, c, d}[[2]]], + EComment["Multi-dimensional indices are supported:"], + ESameTest[{{1, 4, 9, 16, 25}, {2, 8, 18, 32, 50}, {3, 12, 27, 48, 75}, {4, 16, 36, 64, 100}, {5, 20, 45, 80, 125}}, mat = Table[Table[a*b^2, {b, 5}], {a, 5}]], + ESameTest[20, mat[[5, 2]]], + EComment["Use `+"`"+`All`+"`"+` to select along the entire dimension:"], + ESameTest[{5, 20, 45, 80, 125}, mat[[5, All]]] + ], EFurtherExamples[ + EComment["Out of bounds issues will prevent the expression from evaluating:"], + ESameTest[{a}[[2]], Part[{a}, 2]], + EComment["The input need not be a `+"`"+`List`+"`"+`:"], + ESameTest[foo, Part[foo[a], 0]], + EComment["Omitting an index will return the original expression:"], + ESameTest[i, Part[i]], + ESameTest[{2,4,6}, {{1, 2}, {3, 4}, {5, 6}}[[All, 2]]] + ], ETests[ + ESameTest[i, Part[i]], + ESameTest[Part[], Part[]], + ESameTest[{a, b}[[1.5]], Part[{a, b}, 1.5]], + ESameTest[{a, b}[[a, 1.5]], Part[{a, b}, a, 1.5]], + ESameTest[foo, Part[foo[a], 0]], + ESameTest[{{1, 4, 9, 16, 25}, {2, 8, 18, 32, 50}, {3, 12, 27, 48, 75}, {4, 16, 36, 64, 100}, {5, 20, 45, 80, 125}}, mat = Table[Table[a*b^2, {b, 5}], {a, 5}]], + ESameTest[20, mat[[5, 2]]], + ESameTest[{5, 20, 45, 80, 125}, mat[[5, All]]], + ESameTest[foo[a, b, c], foo[a, b, c][[All]]], + ESameTest[1[[5]], Part[1, 5]], + ESameTest[a, Part[{a}, 1]], + ESameTest[{a}[[2]], Part[{a}, 2]], + ESameTest[{5, 20, 45, 80, 125}, mat[[All]][[5]]], + ESameTest[3, {{1, 2}, {3, 4}}[[2, 1]]], + ESameTest[{{1, 2}, {3}}[[2, 2]], {{1, 2}, {3}}[[2, 2]]], + ESameTest[{3,4}, {{1, 2}, {3, 4}, {5, 6}}[[2, All]]], + ESameTest[{25, 50, 75, 100, 125}, mat[[All, 5]]] + ], EKnownFailures[ + ESameTest[Integer[], Part[1, All]], + ESameTest[Symbol[], Part[a, All]] + ] +}; + +Span::usage = "`+"`"+`start ;; end`+"`"+` represents an index span to select using Part."; +Attributes[Span] = {Protected}; +Tests`+"`"+`Span = { + ESimpleExamples[ + ESameTest[{b, c}, {a, b, c, d}[[2 ;; 3]]], + ESameTest[{b, c, d}, {a, b, c, d}[[2 ;; All]]] + ] +}; + +All::usage = "`+"`"+`All`+"`"+` allows selection along a dimension in `+"`"+`Part`+"`"+`."; +Attributes[All] = {Protected}; +Tests`+"`"+`All = { + ESimpleExamples[ + ESameTest[{{1, 4, 9, 16, 25}, {2, 8, 18, 32, 50}, {3, 12, 27, 48, 75}, {4, 16, 36, 64, 100}, {5, 20, 45, 80, 125}}, mat = Table[Table[a*b^2, {b, 5}], {a, 5}]], + EComment["Use `+"`"+`All`+"`"+` to select along the entire dimension:"], + ESameTest[{5, 20, 45, 80, 125}, mat[[5, All]]] + ] +}; + +Thread::usage = "`+"`"+`Thread[f[a1, a2, ...}]]`+"`"+` applies f over the arguments, expanding out any lists."; +Attributes[Thread] = {Protected}; +Tests`+"`"+`Thread = { + ETests[ + ESameTest[{f[x], f[y], f[z]}, Thread[f[{x, y, z}]]], + ESameTest[{f[x, b], f[y, b], f[z, b]}, Thread[f[{x, y, z}, b]]], + ESameTest[f[{x, y, z}, {b}], Thread[f[{x, y, z}, {b}]]], + ESameTest[{f[{x, y, z}, b]}, Thread[f[{{x, y, z}}, {b}]]], + ESameTest[{f[{x, y, z}, b]}, Thread[f[{{x, y, z}}, b]]], + ESameTest[{f[x, b, a], f[y, b, b], f[z, b, c]}, Thread[f[{x, y, z}, b, {a, b, c}]]], + ESameTest[{mypos[-1], mypos[4], mypos[5]}, Thread[mypos[{-1, 4, 5}]]], + ESameTest[f[a, b, c], Thread[f[a, b, c]]], + ESameTest[{f[1]}, Thread[f[{1}]]], + ESameTest[{0, 1, 2}, Thread[{0, 1, 2}]], + ESameTest[{{0, a}, {1, a}, {2, a}}, Thread[{{0, 1, 2}, a}]], + ESameTest[a, Thread[a]], + ESameTest[Thread[], Thread[]] + ] +}; + +Append::usage = "`+"`"+`Append[list, e]`+"`"+` returns `+"`"+`list`+"`"+` with `+"`"+`e`+"`"+` appended."; +Attributes[Append] = {Protected}; +Tests`+"`"+`Append = { + ESimpleExamples[ + ESameTest[{a,b,c}, Append[{a,b},c]], + ESameTest[foo[a,b,c], Append[foo[a,b],c]] + ] +}; + +AppendTo::usage = "`+"`"+`AppendTo[list, e]`+"`"+` appends `+"`"+`e`+"`"+` to `+"`"+`list`+"`"+` and returns the modified `+"`"+`list`+"`"+`."; +AppendTo[list_, e_] := (list = Append[list, e]); +Attributes[AppendTo] = {HoldFirst, Protected}; +Tests`+"`"+`AppendTo = { + ESimpleExamples[ + ESameTest[{a,b,c}, l = {a, b}; AppendTo[l, c]; l] + ] +}; + +Prepend::usage = "`+"`"+`Prepend[list, e]`+"`"+` returns `+"`"+`list`+"`"+` with `+"`"+`e`+"`"+` prepended."; +Attributes[Prepend] = {Protected}; +Tests`+"`"+`Prepend = { + ESimpleExamples[ + ESameTest[{c,a,b}, Prepend[{a,b},c]], + ESameTest[foo[c,a,b], Prepend[foo[a,b],c]] + ] +}; + +PrependTo::usage = "`+"`"+`PrependTo[list, e]`+"`"+` prepends `+"`"+`e`+"`"+` to `+"`"+`list`+"`"+` and returns the modified `+"`"+`list`+"`"+`."; +PrependTo[list_, e_] := (list = Prepend[list, e]); +Attributes[PrependTo] = {HoldFirst, Protected}; +Tests`+"`"+`PrependTo = { + ESimpleExamples[ + ESameTest[{c,a,b}, l = {a, b}; PrependTo[l, c]; l] + ] +}; + +DeleteDuplicates::usage = "`+"`"+`DeleteDuplicates[list]`+"`"+` returns `+"`"+`list`+"`"+` with the duplicates removed."; +Attributes[DeleteDuplicates] = {Protected}; +Tests`+"`"+`DeleteDuplicates = { + ESimpleExamples[ + ESameTest[{b,a}, DeleteDuplicates[{b,a,b}]], + ESameTest[foo[b,a], DeleteDuplicates[foo[b,a,b]]], + ESameTest[{}, DeleteDuplicates[{}]], + ESameTest[10000, Length[DeleteDuplicates[Range[10000]]]] + ] +}; + +Last::usage = "`+"`"+`Last[expr]`+"`"+` returns the last part of `+"`"+`expr`+"`"+`."; +Last[e_?((Length[#]>=1)&)] := e[[Length[e]]]; +Attributes[Last] = {Protected}; +Tests`+"`"+`Last = { + ESimpleExamples[ + ESameTest[6, Last[{1,5,6}]], + ESameTest[b, Last[a+b]] + ], ETests[ + ESameTest[Last[a], Last[a]], + ESameTest[Last[{}], Last[{}]], + ESameTest[a, Last[{a}]] + ] +}; + +First::usage = "`+"`"+`First[expr]`+"`"+` returns the first part of `+"`"+`expr`+"`"+`."; +First[e_?((Length[#]>=1)&)] := e[[1]]; +Attributes[First] = {Protected}; +Tests`+"`"+`First = { + ESimpleExamples[ + ESameTest[1, First[{1,5,6}]], + ESameTest[a, First[a+b]] + ], ETests[ + ESameTest[First[a], First[a]], + ESameTest[First[{}], First[{}]], + ESameTest[a, First[{a}]] + ] +}; + +Rest::usage = "`+"`"+`Rest[expr]`+"`"+` returns all but the first part of `+"`"+`expr`+"`"+`."; +Rest[e_?((Length[#]>=1)&)] := e[[2;;All]]; +Attributes[Rest] = {Protected}; +Tests`+"`"+`Rest = { + ESimpleExamples[ + ESameTest[{5,6}, Rest[{1,5,6}]], + ESameTest[b+c, Rest[a+b+c]] + ], ETests[ + ESameTest[Rest[a], Rest[a]], + ESameTest[Rest[{}], Rest[{}]], + ESameTest[{}, Rest[{a}]] + ] +}; + +Select::usage = "`+"`"+`Select[expr, cond]`+"`"+` selects only parts of `+"`"+`expr`+"`"+` that satisfy `+"`"+`cond`+"`"+`."; +Attributes[Select] = {Protected}; +Tests`+"`"+`Select = { + ESimpleExamples[ + ESameTest[{1,3,5,7,9,11,13,15,17,19}, Select[Range[20],OddQ]], + ESameTest[{1,2,3,4}, Select[{1,2,3,4},(True)&]], + ESameTest[{1,2}, Select[{1,2,3,4},(True)&,2]] + ], ETests[ + ESameTest[{}, Select[{1,2,3,4},(False)&]], + ESameTest[{}, Select[{1,2,3,4},(hello)&]], + ESameTest[foo[2,4], Select[foo[1,2,3,4],(EvenQ[#])&]], + ESameTest[Select[foo[1,2,3,4]], Select[foo[1,2,3,4]]], + ESameTest[foo[], Select[foo[1,2,3,4],notfunction]], + ESameTest[Select[2,EvenQ], Select[2,EvenQ]] + ] +}; + +ListQ::usage = "`+"`"+`ListQ[expr]`+"`"+` checks if `+"`"+`expr`+"`"+` has a head of `+"`"+`List`+"`"+`."; +ListQ[expr_] := Head[expr] === List; +Attributes[ListQ] = {Protected}; +Tests`+"`"+`ListQ = { + ESimpleExamples[ + ESameTest[True, ListQ[{a}]], + ESameTest[False, ListQ[a]], + ESameTest[True, ListQ[{}]] + ] +}; + +Scan::usage = "`+"`"+`Scan[fn, list]`+"`"+` evaluates `+"`"+`fn[elem]`+"`"+` for each element in `+"`"+`list`+"`"+`."; +Attributes[Scan] = {Protected}; +Tests`+"`"+`Scan = { + ESimpleExamples[ + ESameTest[6, Scan[(If[# > 5, Return[#]]) &, {1, 6, 9}]], + ESameTest[False, Catch[Scan[Function[If[IntegerQ[#], Null, Throw[False]]], {a}]; True]], + ESameTest[True, Catch[Scan[Function[If[IntegerQ[#], Null, Throw[False]]], {1, 2}]; True]], + ESameTest[False, Catch[Scan[Function[If[IntegerQ[#], Null, Throw[False]]], {1, a}]; True]] + ] +}; + +Join::usage = "`+"`"+`Join[l1, l2, ...]`+"`"+` joins the lists into a single list."; +Attributes[Join] = {Flat, OneIdentity, Protected}; +Tests`+"`"+`Join = { + ESimpleExamples[ + ESameTest[{a,b,c}, Join[{a},{b,c}]], + ESameTest[{}, Join[]], + ESameTest[foo[a,b,c], Join[foo[a],foo[b,c]]] + ] +}; + +Count::usage = "`+"`"+`Count[l, pattern]`+"`"+` returns the number of expressions in `+"`"+`l`+"`"+` matching `+"`"+`pattern`+"`"+`."; +Count[l_, pattern_] := Count[l, pattern, {1}]; +Attributes[Count] = {Protected}; +Tests`+"`"+`Count = { + ESimpleExamples[ + ESameTest[3, Count[a+b+c^2,_]], + ESameTest[5, Count[a+b+c^2,_,-1]], + ESameTest[5, Count[a+b+c^2,_,Infinity]], + ESameTest[0, Count[a,_,Infinity]], + ESameTest[0, Count[a,_,-1]], + ESameTest[2, Count[a + 2 + c^2, _Integer, Infinity]], + ] +}; + +Tally::usage = "`+"`"+`Tally[list]`+"`"+` creates tallies of the elements in `+"`"+`list`+"`"+`."; +Tally[l_List] := {#, Count[l, #]} & /@ DeleteDuplicates[l]; +Attributes[Tally] = {Protected}; +Tests`+"`"+`Tally = { + ESimpleExamples[ + ESameTest[{{a, 2}, {b, 2}}, Tally[{a, a, b, b}]], + ESameTest[{{b, 2}, {a, 2}}, Tally[{b, b, a, a}]], + ESameTest[{{b, 2}, {a, 1}}, Tally[{b, b, a}]], + ] +}; + +ConstantArray::usage = "`+"`"+`ConstantArray[c, n]`+"`"+` creates a list of `+"`"+`n`+"`"+` copies of `+"`"+`c`+"`"+`."; +Attributes[ConstantArray] = {Protected}; +ConstantArray[c_, n_Integer] := Table[c, n]; +Tests`+"`"+`ConstantArray = { + ESimpleExamples[ + ESameTest[{a, a, a}, ConstantArray[a, 3]], + ] +}; + +Reverse::usage = "`+"`"+`Reverse[list]`+"`"+` evaluates to a reversed copy of `+"`"+`list`+"`"+`."; +Attributes[Reverse] = {Protected}; +Tests`+"`"+`Reverse = { + ESimpleExamples[ + ESameTest[{5, 4, 3, 2, 1}, Reverse[Range[5]]], + ] +}; +`) + +func resourcesListMBytes() ([]byte, error) { + return _resourcesListM, nil +} + +func resourcesListM() (*asset, error) { + bytes, err := resourcesListMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/list.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesManipM = []byte(`ExpreduceDistributeMultiply[e_, multiplicand_] := + If[Head[e] === Plus, (#*multiplicand) & /@ e, e*multiplicand]; + +Together::usage = "`+"`"+`Together[e]`+"`"+` attempts to put the terms in `+"`"+`e`+"`"+` under the same denominator."; + +(*Factor out some operations*) +ExpreduceTogether[a_Plus*b_] := ExpreduceTogether[a]*b; +ExpreduceTogether[e_^p_?NumberQ] := ExpreduceTogether[e]^p; + +(*Process any denominator*) +ExpreduceTogether[c_.*1/d_ + rest_] := ExpreduceTogether[(c+Expand[ExpreduceDistributeMultiply[rest, d]])/d]; +(*Rationals have denominators too. Rational denominators are integers, and +thus automatically distributed through Plus, so no need for the distribute +function.*) +ExpreduceTogether[c_.*Rational[n_,d_] + rest_] := ExpreduceTogether[(c n+Expand[rest d])/d]; +ExpreduceTogether[c_.*Complex[0, Rational[n_,d_]] + rest_] := ExpreduceTogether[(I c n+Expand[rest d])/d]; + +ExpreduceTogether[e_] := e; +Together[e_] := ExpreduceFactorConstant[ExpreduceTogether[e]]; + +Attributes[Together] = {Listable, Protected}; +Tests`+"`"+`Together = { + ESimpleExamples[ + ESameTest[(6+a)/(2 a), 1/2+3/a//Together], + ESameTest[(1/2+3/a)^c, (1/2+3/a)^c//Together], + ESameTest[(6+a)^2/(4 a^2), (1/2+3/a)^2//Together] + ], ETests[ + ESameTest[(c+a d+b d)/d, a+b+c/d//Together], + ESameTest[(a+b+c+d)/((a+b) (c+d)), 1/(a+b)+1/(c+d)//Together], + ESameTest[(a+b+c-a^2*c-a b c+d-a^2*d-a b d)/((a+b) (c+d)), 1/(a+b)+1/(c+d)-a//Together], + (*Only for when we have Cancel:*) + (*ESameTest[-(a/(2+a)), Together[a/(-2+a+a^2)-a^2/(-2+a+a^2)]],*) + ESameTest[(a+b c)/b, a/b+c//Together], + ESameTest[(b c+a d)/(b d), (a+(b c)/d)/b//Together], + ESameTest[(b c+a d)/d, a+(b c)/d//Together], + ESameTest[(c+a d+b d)/d, Together[(c + a*d + b*d)/d]], + ESameTest[(6+a+2 b)/(2 a), (3+a/2+b)/a//Together], + ESameTest[a/(b e), (a/b)/e//Together], + ESameTest[(12+a)/(4 a), (3 + a*1/4)/a//Together], + ESameTest[(1/2) (6+foo[a]), 3 + foo[a]*2/4//Together], + ESameTest[(6+a)/(2 a), 1/2 + 3/a//Together], + ESameTest[(a+b c)/(a b), (c+ a/b)/a//Together], + ESameTest[(a+b c+b e)/(b d), (c+e+ a/b)/d//Together], + ESameTest[(4 a^2*b+36 c+12 a c+a^2*c)/(4 a^2*c), Together[(1/2 + 3/a)^2+b/c]], + ESameTest[(6+a)/(2 a), (3 + a/2)/a//Together], + ESameTest[(a b+c rest)/(c d), (rest + a*b/c)/d//Together], + ESameTest[(1+a c+b c)/(a+b), 1/(a+b)+c//Together], + ESameTest[(c+a d+b d)/d, Together[(c + a*d + b*d)/d]], + ESameTest[(6+a+2 b)/(2 a), (3+a/2+b)/a//Together], + ESameTest[a/(b e), (a/b)/e//Together], + ESameTest[(a+a^2*c+a b c+a d+b d)/(a (a+b)), Together[1/(a + b) + c+d/a]], + ESameTest[(a+b) (c+d), (a+b)(c+d)//Together], + ESameTest[(a+b+c+d)/((a+b) (c+d)), 1/(a + b)+1/(c+d)//Together], + ESameTest[1/(a+b), 1/(a + b)//Together], + ESameTest[(a+b c e f)/(b c), a/(b*c)+(e*f)//Together], + ESameTest[(a+b+c-a^2*c-a b c+d-a^2*d-a b d)/((a+b) (c+d)), Together[1/(a + b) + 1/(c + d) - a]], + ESameTest[(a+b)/(a b), 1/a+1/b//Together], + ESameTest[(a+b+a b c)/(a b), 1/a+1/b+c//Together], + ESameTest[(1+c d)/c, 1/c+d//Together], + ESameTest[1/2*(1+2 a), 1/2+a//Together], + ESameTest[(a+b+c+d)/((a+b) (c+d)), (1+a/(c+d)+b/(c+d))/(a+b)//Together], + ESameTest[(a+b+a b c+a b d)/(a b), 1/a+1/b+c+d//Together], + ESameTest[2(a+b), 2a+2b//Together], + ], EKnownFailures[ + ESameTest[(I (a-b))/(2 Subscript[\[Omega], 0]), (I a)/(2 Subscript[\[Omega], 0])-(I b)/(2 Subscript[\[Omega], 0])//Together], + ] +}; + +Numerator::usage = "`+"`"+`Numerator[e]`+"`"+` returns the numerator of `+"`"+`e`+"`"+`."; +denPattern := b_^n_?Negative | b_^(n : (-_Symbol)); +Numerator[prod_Times] := Times@@Cases[prod, Except[denPattern]]; +Numerator[Rational[n_,d_]] := n; +Numerator[e_] := e; +Attributes[Numerator] = {Listable, Protected}; +Tests`+"`"+`Numerator = { + ESimpleExamples[ + ESameTest[a, Numerator[a/b]], + ESameTest[a^2 b, Numerator[a^2*b]], + ESameTest[a^2, Numerator[a^2*b^-2]], + ESameTest[a^2 c, Numerator[a^2*b^-a*c]], + ESameTest[2, Numerator[2/3]] + ] +}; + +Denominator::usage = "`+"`"+`Denominator[e]`+"`"+` returns the denominator of `+"`"+`e`+"`"+`."; +Denominator[prod_Times] := Times@@Cases[prod, denPattern -> b^(-n)]; +Denominator[Rational[n_,d_]] := d; +Denominator[e_] := 1; +Attributes[Denominator] = {Listable, Protected}; +Tests`+"`"+`Denominator = { + ESimpleExamples[ + ESameTest[b c, Denominator[a/(b*c)]], + ESameTest[1, Denominator[a^2*b]], + ESameTest[b^2 c^3, Denominator[a^2*b^-2*c^-3]], + ESameTest[b^a c^d, Denominator[a^2*b^-a*c^-d]], + ESameTest[3, Denominator[2/3]] + ] +}; + +Apart::usage = "`+"`"+`Apart[e]`+"`"+` attempts to break apart the terms in `+"`"+`e`+"`"+`. Warning: not fully implemented."; +termsWithout[dTerms_, i_] := + Product[If[index === i, 1, dTerms[[index]]], {index, + Length[dTerms]}]; +getPowerApartForm[base_, exp_, denTerms_] := + Module[{b = base, n = exp, holders, form, dTerms = denTerms, + unDivForm, finalForm}, + holders = Table[Unique[], n//Evaluate]; + finalForm = Sum[(holders[[i]]/b^i), {i, n}]; + form = Sum[(holders[[i]]/b^i)*Times @@ dTerms, {i, n}]; + {finalForm, form, holders} + ]; +getApartForm[denTerms_, iVar_] := + Module[{dTerms = denTerms, i = iVar, holder, powerPattern}, + powerPattern := b_^n_Integer; + If[MatchQ[dTerms[[i]], powerPattern], + Replace[dTerms[[i]], + powerPattern :> getPowerApartForm[b, n, dTerms]], + holder = Unique[]; + {holder/dTerms[[i]], + holder*If[Length[dTerms] === 1, dTerms[[1]], + termsWithout[dTerms, i]], {holder}}] + ]; +myLinearApart[num_, denTerms_, var_] := + Module[{n = num, dTerms = denTerms, v = var, lhs, rhs, coeffs, + apartForm, coeffHolders, tmp, toSolve, finalForm}, + coeffHolders = {}; + apartForm = {}; + finalForm = {}; + Do[ + tmp = getApartForm[dTerms, i]; + coeffHolders = Join[coeffHolders, tmp[[3]]]; + AppendTo[apartForm, tmp[[2]]]; + AppendTo[finalForm, tmp[[1]]]; + , {i, Length[dTerms]}]; + apartForm = Plus @@ apartForm; + + lhs = CoefficientList[apartForm, v]; + rhs = PadRight[CoefficientList[n, v], Length[lhs]]; + toSolve = Table[lhs[[i]] == rhs[[i]], {i, Length[lhs]}]; + (*Print[toSolve, coeffHolders];*) + coeffs = Solve[toSolve, coeffHolders]; + If[Length[coeffs] === 0 || Head[coeffs] === Solve, + num/(Times @@ denTerms), + Plus @@ (finalForm /. coeffs[[1]]) + ] + ]; +ClearAll[denTerms]; +getDenTerms[den_Times] := List @@ den; +getDenTerms[den_] := {den}; +validDenTermQ[t_, x_] := + MatchQ[t, (Optional[a_?((FreeQ[#, x]) &)]* + x^Optional[n_Integer]) | (Optional[a_?((FreeQ[#, x]) &)]*x + + Optional[b_?((FreeQ[#, x]) &)])^Optional[n_?((FreeQ[#, x]) &)]]; +myApartUnexpanded[expr_Times, var_] := + Module[{e = expr, v = var, denTerms, num, den, divRes}, + num = Numerator[e]; + den = Denominator[e]; + If[Exponent[num, v] > Exponent[den, v], Return[ + divRes = PolynomialQuotientRemainder[num, den, v]; + divRes[[1]] + myApartUnexpanded[divRes[[2]]/den, v] + ]]; + denTerms = getDenTerms[den]; + (*If[Length[denTerms]<2,Return[e]];*) + + If[! AllTrue[denTerms, validDenTermQ[#, v] &], Return[e]]; + myLinearApart[num, denTerms, v] + ]; +myApartUnexpanded[expr_, var_] := expr; + +Apart[expr_, var_] := myApartUnexpanded[expr, var] // Expand; +Apart[e_] := Expand[e]; + +Attributes[Apart] = {Listable, Protected}; +Tests`+"`"+`Apart = { + ESimpleExamples[ + ESameTest[a^3+3 a^2 b+3 a b^2+b^3, Apart[(a + b)^3]], + ESameTest[7/(6 (-5+x))+11/(6 (1+x)), Apart[(3x-8)/((x+1)*(x-5)),x]], + (*ESameTest[13/(10 (-1+x))-5/(6 (1+x))-7/(15 (4+x)), Apart[(4x+9)/((-1+x) (1+x) (4+x)),x]],*) + (*ESameTest[1/b-a/(b (a+b x)), Apart[(x/(a+(b*x))),x]],*) + ESameTest[(5 a^4)/b^6-(4 a^3 x)/b^5+(3 a^2 x^2)/b^4-(2 a x^3)/b^3+x^4/b^2+a^6/(b^6 (a+b x)^2)-(6 a^5)/(b^6 (a+b x)), Apart[(x^6/((a+(b*x))*(a+(b*x)))),x]], + ESameTest[(5 a^4)/b^6-(4 a^3 x)/b^5+(3 a^2 x^2)/b^4-(2 a x^3)/b^3+x^4/b^2+a^6/(b^6 (a+b x)^2)-(6 a^5)/(b^6 (a+b x)), Apart[(x^6*(a+(b*x))^-2),x]], + (*ESameTest[2/(27 (-2+x))+1/(3 (1+x)^3)+7/(9 (1+x)^2)-2/(27 (1+x)), Apart[(x^2-2)/((x-2)(x+1)^3),x]]*) + ] +}; + +Distribute::usage = "`+"`"+`Distribute[e]`+"`"+` distributes the function over the `+"`"+`Plus`+"`"+` expressions."; +Distribute[e_] := Distribute[e, Plus]; +Attributes[Distribute] = {Protected}; +Tests`+"`"+`Distribute = { + ESimpleExamples[ + ESameTest[a c+b c+a d+b d, Distribute[(a+b)*(c+d)]], + ESameTest[a c+b c, Distribute[(a+b)*c]], + ESameTest[foo[a,c]+foo[b,c], Distribute[foo[(a+b),c]]], + ESameTest[foo[a,b], Distribute[foo[a,b]]], + ESameTest[foo[c]+foo[a,b], Distribute[foo[a,b]+foo[c]]], + ESameTest[a+(a+b) c, Distribute[(a+b)*(c)+a]], + ESameTest[(a+b) c e+d e+(a+b) c f+d f, Distribute[((a+b)*(c)+d)*(e+f)]], + ESameTest[test[foo[a,b]], Distribute[foo[a,b],test]], + ESameTest[test[foo[a,b],foo[a,c]], Distribute[foo[a,test[b,c]],test]], + ESameTest[test[foo[a,b,d],foo[a,b,e],foo[a,c,d],foo[a,c,e]], Distribute[foo[a,test[b,c],test[d,e]],test]], + ESameTest[test[foo[a,b,d,bar[a]],foo[a,b,e,bar[a]],foo[a,c,d,bar[a]],foo[a,c,e,bar[a]]], Distribute[foo[a,test[b,c],test[d,e],bar[a]],test]], + ESameTest[a, Distribute[a,test]], + ESameTest[1[a+b], Distribute[a+b,1]], + ESameTest[test[bar[a,b,d],bar[a,b,e],bar[a,c,d],bar[a,c,e]], Distribute[bar[a,test[b,c],test[d,e]],test]], + ESameTest[test[test[f,g][a,b,d],test[f,g][a,b,e],test[f,g][a,c,d],test[f,g][a,c,e]], Distribute[test[f,g][a,test[b,c],test[d,e]],test]], + ESameTest[test[foo[]], Distribute[foo[],test]], + ESameTest[test[][foo[]], Distribute[foo[],test[]]], + ESameTest[foo, Distribute[foo,test]] + ] +}; +`) + +func resourcesManipMBytes() ([]byte, error) { + return _resourcesManipM, nil +} + +func resourcesManipM() (*asset, error) { + bytes, err := resourcesManipMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/manip.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesMatrixM = []byte(`Inverse::usage = "`+"`"+`Inverse[mat]`+"`"+` finds the inverse of the square matrix `+"`"+`mat`+"`"+`."; +Inverse[{{a_}}] := {{1/a}}; +Inverse[{{a_, b_}, {c_, d_}}] := {{d/(-b c + a d), -(b/(-b c + a d))}, {-(c/(-b c + a d)), a/(-b c + a d)}}; +Inverse[{{a_, b_, c_}, {d_, e_, f_}, {h_, i_, j_}}] := {{(-f i + e j)/(-c e h + b f h + c d i - a f i - b d j + a e j), (c i - b j)/(-c e h + b f h + c d i - a f i - b d j + a e j), (-c e + b f)/(-c e h + b f h + c d i - a f i - b d j + a e j)}, {(f h - d j)/(-c e h + b f h + c d i - a f i - b d j + a e j), (-c h + a j)/(-c e h + b f h + c d i - a f i - b d j + a e j), (c d - a f)/(-c e h + b f h + c d i - a f i - b d j + a e j)}, {(-e h + d i)/(-c e h + b f h + c d i - a f i - b d j + a e j), (b h - a i)/(-c e h + b f h + c d i - a f i - b d j + a e j), (-b d + a e)/(-c e h + b f h + c d i - a f i - b d j + a e j)}}; +Attributes[Inverse] = {Protected}; +Tests`+"`"+`Inverse = { + ESimpleExamples[ + ESameTest[{{-2, 1}, {3/2, -(1/2)}}, Inverse[{{1, 2}, {3, 4}}]], + ESameTest[{{2/5, -(1/5), 0}, {-(1/15), 19/45, -(1/9)}, {-(1/15), -(11/45), 2/9}}, Inverse[{{3, 2, 1}, {1, 4, 2}, {2, 5, 7}}]] + ], EFurtherExamples[ + EComment["Symbolic elements are handled correctly:"], + ESameTest[Inverse[{{b}}], {{1/b}}] + ] +}; + +Dimensions::usage = "`+"`"+`Dimensions[expr]`+"`"+` finds the dimensions of `+"`"+`expr`+"`"+`."; +Attributes[Dimensions] = {Protected}; +Tests`+"`"+`Dimensions = { + ESimpleExamples[ + ESameTest[{2, 2}, Dimensions[{{1, 3}, {1, 2}}]], + ESameTest[{2}, Dimensions[{{{1, 2}, {3, 2}}, {{1, 2}}}]] + ], EFurtherExamples[ + ESameTest[{}, Dimensions[foo]], + EComment["`+"`"+`Dimensions`+"`"+` works with any head, not just `+"`"+`List`+"`"+`:"], + ESameTest[{0}, Dimensions[foo[]]] + ], ETests[ + ESameTest[{2}, Dimensions[{1, 2}]], + ESameTest[{0}, Dimensions[{}]], + ESameTest[{1}, Dimensions[foo[{1}]]], + ESameTest[{1, 1}, Dimensions[{{1}}]], + ESameTest[{2, 1}, Dimensions[{{1}, {1}}]], + ESameTest[{2}, Dimensions[{{1}, {1, 2}}]], + ESameTest[{2, 2}, Dimensions[{{1, 3}, {1, 2}}]], + ESameTest[{2, 2, 1}, Dimensions[{{{1}, {3}}, {{1}, {2}}}]], + ESameTest[{2, 2, 2}, Dimensions[{{{1, 2}, {3, 2}}, {{1, 2}, {2, 2}}}]], + ESameTest[{2}, Dimensions[{{{1, 2}, {3, 2}}, {{1, 2}}}]], + ESameTest[{2, 2, 2}, Dimensions[{{{1, 2}, {3, 2}}, {{1, 2}, {foo, 2}}}]], + ESameTest[{2, 2}, Dimensions[{{{1, 2}, {3, 2}}, {{1, 2}, foo[foo, 2]}}]], + ESameTest[{1, 2, 0}, Dimensions[{{{}, {}}}]], + ESameTest[{1, 2}, Dimensions[{{{}, foo[x]}}]] + ] +}; + +VectorQ::usage = "`+"`"+`VectorQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is a vector, False otherwise."; +Attributes[VectorQ] = {Protected}; +Tests`+"`"+`VectorQ = { + ESimpleExamples[ + ESameTest[True, VectorQ[{1, 2, c}]], + ESameTest[True, VectorQ[{1, 2, foo[a]}]], + ESameTest[False, VectorQ[foo[1, 2, 3]]], + ESameTest[False, VectorQ[{1, 2, 3, {}}]], + ESameTest[True, VectorQ[{f[a], f[b]}]], + ESameTest[True, VectorQ[{a, c}]], + ESameTest[True, VectorQ[{}]] + ] +}; + +MatrixQ::usage = "`+"`"+`MatrixQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is a 2D matrix, False otherwise."; +Attributes[MatrixQ] = {Protected}; +Tests`+"`"+`MatrixQ = { + ESimpleExamples[ + ESameTest[False, MatrixQ[{}]], + ESameTest[True, MatrixQ[{{}}]], + ESameTest[True, MatrixQ[{{a}}]], + ESameTest[False, MatrixQ[{{{}}}]], + ESameTest[False, MatrixQ[{{{a}}}]], + ESameTest[True, MatrixQ[{{a}, {b}}]], + ESameTest[True, MatrixQ[{{a, b}, {c, d}}]], + ESameTest[False, MatrixQ[{{a, b, e}, {c, d}}]], + ESameTest[True, MatrixQ[{{a, b, e}, {c, d, f}}]], + ESameTest[False, MatrixQ[{{{a}, {b}}, {{c}, {d}}}]], + ESameTest[True, MatrixQ[{{a, b, e}}]] + ] +}; + +Dot::usage = "`+"`"+`a.b`+"`"+` computes the product of `+"`"+`a`+"`"+` and `+"`"+`b`+"`"+` for vectors and matrices."; +Dot[m_?MatrixQ,v_?VectorQ]:=Table[m[[idx]].v,{idx,Length[v]}]; +Attributes[Dot] = {Flat, OneIdentity, Protected}; +Tests`+"`"+`Dot = { + ESimpleExamples[ + ESameTest[a c + b d, {a, b}.{c, d}], + ESameTest[{a, b}.{c, d, e}, {a, b}.{c, d, e}], + ESameTest[Dot[1, {c, d, e}], Dot[1, {c, d, e}]], + ESameTest[0, Dot[{}, {}]], + ESameTest[{{a, b}, {c, d}}.{e, f, g}, {{a, b}, {c, d}}.{e, f, g}], + ESameTest[{a, b}, Dot[{a, b}]], + ESameTest[a, Dot[a]], + ESameTest[1, Dot[1]], + ESameTest[{{a e + b g, a f + b h}, {c e + d g, c f + d h}}, {{a, b}, {c, d}}.{{e, f}, {g, h}}], + ESameTest[{{a e + b f}, {c e + d f}}, {{a, b}, {c, d}}.{{e}, {f}}], + ESameTest[{{a, b}, {c, d}}.{{e, f}}, {{a, b}, {c, d}}.{{e, f}}] + ], EKnownFailures[ + ESameTest[{a e + b f, c e + d f}, {{a, b}, {c, d}}.{e, f}], + ESameTest[{a e + c f, b e + d f}, {e, f}.{{a, b}, {c, d}}] + ] +}; + +Transpose::usage = "`+"`"+`Transpose[mat]`+"`"+` transposes the first two levels of `+"`"+`mat`+"`"+`"; +Attributes[Transpose] = {Protected}; +Tests`+"`"+`Transpose = { + ESimpleExamples[ + ESameTest[Transpose[{a, b}], Transpose[{a, b}]], + ESameTest[{{a, b}}, Transpose[{{a}, {b}}]], + ESameTest[{{a}, {b}}, Transpose[{{a, b}}]], + ESameTest[{{{a}}, {{b}}}, Transpose[{{{a}, {b}}}]], + ESameTest[Transpose[a], Transpose[a]] + ] +}; +`) + +func resourcesMatrixMBytes() ([]byte, error) { + return _resourcesMatrixM, nil +} + +func resourcesMatrixM() (*asset, error) { + bytes, err := resourcesMatrixMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/matrix.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesNumbertheoryM = []byte(`PrimeQ::usage = "`+"`"+`PrimeQ[n]`+"`"+` returns True if `+"`"+`n`+"`"+` is prime, False otherwise."; +Attributes[PrimeQ] = {Listable, Protected}; +Tests`+"`"+`PrimeQ = { + ESimpleExamples[ + ESameTest[True, PrimeQ[5]], + ESameTest[False, PrimeQ[100]], + ESameTest[True, PrimeQ[982451653]], + ESameTest[True, PrimeQ[-2]] + ], EFurtherExamples[ + EComment["`+"`"+`PrimeQ`+"`"+` only works for Integers:"], + ESameTest[False, PrimeQ[5.]] + ], ETests[ + ESameTest[False, PrimeQ[0]], + ESameTest[False, PrimeQ[1]], + ESameTest[False, PrimeQ[-1]], + ESameTest[False, PrimeQ[0.5]] + ] +}; + +GCD::usage = "`+"`"+`GCD[n1, n2, ...]`+"`"+` finds the greatest common denominator of the integer inputs."; +(* Eventually we should not need the rest___ term. GCD is Flat. *) +GCD[Rational[a_, b_], Rational[c_, d_], rest___] := + GCD[GCD[a*d, c*b]/(b*d), rest]; +GCD[Rational[a_, b_], c_Integer, rest___] := + GCD[GCD[a, c*b]/b, rest]; +(*This deviates from how it should be done*) +GCD[a_Rational] := Abs[a]; +Attributes[GCD] = {Flat, Listable, OneIdentity, Orderless, Protected}; +Tests`+"`"+`GCD = { + ESimpleExamples[ + ESameTest[3, GCD[9, 6]], + ESameTest[5, GCD[100, 30, 15]] + ], ETests[ + ESameTest[1, GCD[9, 2]], + ESameTest[10, GCD[100, 0, 10]], + ESameTest[3, GCD[9, 3]], + ESameTest[10, GCD[100, 30, 10]], + ESameTest[10, GCD[100, 30]], + ESameTest[1, GCD[100, 30, -1]], + ESameTest[10, GCD[100, 30, -60]], + ESameTest[60, GCD[-60, -60, -60]], + ESameTest[GCD[-60, -60, -0.5], GCD[-60, -60, -0.5]], + ESameTest[GCD[0.5], GCD[0.5]], + ESameTest[GCD[1, a], GCD[1, a]], + ESameTest[GCD[a, a], GCD[a, a]], + ESameTest[0, GCD[]], + ESameTest[1, GCD[1]], + ESameTest[GCD[a], GCD[a]], + ESameTest[1000, GCD[1000]], + ESameTest[5, GCD[5, 15]], + ESameTest[5, GCD[5, 15, 30]], + ESameTest[5, GCD[10, 20, 25]], + ESameTest[1, GCD[5, 14]], + ESameTest[5/2, GCD[5/2, 15/2]], + ESameTest[5/3, GCD[5/3, 5]], + ESameTest[GCD[5/3,a], GCD[5/3, a]], + ESameTest[GCD[a,b,c], GCD[a, b, c]], + ESameTest[0, GCD[0]], + ESameTest[0, GCD[0, 0]], + ESameTest[99, GCD[-99]], + ESameTest[5/2, GCD[-5/2]], + ESameTest[5, GCD[10, -20, 25]], + ESameTest[1, GCD[5, -14]], + ESameTest[5/2, GCD[5/2, -15/2]], + ESameTest[5/2, GCD[5/2, -15/2, -15/2]], + ESameTest[5/3, GCD[-5/3, -5]], + ESameTest[5/3, GCD[-5/3, -5]], + ESameTest[GCD[-(5/3),a], GCD[-5/3, a]] + ] +}; + +LCM::usage = "`+"`"+`LCM[n1, n2, ...]`+"`"+` finds the least common multiple of the inputs."; +LCM[a_?NumberQ, b_?NumberQ] := (a/GCD[a, b])*b; +LCM[a_?NumberQ, b_?NumberQ, rest__?NumberQ] := + LCM[LCM[a, b], rest]; +Attributes[LCM]={Flat,Listable,OneIdentity,Orderless,Protected}; +Tests`+"`"+`LCM = { + ESimpleExamples[ + ESameTest[70, LCM[5, 14]], + ESameTest[2380, LCM[5, 14, 68]], + ESameTest[2/3, LCM[2/3, 1/3]], + ESameTest[10/3, LCM[2/3, 1/3, 5/6]], + ESameTest[30, LCM[2/3, 1/3, 5/6, 3]], + ESameTest[{2/3,10/3,6}, LCM[2/3, {1/3, 5/6, 3}]] + ], ETests[ + ESameTest[{10/3,10/3,30}, LCM[2/3, {1/3, 5/6, 3}, 5/6]], + ESameTest[LCM[a,b], LCM[a, b]], + ESameTest[LCM[a,b,c], LCM[a, b, c]], + ESameTest[LCM[5,6,c], LCM[5, 6, c]] + ] +}; + +Mod::usage = "`+"`"+`Mod[x, y]`+"`"+` finds the remainder when `+"`"+`x`+"`"+` is divided by `+"`"+`y`+"`"+`."; +(* Factor out numeric constants like Pi: *) +Mod[a_Integer*c_?NumericQ,b_Integer*c_?NumericQ] := c * Mod[a,b]; +Mod[Rational[a_,b_],c_Integer] := Mod[a,c*b] / b; +Attributes[Mod] = {Listable, NumericFunction, ReadProtected, Protected}; +Tests`+"`"+`Mod = { + ESimpleExamples[ + ESameTest[2, Mod[5,3]], + ESameTest[0, Mod[0,3]], + ESameTest[Indeterminate, Mod[2,0]], + ESameTest[Pi, Mod[-2 Pi,3 Pi]] + ], ETests[ + ESameTest[1, Mod[-5,3]], + ESameTest[Mod[a,3], Mod[a,3]], + ESameTest[Indeterminate, Mod[0,0]], + ESameTest[Mod[2,a], Mod[2,a]], + ESameTest[Mod[0,a], Mod[0,a]], + ESameTest[1,Mod[-1,2]], + ESameTest[5/4,Mod[-(3/4),2]], + ESameTest[3/2,Mod[-(1/2),2]], + ESameTest[7/4,Mod[-(1/4),2]], + ESameTest[0,Mod[0,2]], + ESameTest[1/4,Mod[1/4,2]], + ESameTest[1/2,Mod[1/2,2]], + ESameTest[3/4,Mod[3/4,2]], + ESameTest[1,Mod[1,2]], + ], EKnownFailures[ + ESameTest[1.5, Mod[1.5,3]], + ESameTest[0., Mod[2,0.5]] + ] +}; + +EvenQ::usage = "`+"`"+`EvenQ[n]`+"`"+` returns True if `+"`"+`n`+"`"+` is an even integer."; +EvenQ[n_Integer] := Mod[n,2]===0; +Attributes[EvenQ] = {Listable, Protected}; +Tests`+"`"+`EvenQ = { + ESimpleExamples[ + ESameTest[True, EvenQ[6]], + ESameTest[True, EvenQ[-2]], + ESameTest[False, EvenQ[1]], + ESameTest[EvenQ[2.], EvenQ[2.]], + ESameTest[EvenQ[a], EvenQ[a]] + ] +}; + +OddQ::usage = "`+"`"+`OddQ[n]`+"`"+` returns True if `+"`"+`n`+"`"+` is an odd integer."; +OddQ[n_Integer] := Mod[n,2]===1; +Attributes[OddQ] = {Listable, Protected}; +Tests`+"`"+`OddQ = { + ESimpleExamples[ + ESameTest[False, OddQ[6]], + ESameTest[False, OddQ[-2]], + ESameTest[True, OddQ[1]], + ESameTest[OddQ[2.], OddQ[2.]], + ESameTest[OddQ[a], OddQ[a]] + ] +}; + +FactorInteger::usage = "`+"`"+`FactorInteger[n]`+"`"+` factors the integer `+"`"+`n`+"`"+`."; +FactorInteger[Rational[n_, d_]] := + DeleteCases[Join[FactorInteger[n], ({#[[1]], -#[[2]]} &) /@ FactorInteger[d]] // Sort, {1,1}]; +FactorInteger[int_Integer?Negative] := DeleteCases[Prepend[FactorInteger[-int], {-1, 1}], {1,1}]; +FactorInteger[0] := {{0, 1}}; +(* TODO: use Pollard's rho algorithm. *) +(* TODO: convert to using the internal primeFactorsTallied function *) + +FactorInteger[int_Integer?Positive] := Module[{n = int, i = 2, factors}, + If[n === 1, Return[{{1, 1}}]]; + factors = {}; + While[n =!= 1, + While[Mod[n, i] =!= 0, i = i + 1]; + AppendTo[factors, i]; + n = n/i; + i = 2 + ]; + Tally[factors] // Sort + ]; +Attributes[FactorInteger] = {Listable, Protected}; +Tests`+"`"+`FactorInteger = { + ESimpleExamples[ + ESameTest[{{2, 3}}, FactorInteger[8]], + ESameTest[{{-1,1},{2,-1},{3,1},{5,1}}, FactorInteger[Rational[-15,2]]], + ], ETests[ + ESameTest[{{-1,1},{2,2},{5,1}}, FactorInteger[-20]], + ESameTest[{{-1,1},{7,1}}, FactorInteger[-7]], + ESameTest[{{-1,1},{3,1}}, FactorInteger[-3]], + ESameTest[{{-1,1},{2,1}}, FactorInteger[-2]], + ESameTest[{{-1,1}}, FactorInteger[-1]], + ESameTest[{{0, 1}}, FactorInteger[0]], + ESameTest[{{1, 1}}, FactorInteger[1]], + ESameTest[{{2, 1}}, FactorInteger[2]], + ESameTest[{{3, 1}}, FactorInteger[3]], + ESameTest[{{2, 2}}, FactorInteger[4]], + ESameTest[{{7, 1}}, FactorInteger[7]], + ESameTest[{{2, -2}, {3, 1}, {7, 1}}, FactorInteger[Rational[21,4]]], + ESameTest[{{-1, 1}, {2, -1}}, FactorInteger[Rational[-1,2]]], + ESameTest[{{2, -1}}, FactorInteger[Rational[1,2]]], + ] +}; + +FractionalPart::usage = "`+"`"+`FractionalPart[n]`+"`"+` gives the fractional part of `+"`"+`n`+"`"+`"; +FractionalPart[n_Rational] := If[n >= 0, Mod[n, 1], -Mod[-n, 1]]; +FractionalPart[n_Integer] := 0; +Attributes[FractionalPart] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`FractionalPart = { + ESimpleExamples[ + ESameTest[0,FractionalPart[-1]], + ESameTest[-(3/4),FractionalPart[-(3/4)]], + ESameTest[-(1/2),FractionalPart[-(1/2)]], + ESameTest[-(1/4),FractionalPart[-(1/4)]], + ESameTest[0,FractionalPart[0]], + ESameTest[1/4,FractionalPart[1/4]], + ESameTest[1/2,FractionalPart[1/2]], + ESameTest[3/4,FractionalPart[3/4]], + ESameTest[0,FractionalPart[1]], + ] +}; + +IntegerPart::usage = "`+"`"+`IntegerPart[n]`+"`"+` gives the integer part of `+"`"+`n`+"`"+`"; +IntegerPart[n_Rational] := n - FractionalPart[n]; +IntegerPart[n_Integer] := n; +Attributes[IntegerPart] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`IntegerPart = { + ESimpleExamples[ + ESameTest[-1,IntegerPart[-1]], + ESameTest[0,IntegerPart[-(3/4)]], + ESameTest[0,IntegerPart[-(1/2)]], + ESameTest[0,IntegerPart[-(1/4)]], + ESameTest[0,IntegerPart[0]], + ESameTest[0,IntegerPart[1/4]], + ESameTest[0,IntegerPart[1/2]], + ESameTest[0,IntegerPart[3/4]], + ESameTest[1,IntegerPart[1]], + ] +}; + +PowerMod::usage = "`+"`"+`PowerMod[x, y, m]`+"`"+` computes `+"`"+`Mod[x^y, m]`+"`"+`"; +(*TODO: use efficient version of this function.*) +PowerMod[x_, y_, m_] := Mod[x^y, m]; +Attributes[PowerMod] = {Listable, Protected, ReadProtected}; +Tests`+"`"+`PowerMod = { + ESimpleExamples[ + ESameTest[6,PowerMod[5, 9999, 7]], + ] +}; + +EulerPhi::usage = "`+"`"+`EulerPhi[n]`+"`"+` computes Euler's totient function for `+"`"+`n`+"`"+`"; +Attributes[EulerPhi] = {Listable, Protected, ReadProtected}; +EulerPhi[0] := 0; +EulerPhi[n_Integer?Positive] := + If[n === 1, 1, + n*Product[1 - 1/p[[1]], {p, FactorInteger[n]}]]; +EulerPhi[n_Integer?Negative] := EulerPhi[-n]; +Tests`+"`"+`EulerPhi = { + ESimpleExamples[ + ESameTest[42,EulerPhi[98]], + ESameTest[0,EulerPhi[0]], + ESameTest[42,EulerPhi[-98]], + ], ETests[ + ESameTest[1,EulerPhi[1]], + ESameTest[1,EulerPhi[-1]], + ] +}; + +Fibonacci::usage = "`+"`"+`Fibonacci[n]`+"`"+` computes the Fibonacci number for `+"`"+`n`+"`"+`"; +Fibonacci[0] = 0; Fibonacci[1] = 1; +(*TODO: implement as RootReduce@(((1 + Sqrt[5])/2)^n - ((1 - Sqrt[5])/2)^n)/Sqrt[5]*) +Fibonacci[n_] := Fibonacci[n] = Fibonacci[n - 1] + Fibonacci[n - 2]; +Attributes[Fibonacci] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`Fibonacci = { + ESimpleExamples[ + ESameTest[6765, Fibonacci[20]] + ] +}; + +IntegerDigits::usage = "`+"`"+`IntegerDigits[n, base]`+"`"+` returns a list of integer digits for `+"`"+`n`+"`"+` under `+"`"+`base`+"`"+`."; +IntegerDigits[n_Integer] := IntegerDigits[n, 10]; +IntegerDigits[signedN_Integer, base_Integer?Positive] := + Module[{n = Abs[signedN], list = {}}, + While[n > 0, + PrependTo[list, Mod[n, base]]; + n = (n - Mod[n, base])/base; + ]; + list + ]; +Attributes[IntegerDigits] = {Listable, Protected}; +Tests`+"`"+`IntegerDigits = { + ESimpleExamples[ + ESameTest[{1, 2, 3}, IntegerDigits[123]], + ESameTest[{1, 1, 1, 1, 0, 1, 1}, IntegerDigits[123, 2]], + ESameTest[{1, 1, 1, 1, 0, 1, 1}, IntegerDigits[-123, 2]] + ] +}; + +Sign::usage = "`+"`"+`Sign[x]`+"`"+` returns the sign of `+"`"+`x`+"`"+`."; +Sign[n_Integer] := Which[ + n < 0, -1, + n > 0, 1, + True, 0]; +Sign[n_Real] := Which[ + n < 0, -1, + n > 0, 1, + True, 0]; +Sign[n_Rational] := Which[ + n < 0, -1, + n > 0, 1, + True, 0]; +Attributes[Sign] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`Sign = { + ESimpleExamples[ + ESameTest[1, Sign[5]], + ESameTest[0, Sign[0]], + ESameTest[-1, Sign[-5]], + ESameTest[1, Sign[5.]], + ESameTest[0, Sign[0.]], + ESameTest[-1, Sign[-5.]], + ESameTest[1, Sign[1/2]], + ] +}; +`) + +func resourcesNumbertheoryMBytes() ([]byte, error) { + return _resourcesNumbertheoryM, nil +} + +func resourcesNumbertheoryM() (*asset, error) { + bytes, err := resourcesNumbertheoryMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/numbertheory.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesPatternM = []byte(`MatchQ::usage = "`+"`"+`MatchQ[expr, form]`+"`"+` returns True if `+"`"+`expr`+"`"+` matches `+"`"+`form`+"`"+`, False otherwise."; +Attributes[MatchQ] = {Protected}; +Tests`+"`"+`MatchQ = { + ESimpleExamples[ + EComment["A `+"`"+`Blank[]`+"`"+` expression matches everything:"], + ESameTest[True, MatchQ[2*x, _]], + EComment["Although a more specific pattern would have matched as well:"], + ESameTest[True, MatchQ[2*x, c1_Integer*a_Symbol]], + EComment["Since `+"`"+`Times`+"`"+` is `+"`"+`Orderless`+"`"+`, this would work as well:"], + ESameTest[True, MatchQ[x*2, c1_Integer*a_Symbol]], + EComment["As would the `+"`"+`FullForm`+"`"+`:"], + ESameTest[True, MatchQ[Times[x, 2], c1_Integer*a_Symbol]], + EComment["Named patterns must match the same expression, or the match will fail:"], + ESameTest[False, MatchQ[a + b, x_Symbol + x_Symbol]] + ], EFurtherExamples[ + ESameTest[True, MatchQ[{2^a, a}, {2^x_Symbol, x_Symbol}]], + ESameTest[False, MatchQ[{2^a, b}, {2^x_Symbol, x_Symbol}]], + EComment["`+"`"+`Blank`+"`"+` sequences allow for the matching of multiple objects. `+"`"+`BlankSequence`+"`"+` (__) matches one or more parts of the expression:"], + ESameTest[True, MatchQ[{a, b}, {a, __}]], + ESameTest[False, MatchQ[{a}, {a, __}]], + EComment["`+"`"+`BlankNullSequence`+"`"+` (___) allows for zero or more matches:"], + ESameTest[True, MatchQ[{a}, {a, ___}]] + ], ETests[ + ESameTest[True, MatchQ[2^x, base_Integer^pow_Symbol]], + ESameTest[True, MatchQ[2+x, c1_Integer+a_Symbol]], + ESameTest[True, MatchQ[a + b, x_Symbol + y_Symbol]], + ESameTest[True, MatchQ[{a,b}, {x_Symbol,y_Symbol}]], + ESameTest[False, MatchQ[{a,b}, {x_Symbol,x_Symbol}]], + (*Test speed of OrderlessIsMatchQ*) + ESameTest[Null, Plus[testa, testb, rest___] := bar + rest], + ESameTest[bar + 1 + a + b + c + d + e + f + g, Plus[testa,1,testb,a,b,c,d,e,f,g]], + ESameTest[True, MatchQ[foo[2*x, x], foo[matcha_Integer*matchx_, matchx_]]], + ESameTest[False, MatchQ[foo[2*x, x], bar[matcha_Integer*matchx_, matchx_]]], + ESameTest[False, MatchQ[foo[2*x, y], foo[matcha_Integer*matchx_, matchx_]]], + ESameTest[False, MatchQ[foo[x, 2*y], foo[matcha_Integer*matchx_, matchx_]]], + ESameTest[True, MatchQ[foo[2 * x,2], foo[(p_ * v_), v_]]], + + ESameTest[True, MatchQ[mysolve[m*x + b == 0, x], mysolve[x_*__ + _ == _, x_]]], + ESameTest[False, MatchQ[mysolve[m*x + b == 0, y], mysolve[x_*__ + _ == _, x_]]], + ESameTest[True, MatchQ[mysolve[m*x+a, m], mysolve[x_*_+a, x_]]], + ESameTest[True, MatchQ[bar[foo[a + b] + c + d, c, d, a, b], bar[w_ + x_ + foo[y_ + z_], w_, x_, y_, z_]]], + ESameTest[True, MatchQ[bar[foo[a + b] + c + d, d, c, b, a], bar[w_ + x_ + foo[y_ + z_], w_, x_, y_, z_]]], + ESameTest[False, MatchQ[bar[foo[a + b] + c + d, d, a, b, c], bar[w_ + x_ + foo[y_ + z_], w_, x_, y_, z_]]], + + (* We disable the tests that use rm because they require the + freezeStateDuringPreMatch flag, which is now turned off by default. *) + (*Test order of pattern checking*) + (*ESameTest[Null, rm[pattern_]:=pattern?((pats=Append[pats,{pattern[[1]],#}];True)&);], + ESameTest[True, pats={};MatchQ[foo[a,b,c],foo[x_//rm,y_//rm,z_//rm]]], + ESameTest[{{x,a},{y,b},{z,c}}, pats],*) + + (*Test pinning in flat*) + (*ESameTest[{{{a},{c}}}, pats={};ReplaceList[ExpreduceFlatFn[a,b,c],ExpreduceFlatFn[x___//rm,b//rm,y___//rm]->{{x},{y}}]], + + ESameTest[{{{a,a,c}},{{a,a,c}},{{a,a,c}},{{a,a,c}}}, pats={};ReplaceList[ExpreduceFlOrOiFn[a,a,c],ExpreduceFlOrOiFn[b___//rm,c//rm,a___//rm]->{{a,b,c}}]], + ESameTest[{{{},{a,c}},{{a},{c}},{{c},{a}},{{a,c},{}}}, pats={};ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[x___//rm,b//rm,y___//rm]->{{x},{y}}]],*) + + (*Test pinning in orderless*) + (*ESameTest[{{b[[1]],b},{y,a},{y,c},{b[[1]],b},{x,a},{y,c},{b[[1]],b},{x,c},{y,a},{b[[1]],b},{x,a},{x,c}}, pats],*) + + ESameTest[True, MatchQ[(a+b)[testsym],Blank[(a+b)]]] + ], EKnownFailures[ + (*Test order of pattern checking*) + (*These probably fail because of my formparsing of PatternTest.*) + (*Try these without the //rm. They will most likely work.*) + ESameTest[{{c[[1]],c},{b,a},{b,a},{c[[1]],c},{a,a},{b,a},{c[[1]],c},{a,a},{b,a},{c[[1]],c},{a,a},{a,a}}, pats], + ESameTest[{{x,a},{b[[1]],b},{y,c}}, pats] + ] +}; + +Pattern::usage = "`+"`"+`name{BLANKFORM}`+"`"+` is equivalent to `+"`"+`Pattern[name, {BLANKFORM}]`+"`"+` and can be used in pattern matching to refer to the matched expression as `+"`"+`name`+"`"+`, where `+"`"+`{BLANKFORM}`+"`"+` is one of `+"`"+`{_, __, ___}`+"`"+`. + +`+"`"+`name{BLANKFORM}head`+"`"+` is equivalent to `+"`"+`Pattern[name, {BLANKFORM}head]`+"`"+` and can be used in pattern matching to refer to the matched expression as `+"`"+`name`+"`"+`, where `+"`"+`{BLANKFORM}`+"`"+` is one of `+"`"+`{_, __, ___}`+"`"+`."; +Attributes[Pattern] = {HoldFirst, Protected}; +Tests`+"`"+`Pattern = { + ESimpleExamples[ + EComment["To demonstrate referencing `+"`"+`name`+"`"+` in the replacement RHS:"], + ESameTest[2, foo[2, 1] /. foo[a_, b_] -> a], + EComment["If two matches share the same name, they must be equivalent:"], + ESameTest[foo[2, 1], foo[2, 1] /. foo[a_, a_] -> a], + ESameTest[2, foo[2, 2] /. foo[a_, a_] -> a], + EComment["To demonstrate the head matching capability:"], + ESameTest[True, MatchQ[2, a_Integer]], + ESameTest[False, MatchQ[2, a_Real]] + ], EFurtherExamples[ + EComment["To demonstrate patterns matching a sequence of expressions:"], + ESameTest[bar[2, 1], foo[2, 1] /. foo[a___Integer] -> bar[a]] + ] +}; + +Blank::usage = "`+"`"+`_`+"`"+` matches any expression. + +`+"`"+`_head`+"`"+` matches any expression with a `+"`"+`Head`+"`"+` of `+"`"+`head`+"`"+`."; +Attributes[Blank] = {Protected}; +Tests`+"`"+`Blank = { + ESimpleExamples[ + ESameTest[True, MatchQ[a + b, _]], + ESameTest[True, MatchQ[1, _Integer]], + ESameTest[False, MatchQ[s, _Integer]], + EComment["`+"`"+`Blank`+"`"+` works with nonatomic `+"`"+`head`+"`"+`s:"], + ESameTest[2, a*b*c*d /. _Times -> 2] + ], EFurtherExamples[ + EComment["For `+"`"+`Orderless`+"`"+` functions, the match engine will attempt to find a match in any order:"], + ESameTest[True, MatchQ[x+3., c1match_Real+matcha_]] + ], ETests[ + ESameTest[True, MatchQ[s, _Symbol]], + ESameTest[False, MatchQ[1, _Symbol]], + ESameTest[False, MatchQ[_Symbol, _Symbol]], + ESameTest[False, MatchQ[_Integer, _Integer]], + ESameTest[True, MatchQ[_Symbol, _Blank]], + ESameTest[True, MatchQ[_Symbol, test_Blank]], + ESameTest[True, MatchQ[_Symbol, test_Blank]], + ESameTest[False, MatchQ[_Symbol, test_Symbol]], + ESameTest[False, MatchQ[name_Symbol, test_Blank]], + ESameTest[True, MatchQ[name_Symbol, test_Pattern]], + ESameTest[True, MatchQ[_Symbol, test_Blank]], + ESameTest[False, MatchQ[_Symbol, test_Pattern]], + ESameTest[False, MatchQ[1.5, _Integer]], + ESameTest[True, MatchQ[1.5, _Real]], + ESameTest[True, _Symbol == _Symbol], + ESameTest[_Symbol == _Integer, _Symbol == _Integer], + ESameTest[False, MatchQ[_Symbol, s]], + ESameTest[False, MatchQ[_Integer, 1]], + ESameTest[_Integer == 1, _Integer == 1], + ESameTest[1 == _Integer, 1 == _Integer], + ESameTest[False, _Integer === 1], + ESameTest[False, 1 === _Integer], + ESameTest[True, _Integer === _Integer], + ESameTest[False, _Symbol === a], + ESameTest[False, a === _Symbol], + ESameTest[True, _Symbol === _Symbol], + ESameTest[a == b, a == b], + ESameTest[2, a == b /. _Equal -> 2], + ESameTest[If[1 == k, itstrue, itsfalse], If[1 == k, itstrue, itsfalse]], + ESameTest[99, If[1 == k, itstrue, itsfalse] /. _If -> 99], + ESameTest[False, MatchQ[kfdsfdsf[], _Function]], + ESameTest[True, MatchQ[kfdsfdsf[], _kfdsfdsf]], + ESameTest[99, kfdsfdsf[] /. _kfdsfdsf -> 99], + ESameTest[a + b, a + b], + ESameTest[2, a + b /. _Plus -> 2], + ESameTest[2, a*b /. _Times -> 2], + ESameTest[2, a^b /. _Power -> 2], + ESameTest[2, a -> b /. _Rule -> 2], + ESameTest[2, a*b*c*d /. _Times -> 2], + ESameTest[True, MatchQ[x*3., c1match_Real*matcha_]], + ESameTest[True, MatchQ[3.*x, c1match_Real*matcha_]], + ESameTest[True, MatchQ[x+3., c1match_Real+matcha_]], + ESameTest[True, MatchQ[3.+x, c1match_Real+matcha_]], + ESameTest[True, MatchQ[y + x, matcha_]], + ESameTest[True, MatchQ[y*x, matcha_]] + ] +}; + +BlankSequence::usage = "`+"`"+`__`+"`"+` matches any sequence of one or more expressions. + +`+"`"+`__head`+"`"+` matches any sequence of one or more expressions, each with a `+"`"+`Head`+"`"+` of `+"`"+`head`+"`"+`."; +Attributes[BlankSequence] = {Protected}; +Tests`+"`"+`BlankSequence = { + ESimpleExamples[ + ESameTest[True, MatchQ[a + b + c, a + b + __]], + ESameTest[False, MatchQ[a + b + c, a + b + c + __]] + ], EFurtherExamples[ + EComment["With head assertions:"], + ESameTest[False, MatchQ[a * b, __Symbol]], + ESameTest[False, MatchQ[a * b, x__Symbol]], + ESameTest[True, MatchQ[a, __Symbol]], + ESameTest[True, MatchQ[a * b, x__Times]], + ESameTest[False, MatchQ[a * b, x__Plus]], + ESameTest[True, MatchQ[a + b, x__Plus]] + ], ETests[ + (*Be wary of the false matches - the default is usually false.*) + ESameTest[True, MatchQ[a, __]], + ESameTest[True, MatchQ[a + b, __]], + ESameTest[True, MatchQ[a*b, __]], + ESameTest[False, MatchQ[a, a*__]], + ESameTest[True, MatchQ[a + b + c, a + b + __]], + ESameTest[True, MatchQ[a + b + c + d, a + b + __]], + ESameTest[False, MatchQ[a*b, __Symbol]], + ESameTest[False, MatchQ[a*b, x__Symbol]], + ESameTest[True, MatchQ[a, __Symbol]], + ESameTest[True, MatchQ[a*b, x__Times]], + ESameTest[False, MatchQ[a*b, x__Plus]], + ESameTest[True, MatchQ[a + b, x__Plus]], + ESameTest[True, MatchQ[a + b + c, a + x__Symbol]], + ESameTest[False, MatchQ[a + b + c, a + x__Plus]], + ESameTest[True, MatchQ[a + b, a + x__Symbol]], + ESameTest[False, MatchQ[a + b, a + x__Plus]], + ESameTest[False, MatchQ[a + b, a + b + x__Symbol]], + ESameTest[False, MatchQ[a + b, a + b + x__Plus]], + ESameTest[True, MatchQ[4*a*b*c*d*e*f, __]], + ESameTest[True, MatchQ[4*a*b*c*d*e*f, 4*__]], + ESameTest[False, MatchQ[4*a*b*c*4*d*e*f, 4*__]], + ESameTest[False, MatchQ[4*a*b*c*4*d*e*f, 4*__]], + ESameTest[True, MatchQ[a*b*c*4*d*e*f, 4*__]], + ESameTest[True, MatchQ[a*b*c*4*d*e*f, 4*__]], + ESameTest[False, MatchQ[a*b*c*4*d*e*f, 5*__]], + ESameTest[False, MatchQ[a*b*c*4*d*e*f + 5, 4*__]], + ESameTest[False, MatchQ[a*b*c*4*d*e*f + 5*k, 4*__]], + ESameTest[False, MatchQ[a*b*c*4*d*e*f + 5*k, 4*__]], + ESameTest[True, MatchQ[a*b*c*4*d*e*f + 5*k, 4*__ + 5*k]], + ESameTest[False, MatchQ[a*b*c*4*d*e*f + 2 + 5*k, 4*__ + 5*k]], + ESameTest[True, MatchQ[(a*b*c)^e, __^e]], + ESameTest[False, MatchQ[(a*b*c)^e, __^f + __^e]], + ESameTest[True, MatchQ[(a*b*c)^e + (a*b*c)^f, __^f + __^e]], + ESameTest[True, MatchQ[(a*b*c)^e + (a + b + c)^f, __^f + __^e]], + ESameTest[False, MatchQ[(a*b*c)^e + (a + b + c)^f, amatch__^f + amatch__^e]], + ESameTest[True, MatchQ[(a*b*c)^e + (a*b*c)^f, amatch__^f + amatch__^e]], + + (*Warm up for combining like terms*) + ESameTest[True, MatchQ[bar[1, foo[a, b]], bar[amatch_Integer, foo[cmatch__]]]], + ESameTest[True, MatchQ[bar[1, foo[a, b, c]], bar[amatch_Integer, foo[cmatch__]]]], + ESameTest[False, MatchQ[bar[1, foo[]], bar[amatch_Integer, foo[cmatch__]]]], + ESameTest[2, bar[1, foo[a, b]] /. bar[amatch_Integer, foo[cmatch__]] -> 2], + ESameTest[4, bar[1, foo[a, b]] + bar[5, foo[a, b]] /. bar[amatch_Integer, foo[cmatch__]] -> 2], + ESameTest[6 * buzz[a, b], bar[1, foo[a, b]] + bar[5, foo[a, b]] /. bar[amatch_Integer, foo[cmatch__]] -> amatch*buzz[cmatch]], + ESameTest[bar[3, foo[a, b]], bar[1, foo[a, b]] + bar[2, foo[a, b]] /. bar[amatch_Integer, foo[cmatch__]] + bar[bmatch_Integer, foo[cmatch__]] -> bar[amatch + bmatch, foo[cmatch]]], + + (*Test special case of Orderless sequence matches*) + ESameTest[Null, fooPlus[Plus[addends__]] := Hold[addends]], + ESameTest[Null, fooList[List[addends__]] := Hold[addends]], + ESameTest[Null, fooBlank[_[addends__]] := Hold[addends]], + ESameTest[Hold[a, b, c], fooList[List[a, b, c]]], + ESameTest[Hold[a, b, c], fooBlank[Plus[a, b, c]]], + + ESameTest[True, MatchQ[foo[1, 2, 3], foo[__Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[__]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[__Real]]], + ESameTest[True, MatchQ[foo[1.], foo[__Real]]], + ESameTest[False, MatchQ[foo[], foo[__Real]]], + ESameTest[True, MatchQ[foo[1.], foo[__]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, __Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, 2, __Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, 2, 3, ___Integer]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, 2, 3, 4, ___Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, ___Integer, 3]]], + + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, 2, 3, a__Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, a__Integer, 3]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, a__Integer, 5]]] + ] +}; + +BlankNullSequence::usage = "`+"`"+`___`+"`"+` matches any sequence of zero or more expressions. + +`+"`"+`___head`+"`"+` matches any sequence of zero or more expressions, each with a `+"`"+`Head`+"`"+` of `+"`"+`head`+"`"+`."; +Attributes[BlankNullSequence] = {Protected}; +Tests`+"`"+`BlankNullSequence = { + ESimpleExamples[ + ESameTest[True, MatchQ[a*b, ___]], + ESameTest[True, MatchQ[a + b, a + b + ___]] + ], EFurtherExamples[ + EComment["With head assertions:"], + ESameTest[True, MatchQ[a + b + c, a + x___Symbol]], + ESameTest[False, MatchQ[a + b + c, a + x___Plus]] + ], ETests[ + ESameTest[True, MatchQ[a*b, ___]], + ESameTest[False, MatchQ[a, a*___]], + ESameTest[False, MatchQ[a, a + ___]], + ESameTest[True, MatchQ[a + b, a + b + ___]], + ESameTest[False, MatchQ[a*b, ___Integer]], + ESameTest[False, MatchQ[a*b, ___Symbol]], + ESameTest[True, MatchQ[a, ___Symbol]], + ESameTest[False, MatchQ[a + b, ___Symbol]], + ESameTest[True, MatchQ[a + b + c, a + x___Symbol]], + ESameTest[False, MatchQ[a + b + c, a + x___Plus]], + ESameTest[True, MatchQ[a + b, a + b + x___Symbol]], + ESameTest[True, MatchQ[foo[1.], foo[___]]], + ESameTest[True, MatchQ[foo[1.], foo[___Real]]], + ESameTest[False, MatchQ[foo[1.], foo[___Integer]]], + ESameTest[True, MatchQ[foo[], foo[___Integer]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, 2]]], + ESameTest[False, MatchQ[foo[1, 2], foo[1, 2, 3]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, 2, 3, __Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, __Integer, 3]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, __Integer, 5]]], + + (*Make sure some similar cases still work with Patterns, not just Blanks*) + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, 2, 3, a___Integer]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, 2, 3, 4, a___Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, a___Integer, 3]]], + ESameTest[False, MatchQ[foo[1, 2, 3], foo[1, a__Integer, 3, b__Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[1, a__Integer, 3, b___Integer]]], + ESameTest[True, MatchQ[foo[1, 2, 3, 4], foo[1, a__Integer, 3, b___Integer, 4]]] + ] +}; + +Except::usage = "`+"`"+`Except[pat]`+"`"+` matches all expressions except those that match `+"`"+`pat`+"`"+`. + +`+"`"+`Except[pat1, pat2]`+"`"+` matches all expressions that match `+"`"+`pat2`+"`"+` but not `+"`"+`pat1`+"`"+`."; +Attributes[Except] = {Protected}; +Tests`+"`"+`Except = { + ESimpleExamples[ + ESameTest[{5, 2, x, y, 4}, Cases[{5, 2, 3.5, x, y, 4}, Except[_Real]]], + ESameTest[{5, 2, x, y, 4}, Cases[{5, 2, a^b, x, y, 4}, Except[_Symbol^_Symbol]]], + ESameTest[{a, b, 0, foo[1], foo[2], x, y}, {a, b, 0, 1, 2, x, y} /. Except[0, a_Integer] -> foo[a]] + ] +}; + +PatternTest::usage = "`+"`"+`pat?test`+"`"+` matches when the expression matches `+"`"+`pat`+"`"+` and `+"`"+`test[MATCH]`+"`"+` evaluates to `+"`"+`True`+"`"+`."; +Attributes[PatternTest] = {HoldRest, Protected}; +Tests`+"`"+`PatternTest = { + ESimpleExamples[ + ESameTest[True, MatchQ[1, _?NumberQ]], + ESameTest[False, MatchQ[a, _?NumberQ]], + ESameTest[True, MatchQ[1, 1?NumberQ]], + ESameTest[False, MatchQ[1, 1.5?NumberQ]], + ESameTest[True, MatchQ[1.5, 1.5?NumberQ]], + ESameTest[{5,2,4.5}, Cases[{5, 2, a^b, x, y, 4.5}, _?NumberQ]] + ] +}; + +Condition::usage = "`+"`"+`pat /; cond`+"`"+` matches an expression if the expression matches `+"`"+`pat`+"`"+`, and if `+"`"+`cond`+"`"+` evaluates to `+"`"+`True`+"`"+` with all the named patterns substituted in."; +Attributes[Condition] = {HoldAll, Protected}; +Tests`+"`"+`Condition = { + ESimpleExamples[ + ESameTest[True, MatchQ[5, _ /; True]], + ESameTest[False, MatchQ[5, _ /; False]], + ESameTest[True, MatchQ[5, y_ /; True]], + ESameTest[False, MatchQ[5, y_Real /; True]], + ESameTest[True, MatchQ[5, y_Integer /; True]], + ESameTest[False, MatchQ[5, y_ /; y == 0]], + ESameTest[True, MatchQ[5, y_ /; y == 5]], + ESameTest[{1,2,3,5}, {3, 5, 2, 1} //. {x___, y_, z_, k___} /; (Order[y, z] == -1) -> {x, z, y, k}], + ESameTest[myfn[1], Replace[1,a_Integer:>myfn[a]/;a>0]], + ESameTest[-1, Replace[-1,a_Integer:>myfn[a]/;a>0]], + + ESameTest[4, Replace[foo[2,3],foo[x_,y_]:>With[{a=x},x^2/;a==2]/;y==3]], + ESameTest[foo[3,3], Replace[foo[3,3],foo[x_,y_]:>With[{a=x},x^2/;a==2]/;y==3]], + ESameTest[foo[2,4], Replace[foo[2,4],foo[x_,y_]:>With[{a=x},x^2/;a==2]/;y==3]], + ESameTest[4, Replace[bar[2],bar[x_]:>With[{a=x},x^2/;a==2]]], + ESameTest[bar[3], Replace[bar[3],bar[x_]:>With[{a=x},x^2/;a==2]]], + + ESameTest[4, Replace[foo[2,3],foo[x_,y_]:>Module[{a=x},x^2/;a==2]/;y==3]], + ESameTest[foo[3,3], Replace[foo[3,3],foo[x_,y_]:>Module[{a=x},x^2/;a==2]/;y==3]], + ESameTest[foo[2,4], Replace[foo[2,4],foo[x_,y_]:>Module[{a=x},x^2/;a==2]/;y==3]], + ESameTest[4, Replace[bar[2],bar[x_]:>Module[{a=x},x^2/;a==2]]], + ESameTest[bar[3], Replace[bar[3],bar[x_]:>Module[{a=x},x^2/;a==2]]], + + ESameTest[4, Replace[2,x_:>Condition[Condition[Condition[x^2,x==2],x==2],x==2]]], + ESameTest[3, Replace[3,x_:>Condition[Condition[Condition[x^2,x==2],x==2],x==2]]] + ] +}; + +Alternatives::usage = "`+"`"+`alt1 | alt2 | ...`+"`"+` matches an expression if it matches any pattern in the list of alternatives."; +Attributes[Alternatives] = {Protected}; +Tests`+"`"+`Alternatives = { + ESimpleExamples[ + ESameTest[Alternatives[c,d], c | d], + ESameTest[False, MatchQ[b, c | d]], + ESameTest[True, MatchQ[c, c | d]], + ESameTest[True, MatchQ[d, c | d]], + ESameTest[{c, 1, 2}, Cases[{a, b, c, 1, 2}, c | _Integer]], + ESameTest[(1 + List)[1 + a, 1 + b, 1 + c, 1., 3], {a, b, c, 1., 2} /. amatch_Symbol | amatch_Integer -> amatch + 1], + ESameTest[{b, c, d, e}, Cases[{a, b, c, d, e, f}, b | c | d | e]], + ESameTest[False, MatchQ[{a, b}, {a_, k | a_}]] + ], ETests[ + ESameTest[False, MatchQ[c || a || Not[b], Or[___, a_, ___, Not[And[___, a_, ___] | a_], ___]]] + ] +}; + +FreeQ::usage = "`+"`"+`FreeQ[e, var]`+"`"+` returns True if `+"`"+`e`+"`"+` is free from any occurences of `+"`"+`var`+"`"+`."; +FreeQ[expr_, val_] := expr === (expr /. val -> Internal`+"`"+`DummyReplace); +Attributes[FreeQ] = {Protected}; +Tests`+"`"+`FreeQ = { + ETests[ + ESameTest[False, FreeQ[{0, 1, 2}, 1]], + ESameTest[True, FreeQ[{0, 1, 2}, 3]], + ESameTest[False, FreeQ[{0, 1, 2}, _Integer]], + ESameTest[True, FreeQ[{0, 1, 2}, _Real]], + ESameTest[True, FreeQ[x^2, _Real]], + ESameTest[False, FreeQ[x^2, _Integer]], + ESameTest[False, FreeQ[x^2, 2]], + ESameTest[True, FreeQ[x^2, 3]], + ESameTest[True, FreeQ[x^2, y]], + ESameTest[True, FreeQ[x^2, y]], + ESameTest[False, FreeQ[x^2, x^_Integer]], + ESameTest[True, FreeQ[x^2, y^_Integer]], + ESameTest[False, FreeQ[5*foo[x], foo]], + ESameTest[True, FreeQ[5*foo[x], bar]] + ] +}; + +ReplaceList::usage = "`+"`"+`ReplaceList[expr, rule]`+"`"+` returns all the possible replacements using `+"`"+`rule`+"`"+` on `+"`"+`expr`+"`"+`."; +Attributes[ReplaceList] = {Protected}; +Tests`+"`"+`ReplaceList = { + ESimpleExamples[ + ESameTest[{{a, b}, {b, a}}, ReplaceList[a + b, x_ + y_ -> {x, y}]], + ESameTest[{{b, a}}, ReplaceList[foo[a + b, b], foo[j_ + k_, j_] -> {j, k}]], + ESameTest[{{a, b}, {b, a}}, ReplaceList[foo[a + b], foo[x_ + y_] -> {x, y}]], + ESameTest[{{a, b, c}, {b, a, c}}, ReplaceList[bar[foo[a + b] + c], bar[foo[x_ + y_] + z_] -> {x, y, z}]], + ESameTest[{{c, d, a, b}, {c, d, b, a}, {d, c, a, b}, {d, c, b, a}}, ReplaceList[bar[foo[a + b] + c + d], bar[w_ + x_ + foo[y_ + z_]] -> {w, x, y, z}]], + ESameTest[{}, ReplaceList[foo[a + b, c], foo[j_ + k_, j_] -> {j, k}]] + ], ETests[ + ESameTest[{{{a},{b}},{{b},{a}}}, ReplaceList[a+b,x__+y__->{{x},{y}}]], + ESameTest[{{{},{a,b}},{{a},{b}},{{a,b},{}}}, ReplaceList[foo[a,b],foo[a___,b___]->{{a},{b}}]], + ESameTest[{}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a:Repeated[b_,{2}],rest___]->{{a},{rest}}]], + ESameTest[{{c}}, ReplaceList[foo[a,b,c],foo[___,a_]->{a}]], + ESameTest[{{a+b+c}}, ReplaceList[a+b+c,a_->{a}]], + ESameTest[{{{},{a,b},{},{b,c}},{{},{a,b},{b},{c}},{{},{a,b},{b,c},{}},{{a},{b},{},{b,c}},{{a},{b},{b},{c}},{{a},{b},{b,c},{}},{{a,b},{},{},{b,c}},{{a,b},{},{b},{c}},{{a,b},{},{b,c},{}}}, ReplaceList[foo[a,b,foo[b,c]],foo[a___,b___,foo[c___,d___]]->{{a},{b},{c},{d}}]], + ESameTest[{{a+b+c},{b+c},{a+c},{a+b},{c},{b},{a}}, ReplaceList[a+b+c,___+a_->{a}]], + ESameTest[{{{},{a,b}},{{a},{b}},{{b},{a}},{{a,b},{}}}, ReplaceList[a+b,x___+y___->{{x},{y}}]], + ESameTest[{{{a},{b+c}},{{b},{a+c}},{{c},{a+b}},{{a+b},{c}},{{a+c},{b}},{{b+c},{a}}}, ReplaceList[a+b+c,a_+b_->{{a},{b}}]], + ESameTest[{{{a},{b,c}},{{b},{a,c}},{{c},{a,b}},{{a,b},{c}},{{a,c},{b}},{{b,c},{a}}}, ReplaceList[a+b+c,a__+b__->{{a},{b}}]], + ESameTest[{{{a},{b,c}},{{b},{a,c}},{{c},{a,b}},{{a,b},{c}},{{a,c},{b}},{{b,c},{a}}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a__,b__]->{{a},{b}}]], + ESameTest[{{{a,b},{c}},{{a,c},{b}},{{b,c},{a}}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a:Repeated[_,{2}],rest___]->{{a},{rest}}]], + ESameTest[{{{a,b},{c}},{{a,c},{b}},{{b,c},{a}}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a:Repeated[_,{2}],rest___]->{{a},{rest}}]], + ESameTest[{{{a,b},{c,d}},{{a,c},{b,d}},{{a,d},{b,c}},{{b,c},{a,d}},{{b,d},{a,c}},{{c,d},{a,b}}}, ReplaceList[ExpreduceOrderlessFn[a,b,c,d],ExpreduceOrderlessFn[a:Repeated[_,{2}],rest___]->{{a},{rest}}]], + ESameTest[{{a+b+c},{b+c},{a+c},{a+b},{c},{b},{a}}, ReplaceList[a+b+c,___+a_->{a}]], + ESameTest[{{{},{a,b}},{{a},{b}},{{b},{a}},{{a,b},{}}}, ReplaceList[ExpreduceOrderlessFn[a,b],ExpreduceOrderlessFn[a___,b___]->{{a},{b}}]], + ESameTest[{{{a},{b},{c}},{{a},{c},{b}},{{b},{a},{c}},{{b},{c},{a}},{{c},{a},{b}},{{c},{b},{a}},{{a},{b+c},{}},{{b},{a+c},{}},{{c},{a+b},{}},{{a+b},{c},{}},{{a+c},{b},{}},{{b+c},{a},{}}}, ReplaceList[a+b+c,a_+b_+rest___->{{a},{b},{rest}}]], + ESameTest[{{{},{a,b,c}},{{a},{b,c}},{{b},{a,c}},{{c},{a,b}},{{a,b},{c}},{{a,c},{b}},{{b,c},{a}},{{a,b,c},{}}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a___,b___]->{{a},{b}}]], + ESameTest[{{{a,a},{b,b,c}},{{b,b},{a,a,c}}}, ReplaceList[ExpreduceOrderlessFn[a,a,b,b,c],ExpreduceOrderlessFn[a:Repeated[b_,{2}],rest___]->{{a},{rest}}]], + ESameTest[{{a,b,c},{b,c},{a,c},{a,b},{c},{b},{a},{}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[a___,rest___]->{rest}]], + ESameTest[{{{},{a,b},{},{c,d}},{{},{a,b},{c},{d}},{{},{a,b},{d},{c}},{{},{a,b},{c,d},{}},{{a},{b},{},{c,d}},{{a},{b},{c},{d}},{{a},{b},{d},{c}},{{a},{b},{c,d},{}},{{b},{a},{},{c,d}},{{b},{a},{c},{d}},{{b},{a},{d},{c}},{{b},{a},{c,d},{}},{{a,b},{},{},{c,d}},{{a,b},{},{c},{d}},{{a,b},{},{d},{c}},{{a,b},{},{c,d},{}}}, ReplaceList[ExpreduceOrderlessFn[a,b,ExpreduceOrderlessFn[c,d]],ExpreduceOrderlessFn[a___,b___,ExpreduceOrderlessFn[c___,d___]]->{{a},{b},{c},{d}}]], + ESameTest[{}, ReplaceList[a+b+c,___+a_+___->{a}]], + ESameTest[{{{2},{x,y}}}, ReplaceList[ExpreduceFlOrOiFn[2,x,y],ExpreduceFlOrOiFn[c_Integer,e__]->{{c},{e}}]], + ESameTest[{{{2},{x,y}}}, ReplaceList[ExpreduceFlOrOiFn[2,x,y],ExpreduceFlOrOiFn[c_?NumberQ,e__]->{{c},{e}}]] + ], EKnownFailures[ + (*Orderless has issues. Flat seems to work fine. regular ordered matching seems perfect.*) + ESameTest[{{a},{b},{c}}, ReplaceList[ExpreduceOrderlessFn[a,b,c],ExpreduceOrderlessFn[___,a_]->{a}]] + ] +}; + +Repeated::usage = "`+"`"+`Repeated[p_]`+"`"+` matches a sequence of expressions that match the pattern `+"`"+`p`+"`"+`."; +Attributes[Repeated] = {Protected}; +Tests`+"`"+`Repeated = { + ETests[ + ESameTest[True, MatchQ[foo[a, a], foo[Repeated[a]]]], + ESameTest[False, MatchQ[foo[a, b], foo[Repeated[a]]]], + ESameTest[True, MatchQ[foo[a], foo[Repeated[a]]]], + ESameTest[False, MatchQ[foo[], foo[Repeated[a]]]], + ESameTest[True, MatchQ[foo[1, 2, 3], foo[Repeated[_Integer]]]], + ESameTest[False, MatchQ[foo[1, 2, a], foo[Repeated[_Integer]]]], + ESameTest[2, foo[1, 2, 3] /. foo[a : (Repeated[_Integer])] -> 2], + ESameTest[matches[1, 2, 3], foo[1, 2, 3] /. foo[a : (Repeated[_Integer])] -> matches[a]], + ESameTest[foo[1, 2, 3], foo[1, 2, 3] /. foo[a : (Repeated[k_Integer])] -> matches[a]], + ESameTest[matches[1, 1, 1], foo[1, 1, 1] /. foo[a : (Repeated[k_Integer])] -> matches[a]], + ESameTest[True, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[a, Repeated[b, {3}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[a, Repeated[b, {a}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[a, Repeated[b, {4}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[a, Repeated[b, {2}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[a, Repeated[_Integer, {2}]]]], + ESameTest[True, MatchQ[ExpreduceFlOrOiFn[a, 1, 2], ExpreduceFlOrOiFn[a, Repeated[_Integer, {2}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, 1, 2], ExpreduceFlOrOiFn[a, Repeated[k_Integer, {2}]]]], + ESameTest[True, MatchQ[ExpreduceFlOrOiFn[a, 2, 2], ExpreduceFlOrOiFn[a, Repeated[k_Integer, {2}]]]], + ESameTest[True, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[___, Repeated[_Integer, {0}]]]], + ESameTest[False, MatchQ[ExpreduceFlOrOiFn[a, b, b, b], ExpreduceFlOrOiFn[___, Repeated[_Integer, {-1}]]]], + ESameTest[x, foo[x, x] /. foo[Repeated[a_, {2}]] -> a] + ] +}; + +Optional::usage = "`+"`"+`Optional[pat, default]`+"`"+` attempts to match `+"`"+`pat`+"`"+` but uses `+"`"+`default`+"`"+` if not present."; +Attributes[Optional] = {Protected}; +Tests`+"`"+`Optional = { + ETests[ + ESameTest[foo[a], foo[a]/.foo[a,b_.]->{a,b}], + ESameTest[{a,b}, foo[a,b]/.foo[a,b_.]->{a,b}], + ESameTest[{a,c}, foo[a]/.foo[a,b_:c]->{a,b}], + ESameTest[{a,b}, foo[a,b]/.foo[a,b_:c]->{a,b}], + ESameTest[{{a},{b}}, foo[a,b]/.foo[a___,b_.]->{{a},{b}}], + ESameTest[{{a},{b}}, foo[a,b]/.foo[a___,b_:c]->{{a},{b}}], + ESameTest[{{},{a},{b}}, foo[a,b]/.foo[a___,b_:c,d_:e]->{{a},{b},{d}}], + ESameTest[{{x},{y},{z}}, foo[x]/.foo[a_,b_:y,c_:z]->{{a},{b},{c}}], + ESameTest[foo[], foo[]/.foo[a_,b_:y,c_:z]->{{a},{b},{c}}], + ESameTest[{{x},{i},{j}}, foo[x,i,j]/.foo[a_,b_:y,c_:z]->{{a},{b},{c}}], + ESameTest[a, a /. foo[a, c_.] -> {{c}}], + (*This match succeeds because Plus has both a Default and has*) + (*OneIdentity*) + ESameTest[{{0}}, a /. a + c_. -> {{c}}], + ESameTest[False, MatchQ[a,foo[a,c_.]]], + ESameTest[True, MatchQ[a,a+c_.]], + ESameTest[False, MatchQ[foo[a],a+c_.]], + ESameTest[{{0},{0}}, a/.a+c_.+d_.->{{c},{d}}], + ESameTest[{{0},{0}}, Cos[x]/.(_+c_.+d_.)->{{c},{d}}], + ESameTest[{{5},{a}}, 5*a/.Optional[c1_?NumberQ]*a_->{{c1},{a}}], + ESameTest[{{1},{a}}, a/.Optional[c1_?NumberQ]*a_->{{c1},{a}}], + ESameTest[False, MatchQ[foo[a,b],foo[c1__?NumberQ]]], + ESameTest[True, MatchQ[foo[1,2],foo[c1__?NumberQ]]], + ESameTest[False, MatchQ[foo[1,2],foo[Optional[c1__?NumberQ]]]], + ESameTest[True, MatchQ[foo[1],foo[Optional[c1__?NumberQ]]]], + + (*Ensure that we attempt to fill optionals before using the*) + (*default.*) + ESameTest[{{{a},{b,c}},{{5},{a,b,c}}}, ReplaceList[{a,b,c},{a_:5,b__}->{{a},{b}}]], + ESameTest[{{{a},{b},{c}},{{a},{6},{b,c}},{{5},{a},{b,c}},{{5},{6},{a,b,c}}}, ReplaceList[{a,b,c},{a_:5,b_:6,c___}->{{a},{b},{c}}]], + ESameTest[True, MatchQ[-x,p_.]], + ESameTest[True, MatchQ[-x*a,p_.*a]], + ESameTest[True, MatchQ[__, Optional[1]*a_]], + ESameTest[True, MatchQ[x^x, x^Optional[exp_]]] + ], EKnownFailures[ + ESameTest[foo[a,b], foo[a,b]/.foo[a___,b_.,d_.]->{{a},{b},{d}}], + ESameTest[True, MatchQ[x^x, (x^x)^Optional[exp_]]], + (*Disabled because issue #79*) + ESameTest[{{0},{5}}, a/.a+c_.+d_:5->{{c},{d}}] + ] +}; + +Verbatim::usage = "`+"`"+`Verbatim[expr]`+"`"+` matches `+"`"+`expr`+"`"+` exactly, even if it has patterns."; +Attributes[Verbatim] = {Protected}; +Tests`+"`"+`Verbatim = { + ETests[ + ESameTest[{{a,b},{b,c},{c,d}}, ReplaceList[a+b+c+d,Verbatim[Plus][___,x_,y_,___]->{x,y}]], + ESameTest[{}, ReplaceList[a+b+c+d,Verbatim[Times][___,x_,y_,___]->{x,y}]], + ESameTest[{}, ReplaceList[a+b*c+d,Verbatim[Times][___,x_,y_,___]->{x,y}]], + ESameTest[{{a,b},{b,c},{c,d}}, ReplaceList[foo[a,b,c,d],Verbatim[foo][___,x_,y_,___]->{x,y}]], + ESameTest[{{a,b},{b,c},{c,d}}, ReplaceList[(a+b)[a,b,c,d],Verbatim[a+b][___,x_,y_,___]->{x,y}]], + ESameTest[{}, ReplaceList[(a+b)[a,b,c,d],Verbatim[a+_][___,x_,y_,___]->{x,y}]] + ] +}; + +HoldPattern::usage = "`+"`"+`HoldPattern[expr]`+"`"+` leaves `+"`"+`expr`+"`"+` unevaluated but is seen as just `+"`"+`expr`+"`"+` for the purposes of pattern matching."; +Attributes[HoldPattern] = {HoldAll, Protected}; +Tests`+"`"+`HoldPattern = { + ESimpleExamples[ + ESameTest[False, MatchQ[2x+2y,2_+2_]], + ESameTest[True, MatchQ[2x+2y,HoldPattern[2_+2_]]], + ESameTest[True, MatchQ[2x+2y,HoldPattern[2_+HoldPattern[2_]]]] + ] +}; +`) + +func resourcesPatternMBytes() ([]byte, error) { + return _resourcesPatternM, nil +} + +func resourcesPatternM() (*asset, error) { + bytes, err := resourcesPatternMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/pattern.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesPlotM = []byte(`ColorData[97, i_] := Switch[i, + 1, RGBColor[0.37, 0.5, 0.71], + 2, RGBColor[0.88, 0.6, 0.14], + True, RGBColor[0.55, 0.7, 0.2] +]; + +ExpreduceFilterImaginaryPoints[unfilteredPoints_] := + Select[unfilteredPoints, (Im[#[[2]]] == 0) &]; + +Attributes[ExpreducePlotPoints] = {HoldAll}; +ExpreduceGetPoints[fn_, range_List] := + Module[{nPoints, stepSize, replacedFn, unfilteredPoints}, + nPoints = 500; + stepSize = (range[[3]] - range[[2]])/nPoints // N; + replacedFn = fn /. range[[1]] -> varOfIteration; + unfilteredPoints = + Table[{varOfIteration, replacedFn // N}, {varOfIteration, + range[[2]], range[[3]], stepSize}]; + ExpreduceFilterImaginaryPoints[unfilteredPoints] + ]; + +Attributes[ExpreduceGetLine] = {HoldAll}; +ExpreduceGetLine[fn_, range_List, + color_] := {Directive[Opacity[1.], color, AbsoluteThickness[1.6]], + Line[ExpreduceGetPoints[fn, range]]}; + +Attributes[ExpreduceGetLines] = {HoldAll}; +ExpreduceGetLines[fn_, range_List] := Module[{fns}, + fns = If[Head[fn] === List, fn, {fn}]; + Table[ExpreduceGetLine[fns[[i]], range, ColorData[97, i]], {i, + Length[fns]}] + ]; + +Plot::usage = "`+"`"+`Plot[fn, {var, min, max}]`+"`"+` plots `+"`"+`fn`+"`"+` over the range specified."; +Attributes[Plot] = {HoldAll, Protected, ReadProtected} +Plot[fn_, range_List] := + Module[{lines, plotPoints, fullRange, displayOptions}, + lines = ExpreduceGetLines[fn, range]; + plotPoints = Join @@ Map[(#[[2]][[1]]) &, lines]; + fullRange = {MinMax[Join[plotPoints[[All, 1]], range[[2 ;; 3]]]], + MinMax[plotPoints[[All, 2]]]}; + displayOptions = {DisplayFunction -> Identity, + AspectRatio -> GoldenRatio^(-1), Axes -> {True, True}, + AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, + DisplayFunction :> Identity, + Frame -> {{False, False}, {False, False}}, + FrameLabel -> {{None, None}, {None, None}}, + FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, + GridLines -> {None, None}, + GridLinesStyle -> Directive[GrayLevel[0.5, 0.4]], + Method -> {"DefaultBoundaryStyle" -> Automatic, + "DefaultMeshStyle" -> AbsolutePointSize[6], + "ScalingFunctions" -> None}, PlotRange -> fullRange, + PlotRangeClipping -> True, + PlotRangePadding -> {{Scaled[0.02], Scaled[0.02]}, {Scaled[0.05], + Scaled[0.05]}}, Ticks -> {Automatic, Automatic}}; + Graphics[lines, displayOptions]]; +Tests`+"`"+`Plot = { + ETests[ + ESameTest[Graphics, Plot[2*Cos[10 t + 1] - Sin[4 t - 1], {t, 0, 10}] // Head], + ] +}; +`) + +func resourcesPlotMBytes() ([]byte, error) { + return _resourcesPlotM, nil +} + +func resourcesPlotM() (*asset, error) { + bytes, err := resourcesPlotMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/plot.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesPowerM = []byte(`Power::usage = "`+"`"+`base^exp`+"`"+` finds `+"`"+`base`+"`"+` raised to the power of `+"`"+`exp`+"`"+`."; +(*Simplify nested exponents*) +Power[Power[a_,b_Integer],c_Integer] := a^(b*c); +Power[Power[a_,b_Real],c_Integer] := a^(b*c); +Power[Power[a_,b_Symbol],c_Integer] := a^(b*c); +Power[Power[a_,b_Complex*c_Symbol],d_Integer] := a^(b*c*d); +Power[Power[a_,b_Rational],c_] := a^(b*c); +Power[Infinity, 0] := Indeterminate; +Power[-Infinity, 0] := Indeterminate; +Power[_, 0] := 1; +Power[Infinity, 0.] := Indeterminate; +Power[-Infinity, 0.] := Indeterminate; +Power[_, 0.] := 1; +Power[Infinity, -1] := 0; +Power[Indeterminate, _] := Indeterminate; +Power[_, Indeterminate] := Indeterminate; +Power[1, a_] := 1; +Power[b_?NumberQ, Infinity] := Which[ + b < -1, + ComplexInfinity, + b === -1, + Indeterminate, + b < 1, + 0, + b === 1, + Indeterminate, + b > 1, + Infinity, + True, + UnexpectedInfinitePowerBase +]; +Power[b_, Infinity] := Indeterminate; +Power[b_?NumberQ, -Infinity] := Which[ + b < -1, + 0, + b === -1, + Indeterminate, + b <= 0, + ComplexInfinity, + b < 1, + Infinity, + b === 1, + Indeterminate, + b > 1, + 0, + True, + UnexpectedInfinitePowerBase +]; +(* Example: 3^(4/3) -> 3*3^(1/3) *) +Power[b_Integer, Rational[n_, d_]] := b^((n-Mod[n,d])/d) * b^(Mod[n,d]/d) /; Or[n > d, -n > d]; +Power[b_, -Infinity] := Indeterminate; +(*Power definitions*) +(*Distribute any kind of power for numeric values in Times:*) +((first:(_Integer | _Real | _Rational)?((#!=-1)&)) * inner__)^pow_ := first^pow * Times[inner]^pow; +(*Otherwise, only distribute integer powers*) +(first_ * inner___)^pow_Integer := first^pow * Times[inner]^pow; +(*Rational simplifications*) +(*These take up time. Possibly convert to Upvalues.*) +Power[Rational[a_,b_], -1] := Rational[b,a]; +Power[Rational[a_,b_], e_Integer?Positive] := Rational[a^e,b^e]; +Power[-1, -1/2] := -I; +Power[-1, 1/2] := I; +Power[Rational[a_?Positive,b_?Positive], 1/2] := Power[a, 1/2] * Power[b, -1/2]; +Power[Power[x_, y_Rational], -1] := Power[x, -y]; +(*We may want to deprecate this in favor of the general definition.*) +Complex[0,1]^e_Integer := Switch[Mod[e, 4], + 0, 1, + 1, I, + 2, -1, + 3, -I]; +Complex[re_,im_]^n_Integer := If[n===-1, + Complex[re/(re^2+im^2), -(im/(re^2+im^2))], + Module[{theta = ArcTan[re,im]}, Sqrt[re^2+im^2]^n*Complex[Cos[n*theta],Sin[n*theta]]]]; +Complex[re_,im_]^n_Real := Module[{theta = ArcTan[re,im]}, Sqrt[re^2+im^2]^n*Complex[Cos[n*theta],Sin[n*theta]]]; +Power[ComplexInfinity+_, -1] := 0; +_^ComplexInfinity := Indeterminate; +(*TODO(corywalker): Remove this as there should be a more general version.*) +E^pow_Real := N[E]^pow; +E^(Log[a_]+rest___) := a * E^rest; +E^Log[a_] := a; +E^(Complex[0, n_Integer]*Pi) := -(Mod[n, 2]*2 - 1); +a_Real ^ Complex[b_Real, c_Real] := Module[{inner}, + inner = b Arg[a]+1/2 c Log[a^2]; + (a^2)^(b/2) E^(-c Arg[a]) * Complex[Cos[inner], Sin[inner]] +]; +Attributes[Power] = {Listable, NumericFunction, OneIdentity, Protected}; +Tests`+"`"+`Power = { + ESimpleExamples[ + EComment["Exponents of integers are computed exactly:"], + ESameTest[-1/125, (-5)^-3], + EComment["Floating point exponents are handled with floating point precision:"], + EStringTest["1.99506e+3010", ".5^-10000."], + EComment["Automatically apply some basic simplification rules:"], + ESameTest[m^4., (m^2.)^2] + ], EFurtherExamples[ + EComment["Expreduce handles problematic exponents accordingly:"], + EStringTest["Indeterminate", "0^0"], + ESameTest[ComplexInfinity, 0^(-1)] + ], ETests[ + (*Test raising expressions to the first power*) + ESameTest[1 + x, (x+1)^1], + EStringTest["0", "0^1"], + EStringTest["0.", "0.^1"], + ESameTest[-5, -5^1], + ESameTest[-5.5, -5.5^1], + ESameTest[1 + x, (x+1)^1.], + EStringTest["0", "0^1."], + EStringTest["0.", "0.^1."], + ESameTest[-5, (-5)^1.], + ESameTest[-5.5, -5.5^1.], + + (*Test raising expressions to the zero power*) + EStringTest["1", "(x+1)^0"], + EStringTest["Indeterminate", "0^0"], + EStringTest["Indeterminate", "0.^0"], + ESameTest[-1, -5^0], + EStringTest["1", "(-5)^0"], + EStringTest["1", "(-5.5)^0"], + EStringTest["1", "(x+1)^0."], + EStringTest["Indeterminate", "0^0."], + EStringTest["Indeterminate", "0.^0."], + ESameTest[-1, -5^0.], + EStringTest["1", "(-5.5)^0."], + ESameTest[-1, -5^0], + EStringTest["1", "99^0"], + + EStringTest["125", "5^3"], + ESameTest[1/125, 5^-3], + ESameTest[-125, (-5)^3], + ESameTest[-1/125, (-5)^-3], + + EStringTest["2.97538e+1589", "39^999."], + EStringTest["3.36092e-1590", "39^-999."], + EStringTest["1.99506e+3010", ".5^-10000."], + EStringTest["1.99506e+3010", ".5^-10000"], + + EStringTest["1", "1^1"], + EStringTest["1", "1^2"], + EStringTest["1", "1^0"], + EStringTest["1", "1^-1"], + EStringTest["1", "1^-2"], + EStringTest["1", "1^99999992"], + EStringTest["1.", "1^2."], + EStringTest["1.", "1^99999992."], + EStringTest["1.", "1.^30"], + EStringTest["4.", "(1.*2*1.)^2"], + ESameTest[-1, (-1)^1], + EStringTest["1", "(-1)^2"], + EStringTest["1", "(-1)^0"], + EStringTest["1", "(-1)^0"], + ESameTest[-1, (-1)^-1], + EStringTest["1", "(-1)^-2"], + EStringTest["1", "(-1)^99999992"], + EStringTest["1.", "(-1.)^30"], + EStringTest["4.", "(1.*2*-1.)^2"], + ESameTest[-0.5, (1.*2*-1.)^(-1)], + + ESameTest[Rational, Power[2, -1] // Head], + ESameTest[Integer, Power[1, -1] // Head], + ESameTest[Integer, Power[2, 2] // Head], + ESameTest[Rational, Power[-2, -1] // Head], + ESameTest[Rational, Power[2, -2] // Head], + + (*Exponent simplifications*) + ESameTest[m^4, m^2*m^2], + ESameTest[m^4, (m^2)^2], + ESameTest[(m^2)^2., (m^2)^2.], + ESameTest[(m^2.)^2., (m^2.)^2.], + ESameTest[m^4., (m^2.)^2], + + ESameTest[ComplexInfinity, 0^(-1)], + + ESameTest[{1}, ReplaceAll[a, a^p_. -> {p}]], + + ESameTest[0, 2^(-Infinity)], + ESameTest[0, (-2)^(-Infinity)], + ESameTest[Indeterminate, (-1)^(-Infinity)], + ESameTest[Indeterminate, (1)^(-Infinity)], + ESameTest[Indeterminate, (d)^(-Infinity)], + ESameTest[Infinity, 2^(Infinity)], + ESameTest[ComplexInfinity, (-2)^(Infinity)], + ESameTest[Indeterminate, (-1)^(Infinity)], + ESameTest[Indeterminate, (1)^(Infinity)], + ESameTest[Indeterminate, (d)^(Infinity)], + + (* Test nth-root algorithm. *) + ESameTest[2, 16^(1/4)], + ESameTest[2., 16^(1/4.)], + ESameTest[True, Head[101^(1/2)]=!=Integer], + ESameTest[True, Head[99^(1/2)]=!=Integer], + ESameTest[0, 0^(1/2)], + ESameTest[8, 2^3], + ESameTest[0, 0^(1/3)], + ESameTest[0, 0^(1/4)], + ESameTest[2, 8^(1/3)], + ESameTest[Power, 7^(1/3)//Head], + ESameTest[Power, 9^(1/3)//Head], + ESameTest[1/2, 16^(-1/4)], + ESameTest[ComplexInfinity, 0^(-1/3)], + ESameTest[ComplexInfinity, 0^(-1/2)], + ESameTest[1/3, 27^(-1/3)], + ESameTest[Power, 7^(-1/3)//Head], + ESameTest[Power, 9^(-1/3)//Head], + ESameTest[27, 9^(3/2)], + ESameTest[1/27, 9^(-3/2)], + + (* Test simplifying radicals *) + ESameTest[4 2^(1/3), (128)^(1/3)], + ESameTest[4 (-2)^(1/3), (-128)^(1/3)], + ESameTest[15 7^(2/3) 57^(1/3), 9426375^(1/3)], + ESameTest[15 7^(2/3) 57^(1/3), 9426375^(1/3)], + ESameTest[3 3^(1/3), (3^4)^(1/3)], + ESameTest[3 3^(1/3), (3)^(4/3)], + ESameTest[15 7^(2/3) 57^(1/3), (3^(4/3)*5^(3/3)*7^(2/3)*19^(1/3))], + ESameTest[15 7^(2/3) 57^(1/3), 5*3^(4/3)*7^(2/3)*19^(1/3)], + ESameTest[3 3^(1/3), 3^(4/3)], + ESameTest[3 3^(1/3), 3^(4/3)], + ESameTest[a^(4/3), a^(4/3)], + ESameTest[-(-1)^(1/3), (-1)^(4/3)], + ESameTest[(-1)^(2/3), (-1)^(-4/3)], + ESameTest[-(I/2), (-4)^(-1/2)], + ESameTest[Indeterminate, (-1)^(-1/0)], + ESameTest[1., (-1.)^0.5 // Im], + ], EKnownFailures[ + ESameTest[(3+I Sqrt[29]) E^(-((2 I \[Pi])/3)), ((3 + I*Sqrt[29])^3)^(1/3)], + ESameTest[{{-5,1/5^(1/3)},{-4,1/2^(2/3)},{-3,1/3^(1/3)},{-2,1/2^(1/3)},{-1,1},{0,ComplexInfinity},{1,-(-1)^(2/3)},{2,-((-1)^(2/3)/2^(1/3))},{3,-((-1)^(2/3)/3^(1/3))},{4,-(-(1/2))^(2/3)},{5,-((-1)^(2/3)/5^(1/3))}}, Table[{n, (-n)^(-1/3)}, {n, -5, 5}] // Quiet], + ] +}; + +Log::usage = "`+"`"+`Log[e]`+"`"+` finds the natural logarithm of `+"`"+`e`+"`"+`."; +Log[n_Integer?Negative] := I*Pi + Log[-n]; +Log[ComplexInfinity] := Infinity; +Log[Infinity] := Infinity; +Log[-ComplexInfinity] := Infinity; +Log[-Infinity] := Infinity; +Log[0] := -Infinity; +Log[1] := 0; +Log[E] := 1; +Log[E^p_?NumberQ] := p; +Log[Rational[1,b_]] := -Log[a]; +Attributes[Log] = {Listable,NumericFunction,Protected}; +Tests`+"`"+`Log = { + ESimpleExamples[ + ESameTest[1, Log[E]], + ESameTest[-2, Log[E^(-2)]], + ESameTest[Log[2], Log[2]] + ] +}; + +Sqrt::usage = "`+"`"+`Sqrt[e]`+"`"+` finds the square root of `+"`"+`e`+"`"+`."; +(*TODO: automatically simplify perfect squares*) +Attributes[Sqrt] = {Listable, NumericFunction, Protected}; +Sqrt[a_Integer?Negative] := I*Sqrt[-a]; +Sqrt[-a_?NumberQ] := I*Sqrt[a]; +Sqrt[a_Integer*b_Plus] := Sqrt[Abs[a]]*Sqrt[(a/Abs[a])*b] /; a != 0; +Sqrt[a_Real?Positive] := a^.5; +Sqrt[x_] := Which[ + (*Normally we would define these directly, but right now "x_" is + considered more specific than 0 or 1. *) + x===0, 0, + x===1, 1, + True, x^(1/2) +]; +Tests`+"`"+`Sqrt = { + ESimpleExamples[ + ESameTest[Sqrt[3], Sqrt[3]], + ESameTest[I, Sqrt[-1]], + ESameTest[I * Sqrt[3], Sqrt[-3]], + ESameTest[1, Sqrt[1]], + ESameTest[0, Sqrt[0]] + ], ETests[ + ESameTest[Sqrt[2] Sqrt[-2-x^y], (-2)(x^y+2)//Sqrt], + ] +}; + +I::usage = "`+"`"+`I`+"`"+` is the imaginary number representing `+"`"+`Sqrt[-1]`+"`"+`."; +I := Complex[0, 1]; +Attributes[I] = {Locked, Protected, ReadProtected}; +Tests`+"`"+`I = { + ESimpleExamples[ + ESameTest[-1, I^2], + ESameTest[1, I^4] + ] +}; + +possibleExponents[n_Integer, m_Integer] := + Flatten[Permutations /@ ((PadRight[#, m]) & /@ + IntegerPartitions[n, m]), 1]; +genVars[addends_List, exponents_List] := Times@@(addends ^ exponents); +genExpand[addends_List, exponents_List] := + Plus@@Table[(Multinomial @@ exponents[[ExpandUnique`+"`"+`i]])* + genVars[addends, exponents[[ExpandUnique`+"`"+`i]]], {ExpandUnique`+"`"+`i, 1, + Length[exponents]}]; +expandRules := { + s_Plus^n_Integer?Positive :> + genExpand[List @@ s, possibleExponents[n, Length[s]]], + c_*s_Plus :> ((c*#) &) /@ s +}; +Expand::usage = "`+"`"+`Expand[expr]`+"`"+` attempts to expand `+"`"+`expr`+"`"+`."; +Expand[a_] := a //. expandRules; +Expand[a_, x_] := (Print["Expand does not support second argument."];Expand[a]); +Attributes[Expand] = {Protected}; +Tests`+"`"+`Expand = { + ESimpleExamples[ + ESameTest[a^3 + 3 a^2 * b + 3 a b^2 + b^3 + 3 a^2 * c + 6 a b c + 3 b^2 * c + 3 a c^2 + 3 b c^2 + c^3, Expand[(a + b + c)^3]], + ESameTest[a c + b c + a d + b d + a e + b e, (a + b) * (c + d + e) // Expand], + ESameTest[a d^2 + b d^2 + c d^2 + 2 a d e + 2 b d e + 2 c d e + a e^2 + b e^2 + c e^2, (a + b + c)*(d + e)^2 // Expand], + ESameTest[a^(2 b) + 2 a^b * c^d + c^(2 d), Expand[(a^b + c^d)^2]], + ESameTest[a/d + b/d + c/d, Expand[(a + b + c)/d]], + ESameTest[1/d + (2 a)/d + a^2/d + b/d + c/d, Expand[((a + 1)^2 + b + c)/d]], + ESameTest[2 + 2 a, 2*(a + 1) // Expand] + ], ETests[ + ESameTest[Null, ((60 * c * a^2 * b^2) + (30 * c * a^2 * b^2) + (30 * c * a^2 * b^2) + a^5 + b^5 + c^5 + (5 * a * b^4) + (5 * a * c^4) + (5 * b * a^4) + (5 * b * c^4) + (5 * c * a^4) + (5 * c * b^4) + (10 * a^2 * b^3) + (10 * a^2 * c^3) + (10 * a^3 * b^2) + (10 * a^3 * c^2) + (10 * b^2 * c^3) + (10 * b^3 * c^2) + (20 * a * b * c^3) + (20 * a * c * b^3) + (20 * b * c * a^3));], + ESameTest[1/3, ((1+I Sqrt[3]) (1/(2 2^(1/3))-(I Sqrt[3])/(2 2^(1/3))))/(3 2^(2/3))//Expand], + ] +}; + +ExpandAll::usage = "`+"`"+`ExpandAll[expr]`+"`"+` attempts to expand `+"`"+`expr`+"`"+` at all levels."; +ExpandAll[a_] := Map[Expand, a, {0, Infinity}]; +Attributes[ExpandAll] = {Protected}; +Tests`+"`"+`ExpandAll = { + ESimpleExamples[ + ESameTest[Log[-1+x^2], Log[(x+1)(x-1)]//ExpandAll], + ] +}; + +PolynomialQ::usage = "`+"`"+`PolynomialQ[e, var]`+"`"+` returns True if `+"`"+`e`+"`"+` is a polynomial in `+"`"+`var`+"`"+`."; +innerPolynomialQ[p_Plus, v_] := + AllTrue[List @@ p, (PolynomialQ[#, v]) &]; +innerPolynomialQ[p_.*v_^Optional[exp_Integer], v_] := + If[FreeQ[p, v] && Positive[exp], True, False]; +innerPolynomialQ[p_, v_] := If[FreeQ[p, v], True, False]; +PolynomialQ[p_, v_] := innerPolynomialQ[p//Expand, v]; +(*Seemingly undocumented version with no variable specification:*) +PolynomialQ[p_] := PolynomialQ[p, Variables[p]]; +Attributes[PolynomialQ] = {Protected}; +Tests`+"`"+`PolynomialQ = { + ETests[ + ESameTest[True, PolynomialQ[2x^2-3x+2, x]], + ESameTest[True, PolynomialQ[2x^2, x]], + ESameTest[False, PolynomialQ[2x^2.5, x]], + ESameTest[False, PolynomialQ[2x^-2, x]], + ESameTest[True, PolynomialQ[2x^0, x]], + ESameTest[False, PolynomialQ[2x^y, x]], + ESameTest[True, PolynomialQ[-3x, x]], + ESameTest[True, PolynomialQ[2, x]], + ESameTest[True, PolynomialQ[2y^2, x]], + ESameTest[True, PolynomialQ[-3y, x]], + ESameTest[False, PolynomialQ[2x^2-3x+2+Cos[x], x]], + ESameTest[False, PolynomialQ[Cos[x], x]], + ESameTest[False, PolynomialQ[2x^2-3x+Cos[x], x]], + ESameTest[False, PolynomialQ[2x^2-x*Cos[x], x]], + ESameTest[True, PolynomialQ[2x^2-x*Cos[y], x]], + ESameTest[True, PolynomialQ[2.5x^2-3x+2.5, 2.5]], + ESameTest[True, PolynomialQ[2x^2-3x+2, "hello"]], + ESameTest[True, PolynomialQ[2x^2-3x+2, y]], + ESameTest[True, PolynomialQ[x, y]], + ESameTest[True, PolynomialQ[y, y]], + ESameTest[True, PolynomialQ[2*x, y]], + ESameTest[False, PolynomialQ[2*x^Sin[y], x]], + ESameTest[True, PolynomialQ[Sin[y]*x^2, x]], + ESameTest[False, PolynomialQ[Sin[y]*x^2.5, x]], + ESameTest[False, PolynomialQ[Sin[y]*x^y, x]], + ESameTest[True, PolynomialQ[2*y, y]], + ESameTest[True, PolynomialQ[y*x, y]], + ESameTest[True, PolynomialQ[y*x, z]], + ESameTest[True, PolynomialQ[y*Sin[x], z]], + ESameTest[False, PolynomialQ[y^x, y]], + ESameTest[False, PolynomialQ[x^y, y]], + ESameTest[True, PolynomialQ[x^y, z]], + ESameTest[True, PolynomialQ[2.5*x^2, 2.5]], + ESameTest[True, PolynomialQ[2.5*x, 2.5]], + ESameTest[False, PolynomialQ[2*x^2, 2]], + ESameTest[True, PolynomialQ[2*x, 2]], + ESameTest[True, PolynomialQ[x, 2]], + ESameTest[True, PolynomialQ[Cos[x*y], Cos[x*y]]], + ESameTest[True, PolynomialQ[x^2,2.]], + ESameTest[False, PolynomialQ[x^a,a]], + ESameTest[False, PolynomialQ[x^n,x]], + ESameTest[True, PolynomialQ[-x*Cos[y],x]], + ESameTest[True, PolynomialQ[x^y, 1]], + ESameTest[True, PolynomialQ[(-280*c^4*d^2*x^2 + -315*c^6*d^2*x^4 + 9*c^2*(63*d^2 + 90*c^2*d^2*x^2 + 35*c^4*d^2*x^4))//Expand,x]], + ESameTest[True, PolynomialQ[(-280*c^4*d^2*x^2 + -315*c^6*d^2*x^4 + 9*c^2*(63*d^2 + 90*c^2*d^2*x^2 + 35*c^4*d^2*x^4)),x]], + ], EKnownFailures[ + ESameTest[True, PolynomialQ[2*x^2-3x+2, 2]], + ESameTest[True, PolynomialQ[2*x^2-3x, 2]], + ESameTest[False, PolynomialQ[x/y]] + ] +}; + +Exponent::usage = "`+"`"+`Exponent[p, var]`+"`"+` returns the degree of `+"`"+`p`+"`"+` with respect to the variable `+"`"+`var`+"`"+`."; +Exponent[expr_/p_Plus, var_, head_] := Exponent[expr, var, head]; +Exponent[expr_, var_, head_] := If[expr === 0, head[], + Module[{e = expr, v = var, h = head, theCases, toCheck}, + toCheck = expr // Expand; + toCheck = If[Head[toCheck] === Plus, toCheck, {toCheck}]; + theCases = + Cases[toCheck, p_.*v^Optional[exp_] -> exp] // DeleteDuplicates; + If[Length[theCases] =!= Length[toCheck], PrependTo[theCases, 0]]; + h @@ theCases + ]]; +Exponent[expr_, var_] := Exponent[expr, var, Max]; +Attributes[Exponent] = {Listable, Protected}; +Tests`+"`"+`Exponent = { + ESimpleExamples[ + EComment["Find the degree of a polynomial:"], + ESameTest[5, Exponent[3 + x^3 + k*x^5, x]] + ], EFurtherExamples[ + EComment["Find the degree of a polynomial:"], + ESameTest[{0,3,5}, Exponent[3 + x^3 + k*x^5, x, List]] + ], ETests[ + (*Sorting of the input addition expression is off here, so we sort*) + (*the result so it does match up.*) + ESameTest[{0,3,5,x^x}, Exponent[3 + "hello" + x^3 + a*x^5 + x^x^x, x, List]//Sort], + ESameTest[{0}, Exponent[1 + x^x^x, x^x, List]], + ESameTest[{0}, Exponent[2 + a, x, List]], + ESameTest[{0}, Exponent[a, x, List]], + ESameTest[{2}, Exponent[x^2, x, List]], + ESameTest[{1}, Exponent[x^2 - x*(a + x), x, List]], + ESameTest[{0,1}, Exponent[(1 + x)/(3 + x), x, List]], + ESameTest[{0,2}, Exponent[(1 + x^2)/(3 + x), x, List]], + ESameTest[{0,1}, Exponent[(1 + x)/(3 + x^3), x, List]], + ESameTest[{0}, Exponent[(3 + x^3)^(-1), x, List]], + ESameTest[{-3}, Exponent[x^(-3), x, List]], + ESameTest[{-3}, Exponent[a/x^3, x, List]], + ESameTest[{-2}, Exponent[x^(-2), x, List]], + ESameTest[{1}, Exponent[(a*x)/(3 + x^3), x, List]], + ESameTest[{0,1}, Exponent[1 + b*x + x^2 - (x*(1 + a*x))/a, x, List]], + ESameTest[{0,1}, Exponent[1 + x + x^2 - (x*(1 + 2*x))/2, x, List]], + ESameTest[{}, Exponent[0, x, List]], + ESameTest[{}, Exponent[0, x+2, List]], + ESameTest[{}, Exponent[0, 0, List]] + ], EKnownFailures[ + ESameTest[{0,1}, Exponent[1 + x^x^x, x^x^x, List]] + ] +}; + +ExpreduceSingleCoefficient[inP_, inTerm_] := + Module[{p = inP, term = inTerm, pat}, + (*If[MatchQ[p,term],Return[1]];*) + pat = If[term === 1, Print["Warning: term of 1 used"]; a_?NumberQ, Optional[a_]*term]; + (*pat=Optional[a_]*term;*) + If[MatchQ[p, pat], + (p) /. pat -> a, 0] + ]; +ExpreduceNonProp[inP_, inTerm_] := + Module[{p = inP, term = inTerm, toMatch, pat}, + toMatch = p // Expand; + pat = Except[Optional[a_]*term^n_.]; + If[Head[toMatch] === Plus, + Plus@@Cases[toMatch, pat], + If[MatchQ[toMatch, pat], toMatch, 0]] + ]; +Coefficient::usage = "`+"`"+`Coefficient[p, form]`+"`"+` returns the coefficient of form `+"`"+`form`+"`"+` in polynomial `+"`"+`p`+"`"+`."; +Coefficient[p_, var_, exp_] := + If[exp === 0, + ExpreduceNonProp[p, var], + Coefficient[p, var^exp] + ]; +Coefficient[inP_, inTerm_] := + Module[{p = inP, term = inTerm, toMatch}, + toMatch = p // Expand; + If[Head[toMatch] === Plus, + Map[ExpreduceSingleCoefficient[#, term] &, toMatch], + ExpreduceSingleCoefficient[toMatch, term]] + ]; +Attributes[Coefficient] = {Listable, Protected}; +Tests`+"`"+`Coefficient = { + ESimpleExamples[ + ESameTest[3, Coefficient[(a + b)^3, a*b^2]] + ], ETests[ + ESameTest[j, Coefficient[c + j*a + k*b, a]], + ESameTest[a, Coefficient[c + k*x + a*x^3, x, 3]], + ESameTest[24, Coefficient[2*b*(2*a + 3*b)*(1 + 2*a + 3*b), a*b^2]], + ESameTest[29, Coefficient[(2 + x)^2 + (5 + x)^2, x, 0]], + ESameTest[1, Coefficient[a + x, x]], + ESameTest[4, Coefficient[2*b*(2*a + 3*b)*(1 + 2*a + 3*b), a*b]], + ESameTest[1, Coefficient[x^2, x^2]], + ESameTest[-a, Coefficient[x^2 - x*(a + x), x]], + ESameTest[-(1/a)+b, Coefficient[1 + b*x + x^2 - (x*(1 + a*x))/a, x]], + ESameTest[1/2, Coefficient[1 + x + x^2 - (x*(1 + 2*x))/2, x]], + ESameTest[a, Coefficient[a,x,0]], + ESameTest[a b, Coefficient[a*b,x,0]], + ESameTest[a b+c^d, Coefficient[a*b+c^d,x,0]], + ESameTest[a b+c^d, Coefficient[a*b+c^d+5x,x,0]], + ESameTest[0, Coefficient[(a*b+c^d)*x^2+5x,x,0]] + ] +}; + +CoefficientList::usage = "`+"`"+`CoefficientList[p, var]`+"`"+` returns the list of coefficients associated with variable `+"`"+`var`+"`"+`."; +CoefficientList[p_, var_] := + Table[Coefficient[p,var,i],{i,0,Exponent[p,var]}]; +Attributes[CoefficientList] = {Protected}; +Tests`+"`"+`CoefficientList = { + ESimpleExamples[ + ESameTest[{b,3,5}, CoefficientList[b+3x+5x^2,x]], + ESameTest[{0,0,5}, CoefficientList[5x^2,x]], + ESameTest[{-(a/b),1/b}, CoefficientList[(-a+x)/b,x]] + ], ETests[ + ESameTest[{b,3,0,5}, CoefficientList[b+3x+5x^3,x]], + ESameTest[{b+3 x+5 x^3}, CoefficientList[b+3x+5x^3,y]], + ESameTest[{3 x+5 x^3,1}, CoefficientList[b+3x+5x^3,b]] + ] +}; + +PolynomialQuotientRemainder::usage = "`+"`"+`PolynomialQuotientRemainder[poly_, div_, var_]`+"`"+` returns the quotient and remainder of `+"`"+`poly`+"`"+` divided by `+"`"+`div`+"`"+` treating `+"`"+`var`+"`"+` as the polynomial variable."; +ExpreduceLeadingCoeff[p_, x_] := Coefficient[p, x^Exponent[p, x]]; +PolynomialQuotientRemainder[inp_, inq_, v_] := + Module[{a = inp, b = inq, x = v, r, d, c, i, s, q, rExp}, + (*I should think carefully about when I use = vs := to avoid unwanted evaluation*) + q = 0; + r = a; + d = Exponent[b, x]; + pow = x^d; + (* This should happen any time that inq is free of v, or if inq === 1. *) + If[pow === 1, Return[{a/b//Distribute, 0}]]; + c = Coefficient[b, pow]; + i = 1; + While[rExp = Exponent[r, x]; rExp >= d && i < 20, + (*Looks like we get the coefficient and the exponent of the leading term here. Perhaps we can just grab the leading term and get both at once. And maybe we can exploit the canonical ordering*) + (*But all of this seems wrong. I think the slowness resides in the expreduce interpreter.*) + (*TODO tomorrow: find out what 'power' function takes up the most time and optimize it by making kernel changes*) + s = (Coefficient[r, x^rExp]/c)*x^(rExp - d); + q = q + s; + r = r - s*b; + i = i + 1; + ]; + {q, r} // Expand + ]; +Attributes[PolynomialQuotientRemainder] = {Protected}; +PolynomialQuotient::usage = "`+"`"+`PolynomialQuotient[poly_, div_, var_]`+"`"+` returns the quotient of `+"`"+`poly`+"`"+` divided by `+"`"+`div`+"`"+` treating `+"`"+`var`+"`"+` as the polynomial variable."; +PolynomialQuotient[inp_, inq_, v_] := + PolynomialQuotientRemainder[inp, inq, v][[1]]; +Attributes[PolynomialQuotient] = {Protected}; +PolynomialRemainder::usage = "`+"`"+`PolynomialRemainder[poly_, div_, var_]`+"`"+` returns the remainder of `+"`"+`poly`+"`"+` divided by `+"`"+`div`+"`"+` treating `+"`"+`var`+"`"+` as the polynomial variable."; +PolynomialRemainder[inp_, inq_, v_] := + PolynomialQuotientRemainder[inp, inq, v][[2]]; +Attributes[PolynomialRemainder] = {Protected}; +Tests`+"`"+`PolynomialQuotientRemainder = { + ESimpleExamples[ + ESameTest[{x^2/2,2}, PolynomialQuotientRemainder[2 + x^2 + x^3, 2 + 2*x, x]], + ESameTest[{x^2-x y+y^2,-y^3}, PolynomialQuotientRemainder[x^3, x + y, x]], + ESameTest[{x/a,1-x/a}, PolynomialQuotientRemainder[1 + x^3, 1 + a*x^2, x]] + ], ETests[ + ESameTest[{b+a/x,0}, PolynomialQuotientRemainder[a+b*x,x,a]], + ESameTest[{a+b x,0}, PolynomialQuotientRemainder[a+b*x,1,a]], + ESameTest[{a/c+(b x)/c,0}, PolynomialQuotientRemainder[a+b*x,c,a]] + ] +}; +Tests`+"`"+`PolynomialQuotient = { + ESimpleExamples[ + ESameTest[x^2/2, PolynomialQuotient[2 + x^2 + x^3, 2 + 2*x, x]], + ESameTest[x^2-x y+y^2, PolynomialQuotient[x^3, x + y, x]], + ESameTest[x/a, PolynomialQuotient[1 + x^3, 1 + a*x^2, x]] + ] +}; +Tests`+"`"+`PolynomialRemainder = { + ESimpleExamples[ + ESameTest[2, PolynomialRemainder[2 + x^2 + x^3, 2 + 2*x, x]], + ESameTest[-y^3, PolynomialRemainder[x^3, x + y, x]], + ESameTest[1-x/a, PolynomialRemainder[1 + x^3, 1 + a*x^2, x]] + ] +}; + +FactorTermsList::usage = "`+"`"+`FactorTermsList[expr]`+"`"+` factors out the constant term of `+"`"+`expr`+"`"+` and puts the factored result into a `+"`"+`List`+"`"+`."; +ExpreduceConstantTerm[c_?NumberQ] := {c, 1}; +ExpreduceConstantTerm[c_?NumberQ*e_] := {c, e}; +ExpreduceConstantTerm[e_] := {1, e}; +FactorTermsList[expr_] := Module[{e = expr, toFactor, cTerms, c}, + toFactor = e // Expand; + If[Head[toFactor] =!= Plus, + Return[ExpreduceConstantTerm[toFactor]] + ]; + (* Parens are necessary due to precedence issue. *) + cTerms = ((ExpreduceConstantTerm /@ (List @@ toFactor)) // + Transpose)[[1]]; + c = GCD @@ cTerms; + If[Last[cTerms] < 0, c = -c]; + {c, toFactor/c // Expand} + ]; +Attributes[FactorTermsList] = {Protected}; +Tests`+"`"+`FactorTermsList = { + ESimpleExamples[ + ESameTest[{2,Sin[8 k]}, FactorTermsList[2*Sin[8*k]]], + ESameTest[{1/2,a+x}, FactorTermsList[a/2 + x/2]], + ESameTest[{1,a+x}, FactorTermsList[a + x]] + ], ETests[ + ESameTest[{1,1}, FactorTermsList[1]], + ESameTest[{5,1}, FactorTermsList[5]], + ESameTest[{5.,1}, FactorTermsList[5.]], + ESameTest[{1,a}, FactorTermsList[a]], + ESameTest[{1/2,a}, FactorTermsList[a/2]], + ESameTest[{-(3/2),x}, FactorTermsList[(-3*x)/2]], + ESameTest[{2,a+x}, FactorTermsList[2*a + 2*x]], + ESameTest[{1/2,a/(2 b+2 c)+x/(2 b+2 c)}, FactorTermsList[(a/2 + x/2)/(2*b + 2*c)]], + ESameTest[{1,2+x^2}, FactorTermsList[(8 + 4*x^2)/4]], + ESameTest[{-(1/2),2+3 x+x^2}, FactorTermsList[(-4 - 6*x - 2*x^2)/4]], + ESameTest[{-(1/2),-2+3 x+x^2}, FactorTermsList[(2 - 3*x - x^2)/2]], + ESameTest[{-(1/2),-2-3 x+x^2}, FactorTermsList[(2 + 3*x - x^2)/2]], + ESameTest[{1/2,2+3 x+x^2}, FactorTermsList[(2 + 3*x + x^2)/2]], + ESameTest[{1/2,-2-3 x+x^2}, FactorTermsList[(-2 - 3*x + x^2)/2]], + ESameTest[{5,2+3 x+x^2}, FactorTermsList[5*(1 + x)*(2 + x)]], + ESameTest[{40,1+3 x+3 x^2+x^3}, FactorTermsList[5*(2 + 2*x)^3]], + ESameTest[{-6,1+x}, FactorTermsList[(-12 - 12*x)/2]], + ESameTest[{2/3,3+x}, FactorTermsList[(18 + 6*x)/9]], + ESameTest[{-(2800301/67344500),1-2 x+x^3}, FactorTermsList[(-2800301/538756 + (2800301*x)/269378 - (2800301*x^3)/538756)/125]] + ] +}; + +FactorTerms::usage = "`+"`"+`FactorTerms[expr]`+"`"+` factors out the constant term of `+"`"+`expr`+"`"+`, if any."; +FactorTerms[expr_] := Module[{e = expr, factored}, + factored = FactorTermsList[e]; + If[factored[[1]] === 1, + factored[[2]], + Times@@factored + ] + ]; +Attributes[FactorTerms] = {Protected}; +Tests`+"`"+`FactorTerms = { + ESimpleExamples[ + ESameTest[2*Sin[8 k], FactorTerms[2*Sin[8*k]]], + ESameTest[(1/2)*(a+x), FactorTerms[a/2 + x/2]], + ESameTest[a+x, FactorTerms[a + x]] + ] +}; + +ExpreduceFactorConstant[p_Plus] := Module[{e = p, cTerms, c}, + (* Parens are necessary due to precedence issue. *) + cTerms = ((ExpreduceConstantTerm /@ (List @@ e)) // Transpose)[[1]]; + c = GCD @@ cTerms; + If[Last[cTerms] < 0, c = -c]; + c * Distribute[e/c] + ]; +ExpreduceFactorConstant[e_] := e; +Attributes[ExpreduceFactorConstant] = {Protected}; + +Varibles::usage = "`+"`"+`Variables[expr]`+"`"+` returns the variables in `+"`"+`expr`+"`"+`."; +Variables[s_Symbol] := {s}; +Variables[s_^p_Integer] := Variables[s]; +Variables[s_^p_Rational] := Variables[s]; +Variables[s_^p_Plus] := + If[NumericQ[s], {}, (((s^#) &) /@ p) // Variables]; +Variables[s_^p_] := If[NumericQ[s], {}, {s^p}]; +Variables[s_^p_Times] := + If[NumericQ[s], {}, {s^DeleteCases[p, _Integer]}]; +Variables[e_] := ( + If[NumericQ[e] || Length[e] === 0, Return[{}]]; + If[MemberQ[{Plus, Times, List}, Head[e]], + Return[Union @@ Variables /@ (List @@ e)]]; + If[Length[e] > 0, Return[{e}]]; + Unknown + ); +Attributes[Variables] = {Protected}; +Tests`+"`"+`Variables = { + ESimpleExamples[ + ESameTest[{x, y}, Variables[x + y + y^2]], + ESameTest[{w^w, x^y, z}, Variables[w^w + x^y + z]], + ESameTest[{a, b^c, b^d}, Variables[a^2*b^(2*c + 2*d)]] + ], ETests[ + ESameTest[{x, y}, Variables[x*y]], + ESameTest[{x, y}, Variables[x + y]], + ESameTest[{x, y, y^2.5}, Variables[x + y + y^2.5]], + ESameTest[{y}, Variables[y^2]], + ESameTest[{x^y}, Variables[x^y]], + ESameTest[{x^y, y^x}, Variables[x^y + y^x]], + ESameTest[{x^y, z}, Variables[x^y + z]], + ESameTest[{w, x^y, z}, Variables[w^2 + x^y + z]], + ESameTest[{}, Variables[2^(x + y)]], + ESameTest[{}, Variables[2^x]], + ESameTest[{}, Variables[foo[]]], + ESameTest[{foo[x]}, Variables[foo[x]]], + ESameTest[{foo[x, y]}, Variables[foo[x, y]]], + ESameTest[{foo[2]}, Variables[foo[2]]], + ESameTest[{}, Variables[Sin[2]]], + ESameTest[{Sin[x]}, Variables[Sin[x]]], + ESameTest[{}, Variables[1]], + ESameTest[{x}, Variables[{x}]], + ESameTest[{}, Variables[{1}]], + ESameTest[{x}, Variables[x]], + ESameTest[{a, b, x, y, z}, Variables[a + (a + b)^2 + x*y^3 + 2*z]], + ESameTest[{a, b}, Variables[(a + b)^2]], + ESameTest[{a, b}, Variables[(a + 2*b)^2]], + ESameTest[{a, b^c, b^d}, Variables[(a + b^(c + d))^2]], + ESameTest[{a, b^c, b^d}, Variables[a + b^(c + d)]], + ESameTest[{(a*b^(c + d))^e}, Variables[(a*b^(c + d))^e]], + ESameTest[{(a + b)^c}, Variables[(a + b)^c]], + ESameTest[{(a + b)^c, (a + b)^d}, Variables[(a + b)^(c + d)]], + ESameTest[{}, Variables[2^(c + d)]], + ESameTest[{Log[b]}, Variables[Log[b]]], + ESameTest[{a^b, a^c}, Variables[a^(b + c)]], + ESameTest[{b^c, b^d}, Variables[b^(2*c + 2*d)]], + ESameTest[{b^(c*d)}, Variables[b^(2*c*d)]], + ESameTest[{b^(c*d)}, Variables[b^(c*d)]], + ESameTest[{b^(2.5*c*d)}, Variables[b^(2.5*c*d)]], + ESameTest[{(a + b)^2.5}, Variables[(a + b)^2.5]], + ESameTest[{(a + b)^(2.5*a)}, Variables[(a + b)^(2.5*a)]], + ESameTest[{(a + b)^2.5, (a + b)^a}, Variables[(a + b)^(2.5 + a)]], + ESameTest[{}, Variables[5.656854249492381]], + ESameTest[{}, Variables[{}]], + ESameTest[{a^"Hello"}, Variables[a^"Hello"]], + ESameTest[{}, Variables[2^"Hello"]], + ESameTest[{}, Variables[2^"Hello"^2]], + ESameTest[{a^"Hello"^2}, Variables[a^"Hello"^2]], + + ESameTest[{}, Variables[Pi^y]], + ESameTest[{a, Log[b]}, Variables[Sqrt[a] + Log[b]]], + ESameTest[{a}, Variables[Sqrt[a]]] + ], EKnownFailures[ + (*I think these have to do with NumericQ.*) + ESameTest[{(a*b)^c, (a*b)^d}, Variables[(a*b)^(c + d)]] + ] +}; + +PolynomialGCD::usage = "`+"`"+`PolynomialGCD[a, b]`+"`"+` calculates the polynomial GCD of `+"`"+`a`+"`"+` and `+"`"+`b`+"`"+`."; +PolySubresultantGCD[inA_, inB_, inX_] := + Module[{u = inA, v = inB, x = inX, h, delta, beta, newU, newV, i}, + h = 1; + i = 1; + While[v =!= 0 && i < 20, + uEx = Exponent[u, x]; + vEx = Exponent[v, x]; + delta = uEx - vEx; + beta = (-1)^(delta + 1)*uEx*h^delta; + h = h*(vEx/h)^delta; + newU := v; + newV = PolynomialRemainder[u, v, x]/beta; + u = newU; + v = newV; + i = i + 1; + ]; + If[Exponent[u, x] == 0, 1, u] + ]; +(* doesn't work with rational functions yet. *) +(* Looks like prefactored inputs remain factored. *) +PolynomialGCD[inA_, inB_] := + FactorTermsList[ + PolySubresultantGCD[inA, inB, Variables[inA][[1]]]][[2]]; +Attributes[PolynomialGCD] = {Listable, Protected}; +Tests`+"`"+`PolynomialGCD = { + ESimpleExamples[ + ESameTest[5+a, PolynomialGCD[15+13 a+2 a^2,10+7 a+a^2]], + ESameTest[5+a+a^2, PolynomialGCD[15+13 a+5 a^2+2 a^3,10+7 a+3 a^2+a^3]], + ESameTest[-5-a+a^2, PolynomialGCD[15+13 a-a^2-2 a^3,5+a-a^2]] + ] +}; + +SquareFreeQ::usage = "`+"`"+`SquareFreeQ[expr]`+"`"+` returns True if `+"`"+`expr`+"`"+` is a square-free polynomial."; +(*only works for univariate polynomials, does not support numbers *) +SquareFreeQ[ex_] := Module[{f = ex, expF, polyvar, fprime}, + If[Length[Variables[f]] != 1, Return[False]]; + expF = Expand[f]; + polyvar = Variables[expF][[1]]; + If[! PolynomialQ[expF, polyvar], Return[False]]; + fprime = D[expF, polyvar]; + PolynomialGCD[expF, fprime] === 1]; +Attributes[SquareFreeQ] = {Protected, ReadProtected}; +Tests`+"`"+`SquareFreeQ = { + ESimpleExamples[ + ESameTest[False, SquareFreeQ[(x+1)(x+2)^2//Expand]], + ESameTest[True, SquareFreeQ[(x + 1) (x + 2)]], + ESameTest[True, SquareFreeQ[(2 x + 3) (x + 2) // Expand]], + ESameTest[False, SquareFreeQ[(2 x + 3)^2]] + ] +}; + +PSimplify[expr_] := expr; +PSimplify[p_?PolynomialQ/q_?PolynomialQ] := + If[Length[Variables[p]] === 1 && Variables[p] === Variables[q], + PolynomialQuotient[p, q, Variables[p][[1]]], p/q]; +Tests`+"`"+`PSimplify = { + ESimpleExamples[ + ESameTest[-1 + x^2, PSimplify[(1 - 2*x^2 + x^4)/(-1 + x^2)]], + ESameTest[4*x, PSimplify[(-4*x + 4*x^3)/(-1 + x^2)]], + ESameTest[-1 - x + x^3 + x^4, PSimplify[(1 - x^2 - x^3 + x^5)/(-1 + x)]], + ESameTest[2*x + 5*x^2 + 5*x^3, PSimplify[(-2*x - 3*x^2 + 5*x^4)/(-1 + x)]], + ESameTest[-6 + 11*x - 6*x^2 + x^3, PSimplify[(18 - 39*x + 29*x^2 - 9*x^3 + x^4)/(-3 + x)]], + ESameTest[13 - 15*x + 4*x^2, PSimplify[(-39 + 58*x - 27*x^2 + 4*x^3)/(-3 + x)]], + ESameTest[-3 - x + 3*x^2 + x^3, PSimplify[(-9 - 6*x + 8*x^2 + 6*x^3 + x^4)/(3 + x)]], + ESameTest[-2 + 6*x + 4*x^2, PSimplify[(-6 + 16*x + 18*x^2 + 4*x^3)/(3 + x)]] + ], ETests[ + ESameTest[12 + 4*x - 15*x^2 - 5*x^3 + 3*x^4 + x^5, PSimplify[(-108 - 108*x + 207*x^2 + 239*x^3 - 81*x^4 - 153*x^5 - 27*x^6 + 21*x^7 + 9*x^8 + x^9)/(-9 - 6*x + 8*x^2 + 6*x^3 + x^4)]], + ESameTest[12 - 54*x - 33*x^2 + 18*x^3 + 9*x^4, PSimplify[(-108 + 414*x + 717*x^2 - 324*x^3 - 765*x^4 - 162*x^5 + 147*x^6 + 72*x^7 + 9*x^8)/(-9 - 6*x + 8*x^2 + 6*x^3 + x^4)]] + ], +}; + +myFactorCommonTerms[a_] := a; +allTimes[p_Plus] := AllTrue[p, (Head[#] === Times) &]; +myFactorCommonTerms[a_Plus] := Module[{commonTerms}, + If[! allTimes[a], Return[a]]; + commonTerms = Intersection @@ a; + commonTerms ((a/commonTerms) // Expand) + ]; + +FactorSquareFree::usage = "`+"`"+`FactorSquareFree[poly]`+"`"+` computes the square free factorization of `+"`"+`poly`+"`"+`."; +FactorSquareFree[poly_] := + Module[{f = poly, a, b, nb, c, d, i, res, fprime, polyvar, vars}, + vars = Variables[f]; + If[Length[vars] != 1, Return[f//myFactorCommonTerms]]; + polyvar = vars[[1]]; + If[! PolynomialQ[f, polyvar], Return[f]]; + fprime = D[f, polyvar]; + a = PolynomialGCD[f, fprime]; + res = If[SquareFreeQ[a], a, a // FactorSquareFree]; + nb = PolynomialQuotient[f,a,polyvar]; + c = PolynomialQuotient[fprime,a,polyvar]; + d = c - D[nb, polyvar]; + i = 1; + b = nb; + While[b =!= 1 && i < 20, + a = PolynomialGCD[b, d]; + nb = PolynomialQuotient[b,a,polyvar]; + c = PolynomialQuotient[d,a,polyvar]; + res = res*If[SquareFreeQ[a], a, a // FactorSquareFree]; + i = i + 1; + b = nb; + d = c - D[b, polyvar] + (*Print[{Subscript[a, i-1],Subscript[b, i],Subscript[c, i], + Subscript[d, i]}]*)]; + res + ]; +Attributes[FactorSquareFree] = {Listable, Protected}; +Tests`+"`"+`FactorSquareFree = { + ESimpleExamples[ + ESameTest[(-1 + x^2)^2, FactorSquareFree[1 - 2*x^2 + x^4]], + ESameTest[(-1 + x)^2*(1 + 2*x + 2*x^2 + x^3), FactorSquareFree[1 - x^2 - x^3 + x^5]], + ESameTest[(-3 + x)^2*(2 - 3*x + x^2), FactorSquareFree[18 - 39*x + 29*x^2 - 9*x^3 + x^4]], + ESameTest[(3 + x)^3*(-4 + x^2)*(-1 + x^2)^2, FactorSquareFree[-108 - 108*x + 207*x^2 + 239*x^3 - 81*x^4 - 153*x^5 - 27*x^6 + 21*x^7 + 9*x^8 + x^9]] + ], ETests[ + ESameTest[a (b+c), FactorSquareFree[a b+a c]], + ] +}; + +Factor::usage = "`+"`"+`Factor[poly]`+"`"+` factors `+"`"+`poly`+"`"+`."; +Factor[poly_] := poly; +Attributes[Factor] = {Listable, Protected}; +Tests`+"`"+`Factor = { + ETests[ + ESameTest[a, Factor[a]], + ] +}; + +PowerExpand[exp_] := exp //. { + Log[x_ y_]:>Log[x]+Log[y], + Log[x_^k_]:>k Log[x], + Sqrt[-a_]:>I*Sqrt[a], + Sqrt[a_^2]:>a, + Sqrt[a_/b_]:>Sqrt[a]/Sqrt[b], + (a_^b_Integer)^c_Rational:>a^(b*c) +}; +Attributes[PowerExpand] = {Protected}; +Tests`+"`"+`PowerExpand = { + ESimpleExamples[ + EComment["`+"`"+`PowerExpand`+"`"+` can expand nested log expressions:"], + ESameTest[Log[a] + e (Log[b] + d Log[c]), PowerExpand[Log[a (b c^d)^e]]], + ESameTest[{I Sqrt[a],a,Sqrt[a]/Sqrt[b]}, {Sqrt[-a],Sqrt[a^2],Sqrt[a/b]}//PowerExpand] + ] +}; + +Arg::usage = "`+"`"+`Arg[x]`+"`"+` computes the argument of `+"`"+`x`+"`"+`."; +Attributes[Arg] = {Listable, NumericFunction, Protected}; +Arg[a_?NumberQ] := ArcTan[Re[a], Im[a]]; +Arg[a_] := ArcTan[Re[a], Im[a]] ;/ (FreeQ[ReIm[a], Re] && FreeQ[ReIm[a], Im]); +Tests`+"`"+`Arg = { + ESimpleExamples[ + ESameTest[Pi/4, Arg[1/2 E^(I*Pi/4)]], + ] +}; + +Conjugate::usage = "`+"`"+`Conjugate[x]`+"`"+` computes the complex conjugate of `+"`"+`x`+"`"+`."; +Conjugate[a_] := a - 2 Im[a] I ;/ (FreeQ[Im[a], Re] && FreeQ[Im[a], Im]); +Attributes[Conjugate] = {Listable, NumericFunction, Protected, ReadProtected}; +Tests`+"`"+`Conjugate = { + ESimpleExamples[ + ESameTest[4-4I, Conjugate[4+4I]], + ESameTest[-4I, Conjugate[4I]], + ESameTest[4, Conjugate[4]], + ] +}; + +ComplexExpand::usage = "`+"`"+`ComplexExpand[e]`+"`"+` returns a complex expansion of `+"`"+`e`+"`"+`."; +Attributes[ComplexExpand] = {Protected}; +complexExpandInner[e_] := e; +complexExpandInner[(a_Integer?Negative)^b_Rational] := + Module[{coeff, inner}, + coeff = ((a^2)^(b/2)); + inner = b*Arg[a]; + coeff*Cos[inner] + I*coeff*Sin[inner]]; +complexExpandInner[E^((a_ + I*b_)*c_)] := E^(a c) Cos[b c]+I E^(a c) Sin[b c]; +complexExpandInner[E^(Complex[a_, b_]*c_)] := E^(a c) Cos[b c]+I E^(a c) Sin[b c]; +ComplexExpand[exp_] := + Map[complexExpandInner, exp, {0, Infinity}] // Expand; +Tests`+"`"+`ComplexExpand = { + ESimpleExamples[ + ESameTest[a, ComplexExpand[a]], + ESameTest[1, ComplexExpand[1]], + ESameTest[a b+a c, ComplexExpand[a*(b+c)]], + ESameTest[1/2+(I Sqrt[3])/2, ComplexExpand[(-1)^(1/3)]], + ESameTest[-(1/2)-(I Sqrt[3])/2, ComplexExpand[(-1)^(4/3)]], + ESameTest[2 2^(1/3), ComplexExpand[(2)^(4/3)]], + ESameTest[-1+I Sqrt[3], ComplexExpand[(-1)^(1/3) (1+I Sqrt[3])]], + ESameTest[d E^(a c) Cos[b c]+d I E^(a c) Sin[b c], ComplexExpand[d*E^((a+I*b)*c)]], + ESameTest[(1/2 + I/2)/Sqrt[2], ComplexExpand[1/2 E^(\[ImaginaryJ]*\[Pi]/4)]], + ], EKnownFailures[ + ESameTest[-2^(1/3)-I 2^(1/3) Sqrt[3], ComplexExpand[(-2)^(4/3)]], + ] +}; +`) + +func resourcesPowerMBytes() ([]byte, error) { + return _resourcesPowerM, nil +} + +func resourcesPowerM() (*asset, error) { + bytes, err := resourcesPowerMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/power.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRandomM = []byte(`RandomReal::usage = "`+"`"+`RandomReal[]`+"`"+` generates a random floating point from 0 to 1. + +`+"`"+`RandomReal[max]`+"`"+` generates a random floating point from 0 to `+"`"+`max`+"`"+`. + +`+"`"+`RandomReal[min, max]`+"`"+` generates a random floating point from `+"`"+`min`+"`"+` to `+"`"+`max."; +RandomReal[{min_, max_}] := RandomReal[]*(max - min) + min; +RandomReal[max_] := RandomReal[]*max; +Attributes[RandomReal] = {Protected}; +Tests`+"`"+`RandomReal = { + ESimpleExamples[ + EExampleOnlyInstruction["0.0750914", "RandomReal[]"] + ], EFurtherExamples[ + EComment["Use `+"`"+`SeedRandom`+"`"+` to seed the RNG:"], + EExampleOnlyInstruction["Null", "SeedRandom[3]"], + EExampleOnlyInstruction["0.719983", "RandomReal[]"], + EExampleOnlyInstruction["0.652631", "RandomReal[]"], + EExampleOnlyInstruction["Null", "SeedRandom[3]"], + EExampleOnlyInstruction["0.719983", "RandomReal[]"] + ] +}; + +SeedRandom::usage = "`+"`"+`SeedRandom[seed]`+"`"+` seeds the internal random number generator with a given integer `+"`"+`seed`+"`"+`."; +Attributes[SeedRandom] = {Protected}; +Tests`+"`"+`SeedRandom = { + ESimpleExamples[ + EExampleOnlyInstruction["0.0750914", "RandomReal[]"], + EExampleOnlyInstruction["Null", "SeedRandom[3]"], + EExampleOnlyInstruction["0.719983", "RandomReal[]"], + EExampleOnlyInstruction["0.652631", "RandomReal[]"], + EExampleOnlyInstruction["Null", "SeedRandom[3]"], + EExampleOnlyInstruction["0.719983", "RandomReal[]"] + ] +}; +`) + +func resourcesRandomMBytes() ([]byte, error) { + return _resourcesRandomM, nil +} + +func resourcesRandomM() (*asset, error) { + bytes, err := resourcesRandomMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/random.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesReplacementM = []byte(`ReplaceAll::usage = "`+"`"+`expr /. rule`+"`"+` replaces all occurences of the LHS of `+"`"+`rule`+"`"+` with the RHS of `+"`"+`rule`+"`"+` in `+"`"+`expr`+"`"+`. + +`+"`"+`expr /. {r1, r2, ...}`+"`"+` performes the same operation as `+"`"+`expr /. rule`+"`"+`, but evaluating each `+"`"+`r_n`+"`"+` in sequence."; +Attributes[ReplaceAll] = {Protected}; +Tests`+"`"+`ReplaceAll = { + ESimpleExamples[ + ESameTest[2^(y+1) + y, 2^(x^2+1) + x^2 /. x^2->y], + EComment["If no match is found, `+"`"+`ReplaceAll`+"`"+` evaluates to an unchanged `+"`"+`expr`+"`"+`:"], + ESameTest[2^(x^2+1) + x^2, 2^(x^2+1) + x^2 /. z^2->y], + EComment["`+"`"+`ReplaceAll`+"`"+` works within Orderless expressions as well (such as `+"`"+`Plus`+"`"+`):"], + ESameTest[b + c + d, a + b + c + c^2 /. c^2 + a -> d], + EComment["`+"`"+`ReplaceAll`+"`"+` can use named patterns:"], + ESameTest[a^b + c + d, a + b + c + d/. x_Symbol + y_Symbol -> x^y], + ESameTest[a + 99 * b + 99 * c, a + 2*b + 5*c /. (c1_Integer*a_Symbol) -> 99*a] + ], EFurtherExamples[ + EComment["`+"`"+`ReplaceAll`+"`"+` can be used to replace sequences of expressions:"], + ESameTest[foo[b, c, d], a + b + c + d /. a + amatch___ -> foo[amatch]], + EComment["The `+"`"+`Head`+"`"+` of functions can be replaced just as the subexpressions:"], + ESameTest[11, (x + 2)[5, 6] /. (2 + x) -> Plus], + ESameTest[2[2, 2, 2, 2], a*b*c*d /. _Symbol -> 2] + ], ETests[ + ESameTest[a * b * c, a*b*c /. c + a -> d], + ESameTest[b * d, a*b*c /. c*a -> d], + ESameTest[2 * a + b + c + c^2, 2 * a + b + c + c^2 /. c^2 + a -> d], + ESameTest[a^2 + b + c + d, a^2 + a + b + c + c^2 /. c^2 + a -> d], + ESameTest[a * b * c + a * b^2 * c, (a*b*c) + (a*b^2*c)], + ESameTest[b * d + b^2 * d, (a*b*c) + (a*b^2*c) /. c*a -> d], + ESameTest[b * d + b^2 * d, (a*b*c) + (a*b^2*c) /. a*c -> d], + ESameTest[a + b + c, a + b + c /. c + a -> c + a], + ESameTest[d, a*b*c /. c*a*b -> d], + ESameTest[a * b * c, a*b*c /. c*a*b*d -> d], + ESameTest[a*b*c*d*e, a*b*c*d*e /. a*b*f -> z], + ESameTest[z*d*e, a*b*c*d*e /. a*b*c -> z], + ESameTest[z*a*b, a*b*c*d*e /. e*d*c -> z], + ESameTest[z*a*b, a*b*c*d*e /. c*e*d -> z], + ESameTest[a^b, a + b /. x_Symbol + y_Symbol -> x^y], + ESameTest[2, x = 2], + ESameTest[2^b, a + b /. x_Symbol + y_Symbol -> x^y], + ESameTest[2, x], + ESameTest[a^b, a == b /. j_Symbol == k_Symbol -> j^k], + ESameTest[2, a == b /. j_Equal -> 2], + ESameTest[(a == b)^k, a == b /. j_Equal -> j^k], + ESameTest[3^k, 2^k /. base_Integer -> base + 1], + ESameTest[3^k, 2^k /. base_Integer^exp_ -> (base + 1)^exp], + ESameTest[(2 + k)^k, 2^k /. base_Integer^exp_ -> (base + exp)^exp], + ESameTest[(2 + k)^k, 2^k /. base_Integer^exp_Symbol -> (base + exp)^exp], + ESameTest[1 + (2 + k)^k, 2^k + 1 /. base_Integer^exp_Symbol -> (base + exp)^exp], + ESameTest[a^c + b, a^c + b /. test_Symbol^test_Symbol + test_Symbol -> test + 1], + ESameTest[1 + a, a^a + a /. test_Symbol^test_Symbol + test_Symbol -> test + 1], + ESameTest[a^a, a^a /. (test_Symbol^test) -> 2], + ESameTest[2, a^a /. (test_Symbol^test_Symbol) -> 2], + ESameTest[a^a, a^a /. (test^test_Symbol) -> 2], + ESameTest[2, test^a /. (test^test_Symbol) -> 2], + ESameTest[2, a^test /. (test_Symbol^test) -> 2], + EResetState[], + ESameTest[testa*testb, testa*testb /. a_Symbol*a_Symbol -> 5], + ESameTest[False, MatchQ[testa*testb, a_Symbol*a_Symbol]], + ESameTest[testa+testb, testa+testb /. a_Symbol+a_Symbol -> 5], + ESameTest[5, testa*testb /. a_Symbol*b_Symbol -> 5], + ESameTest[a+b, a + b /. (b_Symbol + b_Symbol) -> 2], + EResetState[], + + (*Test matching/replacement contexts*) + ESameTest[99^k, test = 99^k], + ESameTest[2, 99^k /. test -> 2], + ESameTest[2, 99^k /. test_ -> 2], + ESameTest[3, test2 = 3], + ESameTest[3, 99 /. test2_Integer -> test2], + ESameTest[a^b, a^b /. test3_Symbol^test3_Symbol -> k], + ESameTest[5, test3 = 5], + ESameTest[a^b, a^b /. test3_Symbol^test3_Symbol -> k], + + EResetState[], + ESameTest[a + 99 * b + 99 * c, a + 2*b + 5*c /. (c1_Integer*a_Symbol) -> 99*a], + ESameTest[a + 99 * b + 5 * c, a + 2*b + 5*c /. (2*a_Symbol) -> 99*a], + ESameTest[a + 99 * b + 99 * c, a + 2*b + 2*c /. (2*a_Symbol) -> 99*a], + ESameTest[a + 99 * b + 99 * c + 99 * d, a + 2*b + 3*c + 3*d /. (cl_Integer*a_Symbol) -> 99*a], + + (*Work way up to combining like terms*) + EResetState[], + ESameTest[a + 99 * b + 99 * c, a + 2*b + 5*c /. (c1_Integer*a_Symbol) -> 99*a], + ESameTest[a + 99 * b, a + 2*b + 5*c /. (c1_Integer*matcha_Symbol) + (c2_Integer*matchb_Symbol) -> 99*matcha], + ESameTest[a + (2 * b) + (5 * c), a + 2*b + 5*c /. (c1_Integer*matcha_Symbol) + (c2_Integer*matcha_Symbol) -> (c1+c2)*matcha], + ESameTest[(a + (7 * b)), a + 2*b + 5*b /. (c1_Integer*matcha_Symbol) + (c2_Integer*matcha_Symbol) -> (c1+c2)*matcha], + + EResetState[], + ESameTest[2, a + b /. (d_Symbol + c_Symbol) -> 2], + ESameTest[2 + c, a + b + c /. (d_Symbol + c_Symbol) -> 2], + ESameTest[2 + c + d, a + b + c + d /. (d_Symbol + c_Symbol) -> 2], + ESameTest[a+99+c+d, a + b + c + d /. (dmatch_Symbol + cmatch_Symbol) -> cmatch + 99], + ESameTest[a * b + c + d, a + b + c + d /. (d_Symbol + c_Symbol) -> c*d], + ESameTest[98, d = 98], + ESameTest[c+98+(b*a), a + b + c + d /. (dmatch_Symbol + cmatch_Symbol) -> cmatch*dmatch], + + EResetState[], + ESameTest[2 * a^2 - 2 * b^2, 2 * a^2 - 2 * b^2 /. matcha_ - matchb_ -> 2], + ESameTest[3 * a^2 + 5 * b^2, 2 * a^2 - 2 * b^2 /. 2*matcha_ - 2*matchb_ -> 3*matcha + 5*matchb], + ESameTest[2 * a^2 - 2 * b^2, 2 * a^2 - 2 * b^2 /. _Integer*matcha_ - _Integer*matchb_ -> 2], + ESameTest[2 * a^2 - 2 * b^2, 2 * a^2 - 2 * b^2 /. _*matcha_ - _*matchb_ -> 2], + ESameTest[2 * a^2 - 2 * b^2, 2 * a^2 - 2 * b^2 /. _ - _ -> 2], + ESameTest[2 * a^2 - 2 * b^2, 2 * a^2 - 2 * b^2 /. _ - 2*_ -> 2], + + (*Test replacing functions*) + ESameTest[test[], kfdsfdsf[] /. _Symbol -> test], + ESameTest[11, (x + 2)[5, 6] /. (2 + x) -> Plus], + ESameTest[2[2, 2, 2, 2], a*b*c*d /. _Symbol -> 2], + ESameTest[2, foo[2*x, x] /. foo[matcha_Integer*matchx_, matchx_] -> matcha], + ESameTest[foo[], a + b /. a + b + amatch___ -> foo[amatch]], + ESameTest[foo[b, c, d], a + b + c + d /. a + amatch___ -> foo[amatch]], + ESameTest[foo[a + b + c + d], a + b + c + d /. amatch___ -> foo[amatch]], + ESameTest[a + b, a + b /. a + b + amatch__ -> foo[amatch]], + ESameTest[foo[b, c, d], a + b + c + d /. a + amatch__ -> foo[amatch]], + ESameTest[foo[a + b + c + d], a + b + c + d /. amatch__ -> foo[amatch]], + + (*Test replacement within Hold parts*) + ESameTest[3, {a, b, c} /. {n__} :> Length[{n}]], + ESameTest[1, {a, b, c} /. {n__} -> Length[{n}]], + ESameTest[bar[m,n], foo[m, n] /. foo[a_, m_] -> bar[a, m]], + + (*Test replacement of functions and arguments*) + ESameTest[foo[False, y, 5], foo[x == 2, y, x] /. x -> 5], + ESameTest[foo[5, y, x], foo[x * 2, y, x] /. x * 2 -> 5], + ESameTest[k, foo[k] /. foo[k] -> k], + ESameTest[foo[k], foo[foo[k]] /. foo[k] -> k], + ESameTest[k, (foo[foo[k]] /. foo[k] -> k) /. foo[k] -> k], + ESameTest[foo[bla], foo[foo[k]] /. foo[k] -> bla], + + ESameTest[2 * a + 12 * b, foo[1, 2, 3, 4] /. foo[1, amatch__Integer, bmatch___Integer] -> a*Times[amatch] + b*Times[bmatch]], + ESameTest[a + 24 * b, foo[1, 2, 3, 4] /. foo[1, amatch___Integer, bmatch___Integer] -> a*Times[amatch] + b*Times[bmatch]], + + (*Test handling of orderless and Flat attributes.*) + ESameTest[False, MatchQ[Plus[a, b], Plus[a, b, c]]], + ESameTest[f && c, And[a, b, c] /. And[a, b] -> f], + ESameTest[False, MatchQ[And[a, b, c], And[b, c]]], + ESameTest[False, MatchQ[And[a, b, c], And[a, b]]], + ESameTest[jjj && eee && c, And[a, b, c] /. And[a, b] -> Sequence[jjj, eee]], + ESameTest[myand[a, b, c], myand[a, b, c] /. myand[a, b] -> f], + ESameTest[a && b && c, And[a, b, c] /. And[b, a] -> Sequence[jjj, eee]], + ESameTest[c + eee + jjj, Plus[a, b, c] /. Plus[b, a] -> Sequence[jjj, eee]], + ESameTest[c + eee + jjj, Plus[a, b, c] /. Plus[a, b] -> Sequence[jjj, eee]], + ESameTest[a && b && c, And[a, b, c] /. And[a, c] -> Sequence[jjj, eee]], + ESameTest[1 && 2 && a && b && jjj && eee, And[1, 2, a, b, c] /. And[___Integer, c] -> Sequence[jjj, eee]], + ESameTest[1 && 2 && a && b && c, And[1, 2, a, b, c] /. And[__Integer, c] -> Sequence[jjj, eee]], + ESameTest[jjj && eee && b && c, And[1, 2, a, b, c] /. And[__Integer, a] -> Sequence[jjj, eee]], + ESameTest[Sequence[jjj, eee], And[1, 2, a, b, c] /. And[__Integer, a, __Symbol] -> Sequence[jjj, eee]], + ESameTest[1 && 2 && a && b && c, And[1, 2, a, b, c] /. And[__Symbol, a, __Integer] -> Sequence[jjj, eee]], + ESameTest[1 && 2 && jjj && eee && b && c, And[1, 2, a, b, c] /. And[___Symbol, a, ___Integer] -> Sequence[jjj, eee]] + ], EKnownDangerous[ + (*Causes stack overflow*) + ESameTest[99 + a + b + c + d, a + b + c + d /. (d_Symbol + c_Symbol) -> c + 99 + d] + ] +}; + +Replace::usage = "`+"`"+`Replace[expr, rules]`+"`"+` applies `+"`"+`rules`+"`"+` to `+"`"+`expr`+"`"+` if they match at the base level."; +Attributes[Replace] = {Protected}; +Tests`+"`"+`Replace = { + ESimpleExamples[ + ESameTest[2, Replace[a+b,a+b->2]], + ESameTest[a+b, Replace[a+b,a->2]], + ESameTest[2, Replace[a+b,_->2]], + ESameTest[c+d, Replace[a+b,{a+b->c+d,c+d->3}]] + ] +}; + +ReplaceRepeated::usage = "`+"`"+`expr //. rule`+"`"+` replaces all occurences of the LHS of `+"`"+`rule`+"`"+` with the RHS of `+"`"+`rule`+"`"+` in `+"`"+`expr`+"`"+` repeatedly until the expression stabilizes. + +`+"`"+`expr //. {r1, r2, ...}`+"`"+` performes the same operation as `+"`"+`expr //. rule`+"`"+`, but evaluating each `+"`"+`r_n`+"`"+` in sequence."; +Attributes[ReplaceRepeated] = {Protected}; +Tests`+"`"+`ReplaceRepeated = { + ESimpleExamples[ + EComment["`+"`"+`ReplaceRepeated`+"`"+` can be used to implement logarithm expansion:"], + ESameTest[Null, logRules := {Log[x_ y_] :> Log[x] + Log[y], Log[x_^k_] :> k Log[x]}], + ESameTest[b Log[a] + (c^d) Log[b], Log[a^b*b^(c^d)] //. logRules] + ] +}; + +Rule::usage = "`+"`"+`lhs -> rhs`+"`"+` can be used in replacement functions to say that instances of `+"`"+`lhs`+"`"+` should be replaced with `+"`"+`rhs`+"`"+`."; +Attributes[Rule] = {SequenceHold, Protected}; +Tests`+"`"+`Rule = { + ESimpleExamples[ + ESameTest[2^(y+1) + y, 2^(x^2+1) + x^2 /. x^2 -> y], + EComment["To demonstrate the difference between `+"`"+`Rule`+"`"+` and `+"`"+`RuleDelayed`+"`"+`:"], + ESameTest[True, Equal @@ ({1, 1} /. 1 -> RandomReal[])], + ESameTest[False, Equal @@ ({1, 1} /. 1 :> RandomReal[])] + ] +}; + +RuleDelayed::usage = "`+"`"+`lhs :> rhs`+"`"+` can be used in replacement functions to say that instances of `+"`"+`lhs`+"`"+` should be replaced with `+"`"+`rhs`+"`"+`, evaluating `+"`"+`rhs`+"`"+` only after replacement."; +Attributes[RuleDelayed] = {HoldRest, SequenceHold, Protected}; +Tests`+"`"+`RuleDelayed = { + ESimpleExamples[ + ESameTest[2^(y+1) + y, 2^(x^2+1) + x^2 /. x^2 :> y], + EComment["To demonstrate the difference between `+"`"+`Rule`+"`"+` and `+"`"+`RuleDelayed`+"`"+`:"], + ESameTest[True, Equal @@ ({1, 1} /. 1 -> RandomReal[])], + ESameTest[False, Equal @@ ({1, 1} /. 1 :> RandomReal[])] + ] +}; + +ReplacePart::usage = "`+"`"+`ReplacePart[e, {loc1 -> newval1, ...}]`+"`"+` replaces the value at the locations with their corresponding new values in `+"`"+`e`+"`"+`."; +Attributes[ReplacePart] = {Protected}; +ReplacePart[e_?((! AtomQ[#]) &), r_, i_Integer?Positive] := + + If[i <= Length[e] === True, + Join[e[[1 ;; i - 1]], Head[e][r], e[[i + 1 ;; Length[e]]]], + Print["Index too large for ReplacePart!"]]; +ReplacePart[e_, r_Rule] := ReplacePart[e, {r}]; +Tests`+"`"+`ReplacePart = { + ESimpleExamples[ + ESameTest[{1,foo,3,4}, ReplacePart[Range[4],foo,2]], + ESameTest[{1,2,foo,4}, ReplacePart[Range[4],3->foo]], + ESameTest[{1,2,foo,4}, ReplacePart[Range[4],{3}->foo]], + ESameTest[{1,2,3,4}, ReplacePart[Range[4],{3,1}->foo]], + ], EFurtherExamples[ + ESameTest[{foo,foo,foo,foo}, ReplacePart[Range[4],i_->foo]], + ESameTest[{1,2,3,4}, ReplacePart[Range[4],7->foo]], + ESameTest[a+b^foo, ReplacePart[a+b^c,{2,2}->foo]], + ESameTest[a+b^c, ReplacePart[a+b^c,{2,2,1}->foo]], + ], ETests[ + ESameTest[a+foo^foo, ReplacePart[a+b^c,{2,_}->foo]], + ESameTest[a+b^foo, ReplacePart[a+b^c,{{2,2}->foo}]], + ESameTest[b^foo+foo, ReplacePart[a+b^c,{{2,2}->foo,{1}->foo}]], + ESameTest[b^foo+foo, ReplacePart[a+b^c,{{1}->foo,{2,2}->foo}]], + ESameTest[ReplacePart[a+b^c,{{1}->foo,bar}], ReplacePart[a+b^c,{{1}->foo,bar}]], + ESameTest[3, ReplacePart[a+b^c,{{a_}->a}]], + ESameTest[hi, ReplacePart[hi,{{a_}->a}]], + ESameTest[a+foo[1]^foo[2], ReplacePart[a+b^c,{{_,a_}->foo[a]}]], + ], EKnownFailures[ + ESameTest[foo[a,b^c], ReplacePart[a+b^c,{{0}->foo}]], + ESameTest[a+foo, ReplacePart[a+b^c,{{-1}->foo}]], + ] +}; +`) + +func resourcesReplacementMBytes() ([]byte, error) { + return _resourcesReplacementM, nil +} + +func resourcesReplacementM() (*asset, error) { + bytes, err := resourcesReplacementMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/replacement.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi111LinearBinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.1.1 Linear Binomial Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*1.1.1.1 (a+b x)^m*) + + +Int[1/x_,x_Symbol] := + Log[x] + + +Int[x_^m_.,x_Symbol] := + x^(m+1)/(m+1) /; +FreeQ[m,x] && NeQ[m,-1] + + +Int[1/(a_+b_.*x_),x_Symbol] := + Log[RemoveContent[a+b*x,x]]/b /; +FreeQ[{a,b},x] + + +Int[(a_.+b_.*x_)^m_,x_Symbol] := + (a+b*x)^(m+1)/(b*(m+1)) /; +FreeQ[{a,b,m},x] && NeQ[m,-1] + + +Int[(a_.+b_.*u_)^m_,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x)^m,x],x,u] /; +FreeQ[{a,b,m},x] && LinearQ[u,x] && NeQ[u-x,0] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.1.2 (a+b x)^m (c+d x)^n*) + + +Int[1/((a_+b_.*x_)*(c_+d_.*x_)),x_Symbol] := + Int[1/(a*c+b*d*x^2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] := + b/(b*c-a*d)*Int[1/(a+b*x),x] - d/(b*c-a*d)*Int[1/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^(n+1)/((b*c-a*d)*(m+1)) /; +FreeQ[{a,b,c,d,m,n},x] && NeQ[b*c-a*d,0] && EqQ[m+n+2,0] && NeQ[m,-1] + + +Int[(a_+b_.*x_)^m_*(c_+d_.*x_)^m_,x_Symbol] := + x*(a+b*x)^m*(c+d*x)^m/(2*m+1) + 2*a*c*m/(2*m+1)*Int[(a+b*x)^(m-1)*(c+d*x)^(m-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && PositiveIntegerQ[m+1/2] + + +Int[1/((a_+b_.*x_)^(3/2)*(c_+d_.*x_)^(3/2)),x_Symbol] := + x/(a*c*Sqrt[a+b*x]*Sqrt[c+d*x]) /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] + + +Int[(a_+b_.*x_)^m_*(c_+d_.*x_)^m_,x_Symbol] := + -x*(a+b*x)^(m+1)*(c+d*x)^(m+1)/(2*a*c*(m+1)) + + (2*m+3)/(2*a*c*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(m+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && NegativeIntegerQ[m+3/2] + + +Int[(a_+b_.*x_)^m_.*(c_+d_.*x_)^m_.,x_Symbol] := + Int[(a*c+b*d*x^2)^m,x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[b*c+a*d,0] && (IntegerQ[m] || PositiveQ[a] && PositiveQ[c]) + + +Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + ArcCosh[b*x/a]/b /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && PositiveQ[a] && EqQ[a+c,0] + + +Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + 2*Subst[Int[1/(b-d*x^2),x],x,Sqrt[a+b*x]/Sqrt[c+d*x]] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] + + +Int[(a_+b_.*x_)^m_*(c_+d_.*x_)^m_,x_Symbol] := + (a+b*x)^FracPart[m]*(c+d*x)^FracPart[m]/(a*c+b*d*x^2)^FracPart[m]*Int[(a*c+b*d*x^2)^m,x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[b*c+a*d,0] && Not[IntegerQ[2*m]] + + +Int[1/((a_+b_.*x_)^(5/4)*(c_+d_.*x_)^(1/4)),x_Symbol] := + -2/(b*(a+b*x)^(1/4)*(c+d*x)^(1/4)) + c*Int[1/((a+b*x)^(5/4)*(c+d*x)^(5/4)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && NegQ[a^2*b^2] + + +Int[1/((a_+b_.*x_)^(9/4)*(c_+d_.*x_)^(1/4)),x_Symbol] := + -4/(5*b*(a+b*x)^(5/4)*(c+d*x)^(1/4)) - d/(5*b)*Int[1/((a+b*x)^(5/4)*(c+d*x)^(5/4)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && NegQ[a^2*b^2] + + +Int[(a_+b_.*x_)^m_*(c_+d_.*x_)^n_,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^n/(b*(m+n+1)) + + 2*c*n/(m+n+1)*Int[(a+b*x)^m*(c+d*x)^(n-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d,0] && IntegerQ[m+1/2] && IntegerQ[n+1/2] && 00) + + +Int[(a_+b_.*x_)^m_.*(c_.+d_.*x_)^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n,x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && ILtQ[m,0] && IntegerQ[n] && Not[IGtQ[n,0] && m+n+2<0] + + +Int[(c_.+d_.*x_)^n_/(a_.+b_.*x_),x_Symbol] := + (c+d*x)^n/(b*n) + + (b*c-a*d)/b*Int[(c+d*x)^(n-1)/(a+b*x),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[n] && n>0 + + +Int[(c_.+d_.*x_)^n_/(a_.+b_.*x_),x_Symbol] := + -(c+d*x)^(n+1)/((n+1)*(b*c-a*d)) + + b*(n+1)/((n+1)*(b*c-a*d))*Int[(c+d*x)^(n+1)/(a+b*x),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[n] && n<-1 + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(1/3)),x_Symbol] := + With[{q=Rt[(b*c-a*d)/b,3]}, + -Log[RemoveContent[a+b*x,x]]/(2*b*q) - + 3/(2*b*q)*Subst[Int[1/(q-x),x],x,(c+d*x)^(1/3)] + + 3/(2*b)*Subst[Int[1/(q^2+q*x+x^2),x],x,(c+d*x)^(1/3)]] /; +FreeQ[{a,b,c,d},x] && PosQ[(b*c-a*d)/b] + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(1/3)),x_Symbol] := + With[{q=Rt[-(b*c-a*d)/b,3]}, + Log[RemoveContent[a+b*x,x]]/(2*b*q) - + 3/(2*b*q)*Subst[Int[1/(q+x),x],x,(c+d*x)^(1/3)] + + 3/(2*b)*Subst[Int[1/(q^2-q*x+x^2),x],x,(c+d*x)^(1/3)]] /; +FreeQ[{a,b,c,d},x] && NegQ[(b*c-a*d)/b] + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(2/3)),x_Symbol] := + With[{q=Rt[(b*c-a*d)/b,3]}, + -Log[RemoveContent[a+b*x,x]]/(2*b*q^2) - + 3/(2*b*q^2)*Subst[Int[1/(q-x),x],x,(c+d*x)^(1/3)] - + 3/(2*b*q)*Subst[Int[1/(q^2+q*x+x^2),x],x,(c+d*x)^(1/3)]] /; +FreeQ[{a,b,c,d},x] && PosQ[(b*c-a*d)/b] + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(2/3)),x_Symbol] := + With[{q=Rt[-(b*c-a*d)/b,3]}, + -Log[RemoveContent[a+b*x,x]]/(2*b*q^2) + + 3/(2*b*q^2)*Subst[Int[1/(q+x),x],x,(c+d*x)^(1/3)] + + 3/(2*b*q)*Subst[Int[1/(q^2-q*x+x^2),x],x,(c+d*x)^(1/3)]] /; +FreeQ[{a,b,c,d},x] && NegQ[(b*c-a*d)/b] + + +Int[(c_.+d_.*x_)^n_/(a_.+b_.*x_),x_Symbol] := + With[{p=Denominator[n]}, + p*Subst[Int[x^(p*(n+1)-1)/(a*d-b*c+b*x^p),x],x,(c+d*x)^(1/p)]] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[n] && -10 && Not[IntegerQ[n] && Not[IntegerQ[m]]] && + Not[IntegerQ[m+n] && m+n+2<=0 && (FractionQ[m] || 2*n+m+1>=0)] && IntLinearcQ[a,b,c,d,m,n,x] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^(n+1)/((b*c-a*d)*(m+1)) - + d*(m+n+2)/((b*c-a*d)*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[m,n] && m<-1 && + Not[n<-1 && (EqQ[a,0] || NeQ[c,0] && m0 && m+n+1!=0 && + Not[PositiveIntegerQ[m] && (Not[IntegerQ[n]] || 00] + + +Int[(a_+b_.*x_)*(d_.*x_)^n_.*(e_+f_.*x_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)*(d*x)^n*(e+f*x)^p,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && IGtQ[p,0] && (NeQ[n,-1] || EqQ[p,1]) && NeQ[b*e+a*f,0] && + (Not[IntegerQ[n]] || 9*p+5*n<0 || n+p+1>=0 || n+p+2>=0 && RationalQ[a,b,d,e,f]) && (NeQ[n+p+3,0] || EqQ[p,1]) + + +Int[(a_.+b_.*x_)*(c_+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)*(c+d*x)^n*(e+f*x)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d,0] && + (NegativeIntegerQ[n,p] || EqQ[p,1] || + IGtQ[p,0] && (Not[IntegerQ[n]] || 9*p+5*(n+2)<=0 || n+p+1>=0 || n+p+2>=0 && RationalQ[a,b,c,d,e,f])) + + +Int[(a_.+b_.*x_)*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + -(b*e-a*f)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(f*(p+1)*(c*f-d*e)) - + (a*d*f*(n+p+2)-b*(d*e*(n+1)+c*f*(p+1)))/(f*(p+1)*(c*f-d*e))*Int[(c+d*x)^n*(e+f*x)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && LtQ[p,-1] && + (Not[LtQ[n,-1]] || IntegerQ[p] || Not[IntegerQ[n] || Not[EqQ[e,0] || Not[EqQ[c,0] || p1 + + +Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] := + f*(e+f*x)^(p+1)/((p+1)*(b*e-a*f)*(d*e-c*f)) + + 1/((b*e-a*f)*(d*e-c*f))*Int[(b*d*e-b*c*f-a*d*f-b*d*f*x)*(e+f*x)^(p+1)/((a+b*x)*(c+d*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && RationalQ[p] && p<-1 + + +Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] := + b/(b*c-a*d)*Int[(e+f*x)^p/(a+b*x),x] - + d/(b*c-a*d)*Int[(e+f*x)^p/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && Not[IntegerQ[p]] + + +Int[(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_/(a_.+b_.*x_),x_Symbol] := + Int[ExpandIntegrand[(e+f*x)^FractionalPart[p],(c+d*x)^n*(e+f*x)^IntegerPart[p]/(a+b*x),x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[n] && FractionQ[p] && p<-1 + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && IntegersQ[m,n] && (IntegerQ[p] || m>0 && n>=-1) + + +Int[(a_.+b_.*x_)^2*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + (b*c-a*d)^2*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d^2*(d*e-c*f)*(n+1)) - + 1/(d^2*(d*e-c*f)*(n+1))*Int[(c+d*x)^(n+1)*(e+f*x)^p* + Simp[a^2*d^2*f*(n+p+2)+b^2*c*(d*e*(n+1)+c*f*(p+1))-2*a*b*d*(d*e*(n+1)+c*f*(p+1))-b^2*d*(d*e-c*f)*(n+1)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && (RationalQ[n] && n<-1 || EqQ[n+p+3,0] && NeQ[n,-1] && (SumSimplerQ[n,1] || Not[SumSimplerQ[p,1]])) + + +Int[(a_.+b_.*x_)^2*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(n+p+3)) + + 1/(d*f*(n+p+3))*Int[(c+d*x)^n*(e+f*x)^p* + Simp[a^2*d*f*(n+p+3)-b*(b*c*e+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(n+p+4)-b*(d*e*(n+2)+c*f*(p+2)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && NeQ[n+p+3,0] + + +Int[1/((a_.+b_.*x_)^(1/3)*(c_.+d_.*x_)^(2/3)*(e_.+f_.*x_)),x_Symbol] := + With[{q=Rt[(d*e-c*f)/(b*e-a*f),3]}, + -Sqrt[3]*q*ArcTan[1/Sqrt[3]+2*q*(a+b*x)^(1/3)/(Sqrt[3]*(c+d*x)^(1/3))]/(d*e-c*f) + + q*Log[e+f*x]/(2*(d*e-c*f)) - + 3*q*Log[q*(a+b*x)^(1/3)-(c+d*x)^(1/3)]/(2*(d*e-c*f))] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[1/(Sqrt[a_.+b_.*x_]*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)),x_Symbol] := + b*f*Subst[Int[1/(d*(b*e-a*f)^2+b*f^2*x^2),x],x,Sqrt[a+b*x]*Sqrt[c+d*x]] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-f*(b*c+a*d),0] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_/(e_.+f_.*x_),x_Symbol] := + With[{q=Denominator[m]}, + q*Subst[Int[x^(q*(m+1)-1)/(b*e-a*f-(d*e-c*f)*x^q),x],x,(a+b*x)^(1/q)/(c+d*x)^(1/q)]] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[m+n+1,0] && RationalQ[m,n] && -10 && Not[SumSimplerQ[p,1] && Not[SumSimplerQ[m,1]]] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[Simplify[m+n+p+3],0] && EqQ[a*d*f*(m+1)+b*c*f*(n+1)+b*d*e*(p+1),0] && NeQ[m,-1] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + (a*d*f*(m+1)+b*c*f*(n+1)+b*d*e*(p+1))/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[Simplify[m+n+p+3],0] && (RationalQ[m] && m<-1 || SumSimplerQ[m,1]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p/(b*(m+1)) - + 1/(b*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^(p-1)*Simp[d*e*n+c*f*p+d*f*(n+p)*x,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && RationalQ[m,n,p] && m<-1 && n>0 && p>0 && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + (b*c-a*d)*(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^(p+1)/(b*(b*e-a*f)*(m+1)) + + 1/(b*(b*e-a*f)*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-2)*(e+f*x)^p* + Simp[a*d*(d*e*(n-1)+c*f*(p+1))+b*c*(d*e*(m-n+2)-c*f*(m+p+2))+d*(a*d*f*(n+p)+b*(d*e*(m+1)-c*f*(m+n+p+1)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && RationalQ[m,n,p] && m<-1 && n>1 && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/((m+1)*(b*e-a*f)) - + 1/((m+1)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^p* + Simp[d*e*n+c*f*(m+p+2)+d*f*(m+n+p+2)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && RationalQ[m,n,p] && m<-1 && n>0 && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m-1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+1)) + + 1/(d*f*(m+n+p+1))*Int[(a+b*x)^(m-2)*(c+d*x)^n*(e+f*x)^p* + Simp[a^2*d*f*(m+n+p+1)-b*(b*c*e*(m-1)+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(2*m+n+p)-b*(d*e*(m+n)+c*f*(m+p)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && RationalQ[m] && m>1 && NeQ[m+n+p+1,0] && IntegerQ[m] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + (a+b*x)^m*(c+d*x)^n*(e+f*x)^(p+1)/(f*(m+n+p+1)) - + 1/(f*(m+n+p+1))*Int[(a+b*x)^(m-1)*(c+d*x)^(n-1)*(e+f*x)^p* + Simp[c*m*(b*e-a*f)+a*n*(d*e-c*f)+(d*m*(b*e-a*f)+b*n*(d*e-c*f))*x,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && RationalQ[m,n,p] && m>0 && n>0 && NeQ[m+n+p+1,0] && + (IntegersQ[2*m,2*n,2*p] || (IntegersQ[m,n+p] || IntegersQ[p,m+n])) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m-1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+1)) + + 1/(d*f*(m+n+p+1))*Int[(a+b*x)^(m-2)*(c+d*x)^n*(e+f*x)^p* + Simp[a^2*d*f*(m+n+p+1)-b*(b*c*e*(m-1)+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(2*m+n+p)-b*(d*e*(m+n)+c*f*(m+p)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && RationalQ[m] && m>1 && NeQ[m+n+p+1,0] && IntegersQ[2*m,2*n,2*p] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && RationalQ[m] && m<-1 && IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n,2*p]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && RationalQ[m] && m<-1 && IntegersQ[2*m,2*n,2*p] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_/(e_.+f_.*x_),x_Symbol] := + b/f*Int[(a+b*x)^(m-1)*(c+d*x)^n,x] - (b*e-a*f)/f*Int[(a+b*x)^(m-1)*(c+d*x)^n/(e+f*x),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && PositiveIntegerQ[Simplify[m+n+1]] && + (RationalQ[m] && m>0 || Not[RationalQ[m]] && (SumSimplerQ[m,-1] || Not[SumSimplerQ[n,-1]])) + + +(* Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + b/f*Int[(a+b*x)^(m-1)*(c+d*x)^n*(e+f*x)^(p+1),x] - (b*e-a*f)/f*Int[(a+b*x)^(m-1)*(c+d*x)^n*(e+f*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NegativeIntegerQ[p] && PositiveIntegerQ[m+n+p+2] && Not[SimplerQ[c+d*x,a+b*x]] *) + + +Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(1/4)),x_Symbol] := + -4*Subst[Int[x^2/((b*e-a*f-b*x^4)*Sqrt[c-d*e/f+d*x^4/f]),x],x,(e+f*x)^(1/4)] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[-f/(d*e-c*f)] + + +Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(1/4)),x_Symbol] := + Sqrt[-f*(c+d*x)/(d*e-c*f)]/Sqrt[c+d*x]*Int[1/((a+b*x)*Sqrt[-c*f/(d*e-c*f)-d*f*x/(d*e-c*f)]*(e+f*x)^(1/4)),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[PositiveQ[-f/(d*e-c*f)]] + + +Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(3/4)),x_Symbol] := + -4*Subst[Int[1/((b*e-a*f-b*x^4)*Sqrt[c-d*e/f+d*x^4/f]),x],x,(e+f*x)^(1/4)] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[-f/(d*e-c*f)] + + +Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(3/4)),x_Symbol] := + Sqrt[-f*(c+d*x)/(d*e-c*f)]/Sqrt[c+d*x]*Int[1/((a+b*x)*Sqrt[-c*f/(d*e-c*f)-d*f*x/(d*e-c*f)]*(e+f*x)^(3/4)),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[PositiveQ[-f/(d*e-c*f)]] + + +Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + 2*Sqrt[e]/b*Rt[-b/d,2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /; +FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && PositiveQ[c] && PositiveQ[e] && Not[NegativeQ[-b/d]] + + +Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + Sqrt[-b*x]/Sqrt[b*x]*Int[Sqrt[e+f*x]/(Sqrt[-b*x]*Sqrt[c+d*x]),x] /; +FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && PositiveQ[c] && PositiveQ[e] && NegativeQ[-b/d] + + +Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + Sqrt[e+f*x]*Sqrt[1+d*x/c]/(Sqrt[c+d*x]*Sqrt[1+f*x/e])*Int[Sqrt[1+f*x/e]/(Sqrt[b*x]*Sqrt[1+d*x/c]),x] /; +FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && Not[PositiveQ[c] && PositiveQ[e]] + + +(* Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + f/b*Int[Sqrt[a+b*x]/(Sqrt[c+d*x]*Sqrt[e+f*x]),x] - + f/b*Int[1/(Sqrt[a+b*x]*Sqrt[c+d*x]*Sqrt[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*e-f*(a-1),0] *) + + +(* Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + 2/b*Rt[-(b*c-a*d)/d,2]*Sqrt[(b*e-a*f)/(b*c-a*d)]* + EllipticE[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)] && Not[NegativeQ[-(b*c-a*d)/d]] && + Not[SimplerQ[c+d*x,a+b*x] && PositiveQ[-d/(b*c-a*d)] && PositiveQ[d/(d*e-c*f)] && Not[NegativeQ[(b*c-a*d)/b]]] *) + + +Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + 2/b*Rt[-(b*e-a*f)/d,2]*EllipticE[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)] && Not[NegativeQ[-(b*c-a*d)/d]] && + Not[SimplerQ[c+d*x,a+b*x] && PositiveQ[-d/(b*c-a*d)] && PositiveQ[d/(d*e-c*f)] && Not[NegativeQ[(b*c-a*d)/b]]] + + +Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] := + Sqrt[e+f*x]*Sqrt[b*(c+d*x)/(b*c-a*d)]/(Sqrt[c+d*x]*Sqrt[b*(e+f*x)/(b*e-a*f)])* + Int[Sqrt[b*e/(b*e-a*f)+b*f*x/(b*e-a*f)]/(Sqrt[a+b*x]*Sqrt[b*c/(b*c-a*d)+b*d*x/(b*c-a*d)]),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)]] && Not[NegativeQ[-(b*c-a*d)/d]] + + +Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + 2/(b*Sqrt[e])*Rt[-b/d,2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /; +FreeQ[{b,c,d,e,f},x] && PositiveQ[c] && PositiveQ[e] && (PositiveQ[-b/d] || NegativeQ[-b/f]) + + +Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + 2/(b*Sqrt[e])*Rt[-b/d,2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /; +FreeQ[{b,c,d,e,f},x] && PositiveQ[c] && PositiveQ[e] && (PosQ[-b/d] || NegQ[-b/f]) + + +Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + Sqrt[1+d*x/c]*Sqrt[1+f*x/e]/(Sqrt[c+d*x]*Sqrt[e+f*x])*Int[1/(Sqrt[b*x]*Sqrt[1+d*x/c]*Sqrt[1+f*x/e]),x] /; +FreeQ[{b,c,d,e,f},x] && Not[PositiveQ[c] && PositiveQ[e]] + + +Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + 2/b*Rt[-(b*c-a*d)/d,2]*Sqrt[b^2/((b*c-a*d)*(b*e-a*f))]* + EllipticF[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)] && SimplerQ[a+b*x,c+d*x] && SimplerQ[a+b*x,e+f*x] && + (PositiveQ[-(b*c-a*d)/d] || NegativeQ[-(b*e-a*f)/f]) + + +Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + 2/b*Rt[-(b*c-a*d)/d,2]*Sqrt[b^2/((b*c-a*d)*(b*e-a*f))]* + EllipticF[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)] && SimplerQ[a+b*x,c+d*x] && SimplerQ[a+b*x,e+f*x] && + (PosQ[-(b*c-a*d)/d] || NegQ[-(b*e-a*f)/f]) + + +Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + Sqrt[b*(c+d*x)/(b*c-a*d)]*Sqrt[b*(e+f*x)/(b*e-a*f)]/(Sqrt[c+d*x]*Sqrt[e+f*x])* + Int[1/(Sqrt[a+b*x]*Sqrt[b*c/(b*c-a*d)+b*d*x/(b*c-a*d)]*Sqrt[b*e/(b*e-a*f)+b*f*x/(b*e-a*f)]),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)]] && SimplerQ[a+b*x,c+d*x] && SimplerQ[a+b*x,e+f*x] + + +Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(1/3)*(e_.+f_.*x_)^(1/3)),x_Symbol] := + With[{q=Rt[b*(b*e-a*f)/(b*c-a*d)^2,3]}, + -Log[a+b*x]/(2*q*(b*c-a*d)) - + Sqrt[3]*ArcTan[1/Sqrt[3]+2*q*(c+d*x)^(2/3)/(Sqrt[3]*(e+f*x)^(1/3))]/(2*q*(b*c-a*d)) + + 3*Log[q*(c+d*x)^(2/3)-(e+f*x)^(1/3)]/(4*q*(b*c-a*d))] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-b*c*f-a*d*f,0] + + +Int[(a_.+b_.*x_)^m_/((c_.+d_.*x_)^(1/3)*(e_.+f_.*x_)^(1/3)),x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(2/3)*(e+f*x)^(2/3)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + f/(6*(m+1)*(b*c-a*d)*(b*e-a*f))* + Int[(a+b*x)^(m+1)*(a*d*(3*m+1)-3*b*c*(3*m+5)-2*b*d*(3*m+7)*x)/((c+d*x)^(1/3)*(e+f*x)^(1/3)),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-b*c*f-a*d*f,0] && IntegerQ[m] && m<-1 + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] := + Int[(a*c+b*d*x^2)^m*(f*x)^p,x] /; +FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[m-n,0] && PositiveQ[a] && PositiveQ[c] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] := + (a+b*x)^FracPart[m]*(c+d*x)^FracPart[m]/(a*c+b*d*x^2)^FracPart[m]*Int[(a*c+b*d*x^2)^m*(f*x)^p,x] /; +FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[m-n,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^n*(c+d*x)^n*(f*x)^p,(a+b*x)^(m-n),x],x] /; +FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && PositiveIntegerQ[m-n] && NeQ[m+n+p+2,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && (PositiveIntegerQ[m] || NegativeIntegerQ[m,n]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] := + b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && NegativeIntegerQ[m+n+p+2] && NeQ[m,-1] && + (SumSimplerQ[m,1] || Not[NeQ[n,-1] && SumSimplerQ[n,1]] && Not[NeQ[p,-1] && SumSimplerQ[p,1]]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_,x_Symbol] := + (b*c-a*d)^n*(a+b*x)^(m+1)/((m+1)*(b*e-a*f)^(n+1)*(e+f*x)^(m+1))* + Hypergeometric2F1[m+1,-n,m+2,-(d*e-c*f)*(a+b*x)/((b*c-a*d)*(e+f*x))] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[m+n+p+2,0] && NegativeIntegerQ[n] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/((b*e-a*f)*(m+1))*((b*e-a*f)*(c+d*x)/((b*c-a*d)*(e+f*x)))^(-n)* + Hypergeometric2F1[m+1,-n,m+2,-(d*e-c*f)*(a+b*x)/((b*c-a*d)*(e+f*x))] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[m+n+p+2,0] && Not[IntegerQ[n]] + + +Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] := + c^n*e^p*(b*x)^(m+1)/(b*(m+1))*AppellF1[m+1,-n,-p,m+2,-d*x/c,-f*x/e] /; +FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && PositiveQ[c] && (IntegerQ[p] || PositiveQ[e]) + + +Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] := + (c+d*x)^(n+1)/(d*(n+1)*(-d/(b*c))^m*(d/(d*e-c*f))^p)*AppellF1[n+1,-m,-p,n+2,1+d*x/c,-f*(c+d*x)/(d*e-c*f)] /; +FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && PositiveQ[-d/(b*c)] && (IntegerQ[p] || PositiveQ[d/(d*e-c*f)]) + + +Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] := + c^IntPart[n]*(c+d*x)^FracPart[n]/(1+d*x/c)^FracPart[n]*Int[(b*x)^m*(1+d*x/c)^n*(e+f*x)^p,x] /; +FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[PositiveQ[c]] + + +Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (b*e-a*f)^p*(a+b*x)^(m+1)/(b^(p+1)*(m+1)*(b/(b*c-a*d))^n)* + AppellF1[m+1,-n,-p,m+2,-d*(a+b*x)/(b*c-a*d),-f*(a+b*x)/(b*e-a*f)] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && PositiveQ[b/(b*c-a*d)] && + Not[PositiveQ[d/(d*a-c*b)] && SimplerQ[c+d*x,a+b*x]] + + +Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (c+d*x)^FracPart[n]/((b/(b*c-a*d))^IntPart[n]*(b*(c+d*x)/(b*c-a*d))^FracPart[n])* + Int[(a+b*x)^m*(b*c/(b*c-a*d)+b*d*x/(b*c-a*d))^n*(e+f*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && Not[PositiveQ[b/(b*c-a*d)]] && + Not[SimplerQ[c+d*x,a+b*x]] + + +Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (a+b*x)^(m+1)/(b*(m+1)*(b/(b*c-a*d))^n*(b/(b*e-a*f))^p)*AppellF1[m+1,-n,-p,m+2,-d*(a+b*x)/(b*c-a*d),-f*(a+b*x)/(b*e-a*f)] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && + PositiveQ[b/(b*c-a*d)] && PositiveQ[b/(b*e-a*f)] && + Not[PositiveQ[d/(d*a-c*b)] && PositiveQ[d/(d*e-c*f)] && SimplerQ[c+d*x,a+b*x]] && + Not[PositiveQ[f/(f*a-e*b)] && PositiveQ[f/(f*c-e*d)] && SimplerQ[e+f*x,a+b*x]] + + +Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (e+f*x)^FracPart[p]/((b/(b*e-a*f))^IntPart[p]*(b*(e+f*x)/(b*e-a*f))^FracPart[p])* + Int[(a+b*x)^m*(c+d*x)^n*(b*e/(b*e-a*f)+b*f*x/(b*e-a*f))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && + PositiveQ[b/(b*c-a*d)] && Not[PositiveQ[b/(b*e-a*f)]] + + +Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] := + (c+d*x)^FracPart[n]/((b/(b*c-a*d))^IntPart[n]*(b*(c+d*x)/(b*c-a*d))^FracPart[n])* + Int[(a+b*x)^m*(b*c/(b*c-a*d)+b*d*x/(b*c-a*d))^n*(e+f*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && Not[PositiveQ[b/(b*c-a*d)]] && + Not[SimplerQ[c+d*x,a+b*x]] && Not[SimplerQ[e+f*x,a+b*x]] + + +Int[(a_.+b_.*u_)^m_.*(c_.+d_.*u_)^n_.*(e_+f_.*u_)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x,u] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && LinearQ[u,x] && NeQ[u-x,0] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q*) + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)*(g+h*x),x],x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && (PositiveIntegerQ[m] || IntegersQ[m,n]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + (b^2*d*e*g-a^2*d*f*h*m-a*b*(d*(f*g+e*h)-c*f*h*(m+1))+b*f*h*(b*c-a*d)*(m+1)*x)*(a+b*x)^(m+1)*(c+d*x)^(n+1)/ + (b^2*d*(b*c-a*d)*(m+1)) + + (a*d*f*h*m+b*(d*(f*g+e*h)-c*f*h*(m+2)))/(b^2*d)*Int[(a+b*x)^(m+1)*(c+d*x)^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[m+n+2,0] && NeQ[m,-1] && Not[SumSimplerQ[n,1] && Not[SumSimplerQ[m,1]]] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + (b^2*c*d*e*g*(n+1)+a^2*c*d*f*h*(n+1)+a*b*(d^2*e*g*(m+1)+c^2*f*h*(m+1)-c*d*(f*g+e*h)*(m+n+2))+ + (a^2*d^2*f*h*(n+1)-a*b*d^2*(f*g+e*h)*(n+1)+b^2*(c^2*f*h*(m+1)-c*d*(f*g+e*h)*(m+1)+d^2*e*g*(m+n+2)))*x)/ + (b*d*(b*c-a*d)^2*(m+1)*(n+1))*(a+b*x)^(m+1)*(c+d*x)^(n+1) - + (a^2*d^2*f*h*(2+3*n+n^2)+a*b*d*(n+1)*(2*c*f*h*(m+1)-d*(f*g+e*h)*(m+n+3))+ + b^2*(c^2*f*h*(2+3*m+m^2)-c*d*(f*g+e*h)*(m+1)*(m+n+3)+d^2*e*g*(6+m^2+5*n+n^2+m*(2*n+5))))/ + (b*d*(b*c-a*d)^2*(m+1)*(n+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && RationalQ[m,n] && m<-1 && n<-1 + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + (b^3*c*e*g*(m+2)-a^3*d*f*h*(n+2)-a^2*b*(c*f*h*m-d*(f*g+e*h)*(m+n+3))-a*b^2*(c*(f*g+e*h)+d*e*g*(2*m+n+4))+ + b*(a^2*d*f*h*(m-n)-a*b*(2*c*f*h*(m+1)-d*(f*g+e*h)*(n+1))+b^2*(c*(f*g+e*h)*(m+1)-d*e*g*(m+n+2)))*x)/ + (b^2*(b*c-a*d)^2*(m+1)*(m+2))*(a+b*x)^(m+1)*(c+d*x)^(n+1) + + (f*h/b^2-(d*(m+n+3)*(a^2*d*f*h*(m-n)-a*b*(2*c*f*h*(m+1)-d*(f*g+e*h)*(n+1))+b^2*(c*(f*g+e*h)*(m+1)-d*e*g*(m+n+2))))/ + (b^2*(b*c-a*d)^2*(m+1)*(m+2)))* + Int[(a+b*x)^(m+2)*(c+d*x)^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && (RationalQ[m] && m<-2 || EqQ[m+n+3,0] && Not[RationalQ[n] && n<-2]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + (a^2*d*f*h*(n+2)+b^2*d*e*g*(m+n+3)+a*b*(c*f*h*(m+1)-d*(f*g+e*h)*(m+n+3))+b*f*h*(b*c-a*d)*(m+1)*x)/ + (b^2*d*(b*c-a*d)*(m+1)*(m+n+3))*(a+b*x)^(m+1)*(c+d*x)^(n+1) - + (a^2*d^2*f*h*(n+1)*(n+2)+a*b*d*(n+1)*(2*c*f*h*(m+1)-d*(f*g+e*h)*(m+n+3))+ + b^2*(c^2*f*h*(m+1)*(m+2)-c*d*(f*g+e*h)*(m+1)*(m+n+3)+d^2*e*g*(m+n+2)*(m+n+3)))/ + (b^2*d*(b*c-a*d)*(m+1)*(m+n+3))*Int[(a+b*x)^(m+1)*(c+d*x)^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && (RationalQ[m] && -2<=m<-1 || SumSimplerQ[m,1]) && NeQ[m,-1] && NeQ[m+n+3,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_+f_.*x_)*(g_.+h_.*x_),x_Symbol] := + -(a*d*f*h*(n+2)+b*c*f*h*(m+2)-b*d*(f*g+e*h)*(m+n+3)-b*d*f*h*(m+n+2)*x)*(a+b*x)^(m+1)*(c+d*x)^(n+1)/ + (b^2*d^2*(m+n+2)*(m+n+3)) + + (a^2*d^2*f*h*(n+1)*(n+2)+a*b*d*(n+1)*(2*c*f*h*(m+1)-d*(f*g+e*h)*(m+n+3))+ + b^2*(c^2*f*h*(m+1)*(m+2)-c*d*(f*g+e*h)*(m+1)*(m+n+3)+d^2*e*g*(m+n+2)*(m+n+3)))/ + (b^2*d^2*(m+n+2)*(m+n+3))*Int[(a+b*x)^m*(c+d*x)^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && NeQ[m+n+2,0] && NeQ[m+n+3,0] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p*(g+h*x),x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,m},x] && (IntegersQ[m,n,p] || PositiveIntegerQ[n,p]) + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + (b*g-a*h)*(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/(b*(b*e-a*f)*(m+1)) - + 1/(b*(b*e-a*f)*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^p* + Simp[b*c*(f*g-e*h)*(m+1)+(b*g-a*h)*(d*e*n+c*f*(p+1))+d*(b*(f*g-e*h)*(m+1)+f*(b*g-a*h)*(n+p+1))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,p},x] && RationalQ[m,n] && m<-1 && n>0 && IntegerQ[m] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + (b*g-a*h)*(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/(b*(b*e-a*f)*(m+1)) - + 1/(b*(b*e-a*f)*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^p* + Simp[b*c*(f*g-e*h)*(m+1)+(b*g-a*h)*(d*e*n+c*f*(p+1))+d*(b*(f*g-e*h)*(m+1)+f*(b*g-a*h)*(n+p+1))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,p},x] && RationalQ[m,n] && m<-1 && n>0 && IntegersQ[2*m,2*n,2*p] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + (b*g-a*h)*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[(a*d*f*g-b*(d*e+c*f)*g+b*c*e*h)*(m+1)-(b*g-a*h)*(d*e*(n+1)+c*f*(p+1))-d*f*(b*g-a*h)*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] && RationalQ[m] && m<-1 && IntegerQ[m] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + (b*g-a*h)*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[(a*d*f*g-b*(d*e+c*f)*g+b*c*e*h)*(m+1)-(b*g-a*h)*(d*e*(n+1)+c*f*(p+1))-d*f*(b*g-a*h)*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] && RationalQ[m] && m<-1 && IntegersQ[2*m,2*n,2*p] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + h*(a+b*x)^m*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+2)) + + 1/(d*f*(m+n+p+2))*Int[(a+b*x)^(m-1)*(c+d*x)^n*(e+f*x)^p* + Simp[a*d*f*g*(m+n+p+2)-h*(b*c*e*m+a*(d*e*(n+1)+c*f*(p+1)))+(b*d*f*g*(m+n+p+2)+h*(a*d*f*m-b*(d*e*(m+n+1)+c*f*(m+p+1))))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] && RationalQ[m] && m>0 && NeQ[m+n+p+2,0] && IntegerQ[m] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + h*(a+b*x)^m*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+2)) + + 1/(d*f*(m+n+p+2))*Int[(a+b*x)^(m-1)*(c+d*x)^n*(e+f*x)^p* + Simp[a*d*f*g*(m+n+p+2)-h*(b*c*e*m+a*(d*e*(n+1)+c*f*(p+1)))+(b*d*f*g*(m+n+p+2)+h*(a*d*f*m-b*(d*e*(m+n+1)+c*f*(m+p+1))))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] && RationalQ[m] && m>0 && NeQ[m+n+p+2,0] && IntegersQ[2*m,2*n,2*p] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + (b*g-a*h)*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + + 1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p* + Simp[(a*d*f*g-b*(d*e+c*f)*g+b*c*e*h)*(m+1)-(b*g-a*h)*(d*e*(n+1)+c*f*(p+1))-d*f*(b*g-a*h)*(m+n+p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] && NegativeIntegerQ[m+n+p+2] && NeQ[m,-1] && + (SumSimplerQ[m,1] || Not[NeQ[n,-1] && SumSimplerQ[n,1]] && Not[NeQ[p,-1] && SumSimplerQ[p,1]]) + + +Int[(e_.+f_.*x_)^p_*(g_.+h_.*x_)/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] := + (b*g-a*h)/(b*c-a*d)*Int[(e+f*x)^p/(a+b*x),x] - + (d*g-c*h)/(b*c-a*d)*Int[(e+f*x)^p/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] + + +Int[(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_)/(a_.+b_.*x_),x_Symbol] := + h/b*Int[(c+d*x)^n*(e+f*x)^p,x] + (b*g-a*h)/b*Int[(c+d*x)^n*(e+f*x)^p/(a+b*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p},x] + + +Int[(g_.+h_.*x_)/(Sqrt[a_.+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] := + h/f*Int[Sqrt[e+f*x]/(Sqrt[a+b*x]*Sqrt[c+d*x]),x] + (f*g-e*h)/f*Int[1/(Sqrt[a+b*x]*Sqrt[c+d*x]*Sqrt[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && SimplerQ[a+b*x,e+f*x] && SimplerQ[c+d*x,e+f*x] + + +Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_*(g_.+h_.*x_),x_Symbol] := + h/b*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p,x] + (b*g-a*h)/b*Int[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && (SumSimplerQ[m,1] || Not[SumSimplerQ[n,1]] && Not[SumSimplerQ[p,1]]) + + +Int[(e_.+f_.*x_)^p_*(g_.+h_.*x_)^q_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] := + (b*e-a*f)/(b*c-a*d)*Int[(e+f*x)^(p-1)*(g+h*x)^q/(a+b*x),x] - + (d*e-c*f)/(b*c-a*d)*Int[(e+f*x)^(p-1)*(g+h*x)^q/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,q},x] && RationalQ[p] && 00 && m<-1 && Not[NegativeIntegerQ[(m+n*p+n+1)/n]] && + IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a1+b1*x^n)^p*(a2+b2*x^n)^p/(c*(m+1)) - + 2*b1*b2*n*p/(c^(2*n)*(m+1))*Int[(c*x)^(m+2*n)*(a1+b1*x^n)^(p-1)*(a2+b2*x^n)^(p-1),x] /; +FreeQ[{a1,b1,a2,b2,c,m},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m,p] && p>0 && NeQ[m+2*n*p+1] && + IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[(c_.*x_)^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a+b*x^n)^p/(c*(m+n*p+1)) + + a*n*p/(m+n*p+1)*Int[(c*x)^m*(a+b*x^n)^(p-1),x] /; +FreeQ[{a,b,c,m},x] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && NeQ[m+n*p+1] && IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a1+b1*x^n)^p*(a2+b2*x^n)^p/(c*(m+2*n*p+1)) + + 2*a1*a2*n*p/(m+2*n*p+1)*Int[(c*x)^m*(a1+b1*x^n)^(p-1)*(a2+b2*x^n)^(p-1),x] /; +FreeQ[{a1,b1,a2,b2,c,m},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m,p] && p>0 && NeQ[m+2*n*p+1] && + IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[x_^2/(a_+b_.*x_^4)^(5/4),x_Symbol] := + x*(1+a/(b*x^4))^(1/4)/(b*(a+b*x^4)^(1/4))*Int[1/(x^3*(1+a/(b*x^4))^(5/4)),x] /; +FreeQ[{a,b},x] && PosQ[b/a] + + +Int[x_^m_/(a_+b_.*x_^4)^(5/4),x_Symbol] := + x^(m-3)/(b*(m-4)*(a+b*x^4)^(1/4)) - a*(m-3)/(b*(m-4))*Int[x^(m-4)/(a+b*x^4)^(5/4),x] /; +FreeQ[{a,b},x] && PosQ[b/a] && PositiveIntegerQ[(m-2)/4] + + +Int[x_^m_/(a_+b_.*x_^4)^(5/4),x_Symbol] := + x^(m+1)/(a*(m+1)*(a+b*x^4)^(1/4)) - b*m/(a*(m+1))*Int[x^(m+4)/(a+b*x^4)^(5/4),x] /; +FreeQ[{a,b},x] && PosQ[b/a] && NegativeIntegerQ[(m-2)/4] + + +Int[Sqrt[c_.*x_]/(a_+b_.*x_^2)^(5/4),x_Symbol] := + Sqrt[c*x]*(1+a/(b*x^2))^(1/4)/(b*(a+b*x^2)^(1/4))*Int[1/(x^2*(1+a/(b*x^2))^(5/4)),x] /; +FreeQ[{a,b,c},x] && PosQ[b/a] + + +Int[(c_.*x_)^m_/(a_+b_.*x_^2)^(5/4),x_Symbol] := + 2*c*(c*x)^(m-1)/(b*(2*m-3)*(a+b*x^2)^(1/4)) - 2*a*c^2*(m-1)/(b*(2*m-3))*Int[(c*x)^(m-2)/(a+b*x^2)^(5/4),x] /; +FreeQ[{a,b,c},x] && PosQ[b/a] && IntegerQ[2*m] && m>3/2 + + +Int[(c_.*x_)^m_/(a_+b_.*x_^2)^(5/4),x_Symbol] := + (c*x)^(m+1)/(a*c*(m+1)*(a+b*x^2)^(1/4)) - b*(2*m+1)/(2*a*c^2*(m+1))*Int[(c*x)^(m+2)/(a+b*x^2)^(5/4),x] /; +FreeQ[{a,b,c},x] && PosQ[b/a] && IntegerQ[2*m] && m<-1 + + +Int[x_^2/(a_+b_.*x_^4)^(5/4),x_Symbol] := + -1/(b*x*(a+b*x^4)^(1/4)) - 1/b*Int[1/(x^2*(a+b*x^4)^(1/4)),x] /; +FreeQ[{a,b},x] && NegQ[b/a] + + +Int[(c_.*x_)^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^(n-1)*(c*x)^(m-n+1)*(a+b*x^n)^(p+1)/(b*n*(p+1)) - + c^n*(m-n+1)/(b*n*(p+1))*Int[(c*x)^(m-n)*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m+1>n && + Not[NegativeIntegerQ[(m+n*(p+1)+1)/n]] && IntBinomialQ[a,b,c,n,m,p,x] + + +(* Int[(c_.*x_)^m_.*u_^p_*v_^p_,x_Symbol] := + With[{a=BinomialParts[u,x][[1]],b=BinomialParts[u,x][[2]],n=BinomialParts[u,x][[3]]}, + c^(n-1)*(c*x)^(m-n+1)*u^(p+1)*v^(p+1)/(b*n*(p+1)) - + c^n*(m-n+1)/(b*n*(p+1))*Int[(c*x)^(m-n)*u^(p+1)*v^(p+1),x] /; + PositiveIntegerQ[n] && m+1>n && Not[NegativeIntegerQ[(m+n*(p+1)+1)/n]] && + IntBinomialQ[a,b,c,n,m,p,x]] /; +FreeQ[c,x] && BinomialQ[u*v,x] && RationalQ[m,p] && p<-1 *) + + +Int[(c_.*x_)^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + c^(2*n-1)*(c*x)^(m-2*n+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(2*b1*b2*n*(p+1)) - + c^(2*n)*(m-2*n+1)/(2*b1*b2*n*(p+1))*Int[(c*x)^(m-2*n)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1),x] /; +FreeQ[{a1,b1,a2,b2,c},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m,p] && p<-1 && m+1>2*n && + Not[NegativeIntegerQ[(m+2*n*(p+1)+1)/(2*n)]] && IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[(c_.*x_)^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + -(c*x)^(m+1)*(a+b*x^n)^(p+1)/(a*c*n*(p+1)) + + (m+n*(p+1)+1)/(a*n*(p+1))*Int[(c*x)^m*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c,m},x] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + -(c*x)^(m+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(2*a1*a2*c*n*(p+1)) + + (m+2*n*(p+1)+1)/(2*a1*a2*n*(p+1))*Int[(c*x)^m*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1),x] /; +FreeQ[{a1,b1,a2,b2,c,m},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m,p] && p<-1 && + IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[x_/(a_+b_.*x_^3),x_Symbol] := + -1/(3*Rt[a,3]*Rt[b,3])*Int[1/(Rt[a,3]+Rt[b,3]*x),x] + + 1/(3*Rt[a,3]*Rt[b,3])*Int[(Rt[a,3]+Rt[b,3]*x)/(Rt[a,3]^2-Rt[a,3]*Rt[b,3]*x+Rt[b,3]^2*x^2),x] /; +FreeQ[{a,b},x] + + +(* Int[x_^m_./(a_+b_.*x_^5),x_Symbol] := + With[{r=Numerator[Rt[a/b,5]], s=Denominator[Rt[a/b,5]]}, + (-1)^m*r^(m+1)/(5*a*s^m)*Int[1/(r+s*x),x] + + 2*r^(m+1)/(5*a*s^m)*Int[(r*Cos[m*Pi/5]-s*Cos[(m+1)*Pi/5]*x)/(r^2-1/2*(1+Sqrt[5])*r*s*x+s^2*x^2),x] + + 2*r^(m+1)/(5*a*s^m)*Int[(r*Cos[3*m*Pi/5]-s*Cos[3*(m+1)*Pi/5]*x)/(r^2-1/2*(1-Sqrt[5])*r*s*x+s^2*x^2),x]] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] && m<4 && PosQ[a/b] *) + + +(* Int[x_^m_./(a_+b_.*x_^5),x_Symbol] := + With[{r=Numerator[Rt[-a/b,5]], s=Denominator[Rt[-a/b,5]]}, + (r^(m+1)/(5*a*s^m))*Int[1/(r-s*x),x] + + 2*(-1)^m*r^(m+1)/(5*a*s^m)*Int[(r*Cos[m*Pi/5]+s*Cos[(m+1)*Pi/5]*x)/(r^2+1/2*(1+Sqrt[5])*r*s*x+s^2*x^2),x] + + 2*(-1)^m*r^(m+1)/(5*a*s^m)*Int[(r*Cos[3*m*Pi/5]+s*Cos[3*(m+1)*Pi/5]*x)/(r^2+1/2*(1-Sqrt[5])*r*s*x+s^2*x^2),x]] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] && m<4 && NegQ[a/b] *) + + +Int[x_^m_./(a_+b_.*x_^n_),x_Symbol] := + Module[{r=Numerator[Rt[a/b,n]], s=Denominator[Rt[a/b,n]], k, u}, + u=Int[(r*Cos[(2*k-1)*m*Pi/n]-s*Cos[(2*k-1)*(m+1)*Pi/n]*x)/(r^2-2*r*s*Cos[(2*k-1)*Pi/n]*x+s^2*x^2),x]; + -(-r)^(m+1)/(a*n*s^m)*Int[1/(r+s*x),x] + Dist[2*r^(m+1)/(a*n*s^m),Sum[u,{k,1,(n-1)/2}],x]] /; +FreeQ[{a,b},x] && PositiveIntegerQ[(n-1)/2] && PositiveIntegerQ[m] && m2*n-1 + + +Int[x_/Sqrt[a_+b_.*x_^3],x_Symbol] := + With[{r=Numer[Rt[b/a,3]], s=Denom[Rt[b/a,3]]}, + Sqrt[2]*s/(Sqrt[2+Sqrt[3]]*r)*Int[1/Sqrt[a+b*x^3],x] + 1/r*Int[((1-Sqrt[3])*s+r*x)/Sqrt[a+b*x^3],x]] /; +FreeQ[{a,b},x] && PosQ[a] + + +Int[x_/Sqrt[a_+b_.*x_^3],x_Symbol] := + With[{r=Numer[Rt[b/a,3]], s=Denom[Rt[b/a,3]]}, + -Sqrt[2]*s/(Sqrt[2-Sqrt[3]]*r)*Int[1/Sqrt[a+b*x^3],x] + 1/r*Int[((1+Sqrt[3])*s+r*x)/Sqrt[a+b*x^3],x]] /; +FreeQ[{a,b},x] && NegQ[a] + + +Int[x_^2/Sqrt[a_+b_.*x_^4],x_Symbol] := + With[{q=Rt[b/a,2]}, + 1/q*Int[1/Sqrt[a+b*x^4],x] - 1/q*Int[(1-q*x^2)/Sqrt[a+b*x^4],x]] /; +FreeQ[{a,b},x] && PosQ[b/a] + + +Int[x_^2/Sqrt[a_+b_.*x_^4],x_Symbol] := + With[{q=Rt[-b/a,2]}, + 1/q*Int[1/Sqrt[a+b*x^4],x] - 1/q*Int[(1-q*x^2)/Sqrt[a+b*x^4],x]] /; +FreeQ[{a,b},x] && NegativeQ[a] && PositiveQ[b] + + +Int[x_^2/Sqrt[a_+b_.*x_^4],x_Symbol] := + With[{q=Rt[-b/a,2]}, + -1/q*Int[1/Sqrt[a+b*x^4],x] + 1/q*Int[(1+q*x^2)/Sqrt[a+b*x^4],x]] /; +FreeQ[{a,b},x] && NegQ[b/a] + + +Int[x_^4/Sqrt[a_+b_.*x_^6],x_Symbol] := + With[{r=Numer[Rt[b/a,3]], s=Denom[Rt[b/a,3]]}, + (Sqrt[3]-1)*s^2/(2*r^2)*Int[1/Sqrt[a+b*x^6],x] - 1/(2*r^2)*Int[((Sqrt[3]-1)*s^2-2*r^2*x^4)/Sqrt[a+b*x^6],x]] /; +FreeQ[{a,b},x] + + +(* Int[x_^4/Sqrt[a_+b_.*x_^6],x_Symbol] := + With[{r=Numer[Rt[b/a,3]], s=Denom[Rt[b/a,3]]}, + (1+Sqrt[3])*r*x*Sqrt[a+b*x^6]/(2*b*(s+(1+Sqrt[3])*r*x^2)) - + 3^(1/4)*s*x*(s+r*x^2)*Sqrt[(s^2-r*s*x^2+r^2*x^4)/(s+(1+Sqrt[3])*r*x^2)^2]/ + (2*r^2*Sqrt[a+b*x^6]*Sqrt[r*x^2*(s+r*x^2)/(s+(1+Sqrt[3])*r*x^2)^2])* + EllipticE[ArcCos[(s+(1-Sqrt[3])*r*x^2)/(s+(1+Sqrt[3])*r*x^2)],(2+Sqrt[3])/4] - + (1-Sqrt[3])*s*x*(s+r*x^2)*Sqrt[(s^2-r*s*x^2+r^2*x^4)/(s+(1+Sqrt[3])*r*x^2)^2]/ + (4*3^(1/4)*r^2*Sqrt[a+b*x^6]*Sqrt[r*x^2*(s+r*x^2)/(s+(1+Sqrt[3])*r*x^2)^2])* + EllipticF[ArcCos[(s+(1-Sqrt[3])*r*x^2)/(s+(1+Sqrt[3])*r*x^2)],(2+Sqrt[3])/4]] /; +FreeQ[{a,b},x] *) + + +Int[x_^2/Sqrt[a_+b_.*x_^8],x_Symbol] := + 1/(2*Rt[b/a,4])*Int[(1+Rt[b/a,4]*x^2)/Sqrt[a+b*x^8],x] - + 1/(2*Rt[b/a,4])*Int[(1-Rt[b/a,4]*x^2)/Sqrt[a+b*x^8],x] /; +FreeQ[{a,b},x] + + +Int[x_^2/(a_+b_.*x_^4)^(1/4),x_Symbol] := + x^3/(2*(a+b*x^4)^(1/4)) - a/2*Int[x^2/(a+b*x^4)^(5/4),x] /; +FreeQ[{a,b},x] && PosQ[b/a] + + +Int[x_^2/(a_+b_.*x_^4)^(1/4),x_Symbol] := + (a+b*x^4)^(3/4)/(2*b*x) + a/(2*b)*Int[1/(x^2*(a+b*x^4)^(1/4)),x] /; +FreeQ[{a,b},x] && NegQ[b/a] + + +Int[1/(x_^2*(a_+b_.*x_^4)^(1/4)),x_Symbol] := + -1/(x*(a+b*x^4)^(1/4)) - b*Int[x^2/(a+b*x^4)^(5/4),x] /; +FreeQ[{a,b},x] && PosQ[b/a] + + +Int[1/(x_^2*(a_+b_.*x_^4)^(1/4)),x_Symbol] := + x*(1+a/(b*x^4))^(1/4)/(a+b*x^4)^(1/4)*Int[1/(x^3*(1+a/(b*x^4))^(1/4)),x] /; +FreeQ[{a,b},x] && NegQ[b/a] + + +Int[Sqrt[c_*x_]/(a_+b_.*x_^2)^(1/4),x_Symbol] := + x*Sqrt[c*x]/(a+b*x^2)^(1/4) - a/2*Int[Sqrt[c*x]/(a+b*x^2)^(5/4),x] /; +FreeQ[{a,b,c},x] && PosQ[b/a] + + +Int[Sqrt[c_*x_]/(a_+b_.*x_^2)^(1/4),x_Symbol] := + c*(a+b*x^2)^(3/4)/(b*Sqrt[c*x]) + a*c^2/(2*b)*Int[1/((c*x)^(3/2)*(a+b*x^2)^(1/4)),x] /; +FreeQ[{a,b,c},x] && NegQ[b/a] + + +Int[1/((c_.*x_)^(3/2)*(a_+b_.*x_^2)^(1/4)),x_Symbol] := + -2/(c*Sqrt[c*x]*(a+b*x^2)^(1/4)) - b/c^2*Int[Sqrt[c*x]/(a+b*x^2)^(5/4),x] /; +FreeQ[{a,b,c},x] && PosQ[b/a] + + +Int[1/((c_.*x_)^(3/2)*(a_+b_.*x_^2)^(1/4)),x_Symbol] := + Sqrt[c*x]*(1+a/(b*x^2))^(1/4)/(c^2*(a+b*x^2)^(1/4))*Int[1/(x^2*(1+a/(b*x^2))^(1/4)),x] /; +FreeQ[{a,b,c},x] && NegQ[b/a] + + +Int[(c_.*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^(n-1)*(c*x)^(m-n+1)*(a+b*x^n)^(p+1)/(b*(m+n*p+1)) - + a*c^n*(m-n+1)/(b*(m+n*p+1))*Int[(c*x)^(m-n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,p},x] && PositiveIntegerQ[n] && RationalQ[m] && m>n-1 && NeQ[m+n*p+1] && IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^(n-1)*(c*x)^(m-n+1)*(a+b*x^n)^(p+1)/(b*(m+n*p+1)) - + a*c^n*(m-n+1)/(b*(m+n*p+1))*Int[(c*x)^(m-n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,p},x] && PositiveIntegerQ[n] && SumSimplerQ[m,-n] && NeQ[m+n*p+1] && NegativeIntegerQ[Simplify[(m+1)/n+p]] + + +Int[(c_.*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + c^(2*n-1)*(c*x)^(m-2*n+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(b1*b2*(m+2*n*p+1)) - + a1*a2*c^(2*n)*(m-2*n+1)/(b1*b2*(m+2*n*p+1))*Int[(c*x)^(m-2*n)*(a1+b1*x^n)^p*(a2+b2*x^n)^p,x] /; +FreeQ[{a1,b1,a2,b2,c,p},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m] && m>2*n-1 && + NeQ[m+2*n*p+1] && IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[(c_.*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + c^(2*n-1)*(c*x)^(m-2*n+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(b1*b2*(m+2*n*p+1)) - + a1*a2*c^(2*n)*(m-2*n+1)/(b1*b2*(m+2*n*p+1))*Int[(c*x)^(m-2*n)*(a1+b1*x^n)^p*(a2+b2*x^n)^p,x] /; +FreeQ[{a1,b1,a2,b2,c,m,p},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && SumSimplerQ[m,-2*n] && NeQ[m+2*n*p+1] && + NegativeIntegerQ[Simplify[(m+1)/(2*n)+p]] + + +Int[(c_.*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a+b*x^n)^(p+1)/(a*c*(m+1)) - + b*(m+n*(p+1)+1)/(a*c^n*(m+1))*Int[(c*x)^(m+n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,p},x] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 && IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a+b*x^n)^(p+1)/(a*c*(m+1)) - + b*(m+n*(p+1)+1)/(a*c^n*(m+1))*Int[(c*x)^(m+n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,p},x] && PositiveIntegerQ[n] && SumSimplerQ[m,n] && NegativeIntegerQ[Simplify[(m+1)/n+p]] + + +Int[(c_.*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(a1*a2*c*(m+1)) - + b1*b2*(m+2*n*(p+1)+1)/(a1*a2*c^(2*n)*(m+1))*Int[(c*x)^(m+2*n)*(a1+b1*x^n)^p*(a2+b2*x^n)^p,x] /; +FreeQ[{a1,b1,a2,b2,c,p},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && RationalQ[m] && m<-1 && + IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[(c_.*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(a1*a2*c*(m+1)) - + b1*b2*(m+2*n*(p+1)+1)/(a1*a2*c^(2*n)*(m+1))*Int[(c*x)^(m+2*n)*(a1+b1*x^n)^p*(a2+b2*x^n)^p,x] /; +FreeQ[{a1,b1,a2,b2,c,m,p},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && SumSimplerQ[m,2*n] && + NegativeIntegerQ[Simplify[(m+1)/(2*n)+p]] + + +Int[(c_.*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/c*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n)/c^n)^p,x],x,(c*x)^(1/k)]] /; +FreeQ[{a,b,c,p},x] && PositiveIntegerQ[n] && FractionQ[m] && IntBinomialQ[a,b,c,n,m,p,x] + + +Int[(c_.*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/c*Subst[Int[x^(k*(m+1)-1)*(a1+b1*x^(k*n)/c^n)^p*(a2+b2*x^(k*n)/c^n)^p,x],x,(c*x)^(1/k)]] /; +FreeQ[{a1,b1,a2,b2,c,p},x] && EqQ[a2*b1+a1*b2] && PositiveIntegerQ[2*n] && FractionQ[m] && IntBinomialQ[a1*a2,b1*b2,c,n,m,p,x] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + a^(p+(m+1)/n)*Subst[Int[x^m/(1-b*x^n)^(p+(m+1)/n+1),x],x,x/(a+b*x^n)^(1/n)] /; +FreeQ[{a,b},x] && PositiveIntegerQ[n] && RationalQ[p] && -10 + + +Int[x_^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + x^(m+1)*(a1+b1*x^n)^p*(a2+b2*x^n)^p/(m+1) - + 2*b1*b2*n*p/(m+1)*Int[x^(m+n)*(a1+b1*x^n)^(p-1)*(a2+b2*x^n)^(p-1),x] /; +FreeQ[{a1,b1,a2,b2,m,n},x] && EqQ[a2*b1+a1*b2] && EqQ[(m+1)/(2*n)+p] && RationalQ[p] && p>0 + + +Int[(c_*x_)^m_*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[(m+1)/n+p] && RationalQ[p] && p>0 + + +Int[(c_*x_)^m_*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a1+b1*x^n)^p*(a2+b2*x^n)^p,x] /; +FreeQ[{a1,b1,a2,b2,c,m,n},x] && EqQ[a2*b1+a1*b2] && EqQ[(m+1)/(2*n)+p] && RationalQ[p] && p>0 + + +Int[(c_.*x_)^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a+b*x^n)^p/(c*(m+n*p+1)) + + a*n*p/(m+n*p+1)*Int[(c*x)^m*(a+b*x^n)^(p-1),x] /; +FreeQ[{a,b,c,m,n},x] && IntegerQ[p+Simplify[(m+1)/n]] && RationalQ[p] && p>0 && NeQ[m+n*p+1] + + +Int[(c_.*x_)^m_.*(a1_+b1_.*x_^n_)^p_*(a2_+b2_.*x_^n_)^p_,x_Symbol] := + (c*x)^(m+1)*(a1+b1*x^n)^p*(a2+b2*x^n)^p/(c*(m+2*n*p+1)) + + 2*a1*a2*n*p/(m+2*n*p+1)*Int[(c*x)^m*(a1+b1*x^n)^(p-1)*(a2+b2*x^n)^(p-1),x] /; +FreeQ[{a1,b1,a2,b2,c,m,n},x] && EqQ[a2*b1+a1*b2] && IntegerQ[p+Simplify[(m+1)/(2*n)]] && RationalQ[p] && p>0 && + NeQ[m+2*n*p+1] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{k=Denominator[p]}, + k*a^(p+Simplify[(m+1)/n])/n* + Subst[Int[x^(k*Simplify[(m+1)/n]-1)/(1-b*x^k)^(p+Simplify[(m+1)/n]+1),x],x,x^(n/k)/(a+b*x^n)^(1/k)]] /; +FreeQ[{a,b,m,n},x] && IntegerQ[p+Simplify[(m+1)/n]] && RationalQ[p] && -10 && NeQ[p+1] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + a^p*x/(c^(p+1)*(c+d*x^n)^(1/n))*Hypergeometric2F1[1/n,-p,1+1/n,-(b*c-a*d)*x^n/(a*(c+d*x^n))] /; +FreeQ[{a,b,c,d,n,q},x] && NeQ[b*c-a*d,0] && EqQ[n*(p+q+1)+1] && NegativeIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + x*(a+b*x^n)^p/(c*(c*(a+b*x^n)/(a*(c+d*x^n)))^p*(c+d*x^n)^(1/n+p))* + Hypergeometric2F1[1/n,-p,1+1/n,-(b*c-a*d)*x^n/(a*(c+d*x^n))] /; +FreeQ[{a,b,c,d,n,p,q},x] && NeQ[b*c-a*d,0] && EqQ[n*(p+q+1)+1] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + x*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*c) /; +FreeQ[{a,b,c,d,n,p,q},x] && NeQ[b*c-a*d,0] && EqQ[n*(p+q+2)+1] && EqQ[a*d*(p+1)+b*c*(q+1)] + + +(* Int[(a1_+b1_.*x_^n2_.)^p_*(a2_+b2_.*x_^n2_.)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + x*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)*(c+d*x^n)^(q+1)/(a1*a2*c) /; +FreeQ[{a1,b1,a2,b2,c,d,n,p,q},x] && EqQ[n2-n/2] && EqQ[a2*b1+a1*b2] && EqQ[n*(p+q+2)+1] && EqQ[a1*a2*d*(p+1)+b1*b2*c*(q+1)] *) + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -b*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*n*(p+1)*(b*c-a*d)) + + (b*c+n*(p+1)*(b*c-a*d))/(a*n*(p+1)*(b*c-a*d))*Int[(a+b*x^n)^(p+1)*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,n,q},x] && NeQ[b*c-a*d,0] && EqQ[n*(p+q+2)+1] && (RationalQ[p] && p<-1 || Not[RationalQ[q] && q<-1]) && NeQ[p+1] + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*x*(a+b*x^n)^(p+1)/a /; +FreeQ[{a,b,c,d,n,p},x] && NeQ[b*c-a*d,0] && EqQ[a*d-b*c*(n*(p+1)+1),0] + + +Int[(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*x*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(a1*a2) /; +FreeQ[{a1,b1,a2,b2,c,d,n,p},x] && EqQ[non2,n/2] && EqQ[a2*b1+a1*b2,0] && EqQ[a1*a2*d-b1*b2*c*(n*(p+1)+1),0] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_),x_Symbol] := + -(b*c-a*d)*x*(a+b*x^n)^(p+1)/(a*b*n*(p+1)) - + (a*d-b*c*(n*(p+1)+1))/(a*b*n*(p+1))*Int[(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c,d,n,p},x] && NeQ[b*c-a*d,0] && (LtQ[p,-1] || ILtQ[1/n+p,0]) + + +Int[(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + -(b1*b2*c-a1*a2*d)*x*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(a1*a2*b1*b2*n*(p+1)) - + (a1*a2*d-b1*b2*c*(n*(p+1)+1))/(a1*a2*b1*b2*n*(p+1))*Int[(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1),x] /; +FreeQ[{a1,b1,a2,b2,c,d,n},x] && EqQ[non2,n/2] && EqQ[a2*b1+a1*b2,0] && (LtQ[p,-1] || ILtQ[1/n+p,0]) + + +Int[(c_+d_.*x_^n_)/(a_+b_.*x_^n_),x_Symbol] := + c*x/a - (b*c-a*d)/a*Int[1/(b+a*x^(-n)),x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[b*c-a*d,0] && RationalQ[n] && n<0 + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_),x_Symbol] := + d*x*(a+b*x^n)^(p+1)/(b*(n*(p+1)+1)) - + (a*d-b*c*(n*(p+1)+1))/(b*(n*(p+1)+1))*Int[(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[b*c-a*d,0] && NeQ[n*(p+1)+1,0] + + +Int[(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + d*x*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(b1*b2*(n*(p+1)+1)) - + (a1*a2*d-b1*b2*c*(n*(p+1)+1))/(b1*b2*(n*(p+1)+1))*Int[(a1+b1*x^(n/2))^p*(a2+b2*x^(n/2))^p,x] /; +FreeQ[{a1,b1,a2,b2,c,d,n,p},x] && EqQ[non2,n/2] && EqQ[a2*b1+a1*b2,0] && NeQ[n*(p+1)+1,0] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + Int[PolynomialDivide[(a+b*x^n)^p,(c+d*x^n)^(-q),x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && PositiveIntegerQ[n,p] && NegativeIntegerQ[q] && p>=-q + + +Int[1/((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + b/(b*c-a*d)*Int[1/(a+b*x^n),x] - d/(b*c-a*d)*Int[1/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[b*c-a*d,0] + + +Int[1/((a_+b_.*x_^2)^(1/3)*(c_+d_.*x_^2)),x_Symbol] := + Sqrt[3]/(2*c)*Int[1/((a+b*x^2)^(1/3)*(Sqrt[3]-Rt[b/a,2]*x)),x] + + Sqrt[3]/(2*c)*Int[1/((a+b*x^2)^(1/3)*(Sqrt[3]+Rt[b/a,2]*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && EqQ[b*c+3*a*d] && PosQ[b/a] + + +Int[1/((a_+b_.*x_^2)^(1/3)*(c_+d_.*x_^2)),x_Symbol] := + 1/6*Int[(3+Rt[-b/a,2]*x)/((a+b*x^2)^(1/3)*(c+d*x^2)),x] + + 1/6*Int[(3-Rt[-b/a,2]*x)/((a+b*x^2)^(1/3)*(c+d*x^2)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && EqQ[b*c+3*a*d] && NegQ[b/a] + + +Int[(a_+b_.*x_^2)^(2/3)/(c_+d_.*x_^2),x_Symbol] := + b/d*Int[1/(a+b*x^2)^(1/3),x] - (b*c-a*d)/d*Int[1/((a+b*x^2)^(1/3)*(c+d*x^2)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && EqQ[b*c+3*a*d] + + +Int[1/((a_+b_.*x_^2)^(1/4)*(c_+d_.*x_^2)),x_Symbol] := + Sqrt[-b*x^2/a]/(2*x)*Subst[Int[1/(Sqrt[-b*x/a]*(a+b*x)^(1/4)*(c+d*x)),x],x,x^2] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[1/((a_+b_.*x_^2)^(3/4)*(c_+d_.*x_^2)),x_Symbol] := + Sqrt[-b*x^2/a]/(2*x)*Subst[Int[1/(Sqrt[-b*x/a]*(a+b*x)^(3/4)*(c+d*x)),x],x,x^2] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[(a_+b_.*x_^2)^p_./(c_+d_.*x_^2),x_Symbol] := + b/d*Int[(a+b*x^2)^(p-1),x] - (b*c-a*d)/d*Int[(a+b*x^2)^(p-1)/(c+d*x^2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[p] && p>0 && (p==1/2 || Denominator[p]==4) + + +Int[(a_+b_.*x_^2)^p_/(c_+d_.*x_^2),x_Symbol] := + b/(b*c-a*d)*Int[(a+b*x^2)^p,x] - d/(b*c-a*d)*Int[(a+b*x^2)^(p+1)/(c+d*x^2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[p] && p<-1 && Denominator[p]==4 && (p==-5/4 || p==-7/4) + + +Int[Sqrt[a_+b_.*x_^4]/(c_+d_.*x_^4),x_Symbol] := + a/c*Subst[Int[1/(1-4*a*b*x^4),x],x,x/Sqrt[a+b*x^4]] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d] && PosQ[a*b] + + +Int[Sqrt[a_+b_.*x_^4]/(c_+d_.*x_^4),x_Symbol] := + With[{q=Rt[-a*b,4]}, + a/(2*c*q)*ArcTan[q*x*(a+q^2*x^2)/(a*Sqrt[a+b*x^4])] + a/(2*c*q)*ArcTanh[q*x*(a-q^2*x^2)/(a*Sqrt[a+b*x^4])]] /; +FreeQ[{a,b,c,d},x] && EqQ[b*c+a*d] && NegQ[a*b] + + +Int[Sqrt[a_+b_.*x_^4]/(c_+d_.*x_^4),x_Symbol] := + b/d*Int[1/Sqrt[a+b*x^4],x] - (b*c-a*d)/d*Int[1/(Sqrt[a+b*x^4]*(c+d*x^4)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[(a_+b_.*x_^4)^(1/4)/(c_+d_.*x_^4),x_Symbol] := + Sqrt[a+b*x^4]*Sqrt[a/(a+b*x^4)]*Subst[Int[1/(Sqrt[1-b*x^4]*(c-(b*c-a*d)*x^4)),x],x,x/(a+b*x^4)^(1/4)] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[(a_+b_.*x_^4)^p_/(c_+d_.*x_^4),x_Symbol] := + b/d*Int[(a+b*x^4)^(p-1),x] - (b*c-a*d)/d*Int[(a+b*x^4)^(p-1)/(c+d*x^4),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && RationalQ[p] && (p==3/4 || p==5/4) + + +Int[1/(Sqrt[a_+b_.*x_^4]*(c_+d_.*x_^4)),x_Symbol] := + 1/(2*c)*Int[1/(Sqrt[a+b*x^4]*(1-Rt[-d/c,2]*x^2)),x] + 1/(2*c)*Int[1/(Sqrt[a+b*x^4]*(1+Rt[-d/c,2]*x^2)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[1/((a_+b_.*x_^4)^(3/4)*(c_+d_.*x_^4)),x_Symbol] := + b/(b*c-a*d)*Int[1/(a+b*x^4)^(3/4),x] - d/(b*c-a*d)*Int[(a+b*x^4)^(1/4)/(c+d*x^4),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] + + +Int[Sqrt[a_+b_.*x_^2]/(c_+d_.*x_^2)^(3/2),x_Symbol] := + Sqrt[a+b*x^2]/(c*Rt[d/c,2]*Sqrt[c+d*x^2]*Sqrt[c*(a+b*x^2)/(a*(c+d*x^2))])*EllipticE[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && PosQ[b/a] && PosQ[d/c] + + +(* Int[Sqrt[a_+b_.*x_^2]/(c_+d_.*x_^2)^(3/2),x_Symbol] := + a*Sqrt[c+d*x^2]*Sqrt[(c*(a+b*x^2))/(a*(c+d*x^2))]/(c^2*Rt[d/c,2]*Sqrt[a+b*x^2])*EllipticE[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && PosQ[b/a] && PosQ[d/c] *) + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -x*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*n*(p+1)) + + 1/(a*n*(p+1))*Int[(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*Simp[c*(n*(p+1)+1)+d*(n*(p+q+1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[b*c-a*d,0] && RationalQ[p,q] && p<-1 && 01 && IntBinomialQ[a,b,c,d,n,p,q,x] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -b*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*n*(p+1)*(b*c-a*d)) + + 1/(a*n*(p+1)*(b*c-a*d))* + Int[(a+b*x^n)^(p+1)*(c+d*x^n)^q*Simp[b*c+n*(p+1)*(b*c-a*d)+d*b*(n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,n,q},x] && NeQ[b*c-a*d,0] && RationalQ[p] && p<-1 && Not[Not[IntegerQ[p]] && IntegerQ[q] && q<-1] && + IntBinomialQ[a,b,c,d,n,p,q,x] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^n)^p*(c+d*x^n)^q,x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0] && PositiveIntegerQ[n] && IntegersQ[p,q] && p+q>0 + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + d*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)/(b*(n*(p+q)+1)) + + 1/(b*(n*(p+q)+1))* + Int[(a+b*x^n)^p*(c+d*x^n)^(q-2)*Simp[c*(b*c*(n*(p+q)+1)-a*d)+d*(b*c*(n*(p+2*q-1)+1)-a*d*(n*(q-1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,n,p},x] && NeQ[b*c-a*d,0] && RationalQ[q] && q>1 && NeQ[n*(p+q)+1] && Not[IntegerQ[p] && p>1] && + IntBinomialQ[a,b,c,d,n,p,q,x] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + x*(a+b*x^n)^p*(c+d*x^n)^q/(n*(p+q)+1) + + n/(n*(p+q)+1)*Int[(a+b*x^n)^(p-1)*(c+d*x^n)^(q-1)*Simp[a*c*(p+q)+(q*(b*c-a*d)+a*d*(p+q))*x^n,x],x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[b*c-a*d,0] && RationalQ[p,q] && q>0 && p>0 && IntBinomialQ[a,b,c,d,n,p,q,x] + + +Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol] := + Sqrt[a+b*x^2]/(a*Rt[d/c,2]*Sqrt[c+d*x^2]*Sqrt[c*(a+b*x^2)/(a*(c+d*x^2))])*EllipticF[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && PosQ[d/c] && PosQ[b/a] && Not[SimplerSqrtQ[b/a,d/c]] + + +(* Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol] := + Sqrt[c+d*x^2]*Sqrt[c*(a+b*x^2)/(a*(c+d*x^2))]/(c*Rt[d/c,2]*Sqrt[a+b*x^2])*EllipticF[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && PosQ[d/c] && PosQ[b/a] && Not[SimplerSqrtQ[b/a,d/c]] *) + + +Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol] := + 1/(Sqrt[a]*Sqrt[c]*Rt[-d/c,2])*EllipticF[ArcSin[Rt[-d/c,2]*x],b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && PositiveQ[c] && PositiveQ[a] && Not[NegQ[b/a] && SimplerSqrtQ[-b/a,-d/c]] + + +Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol] := + -1/(Sqrt[c]*Rt[-d/c,2]*Sqrt[a-b*c/d])*EllipticF[ArcCos[Rt[-d/c,2]*x],b*c/(b*c-a*d)] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && PositiveQ[c] && PositiveQ[a-b*c/d] + + +Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol] := + Sqrt[1+d/c*x^2]/Sqrt[c+d*x^2]*Int[1/(Sqrt[a+b*x^2]*Sqrt[1+d/c*x^2]),x] /; +FreeQ[{a,b,c,d},x] && Not[PositiveQ[c]] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + a*Int[1/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]),x] + b*Int[x^2/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d},x] && PosQ[d/c] && PosQ[b/a] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + b/d*Int[Sqrt[c+d*x^2]/Sqrt[a+b*x^2],x] - (b*c-a*d)/d*Int[1/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d},x] && PosQ[d/c] && NegQ[b/a] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + Sqrt[a]/(Sqrt[c]*Rt[-d/c,2])*EllipticE[ArcSin[Rt[-d/c,2]*x],b*c/(a*d)] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && PositiveQ[c] && PositiveQ[a] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + -Sqrt[a-b*c/d]/(Sqrt[c]*Rt[-d/c,2])*EllipticE[ArcCos[Rt[-d/c,2]*x],b*c/(b*c-a*d)] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && PositiveQ[c] && PositiveQ[a-b*c/d] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2]/Sqrt[1+b/a*x^2]*Int[Sqrt[1+b/a*x^2]/Sqrt[c+d*x^2],x] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && PositiveQ[c] && Not[PositiveQ[a]] + + +Int[Sqrt[a_+b_.*x_^2]/Sqrt[c_+d_.*x_^2],x_Symbol] := + Sqrt[1+d/c*x^2]/Sqrt[c+d*x^2]*Int[Sqrt[a+b*x^2]/Sqrt[1+d/c*x^2],x] /; +FreeQ[{a,b,c,d},x] && NegQ[d/c] && Not[PositiveQ[c]] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^n)^p*(c+d*x^n)^q,x],x] /; +FreeQ[{a,b,c,d,n,q},x] && NeQ[b*c-a*d,0] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + a^p*c^q*x*AppellF1[1/n,-p,-q,1+1/n,-b*x^n/a,-d*x^n/c] /; +FreeQ[{a,b,c,d,n,p,q},x] && NeQ[b*c-a*d,0] && NeQ[n+1] && PositiveQ[a] && PositiveQ[c] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + a^IntPart[p]*(a+b*x^n)^FracPart[p]/(1+b*x^n/a)^FracPart[p]*Int[(1+b*x^n/a)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,n,p,q},x] && NeQ[b*c-a*d,0] && NeQ[n+1] && Not[PositiveQ[a]] + + +Int[(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.,x_Symbol] := + Int[(a+b*x^n)^p*(d+c*x^n)^q/x^(n*q),x] /; +FreeQ[{a,b,c,d,n,p},x] && EqQ[mn,-n] && IntegerQ[q] && (PosQ[n] || Not[IntegerQ[p]]) + + +Int[(a_+b_.*x_^n_.)^p_*(c_+d_.*x_^mn_.)^q_,x_Symbol] := + x^(n*FracPart[q])*(c+d*x^(-n))^FracPart[q]/(d+c*x^n)^FracPart[q]*Int[(a+b*x^n)^p*(d+c*x^n)^q/x^(n*q),x] /; +FreeQ[{a,b,c,d,n,p,q},x] && EqQ[mn,-n] && Not[IntegerQ[q]] && Not[IntegerQ[p]] + + +Int[(a_.+b_.*u_^n_)^p_.*(c_.+d_.*u_^n_)^q_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x^n)^p*(c+d*x^n)^q,x],x,u] /; +FreeQ[{a,b,c,d,n,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[u_^p_.*v_^q_.,x_Symbol] := + Int[NormalizePseudoBinomial[u,x]^p*NormalizePseudoBinomial[v,x]^q,x] /; +FreeQ[{p,q},x] && PseudoBinomialPairQ[u,v,x] + + +Int[x_^m_.*u_^p_.*v_^q_.,x_Symbol] := + Int[NormalizePseudoBinomial[x^(m/p)*u,x]^p*NormalizePseudoBinomial[v,x]^q,x] /; +FreeQ[{p,q},x] && IntegersQ[p,m/p] && PseudoBinomialPairQ[x^(m/p)*u,v,x] + + +(* IntBinomialQ[a,b,c,d,n,p,q,x] returns True iff (a+b*x^n)^p*(c+d*x^n)^q is integrable wrt x in terms of non-Appell functions. *) +IntBinomialQ[a_,b_,c_,d_,n_,p_,q_,x_Symbol] := + IntegersQ[p,q] || PositiveIntegerQ[p] || PositiveIntegerQ[q] || + (EqQ[n-2] || EqQ[n-4]) && (IntegersQ[p,4*q] || IntegersQ[4*p,q]) || + EqQ[n-2] && (IntegersQ[2*p,2*q] || IntegersQ[3*p,q] && EqQ[b*c+3*a*d] || IntegersQ[p,3*q] && EqQ[3*b*c+a*d]) + + +(* ::Subsection::Closed:: *) +(*1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q*) + + +Int[(e_.*x_)^m_.*(b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.,x_Symbol] := + e^m/(n*b^(Simplify[(m+1)/n]-1))*Subst[Int[(b*x)^(p+Simplify[(m+1)/n]-1)*(c+d*x)^q,x],x,x^n] /; +FreeQ[{b,c,d,e,m,n,p,q},x] && (IntegerQ[m] || PositiveQ[e]) && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_.*x_)^m_.*(b_.*x_^n_.)^p_*(c_+d_.*x_^n_)^q_.,x_Symbol] := + e^m*b^IntPart[p]*(b*x^n)^FracPart[p]/x^(n*FracPart[p])*Int[x^(m+n*p)*(c+d*x^n)^q,x] /; +FreeQ[{b,c,d,e,m,n,p,q},x] && (IntegerQ[m] || PositiveQ[e]) && Not[IntegerQ[Simplify[(m+1)/n]]] + + +Int[(e_*x_)^m_*(b_.*x_^n_.)^p_*(c_+d_.*x_^n_)^q_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{b,c,d,e,m,n,p,q},x] && Not[IntegerQ[m]] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + 1/n*Subst[Int[(a+b*x)^p*(c+d*x)^q,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && NeQ[b*c-a*d] && EqQ[m-n+1] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + Int[x^(m+n*(p+q))*(b+a*x^(-n))^p*(d+c*x^(-n))^q,x] /; +FreeQ[{a,b,c,d,m,n},x] && NeQ[b*c-a*d] && IntegersQ[p,q] && NegQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*x)^p*(c+d*x)^q,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && NeQ[b*c-a*d] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && NeQ[b*c-a*d] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + Int[ExpandIntegrand[(e*x)^m*(a+b*x^n)^p*(c+d*x^n)^q,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d] && PositiveIntegerQ[p,q] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*e*(m+1)) /; +FreeQ[{a,b,c,d,e,m,n,p},x] && NeQ[b*c-a*d] && EqQ[a*d*(m+1)-b*c*(m+n*(p+1)+1)] && NeQ[m+1] + + +Int[(e_.*x_)^m_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*(e*x)^(m+1)*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(a1*a2*e*(m+1)) /; +FreeQ[{a1,b1,a2,b2,c,d,e,m,n,p},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] && EqQ[a1*a2*d*(m+1)-b1*b2*c*(m+n*(p+1)+1)] && + NeQ[m+1] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*e*(m+1)) + d/e^n*Int[(e*x)^(m+n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && EqQ[m+n*(p+1)+1] && (IntegerQ[n] || PositiveQ[e]) && + RationalQ[m,n] && (n>0 && m<-1 || n<0 && m+n>-1) + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + (b*c-a*d)*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*b*e*(m+1)) + d/b*Int[(e*x)^m*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && NeQ[b*c-a*d] && EqQ[m+n*(p+1)+1] && NeQ[m+1] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*e*(m+1)) + + (a*d*(m+1)-b*c*(m+n*(p+1)+1))/(a*e^n*(m+1))*Int[(e*x)^(m+n)*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && (IntegerQ[n] || PositiveQ[e]) && + RationalQ[m,n] && (n>0 && m<-1 || n<0 && m+n>-1) && Not[IntegerQ[p] && p<-1] + + +Int[(e_.*x_)^m_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + c*(e*x)^(m+1)*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(a1*a2*e*(m+1)) + + (a1*a2*d*(m+1)-b1*b2*c*(m+n*(p+1)+1))/(a1*a2*e^n*(m+1))*Int[(e*x)^(m+n)*(a1+b1*x^(n/2))^p*(a2+b2*x^(n/2))^p,x] /; +FreeQ[{a1,b1,a2,b2,c,d,e,p},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] && (IntegerQ[n] || PositiveQ[e]) && + RationalQ[m,n] && (n>0 && m<-1 || n<0 && m+n>-1) && Not[IntegerQ[p] && p<-1] + + +Int[x_^m_*(a_+b_.*x_^2)^p_*(c_+d_.*x_^2),x_Symbol] := + (-a)^(m/2-1)*(b*c-a*d)*x*(a+b*x^2)^(p+1)/(2*b^(m/2+1)*(p+1)) + + 1/(2*b^(m/2+1)*(p+1))*Int[(a+b*x^2)^(p+1)* + ExpandToSum[2*b*(p+1)*x^2*Together[(b^(m/2)*x^(m-2)*(c+d*x^2)-(-a)^(m/2-1)*(b*c-a*d))/(a+b*x^2)]-(-a)^(m/2-1)*(b*c-a*d),x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d] && RationalQ[p] && p<-1 && PositiveIntegerQ[m/2] && (IntegerQ[p] || m+2*p+1==0) + + +Int[x_^m_*(a_+b_.*x_^2)^p_*(c_+d_.*x_^2),x_Symbol] := + (-a)^(m/2-1)*(b*c-a*d)*x*(a+b*x^2)^(p+1)/(2*b^(m/2+1)*(p+1)) + + 1/(2*b^(m/2+1)*(p+1))*Int[x^m*(a+b*x^2)^(p+1)* + ExpandToSum[2*b*(p+1)*Together[(b^(m/2)*(c+d*x^2)-(-a)^(m/2-1)*(b*c-a*d)*x^(-m+2))/(a+b*x^2)]- + (-a)^(m/2-1)*(b*c-a*d)*x^(-m),x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d] && RationalQ[p] && p<-1 && NegativeIntegerQ[m/2] && (IntegerQ[p] || m+2*p+1==0) + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + -(b*c-a*d)*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*b*e*n*(p+1)) - + (a*d*(m+1)-b*c*(m+n*(p+1)+1))/(a*b*n*(p+1))*Int[(e*x)^m*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d] && RationalQ[p] && p<-1 && + (IntegerQ[p] || Not[RationalQ[m]] || PositiveIntegerQ[n] && NegativeIntegerQ[p+1/2] && -1<=m<=-n*(p+1)) + + +Int[(e_.*x_)^m_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + -(b1*b2*c-a1*a2*d)*(e*x)^(m+1)*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(a1*a2*b1*b2*e*n*(p+1)) - + (a1*a2*d*(m+1)-b1*b2*c*(m+n*(p+1)+1))/(a1*a2*b1*b2*n*(p+1))*Int[(e*x)^m*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1),x] /; +FreeQ[{a1,b1,a2,b2,c,d,e,m,n},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] && RationalQ[p] && p<-1 && + (IntegerQ[p] || Not[RationalQ[m]] || PositiveIntegerQ[n] && NegativeIntegerQ[p+1/2] && -1<=m<=-n*(p+1)) + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_),x_Symbol] := + d*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(b*e*(m+n*(p+1)+1)) - + (a*d*(m+1)-b*c*(m+n*(p+1)+1))/(b*(m+n*(p+1)+1))*Int[(e*x)^m*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && NeQ[b*c-a*d] && NeQ[m+n*(p+1)+1] + + +Int[(e_.*x_)^m_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_),x_Symbol] := + d*(e*x)^(m+1)*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)/(b1*b2*e*(m+n*(p+1)+1)) - + (a1*a2*d*(m+1)-b1*b2*c*(m+n*(p+1)+1))/(b1*b2*(m+n*(p+1)+1))*Int[(e*x)^m*(a1+b1*x^(n/2))^p*(a2+b2*x^(n/2))^p,x] /; +FreeQ[{a1,b1,a2,b2,c,d,e,m,n,p},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] && NeQ[m+n*(p+1)+1] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_/(c_+d_.*x_^n_),x_Symbol] := + Int[ExpandIntegrand[(e*x)^m*(a+b*x^n)^p/(c+d*x^n),x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && PositiveIntegerQ[p] && + (IntegerQ[m] || PositiveIntegerQ[2*(m+1)] || Not[RationalQ[m]]) + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^2,x_Symbol] := + c^2*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*e*(m+1)) - + 1/(a*e^n*(m+1))*Int[(e*x)^(m+n)*(a+b*x^n)^p*Simp[b*c^2*n*(p+1)+c*(b*c-2*a*d)*(m+1)-a*(m+1)*d^2*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,n] && m<-1 && n>0 + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^2,x_Symbol] := + -(b*c-a*d)^2*(e*x)^(m+1)*(a+b*x^n)^(p+1)/(a*b^2*e*n*(p+1)) + + 1/(a*b^2*n*(p+1))*Int[(e*x)^m*(a+b*x^n)^(p+1)*Simp[(b*c-a*d)^2*(m+1)+b^2*c^2*n*(p+1)+a*b*d^2*n*(p+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^2,x_Symbol] := + d^2*(e*x)^(m+n+1)*(a+b*x^n)^(p+1)/(b*e^(n+1)*(m+n*(p+2)+1)) + + 1/(b*(m+n*(p+2)+1))*Int[(e*x)^m*(a+b*x^n)^p*Simp[b*c^2*(m+n*(p+2)+1)+d*((2*b*c-a*d)*(m+n+1)+2*b*c*n*(p+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && NeQ[m+n*(p+2)+1] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + With[{k=GCD[m+1,n]}, + 1/k*Subst[Int[x^((m+1)/k-1)*(a+b*x^(n/k))^p*(c+d*x^(n/k))^q,x],x,x^k] /; + k!=1] /; +FreeQ[{a,b,c,d,p,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + With[{k=Denominator[m]}, + k/e*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n)/e^n)^p*(c+d*x^(k*n)/e^n)^q,x],x,(e*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && FractionQ[m] && IntegerQ[p] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + e^(n-1)*(e*x)^(m-n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(b*n*(p+1)) - + e^n/(b*n*(p+1))*Int[(e*x)^(m-n)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*Simp[c*(m-n+1)+d*(m+n*(q-1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,p,q] && p<-1 && q>0 && m-n+1>0 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -(c*b-a*d)*(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)/(a*b*e*n*(p+1)) + + 1/(a*b*n*(p+1))*Int[(e*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-2)* + Simp[c*(c*b*n*(p+1)+(c*b-a*d)*(m+1))+d*(c*b*n*(p+1)+(c*b-a*d)*(m+n*(q-1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[p,q] && p<-1 && q>1 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*e*n*(p+1)) + + 1/(a*n*(p+1))*Int[(e*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*Simp[c*(m+n*(p+1)+1)+d*(m+n*(p+q+1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[p,q] && p<-1 && 0n && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + e^(n-1)*(e*x)^(m-n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(n*(b*c-a*d)*(p+1)) - + e^n/(n*(b*c-a*d)*(p+1))*Int[(e*x)^(m-n)*(a+b*x^n)^(p+1)*(c+d*x^n)^q*Simp[c*(m-n+1)+d*(m+n*(p+q+1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && n>=m-n+1>0 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -b*(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*e*n*(b*c-a*d)*(p+1)) + + 1/(a*n*(b*c-a*d)*(p+1))* + Int[(e*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^q*Simp[c*b*(m+1)+n*(b*c-a*d)*(p+1)+d*b*(m+n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + (e*x)^(m+1)*(a+b*x^n)^p*(c+d*x^n)^q/(e*(m+1)) - + n/(e^n*(m+1))*Int[(e*x)^(m+n)*(a+b*x^n)^(p-1)*(c+d*x^n)^(q-1)*Simp[b*c*p+a*d*q+b*d*(p+q)*x^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,p,q] && q>0 && m<-1 && p>0 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + c*(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)/(a*e*(m+1)) - + 1/(a*e^n*(m+1))*Int[(e*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^(q-2)* + Simp[c*(c*b-a*d)*(m+1)+c*n*(b*c*(p+1)+a*d*(q-1))+d*((c*b-a*d)*(m+1)+c*b*n*(p+q))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,q] && q>1 && m<-1 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + (e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*e*(m+1)) - + 1/(a*e^n*(m+1))*Int[(e*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^(q-1)* + Simp[c*b*(m+1)+n*(b*c*(p+1)+a*d*q)+d*(b*(m+1)+b*n*(p+q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,q] && 00 && p>0 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + d*(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)/(b*e*(m+n*(p+q)+1)) + + 1/(b*(m+n*(p+q)+1))*Int[(e*x)^m*(a+b*x^n)^p*(c+d*x^n)^(q-2)* + Simp[c*((c*b-a*d)*(m+1)+c*b*n*(p+q))+(d*(c*b-a*d)*(m+1)+d*n*(q-1)*(b*c-a*d)+c*b*d*n*(p+q))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[q] && q>1 && IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + e^(n-1)*(e*x)^(m-n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(b*(m+n*(p+q)+1)) - + e^n/(b*(m+n*(p+q)+1))* + Int[(e*x)^(m-n)*(a+b*x^n)^p*(c+d*x^n)^(q-1)*Simp[a*c*(m-n+1)+(a*d*(m-n+1)-n*q*(b*c-a*d))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m,q] && q>0 && m-n+1>0 && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + e^(2*n-1)*(e*x)^(m-2*n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(b*d*(m+n*(p+q)+1)) - + e^(2*n)/(b*d*(m+n*(p+q)+1))* + Int[(e*x)^(m-2*n)*(a+b*x^n)^p*(c+d*x^n)^q*Simp[a*c*(m-2*n+1)+(a*d*(m+n*(q-1)+1)+b*c*(m+n*(p-1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m] && m-n+1>n && + IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + (e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*c*e*(m+1)) - + 1/(a*c*e^n*(m+1))* + Int[(e*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^q*Simp[(b*c+a*d)*(m+n+1)+n*(b*c*p+a*d*q)+b*d*(m+n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 && IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_./((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + -a*e^n/(b*c-a*d)*Int[(e*x)^(m-n)/(a+b*x^n),x] + c*e^n/(b*c-a*d)*Int[(e*x)^(m-n)/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,m,n},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] && RationalQ[m] && n<=m<=2*n-1 + + +Int[(e_.*x_)^m_./((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + b/(b*c-a*d)*Int[(e*x)^m/(a+b*x^n),x] - d/(b*c-a*d)*Int[(e*x)^m/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b*c-a*d] && PositiveIntegerQ[n] + + +Int[x_^m_/((a_+b_.*x_^n_)*Sqrt[c_+d_.*x_^n_]),x_Symbol] := + 1/b*Int[x^(m-n)/Sqrt[c+d*x^n],x] - a/b*Int[x^(m-n)/((a+b*x^n)*Sqrt[c+d*x^n]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d] && IntegersQ[m/2,n/2] && 01 && IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + -(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*e*n*(p+1)) + + 1/(a*n*(p+1))*Int[(e*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*Simp[c*(m+n*(p+1)+1)+d*(m+n*(p+q+1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d] && RationalQ[p,q] && p<-1 && 00 && p>0 && IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + d*(e*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)/(b*e*(m+n*(p+q)+1)) + + 1/(b*(m+n*(p+q)+1))*Int[(e*x)^m*(a+b*x^n)^p*(c+d*x^n)^(q-2)* + Simp[c*((c*b-a*d)*(m+1)+c*b*n*(p+q))+(d*(c*b-a*d)*(m+1)+d*n*(q-1)*(b*c-a*d)+c*b*d*n*(p+q))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && NeQ[b*c-a*d] && RationalQ[q] && q>1 && IntBinomialQ[a,b,c,d,e,m,n,p,q,x] + + +Int[x_^m_/((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + -a/(b*c-a*d)*Int[x^(m-n)/(a+b*x^n),x] + c/(b*c-a*d)*Int[x^(m-n)/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,m,n},x] && NeQ[b*c-a*d] && (EqQ[m-n] || EqQ[m-2*n+1]) + + +Int[(e_.*x_)^m_./((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + b/(b*c-a*d)*Int[(e*x)^m/(a+b*x^n),x] - d/(b*c-a*d)*Int[(e*x)^m/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,e,n,m},x] && NeQ[b*c-a*d] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + Int[ExpandIntegrand[(e*x)^m*(a+b*x^n)^p*(c+d*x^n)^q,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d] && IntegersQ[m,p,q] && p>=-2 && (q>=-2 || q==-3 && IntegerQ[(m-1)/2]) + + +Int[x_^m_.*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.,x_Symbol] := + Int[x^(m-n*q)*(a+b*x^n)^p*(d+c*x^n)^q,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[mn,-n] && IntegerQ[q] && (PosQ[n] || Not[IntegerQ[p]]) + + +Int[x_^m_.*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_,x_Symbol] := + x^(n*FracPart[q])*(c+d*x^(-n))^FracPart[q]/(d+c*x^n)^FracPart[q]*Int[x^(m-n*q)*(a+b*x^n)^p*(d+c*x^n)^q,x] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && EqQ[mn,-n] && Not[IntegerQ[q]] && Not[IntegerQ[p]] + + +Int[(e_*x_)^m_*(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^(-n))^q,x] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[mn,-n] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + a^p*c^q*(e*x)^(m+1)/(e*(m+1))*AppellF1[(m+1)/n,-p,-q,1+(m+1)/n,-b*x^n/a,-d*x^n/c] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && NeQ[b*c-a*d] && NeQ[m+1] && NeQ[m-n+1] && PositiveQ[a] && PositiveQ[c] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol] := + a^IntPart[p]*(a+b*x^n)^FracPart[p]/(1+b*x^n/a)^FracPart[p]*Int[(e*x)^m*(1+b*x^n/a)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && NeQ[b*c-a*d] && NeQ[m+1] && NeQ[m-n+1] && Not[PositiveQ[a]] + + +Int[x_^m_.*(a_.+b_.*v_^n_)^p_.*(c_.+d_.*v_^n_)^q_.,x_Symbol] := + 1/Coefficient[v,x,1]^(m+1)*Subst[Int[SimplifyIntegrand[(x-Coefficient[v,x,0])^m*(a+b*x^n)^p*(c+d*x^n)^q,x],x],x,v] /; +FreeQ[{a,b,c,d,n,p,q},x] && LinearQ[v,x] && IntegerQ[m] && NeQ[v-x] + + +Int[u_^m_.*(a_.+b_.*v_^n_)^p_.*(c_.+d_.*v_^n_)^q_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q,x],x,v] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && LinearPairQ[u,v,x] + + +(* IntBinomialQ[a,b,c,d,e,m,n,p,q,x] returns True iff (e*x)^m*(a+b*x^n)^p*(c+d*x^n)^q is integrable wrt x in terms of non-Appell functions. *) +IntBinomialQ[a_,b_,c_,d_,e_,m_,n_,p_,q_,x_Symbol] := + IntegersQ[p,q] || PositiveIntegerQ[p] || PositiveIntegerQ[q] || + EqQ[n-2] && IntegerQ[m] && IntegersQ[2*p,2*q] || + EqQ[n-4] && (IntegersQ[m,p,2*q] || IntegersQ[m,2*p,q]) + + +Int[u_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_.)^q_.,x_Symbol] := + Int[u*(a1*a2+b1*b2*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a1,b1,a2,b2,c,d,n,p,q},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] && (IntegerQ[p] || PositiveQ[a1] && PositiveQ[a2]) + + +Int[u_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_.+e_.*x_^n2_.)^q_.,x_Symbol] := + Int[u*(a1*a2+b1*b2*x^n)^p*(c+d*x^n+e*x^(2*n))^q,x] /; +FreeQ[{a1,b1,a2,b2,c,d,e,n,p,q},x] && EqQ[non2-n/2] && EqQ[n2-2*n] && EqQ[a2*b1+a1*b2] && + (IntegerQ[p] || PositiveQ[a1] && PositiveQ[a2]) + + +Int[u_.*(a1_+b1_.*x_^non2_.)^p_*(a2_+b2_.*x_^non2_.)^p_*(c_+d_.*x_^n_.)^q_.,x_Symbol] := + (a1+b1*x^(n/2))^FracPart[p]*(a2+b2*x^(n/2))^FracPart[p]/(a1*a2+b1*b2*x^n)^FracPart[p]* + Int[u*(a1*a2+b1*b2*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a1,b1,a2,b2,c,d,n,p,q},x] && EqQ[non2-n/2] && EqQ[a2*b1+a1*b2] + + +Int[u_.*(a1_+b1_.*x_^non2_.)^p_.*(a2_+b2_.*x_^non2_.)^p_.*(c_+d_.*x_^n_.+e_.*x_^n2_.)^q_.,x_Symbol] := + (a1+b1*x^(n/2))^FracPart[p]*(a2+b2*x^(n/2))^FracPart[p]/(a1*a2+b1*b2*x^n)^FracPart[p]* + Int[u*(a1*a2+b1*b2*x^n)^p*(c+d*x^n+e*x^(2*n))^q,x] /; +FreeQ[{a1,b1,a2,b2,c,d,e,n,p,q},x] && EqQ[non2-n/2] && EqQ[n2-2*n] && EqQ[a2*b1+a1*b2] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r*) + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[p,q,r] + + +Int[(e_+f_.*x_^n_)/((a_+b_.*x_^n_)*(c_+d_.*x_^n_)),x_Symbol] := + (b*e-a*f)/(b*c-a*d)*Int[1/(a+b*x^n),x] - + (d*e-c*f)/(b*c-a*d)*Int[1/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,e,f,n},x] + + +Int[(e_+f_.*x_^n_)/((a_+b_.*x_^n_)*Sqrt[c_+d_.*x_^n_]),x_Symbol] := + f/b*Int[1/Sqrt[c+d*x^n],x] + + (b*e-a*f)/b*Int[1/((a+b*x^n)*Sqrt[c+d*x^n]),x] /; +FreeQ[{a,b,c,d,e,f,n},x] + + +Int[(e_+f_.*x_^n_)/(Sqrt[a_+b_.*x_^n_]*Sqrt[c_+d_.*x_^n_]),x_Symbol] := + f/b*Int[Sqrt[a+b*x^n]/Sqrt[c+d*x^n],x] + + (b*e-a*f)/b*Int[1/(Sqrt[a+b*x^n]*Sqrt[c+d*x^n]),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && + Not[EqQ[n-2] && (PosQ[b/a] && PosQ[d/c] || NegQ[b/a] && (PosQ[d/c] || PositiveQ[a] && (Not[PositiveQ[c]] || SimplerSqrtQ[-b/a,-d/c])))] + + +Int[(e_+f_.*x_^2)/(Sqrt[a_+b_.*x_^2]*(c_+d_.*x_^2)^(3/2)),x_Symbol] := + (b*e-a*f)/(b*c-a*d)*Int[1/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]),x] - + (d*e-c*f)/(b*c-a*d)*Int[Sqrt[a+b*x^2]/(c+d*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[b/a] && PosQ[d/c] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*x*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*b*n*(p+1)) + + 1/(a*b*n*(p+1))* + Int[(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*Simp[c*(b*e*n*(p+1)+b*e-a*f)+d*(b*e*n*(p+1)+(b*e-a*f)*(n*q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && RationalQ[p,q] && p<-1 && q>0 + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*n*(b*c-a*d)*(p+1)) + + 1/(a*n*(b*c-a*d)*(p+1))* + Int[(a+b*x^n)^(p+1)*(c+d*x^n)^q*Simp[c*(b*e-a*f)+e*n*(b*c-a*d)*(p+1)+d*(b*e-a*f)*(n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n,q},x] && RationalQ[p] && p<-1 + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + f*x*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(b*(n*(p+q+1)+1)) + + 1/(b*(n*(p+q+1)+1))* + Int[(a+b*x^n)^p*(c+d*x^n)^(q-1)*Simp[c*(b*e-a*f+b*e*n*(p+q+1))+(d*(b*e-a*f)+f*n*q*(b*c-a*d)+b*d*e*n*(p+q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && RationalQ[q] && q>0 && NeQ[n*(p+q+1)+1] + + +Int[(e_+f_.*x_^4)/((a_+b_.*x_^4)^(3/4)*(c_+d_.*x_^4)),x_Symbol] := + (b*e-a*f)/(b*c-a*d)*Int[1/(a+b*x^4)^(3/4),x] - (d*e-c*f)/(b*c-a*d)*Int[(a+b*x^4)^(1/4)/(c+d*x^4),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[(a_+b_.*x_^n_)^p_*(e_+f_.*x_^n_)/(c_+d_.*x_^n_),x_Symbol] := + f/d*Int[(a+b*x^n)^p,x] + (d*e-c*f)/d*Int[(a+b*x^n)^p/(c+d*x^n),x] /; +FreeQ[{a,b,c,d,e,f,p,n},x] + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + e*Int[(a+b*x^n)^p*(c+d*x^n)^q,x] + f*Int[x^n*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,f,n,p,q},x] + + +Int[1/((a_+b_.*x_^2)*(c_+d_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + b/(b*c-a*d)*Int[1/((a+b*x^2)*Sqrt[e+f*x^2]),x] - + d/(b*c-a*d)*Int[1/((c+d*x^2)*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[1/(x_^2*(c_+d_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + 1/c*Int[1/(x^2*Sqrt[e+f*x^2]),x] - + d/c*Int[1/((c+d*x^2)*Sqrt[e+f*x^2]),x] /; +FreeQ[{c,d,e,f},x] && NeQ[d*e-c*f] + + +Int[Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]/(a_+b_.*x_^2),x_Symbol] := + d/b*Int[Sqrt[e+f*x^2]/Sqrt[c+d*x^2],x] + (b*c-a*d)/b*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveQ[d/c] && PositiveQ[f/e] && Not[SimplerSqrtQ[d/c,f/e]] + + +Int[Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]/(a_+b_.*x_^2),x_Symbol] := + d/b*Int[Sqrt[e+f*x^2]/Sqrt[c+d*x^2],x] + (b*c-a*d)/b*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[SimplerSqrtQ[-f/e,-d/c]] + + +Int[1/((a_+b_.*x_^2)*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + -f/(b*e-a*f)*Int[1/(Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] + + b/(b*e-a*f)*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] && PosQ[f/e] && Not[SimplerSqrtQ[d/c,f/e]] + + +Int[1/((a_+b_.*x_^2)*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + 1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c,2])*EllipticPi[b*c/(a*d), ArcSin[Rt[-d/c,2]*x], c*f/(d*e)] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[d/c] && PositiveQ[c] && PositiveQ[e] && Not[NegQ[f/e] && SimplerSqrtQ[-f/e,-d/c]] + + +Int[1/((a_+b_.*x_^2)*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + Sqrt[1+d/c*x^2]/Sqrt[c+d*x^2]*Int[1/((a+b*x^2)*Sqrt[1+d/c*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && Not[PositiveQ[c]] + + +Int[Sqrt[c_+d_.*x_^2]/((a_+b_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + c*Sqrt[e+f*x^2]/(a*e*Rt[d/c,2]*Sqrt[c+d*x^2]*Sqrt[c*(e+f*x^2)/(e*(c+d*x^2))])* + EllipticPi[1-b*c/(a*d),ArcTan[Rt[d/c,2]*x],1-c*f/(d*e)] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] + + +(* Int[Sqrt[c_+d_.*x_^2]/((a_+b_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + Sqrt[c+d*x^2]*Sqrt[c*(e+f*x^2)/(e*(c+d*x^2))]/(a*Rt[d/c,2]*Sqrt[e+f*x^2])* + EllipticPi[1-b*c/(a*d),ArcTan[Rt[d/c,2]*x],1-c*f/(d*e)] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] *) + + +Int[Sqrt[c_+d_.*x_^2]/((a_+b_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + d/b*Int[1/(Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] + + (b*c-a*d)/b*Int[1/((a+b*x^2)*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[d/c] + + +Int[Sqrt[e_+f_.*x_^2]/((a_+b_.*x_^2)*(c_+d_.*x_^2)^(3/2)),x_Symbol] := + b/(b*c-a*d)*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] - + d/(b*c-a*d)*Int[Sqrt[e+f*x^2]/(c+d*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] && PosQ[f/e] + + +Int[(e_+f_.*x_^2)^(3/2)/((a_+b_.*x_^2)*(c_+d_.*x_^2)^(3/2)),x_Symbol] := + (b*e-a*f)/(b*c-a*d)*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] - + (d*e-c*f)/(b*c-a*d)*Int[Sqrt[e+f*x^2]/(c+d*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] && PosQ[f/e] + + +Int[(c_+d_.*x_^2)^(3/2)*Sqrt[e_+f_.*x_^2]/(a_+b_.*x_^2),x_Symbol] := + (b*c-a*d)^2/b^2*Int[Sqrt[e+f*x^2]/((a+b*x^2)*Sqrt[c+d*x^2]),x] + + d/b^2*Int[(2*b*c-a*d+b*d*x^2)*Sqrt[e+f*x^2]/Sqrt[c+d*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[d/c] && PosQ[f/e] + + +Int[(c_+d_.*x_^2)^q_*(e_+f_.*x_^2)^r_/(a_+b_.*x_^2),x_Symbol] := + b*(b*e-a*f)/(b*c-a*d)^2*Int[(c+d*x^2)^(q+2)*(e+f*x^2)^(r-1)/(a+b*x^2),x] - + 1/(b*c-a*d)^2*Int[(c+d*x^2)^q*(e+f*x^2)^(r-1)*(2*b*c*d*e-a*d^2*e-b*c^2*f+d^2*(b*e-a*f)*x^2),x] /; +FreeQ[{a,b,c,d,e,f},x] && RationalQ[q,r] && q<-1 && r>1 + + +Int[(c_+d_.*x_^2)^q_*(e_+f_.*x_^2)^r_/(a_+b_.*x_^2),x_Symbol] := + d/b*Int[(c+d*x^2)^(q-1)*(e+f*x^2)^r,x] + + (b*c-a*d)/b*Int[(c+d*x^2)^(q-1)*(e+f*x^2)^r/(a+b*x^2),x] /; +FreeQ[{a,b,c,d,e,f,r},x] && RationalQ[q] && q>1 + + +Int[(c_+d_.*x_^2)^q_*(e_+f_.*x_^2)^r_/(a_+b_.*x_^2),x_Symbol] := + b^2/(b*c-a*d)^2*Int[(c+d*x^2)^(q+2)*(e+f*x^2)^r/(a+b*x^2),x] - + d/(b*c-a*d)^2*Int[(c+d*x^2)^q*(e+f*x^2)^r*(2*b*c-a*d+b*d*x^2),x] /; +FreeQ[{a,b,c,d,e,f,r},x] && RationalQ[q] && q<-1 + + +Int[(c_+d_.*x_^2)^q_*(e_+f_.*x_^2)^r_/(a_+b_.*x_^2),x_Symbol] := + -d/(b*c-a*d)*Int[(c+d*x^2)^q*(e+f*x^2)^r,x] + + b/(b*c-a*d)*Int[(c+d*x^2)^(q+1)*(e+f*x^2)^r/(a+b*x^2),x] /; +FreeQ[{a,b,c,d,e,f,r},x] && RationalQ[q] && q<=-1 + + +Int[Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]/(a_+b_.*x_^2)^2,x_Symbol] := + x*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]/(2*a*(a+b*x^2)) + + d*f/(2*a*b^2)*Int[(a-b*x^2)/(Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] + + (b^2*c*e-a^2*d*f)/(2*a*b^2)*Int[1/((a+b*x^2)*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[1/((a_+b_.*x_^2)^2*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + b^2*x*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]/(2*a*(b*c-a*d)*(b*e-a*f)*(a+b*x^2)) - + d*f/(2*a*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x^2)/(Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] + + (b^2*c*e+3*a^2*d*f-2*a*b*(d*e+c*f))/(2*a*(b*c-a*d)*(b*e-a*f))*Int[1/((a+b*x^2)*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_)^r_,x_Symbol] := + d/b*Int[(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)*(e+f*x^n)^r,x] + + (b*c-a*d)/b*Int[(a+b*x^n)^p*(c+d*x^n)^(q-1)*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,n,r},x] && NegativeIntegerQ[p] && RationalQ[q] && q>0 + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_)^r_,x_Symbol] := + b/(b*c-a*d)*Int[(a+b*x^n)^p*(c+d*x^n)^(q+1)*(e+f*x^n)^r,x] - + d/(b*c-a*d)*Int[(a+b*x^n)^(p+1)*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,n,q},x] && NegativeIntegerQ[p] && RationalQ[q] && q<=-1 + + +Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + Sqrt[c+d*x^2]*Sqrt[a*(e+f*x^2)/(e*(a+b*x^2))]/(c*Sqrt[e+f*x^2]*Sqrt[a*(c+d*x^2)/(c*(a+b*x^2))])* + Subst[Int[1/(Sqrt[1-(b*c-a*d)*x^2/c]*Sqrt[1-(b*e-a*f)*x^2/e]),x],x,x/Sqrt[a+b*x^2]] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[Sqrt[a_+b_.*x_^2]/(Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + a*Sqrt[c+d*x^2]*Sqrt[a*(e+f*x^2)/(e*(a+b*x^2))]/(c*Sqrt[e+f*x^2]*Sqrt[a*(c+d*x^2)/(c*(a+b*x^2))])* + Subst[Int[1/((1-b*x^2)*Sqrt[1-(b*c-a*d)*x^2/c]*Sqrt[1-(b*e-a*f)*x^2/e]),x],x,x/Sqrt[a+b*x^2]] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[Sqrt[c_+d_.*x_^2]/((a_+b_.*x_^2)^(3/2)*Sqrt[e_+f_.*x_^2]),x_Symbol] := + Sqrt[c+d*x^2]*Sqrt[a*(e+f*x^2)/(e*(a+b*x^2))]/(a*Sqrt[e+f*x^2]*Sqrt[a*(c+d*x^2)/(c*(a+b*x^2))])* + Subst[Int[Sqrt[1-(b*c-a*d)*x^2/c]/Sqrt[1-(b*e-a*f)*x^2/e],x],x,x/Sqrt[a+b*x^2]] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]/Sqrt[e_+f_.*x_^2],x_Symbol] := + d*x*Sqrt[a+b*x^2]*Sqrt[e+f*x^2]/(2*f*Sqrt[c+d*x^2]) - + c*(d*e-c*f)/(2*f)*Int[Sqrt[a+b*x^2]/((c+d*x^2)^(3/2)*Sqrt[e+f*x^2]),x] + + b*c*(d*e-c*f)/(2*d*f)*Int[1/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] - + (b*d*e-b*c*f-a*d*f)/(2*d*f)*Int[Sqrt[c+d*x^2]/(Sqrt[a+b*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[(d*e-c*f)/c] + + +Int[Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]/Sqrt[e_+f_.*x_^2],x_Symbol] := + x*Sqrt[a+b*x^2]*Sqrt[c+d*x^2]/(2*Sqrt[e+f*x^2]) + + e*(b*e-a*f)/(2*f)*Int[Sqrt[c+d*x^2]/(Sqrt[a+b*x^2]*(e+f*x^2)^(3/2)),x] + + (b*e-a*f)*(d*e-2*c*f)/(2*f^2)*Int[1/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] - + (b*d*e-b*c*f-a*d*f)/(2*f^2)*Int[Sqrt[e+f*x^2]/(Sqrt[a+b*x^2]*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[(d*e-c*f)/c] + + +Int[Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]/(e_+f_.*x_^2)^(3/2),x_Symbol] := + b/f*Int[Sqrt[c+d*x^2]/(Sqrt[a+b*x^2]*Sqrt[e+f*x^2]),x] - + (b*e-a*f)/f*Int[Sqrt[c+d*x^2]/(Sqrt[a+b*x^2]*(e+f*x^2)^(3/2)),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_)^r_,x_Symbol] := + With[{u=ExpandIntegrand[(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && PositiveIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_)^r_,x_Symbol] := + -Subst[Int[(a+b*x^(-n))^p*(c+d*x^(-n))^q*(e+f*x^(-n))^r/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && NegativeIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + Defer[Int][(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,n,p,q,r},x] + + +Int[(a_.+b_.*u_^n_)^p_.*(c_.+d_.*v_^n_)^q_.*(e_.+f_.*w_^n_)^r_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x],x,u] /; +FreeQ[{a,b,c,d,e,f,p,n,q,r},x] && EqQ[u-v] && EqQ[u-w] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + Int[(a+b*x^n)^p*(d+c*x^n)^q*(e+f*x^n)^r/x^(n*q),x] /; +FreeQ[{a,b,c,d,e,f,n,p,r},x] && EqQ[mn,-n] && IntegerQ[q] + + +Int[(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + Int[x^(n*(p+r))*(b+a*x^(-n))^p*(c+d*x^(-n))^q*(f+e*x^(-n))^r,x] /; +FreeQ[{a,b,c,d,e,f,n,q},x] && EqQ[mn,-n] && IntegerQ[p] && IntegerQ[r] + + +Int[(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + x^(n*FracPart[q])*(c+d*x^(-n))^FracPart[q]/(d+c*x^n)^FracPart[q]*Int[(a+b*x^n)^p*(d+c*x^n)^q*(e+f*x^n)^r/x^(n*q),x] /; +FreeQ[{a,b,c,d,e,f,n,p,q,r},x] && EqQ[mn,-n] && Not[IntegerQ[q]] + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e1_+f1_.*x_^n2_.)^r_.*(e2_+f2_.*x_^n2_.)^r_.,x_Symbol] := + Int[(a+b*x^n)^p*(c+d*x^n)^q*(e1*e2+f1*f2*x^n)^r,x] /; +FreeQ[{a,b,c,d,e1,f1,e2,f2,n,p,q,r},x] && EqQ[n2-n/2] && EqQ[e2*f1+e1*f2] && (IntegerQ[r] || PositiveQ[e1] && PositiveQ[e2]) + + +Int[(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e1_+f1_.*x_^n2_.)^r_.*(e2_+f2_.*x_^n2_.)^r_.,x_Symbol] := + (e1+f1*x^(n/2))^FracPart[r]*(e2+f2*x^(n/2))^FracPart[r]/(e1*e2+f1*f2*x^n)^FracPart[r]* + Int[(a+b*x^n)^p*(c+d*x^n)^q*(e1*e2+f1*f2*x^n)^r,x] /; +FreeQ[{a,b,c,d,e1,f1,e2,f2,n,p,q,r},x] && EqQ[n2-n/2] && EqQ[e2*f1+e1*f2] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r*) + + +Int[(g_.*x_)^m_.*(b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^m/(n*b^(Simplify[(m+1)/n]-1))*Subst[Int[(b*x)^(p+Simplify[(m+1)/n]-1)*(c+d*x)^q*(e+f*x)^r,x],x,x^n] /; +FreeQ[{b,c,d,e,f,g,m,n,p,q,r},x] && (IntegerQ[m] || PositiveQ[g]) && IntegerQ[Simplify[(m+1)/n]] + + +Int[(g_.*x_)^m_.*(b_.*x_^n_.)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^m*b^IntPart[p]*(b*x^n)^FracPart[p]/x^(n*FracPart[p])*Int[x^(m+n*p)*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{b,c,d,e,f,g,m,n,p,q,r},x] && (IntegerQ[m] || PositiveQ[g]) && Not[IntegerQ[Simplify[(m+1)/n]]] + + +Int[(g_*x_)^m_*(b_.*x_^n_.)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{b,c,d,e,f,g,m,n,p,q,r},x] && Not[IntegerQ[m]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + Int[ExpandIntegrand[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && PositiveIntegerQ[p+2,q,r] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + 1/n*Subst[Int[(a+b*x)^p*(c+d*x)^q*(e+f*x)^r,x],x,x^n] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && EqQ[m-n+1] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + Int[x^(m+n*(p+q+r))*(b+a*x^(-n))^p*(d+c*x^(-n))^q*(f+e*x^(-n))^r,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IntegersQ[p,q,r] && NegQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*x)^p*(c+d*x)^q*(e+f*x)^r,x],x,x^n] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(g_*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q,r},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + With[{k=GCD[m+1,n]}, + 1/k*Subst[Int[x^((m+1)/k-1)*(a+b*x^(n/k))^p*(c+d*x^(n/k))^q*(e+f*x^(n/k))^r,x],x,x^k] /; + k!=1] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(g_.*x_)^m_*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_)^r_,x_Symbol] := + With[{k=Denominator[m]}, + k/g*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n)/g^n)^p*(c+d*x^(k*n)/g^n)^q*(e+f*x^(k*n)/g^n)^r,x],x,(g*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,g,p,q,r},x] && PositiveIntegerQ[n] && FractionQ[m] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*b*g*n*(p+1)) + + 1/(a*b*n*(p+1))*Int[(g*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)* + Simp[c*(b*e*n*(p+1)+(b*e-a*f)*(m+1))+d*(b*e*n*(p+1)+(b*e-a*f)*(m+n*q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && PositiveIntegerQ[n] && RationalQ[p,q] && p<-1 && q>0 && Not[q==1 && SimplerQ[b*c-a*d,b*e-a*f]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_),x_Symbol] := + g^(n-1)*(b*e-a*f)*(g*x)^(m-n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(b*n*(b*c-a*d)*(p+1)) - + g^n/(b*n*(b*c-a*d)*(p+1))*Int[(g*x)^(m-n)*(a+b*x^n)^(p+1)*(c+d*x^n)^q* + Simp[c*(b*e-a*f)*(m-n+1)+(d*(b*e-a*f)*(m+n*q+1)-b*n*(c*f-d*e)*(p+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,q},x] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m-n+1>0 + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*g*n*(b*c-a*d)*(p+1)) + + 1/(a*n*(b*c-a*d)*(p+1))*Int[(g*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^q* + Simp[c*(b*e-a*f)*(m+1)+e*n*(b*c-a*d)*(p+1)+d*(b*e-a*f)*(m+n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,q},x] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[(g_.*x_)^m_*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + e*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*g*(m+1)) - + 1/(a*g^n*(m+1))*Int[(g*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^(q-1)* + Simp[c*(b*e-a*f)*(m+1)+e*n*(b*c*(p+1)+a*d*q)+d*((b*e-a*f)*(m+1)+b*e*n*(p+q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && PositiveIntegerQ[n] && RationalQ[m,q] && q>0 && m<-1 && Not[q==1 && SimplerQ[e+f*x^n,c+d*x^n]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + f*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(b*g*(m+n*(p+q+1)+1)) + + 1/(b*(m+n*(p+q+1)+1))*Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^(q-1)* + Simp[c*((b*e-a*f)*(m+1)+b*e*n*(p+q+1))+(d*(b*e-a*f)*(m+1)+f*n*q*(b*c-a*d)+b*e*d*n*(p+q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && PositiveIntegerQ[n] && RationalQ[q] && q>0 && Not[q==1 && SimplerQ[e+f*x^n,c+d*x^n]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + f*g^(n-1)*(g*x)^(m-n+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(b*d*(m+n*(p+q+1)+1)) - + g^n/(b*d*(m+n*(p+q+1)+1))*Int[(g*x)^(m-n)*(a+b*x^n)^p*(c+d*x^n)^q* + Simp[a*f*c*(m-n+1)+(a*f*d*(m+n*q+1)+b*(f*c*(m+n*p+1)-e*d*(m+n*(p+q+1)+1)))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p,q},x] && PositiveIntegerQ[n] && RationalQ[m] && m>n-1 + + +Int[(g_.*x_)^m_*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + e*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*c*g*(m+1)) + + 1/(a*c*g^n*(m+1))*Int[(g*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^q* + Simp[a*f*c*(m+1)-e*(b*c+a*d)*(m+n+1)-e*n*(b*c*p+a*d*q)-b*e*d*(m+n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p,q},x] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(e_+f_.*x_^n_)/(c_+d_.*x_^n_),x_Symbol] := + Int[ExpandIntegrand[(g*x)^m*(a+b*x^n)^p*(e+f*x^n)/(c+d*x^n),x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && PositiveIntegerQ[n] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + e*Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q,x] + + f/e^n*Int[(g*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p,q},x] && PositiveIntegerQ[n] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + e*Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^(r-1),x] + + f/e^n*Int[(g*x)^(m+n)*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^(r-1),x] /; +FreeQ[{a,b,c,d,e,f,g,m,p,q},x] && PositiveIntegerQ[n] && PositiveIntegerQ[r] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + -Subst[Int[(a+b*x^(-n))^p*(c+d*x^(-n))^q*(e+f*x^(-n))^r/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[(g_.*x_)^m_*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + With[{k=Denominator[m]}, + -k/g*Subst[Int[(a+b*g^(-n)*x^(-k*n))^p*(c+d*g^(-n)*x^(-k*n))^q*(e+f*g^(-n)*x^(-k*n))^r/x^(k*(m+1)+1),x],x,1/(g*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,g,p,q,r},x] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(g_.*x_)^m_*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + -(g*x)^m*(x^(-1))^m*Subst[Int[(a+b*x^(-n))^p*(c+d*x^(-n))^q*(e+f*x^(-n))^r/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,d,e,f,g,m,p,q,r},x] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + With[{k=Denominator[n]}, + k*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n))^p*(c+d*x^(k*n))^q*(e+f*x^(k*n))^r,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,m,p,q,r},x] && FractionQ[n] + + +Int[(g_*x_)^m_*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p,q,r},x] && FractionQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + 1/(m+1)*Subst[Int[(a+b*x^Simplify[n/(m+1)])^p*(c+d*x^Simplify[n/(m+1)])^q*(e+f*x^Simplify[n/(m+1)])^r,x],x,x^(m+1)] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && IntegerQ[Simplify[n/(m+1)]] + + +Int[(g_*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q,r},x] && IntegerQ[Simplify[n/(m+1)]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(a*b*g*n*(p+1)) + + 1/(a*b*n*(p+1))*Int[(g*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^(q-1)* + Simp[c*(b*e*n*(p+1)+(b*e-a*f)*(m+1))+d*(b*e*n*(p+1)+(b*e-a*f)*(m+n*q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && RationalQ[p,q] && p<-1 && q>0 && Not[q==1 && SimplerQ[b*c-a*d,b*e-a*f]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_),x_Symbol] := + -(b*e-a*f)*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(q+1)/(a*g*n*(b*c-a*d)*(p+1)) + + 1/(a*n*(b*c-a*d)*(p+1))*Int[(g*x)^m*(a+b*x^n)^(p+1)*(c+d*x^n)^q* + Simp[c*(b*e-a*f)*(m+1)+e*n*(b*c-a*d)*(p+1)+d*(b*e-a*f)*(m+n*(p+q+2)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,q},x] && RationalQ[p] && p<-1 + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_),x_Symbol] := + f*(g*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^q/(b*g*(m+n*(p+q+1)+1)) + + 1/(b*(m+n*(p+q+1)+1))*Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^(q-1)* + Simp[c*((b*e-a*f)*(m+1)+b*e*n*(p+q+1))+(d*(b*e-a*f)*(m+1)+f*n*q*(b*c-a*d)+b*e*d*n*(p+q+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && RationalQ[q] && q>0 && Not[q==1 && SimplerQ[e+f*x^n,c+d*x^n]] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(e_+f_.*x_^n_)/(c_+d_.*x_^n_),x_Symbol] := + Int[ExpandIntegrand[(g*x)^m*(a+b*x^n)^p*(e+f*x^n)/(c+d*x^n),x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_*(e_+f_.*x_^n_),x_Symbol] := + e*Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q,x] + + f*(g*x)^m/x^m*Int[x^(m+n)*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q},x] + + +Int[x_^m_.*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + Int[x^(m-n*q)*(a+b*x^n)^p*(d+c*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,r},x] && EqQ[mn,-n] && IntegerQ[q] + + +Int[x_^m_.*(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + Int[x^(m+n*(p+r))*(b+a*x^(-n))^p*(c+d*x^(-n))^q*(f+e*x^(-n))^r,x] /; +FreeQ[{a,b,c,d,e,f,m,n,q},x] && EqQ[mn,-n] && IntegerQ[p] && IntegerQ[r] + + +Int[x_^m_.*(a_.+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + x^(n*FracPart[q])*(c+d*x^(-n))^FracPart[q]/(d+c*x^n)^FracPart[q]*Int[x^(m-n*q)*(a+b*x^n)^p*(d+c*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && EqQ[mn,-n] && Not[IntegerQ[q]] + + +Int[(g_*x_)^m_*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^mn_.)^q_.*(e_+f_.*x_^n_.)^r_.,x_Symbol] := + g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n)^p*(c+d*x^(-n))^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q,r},x] && EqQ[mn,-n] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e_+f_.*x_^n_)^r_.,x_Symbol] := + Defer[Int][(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q,r},x] + + +Int[u_^m_.*(a_.+b_.*v_^n_)^p_.*(c_.+d_.*v_^n_)^q_.*(e_+f_.*v_^n_)^r_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^r,x],x,v] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && LinearPairQ[u,v,x] + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e1_+f1_.*x_^n2_.)^r_.*(e2_+f2_.*x_^n2_.)^r_.,x_Symbol] := + Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e1*e2+f1*f2*x^n)^r,x] /; +FreeQ[{a,b,c,d,e1,f1,e2,f2,g,m,n,p,q,r},x] && EqQ[n2-n/2] && EqQ[e2*f1+e1*f2] && (IntegerQ[r] || PositiveQ[e1] && PositiveQ[e2]) + + +Int[(g_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.*(e1_+f1_.*x_^n2_.)^r_.*(e2_+f2_.*x_^n2_.)^r_.,x_Symbol] := + (e1+f1*x^(n/2))^FracPart[r]*(e2+f2*x^(n/2))^FracPart[r]/(e1*e2+f1*f2*x^n)^FracPart[r]* + Int[(g*x)^m*(a+b*x^n)^p*(c+d*x^n)^q*(e1*e2+f1*f2*x^n)^r,x] /; +FreeQ[{a,b,c,d,e1,f1,e2,f2,g,m,n,p,q,r},x] && EqQ[n2-n/2] && EqQ[e2*f1+e1*f2] + + + +`) + +func resourcesRubi113GeneralBinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi113GeneralBinomialProductsM, nil +} + +func resourcesRubi113GeneralBinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi113GeneralBinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.1.3 General binomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi114ImproperBinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.3.4 Normalizing algebraic functions*) + + +(* ::Subsection::Closed:: *) +(*1.3.4 Normalizing algebraic functions*) + + +Int[(a_.+b_.*(c_.*x_^n_)^q_)^p_.,x_Symbol] := + x/(c*x^n)^(1/n)*Subst[Int[(a+b*x^(n*q))^p,x],x,(c*x^n)^(1/n)] /; +FreeQ[{a,b,c,q,n,p},x] && IntegerQ[n*q] + + +Int[x_^m_.*(a_.+b_.*(c_.*x_^n_)^q_)^p_.,x_Symbol] := + x^(m+1)/(c*x^n)^((m+1)/n)*Subst[Int[x^m*(a+b*x^(n*q))^p,x],x,(c*x^n)^(1/n)] /; +FreeQ[{a,b,c,m,n,p,q},x] && IntegerQ[n*q] && IntegerQ[m] + + +Int[x_^m_.*(e_.*(a_+b_.*x_^n_.)^r_.)^p_*(f_.*(c_+d_.*x_^n_.)^s_)^q_,x_Symbol] := + (e*(a+b*x^n)^r)^p*(f*(c+d*x^n)^s)^q/((a+b*x^n)^(p*r)*(c+d*x^n)^(q*s))* + Int[x^m*(a+b*x^n)^(p*r)*(c+d*x^n)^(q*s),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r,s},x] + + +Int[u_.*(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + (b*e/d)^p*Int[u,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[b*c-a*d] + + +Int[u_.*(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + Int[u*(e*(a+b*x^n))^p/(c+d*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && PositiveQ[b*d*e] && PositiveQ[c-a*d/b] + + +Int[(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + With[{q=Denominator[p]}, + q*e*(b*c-a*d)/n*Subst[ + Int[x^(q*(p+1)-1)*(-a*e+c*x^q)^(1/n-1)/(b*e-d*x^q)^(1/n+1),x],x,(e*(a+b*x^n)/(c+d*x^n))^(1/q)]] /; +FreeQ[{a,b,c,d,e},x] && FractionQ[p] && IntegerQ[1/n] + + +Int[x_^m_.*(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + With[{q=Denominator[p]}, + q*e*(b*c-a*d)/n*Subst[ + Int[x^(q*(p+1)-1)*(-a*e+c*x^q)^(Simplify[(m+1)/n]-1)/(b*e-d*x^q)^(Simplify[(m+1)/n]+1),x],x,(e*(a+b*x^n)/(c+d*x^n))^(1/q)]] /; +FreeQ[{a,b,c,d,e,m,n},x] && FractionQ[p] && IntegerQ[Simplify[(m+1)/n]] + + +Int[u_^r_.*(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + With[{q=Denominator[p]}, + q*e*(b*c-a*d)/n*Subst[Int[SimplifyIntegrand[x^(q*(p+1)-1)*(-a*e+c*x^q)^(1/n-1)/(b*e-d*x^q)^(1/n+1)* + ReplaceAll[u,x->(-a*e+c*x^q)^(1/n)/(b*e-d*x^q)^(1/n)]^r,x],x],x,(e*(a+b*x^n)/(c+d*x^n))^(1/q)]] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[u,x] && FractionQ[p] && IntegerQ[1/n] && IntegerQ[r] + + +Int[x_^m_.*u_^r_.*(e_.*(a_.+b_.*x_^n_.)/(c_+d_.*x_^n_.))^p_,x_Symbol] := + With[{q=Denominator[p]}, + q*e*(b*c-a*d)/n*Subst[Int[SimplifyIntegrand[x^(q*(p+1)-1)*(-a*e+c*x^q)^((m+1)/n-1)/(b*e-d*x^q)^((m+1)/n+1)* + ReplaceAll[u,x->(-a*e+c*x^q)^(1/n)/(b*e-d*x^q)^(1/n)]^r,x],x],x,(e*(a+b*x^n)/(c+d*x^n))^(1/q)]] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[u,x] && FractionQ[p] && IntegerQ[1/n] && IntegersQ[m,r] + + +Int[(a_.+b_.*(c_./x_)^n_)^p_,x_Symbol] := + -c*Subst[Int[(a+b*x^n)^p/x^2,x],x,c/x] /; +FreeQ[{a,b,c,n,p},x] + + +Int[x_^m_.*(a_.+b_.*(c_./x_)^n_)^p_.,x_Symbol] := + -c^(m+1)*Subst[Int[(a+b*x^n)^p/x^(m+2),x],x,c/x] /; +FreeQ[{a,b,c,n,p},x] && IntegerQ[m] + + +Int[(d_.*x_)^m_*(a_.+b_.*(c_./x_)^n_)^p_.,x_Symbol] := + -c*(d*x)^m*(c/x)^m*Subst[Int[(a+b*x^n)^p/x^(m+2),x],x,c/x] /; +FreeQ[{a,b,c,d,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(a_.+b_.*(d_./x_)^n_+c_.*(d_./x_)^n2_.)^p_.,x_Symbol] := + -d*Subst[Int[(a+b*x^n+c*x^(2*n))^p/x^2,x],x,d/x] /; +FreeQ[{a,b,c,d,n,p},x] && EqQ[n2-2*n] + + +Int[x_^m_.*(a_+b_.*(d_./x_)^n_+c_.*(d_./x_)^n2_.)^p_.,x_Symbol] := + -d^(m+1)*Subst[Int[(a+b*x^n+c*x^(2*n))^p/x^(m+2),x],x,d/x] /; +FreeQ[{a,b,c,d,n,p},x] && EqQ[n2-2*n] && IntegerQ[m] + + +Int[(e_.*x_)^m_*(a_+b_.*(d_./x_)^n_+c_.*(d_./x_)^n2_.)^p_.,x_Symbol] := + -d*(e*x)^m*(d/x)^m*Subst[Int[(a+b*x^n+c*x^(2*n))^p/x^(m+2),x],x,d/x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && Not[IntegerQ[m]] + + +Int[(a_.+b_.*(d_./x_)^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + -d*Subst[Int[(a+b*x^n+c/d^(2*n)*x^(2*n))^p/x^2,x],x,d/x] /; +FreeQ[{a,b,c,d,n,p},x] && EqQ[n2+2*n] && IntegerQ[2*n] + + +Int[x_^m_.*(a_+b_.*(d_./x_)^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + -d^(m+1)*Subst[Int[(a+b*x^n+c/d^(2*n)*x^(2*n))^p/x^(m+2),x],x,d/x] /; +FreeQ[{a,b,c,d,n,p},x] && EqQ[n2+2*n] && IntegerQ[2*n] && IntegerQ[m] + + +Int[(e_.*x_)^m_*(a_+b_.*(d_./x_)^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + -d*(e*x)^m*(d/x)^m*Subst[Int[(a+b*x^n+c/d^(2*n)*x^(2*n))^p/x^(m+2),x],x,d/x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[n2+2*n] && Not[IntegerQ[m]] && IntegerQ[2*n] + + +Int[u_^m_,x_Symbol] := + Int[ExpandToSum[u,x]^m,x] /; +FreeQ[m,x] && LinearQ[u,x] && Not[LinearMatchQ[u,x]] + + +Int[u_^m_.*v_^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^n,x] /; +FreeQ[{m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*v_^n_.*w_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^n*ExpandToSum[w,x]^p,x] /; +FreeQ[{m,n,p},x] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]] + + +Int[u_^m_.*v_^n_.*w_^p_.*z_^q_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^n*ExpandToSum[w,x]^p*ExpandToSum[z,x]^q,x] /; +FreeQ[{m,n,p,q},x] && LinearQ[{u,v,w,z},x] && Not[LinearMatchQ[{u,v,w,z},x]] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(c_.*x_)^m_.*u_^p_.,x_Symbol] := + Int[(c*x)^m*ExpandToSum[u,x]^p,x] /; +FreeQ[{c,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[u_^p_.*v_^q_.,x_Symbol] := + Int[ExpandToSum[u,x]^p*ExpandToSum[v,x]^q,x] /; +FreeQ[{p,q},x] && BinomialQ[{u,v},x] && EqQ[BinomialDegree[u,x]-BinomialDegree[v,x]] && Not[BinomialMatchQ[{u,v},x]] + + +Int[(e_.*x_)^m_.*u_^p_.*v_^q_.,x_Symbol] := + Int[(e*x)^m*ExpandToSum[u,x]^p*ExpandToSum[v,x]^q,x] /; +FreeQ[{e,m,p,q},x] && BinomialQ[{u,v},x] && EqQ[BinomialDegree[u,x]-BinomialDegree[v,x]] && Not[BinomialMatchQ[{u,v},x]] + + +Int[u_^m_.*v_^p_.*w_^q_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^p*ExpandToSum[w,x]^q,x] /; +FreeQ[{m,p,q},x] && BinomialQ[{u,v,w},x] && EqQ[BinomialDegree[u,x]-BinomialDegree[v,x]] && + EqQ[BinomialDegree[u,x]-BinomialDegree[w,x]] && Not[BinomialMatchQ[{u,v,w},x]] + + +Int[(g_.*x_)^m_.*u_^p_.*v_^q_.*z_^r_.,x_Symbol] := + Int[(g*x)^m*ExpandToSum[u,x]^p*ExpandToSum[v,x]^q*ExpandToSum[z,x]^r,x] /; +FreeQ[{g,m,p,q,r},x] && BinomialQ[{u,v,z},x] && EqQ[BinomialDegree[u,x]-BinomialDegree[v,x]] && + EqQ[BinomialDegree[u,x]-BinomialDegree[z,x]] && Not[BinomialMatchQ[{u,v,z},x]] + + +Int[(c_.*x_)^m_.*Pq_*u_^p_.,x_Symbol] := + Int[(c*x)^m*Pq*ExpandToSum[u,x]^p,x] /; +FreeQ[{c,m,p},x] && PolyQ[Pq,x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && GeneralizedBinomialQ[u,x] && Not[GeneralizedBinomialMatchQ[u,x]] + + +Int[(c_.*x_)^m_.*u_^p_.,x_Symbol] := + Int[(c*x)^m*ExpandToSum[u,x]^p,x] /; +FreeQ[{c,m,p},x] && GeneralizedBinomialQ[u,x] && Not[GeneralizedBinomialMatchQ[u,x]] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && QuadraticQ[u,x] && Not[QuadraticMatchQ[u,x]] + + +Int[u_^m_.*v_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^p,x] /; +FreeQ[{m,p},x] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + +Int[u_^m_.*v_^n_.*w_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^n*ExpandToSum[w,x]^p,x] /; +FreeQ[{m,n,p},x] && LinearQ[{u,v},x] && QuadraticQ[w,x] && Not[LinearMatchQ[{u,v},x] && QuadraticMatchQ[w,x]] + + +Int[u_^p_.*v_^q_.,x_Symbol] := + Int[ExpandToSum[u,x]^p*ExpandToSum[v,x]^q,x] /; +FreeQ[{p,q},x] && QuadraticQ[{u,v},x] && Not[QuadraticMatchQ[{u,v},x]] + + +Int[z_^m_.*u_^p_.*v_^q_.,x_Symbol] := + Int[ExpandToSum[z,x]^m*ExpandToSum[u,x]^p*ExpandToSum[v,x]^q,x] /; +FreeQ[{m,p,q},x] && LinearQ[z,x] && QuadraticQ[{u,v},x] && Not[LinearMatchQ[z,x] && QuadraticMatchQ[{u,v},x]] + + +Int[Pq_*u_^p_.,x_Symbol] := + Int[Pq*ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && PolyQ[Pq,x] && QuadraticQ[u,x] && Not[QuadraticMatchQ[u,x]] + + +Int[u_^m_.*Pq_*v_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Pq*ExpandToSum[u,x]^p,x] /; +FreeQ[{m,p},x] && PolyQ[Pq,x] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && TrinomialQ[u,x] && Not[TrinomialMatchQ[u,x]] + + +Int[(d_.*x_)^m_.*u_^p_.,x_Symbol] := + Int[(d*x)^m*ExpandToSum[u,x]^p,x] /; +FreeQ[{d,m,p},x] && TrinomialQ[u,x] && Not[TrinomialMatchQ[u,x]] + + +Int[u_^q_.*v_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^q*ExpandToSum[v,x]^p,x] /; +FreeQ[{p,q},x] && BinomialQ[u,x] && TrinomialQ[v,x] && Not[BinomialMatchQ[u,x] && TrinomialMatchQ[v,x]] + + +Int[u_^q_.*v_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^q*ExpandToSum[v,x]^p,x] /; +FreeQ[{p,q},x] && BinomialQ[u,x] && BinomialQ[v,x] && Not[BinomialMatchQ[u,x] && BinomialMatchQ[v,x]] + + +Int[(f_.*x_)^m_.*z_^q_.*u_^p_.,x_Symbol] := + Int[(f*x)^m*ExpandToSum[z,x]^q*ExpandToSum[u,x]^p,x] /; +FreeQ[{f,m,p,q},x] && BinomialQ[z,x] && TrinomialQ[u,x] && Not[BinomialMatchQ[z,x] && TrinomialMatchQ[u,x]] + + +Int[(f_.*x_)^m_.*z_^q_.*u_^p_.,x_Symbol] := + Int[(f*x)^m*ExpandToSum[z,x]^q*ExpandToSum[u,x]^p,x] /; +FreeQ[{f,m,p,q},x] && BinomialQ[z,x] && BinomialQ[u,x] && Not[BinomialMatchQ[z,x] && BinomialMatchQ[u,x]] + + +Int[Pq_*u_^p_.,x_Symbol] := + Int[Pq*ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && PolyQ[Pq,x] && TrinomialQ[u,x] && Not[TrinomialMatchQ[u,x]] + + +Int[(d_.*x_)^m_.*Pq_*u_^p_.,x_Symbol] := + Int[(d*x)^m*Pq*ExpandToSum[u,x]^p,x] /; +FreeQ[{d,m,p},x] && PolyQ[Pq,x] && TrinomialQ[u,x] && Not[TrinomialMatchQ[u,x]] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && GeneralizedTrinomialQ[u,x] && Not[GeneralizedTrinomialMatchQ[u,x]] + + +Int[(d_.*x_)^m_.*u_^p_.,x_Symbol] := + Int[(d*x)^m*ExpandToSum[u,x]^p,x] /; +FreeQ[{d,m,p},x] && GeneralizedTrinomialQ[u,x] && Not[GeneralizedTrinomialMatchQ[u,x]] + + +Int[z_*u_^p_.,x_Symbol] := + Int[ExpandToSum[z,x]*ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && BinomialQ[z,x] && GeneralizedTrinomialQ[u,x] && + EqQ[BinomialDegree[z,x]-GeneralizedTrinomialDegree[u,x]] && Not[BinomialMatchQ[z,x] && GeneralizedTrinomialMatchQ[u,x]] + + +Int[(f_.*x_)^m_.*z_*u_^p_.,x_Symbol] := + Int[(f*x)^m*ExpandToSum[z,x]*ExpandToSum[u,x]^p,x] /; +FreeQ[{f,m,p},x] && BinomialQ[z,x] && GeneralizedTrinomialQ[u,x] && + EqQ[BinomialDegree[z,x]-GeneralizedTrinomialDegree[u,x]] && Not[BinomialMatchQ[z,x] && GeneralizedTrinomialMatchQ[u,x]] + + + + + +(* ::Section:: *) +(*1.1.4 Improper Binomial Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*1.1.4.1 (a x^j+b x^n)^p*) + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + (a*x^j+b*x^n)^(p+1)/(b*(n-j)(p+1)*x^(n-1)) /; +FreeQ[{a,b,j,n,p},x] && Not[IntegerQ[p]] && NeQ[n-j] && EqQ[j*p-n+j+1] + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + -(a*x^j+b*x^n)^(p+1)/(a*(n-j)*(p+1)*x^(j-1)) + + (n*p+n-j+1)/(a*(n-j)*(p+1))*Int[(a*x^j+b*x^n)^(p+1)/x^j,x] /; +FreeQ[{a,b,j,n},x] && Not[IntegerQ[p]] && NeQ[n-j] && NegativeIntegerQ[Simplify[(n*p+n-j+1)/(n-j)]] && RationalQ[p] && p<-1 + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + (a*x^j+b*x^n)^(p+1)/(a*(j*p+1)*x^(j-1)) - + b*(n*p+n-j+1)/(a*(j*p+1))*Int[x^(n-j)*(a*x^j+b*x^n)^p,x] /; +FreeQ[{a,b,j,n,p},x] && Not[IntegerQ[p]] && NeQ[n-j] && NegativeIntegerQ[Simplify[(n*p+n-j+1)/(n-j)]] && NeQ[j*p+1] + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + x*(a*x^j+b*x^n)^p/(j*p+1) - + b*(n-j)*p/(j*p+1)*Int[x^n*(a*x^j+b*x^n)^(p-1),x] /; +FreeQ[{a,b},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 00 && j*p+1<0 + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + x*(a*x^j+b*x^n)^p/(n*p+1) + + a*(n-j)*p/(n*p+1)*Int[x^j*(a*x^j+b*x^n)^(p-1),x] /; +FreeQ[{a,b},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 00 && NeQ[n*p+1] + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + (a*x^j+b*x^n)^(p+1)/(b*(n-j)*(p+1)*x^(n-1)) - + (j*p-n+j+1)/(b*(n-j)*(p+1))*Int[(a*x^j+b*x^n)^(p+1)/x^n,x] /; +FreeQ[{a,b},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 0n-j + + +Int[(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + -(a*x^j+b*x^n)^(p+1)/(a*(n-j)*(p+1)*x^(j-1)) + + (n*p+n-j+1)/(a*(n-j)*(p+1))*Int[(a*x^j+b*x^n)^(p+1)/x^j,x] /; +FreeQ[{a,b},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 00 && m+j*p+1<0 + + +Int[(c_.*x_)^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + (c*x)^(m+1)*(a*x^j+b*x^n)^p/(c*(m+n*p+1)) + + a*(n-j)*p/(c^j*(m+n*p+1))*Int[(c*x)^(m+j)*(a*x^j+b*x^n)^(p-1),x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 00 && NeQ[m+n*p+1] + + +Int[(c_.*x_)^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + c^(n-1)*(c*x)^(m-n+1)*(a*x^j+b*x^n)^(p+1)/(b*(n-j)*(p+1)) - + c^n*(m+j*p-n+j+1)/(b*(n-j)*(p+1))*Int[(c*x)^(m-n)*(a*x^j+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c},x] && Not[IntegerQ[p]] && RationalQ[j,m,n,p] && 0n-j + + +Int[(c_.*x_)^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol] := + -c^(j-1)*(c*x)^(m-j+1)*(a*x^j+b*x^n)^(p+1)/(a*(n-j)*(p+1)) + + c^j*(m+n*p+n-j+1)/(a*(n-j)*(p+1))*Int[(c*x)^(m-j)*(a*x^j+b*x^n)^(p+1),x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[p]] && RationalQ[j,n,p] && 00 && + (m+j*p<-1 || IntegersQ[m-1/2,p-1/2] && p<0 && m<-n*p-1) && + (PositiveQ[e] || IntegersQ[j,n]) && NeQ[m+j*p+1] && NeQ[m-n+j*p+1] + + +Int[(e_.*x_)^m_.*(a_.*x_^j_.+b_.*x_^jn_.)^p_*(c_+d_.*x_^n_.),x_Symbol] := + d*e^(j-1)*(e*x)^(m-j+1)*(a*x^j+b*x^(j+n))^(p+1)/(b*(m+n+p*(j+n)+1)) - + (a*d*(m+j*p+1)-b*c*(m+n+p*(j+n)+1))/(b*(m+n+p*(j+n)+1))*Int[(e*x)^m*(a*x^j+b*x^(j+n))^p,x] /; +FreeQ[{a,b,c,d,e,j,m,n,p},x] && EqQ[jn-j-n] && Not[IntegerQ[p]] && NeQ[b*c-a*d] && + NeQ[m+n+p*(j+n)+1] && (PositiveQ[e] || IntegerQ[j]) + + +Int[x_^m_.*(a_.*x_^j_+b_.*x_^k_.)^p_*(c_+d_.*x_^n_.)^q_.,x_Symbol] := + 1/(m+1)*Subst[Int[(a*x^Simplify[j/(m+1)]+b*x^Simplify[k/(m+1)])^p*(c+d*x^Simplify[n/(m+1)])^q,x],x,x^(m+1)] /; +FreeQ[{a,b,c,d,j,k,m,n,p,q},x] && Not[IntegerQ[p]] && NeQ[k-j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && + NeQ[m+1] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(e_*x_)^m_.*(a_.*x_^j_+b_.*x_^k_.)^p_*(c_+d_.*x_^n_.)^q_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a*x^j+b*x^k)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,j,k,m,n,p,q},x] && Not[IntegerQ[p]] && NeQ[k-j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && + NeQ[m+1] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(e_.*x_)^m_.*(a_.*x_^j_.+b_.*x_^jn_.)^p_*(c_+d_.*x_^n_.)^q_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j+b*x^(j+n))^FracPart[p]/ + (x^(FracPart[m]+j*FracPart[p])*(a+b*x^n)^FracPart[p])* + Int[x^(m+j*p)*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,e,j,m,n,p,q},x] && EqQ[jn-j-n] && Not[IntegerQ[p]] && NeQ[b*c-a*d] && Not[EqQ[n-1] && EqQ[j-1]] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.4.4 (c x)^m Pq(x) (a x^j+b x^n)^p*) + + +Int[Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + With[{d=Denominator[n]}, + d*Subst[Int[x^(d-1)*ReplaceAll[SubstFor[x^n,Pq,x],x->x^(d*n)]*(a*x^(d*j)+b*x^(d*n))^p,x],x,x^(1/d)]] /; +FreeQ[{a,b,j,n,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && NeQ[n-j] && RationalQ[j,n] && IntegerQ[j/n] && -11 + + +Int[(c_*x_)^m_.*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^m/x^m*Int[x^m*Pq*(a*x^j+b*x^n)^p,x] /; +FreeQ[{a,b,c,j,m,n,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && NeQ[n-j] && IntegerQ[Simplify[j/n]] && + IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + With[{g=GCD[m+1,n]}, + 1/g*Subst[Int[x^((m+1)/g-1)*ReplaceAll[Pq,x->x^(1/g)]*(a*x^(j/g)+b*x^(n/g))^p,x],x,x^g] /; + g!=1] /; +FreeQ[{a,b,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && PositiveIntegerQ[j,n,j/n] && IntegerQ[m] + + +Int[(c_.*x_)^m_.*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + With[{Pqq=Coeff[Pq,x,q]}, + Pqq*(c*x)^(m+q-n+1)*(a*x^j+b*x^n)^(p+1)/(b*c^(q-n+1)*(m+q+n*p+1)) + + Int[(c*x)^m*ExpandToSum[Pq-Pqq*x^q-a*Pqq*(m+q-n+1)*x^(q-n)/(b*(m+q+n*p+1)),x]*(a*x^j+b*x^n)^p,x]] /; + q>n-1 && m+q+n*p+1!=0 && (IntegerQ[2*p] || IntegerQ[p+(q+1)/(2*n)])] /; +FreeQ[{a,b,c,m,p},x] && PolyQ[Pq,x] && Not[IntegerQ[p]] && PositiveIntegerQ[j,n] && jx^Simplify[n/(m+1)]]*(a*x^Simplify[j/(m+1)]+b*x^Simplify[n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,j,m,n,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && NeQ[n-j] && IntegerQ[Simplify[j/n]] && + IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(c_*x_)^m_*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + c^(Sign[m]*Quotient[m,Sign[m]])*(c*x)^Mod[m,Sign[m]]/x^Mod[m,Sign[m]]*Int[x^m*Pq*(a*x^j+b*x^n)^p,x] /; +FreeQ[{a,b,c,j,n,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && NeQ[n-j] && IntegerQ[Simplify[j/n]] && + IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] && RationalQ[m] && m^2>1 + + +Int[(c_*x_)^m_*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + (c*x)^m/x^m*Int[x^m*Pq*(a*x^j+b*x^n)^p,x] /; +FreeQ[{a,b,c,j,m,n,p},x] && PolyQ[Pq,x^n] && Not[IntegerQ[p]] && NeQ[n-j] && IntegerQ[Simplify[j/n]] && + IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(c_.*x_)^m_.*Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,j,m,n,p},x] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) && Not[IntegerQ[p]] && NeQ[n-j] + + +Int[Pq_*(a_.*x_^j_.+b_.*x_^n_)^p_,x_Symbol] := + Int[ExpandIntegrand[Pq*(a*x^j+b*x^n)^p,x],x] /; +FreeQ[{a,b,j,n,p},x] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) && Not[IntegerQ[p]] && NeQ[n-j] + + + +`) + +func resourcesRubi114ImproperBinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi114ImproperBinomialProductsM, nil +} + +func resourcesRubi114ImproperBinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi114ImproperBinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.1.4 Improper binomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi121QuadraticTrinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Quadratic Product Rules*) + + +(* ::Subsection::Closed:: *) +(*1.2.1.1 (a+b x+c x^2)^p*) + + +Int[1/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] := + (b/2+c*x)/Sqrt[a+b*x+c*x^2]*Int[1/(b/2+c*x),x] /; +FreeQ[{a,b,c},x] && EqQ[b^2-4*a*c,0] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (b+2*c*x)*(a+b*x+c*x^2)^p/(2*c*(2*p+1)) /; +FreeQ[{a,b,c,p},x] && EqQ[b^2-4*a*c,0] && NeQ[p,-1/2] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 1/c^p*Int[Simp[b/2-q/2+c*x,x]^p*Simp[b/2+q/2+c*x,x]^p,x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && IGtQ[p,0] && PerfectSquareQ[b^2-4*a*c] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[ExpandIntegrand[(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && IGtQ[p,0] && Not[PerfectSquareQ[b^2-4*a*c]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (b+2*c*x)*(a+b*x+c*x^2)^p/(2*c*(2*p+1)) - + p*(b^2-4*a*c)/(2*c*(2*p+1))*Int[(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && GtQ[p,0] && IntegerQ[4*p] + + +Int[1/(a_.+b_.*x_+c_.*x_^2)^(3/2),x_Symbol] := + -2*(b+2*c*x)/((b^2-4*a*c)*Sqrt[a+b*x+c*x^2]) /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (b+2*c*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) - + 2*c*(2*p+3)/((p+1)*(b^2-4*a*c))*Int[(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] && NeQ[p,-3/2] && IntegerQ[4*p] + + +Int[1/(a_+b_.*x_+c_.*x_^2),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + c/q*Int[1/Simp[b/2-q/2+c*x,x],x] - c/q*Int[1/Simp[b/2+q/2+c*x,x],x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && PosQ[b^2-4*a*c] && PerfectSquareQ[b^2-4*a*c] + + +Int[1/(a_+b_.*x_+c_.*x_^2),x_Symbol] := + With[{q=1-4*Simplify[a*c/b^2]}, + -2/b*Subst[Int[1/(q-x^2),x],x,1+2*c*x/b] /; + RationalQ[q] && (EqQ[q^2,1] || Not[RationalQ[b^2-4*a*c]])] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] + + +Int[1/(a_+b_.*x_+c_.*x_^2),x_Symbol] := + -2*Subst[Int[1/Simp[b^2-4*a*c-x^2,x],x],x,b+2*c*x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + 1/(2*c*(-4*c/(b^2-4*a*c))^p)*Subst[Int[Simp[1-x^2/(b^2-4*a*c),x]^p,x],x,b+2*c*x] /; +FreeQ[{a,b,c,p},x] && PositiveQ[4*a-b^2/c] + + +Int[1/Sqrt[b_.*x_+c_.*x_^2],x_Symbol] := + 2*Subst[Int[1/(1-c*x^2),x],x,x/Sqrt[b*x+c*x^2]] /; +FreeQ[{b,c},x] + + +Int[1/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] := + 2*Subst[Int[1/(4*c-x^2),x],x,(b+2*c*x)/Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] + + +Int[(b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (b*x+c*x^2)^p/(-c*(b*x+c*x^2)/(b^2))^p*Int[(-c*x/b-c^2*x^2/b^2)^p,x] /; +FreeQ[{b,c},x] && RationalQ[p] && 3<=Denominator[p]<=4 + + +(* Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^p/(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^p*Int[(-a*c/(b^2-4*a*c)-b*c*x/(b^2-4*a*c)-c^2*x^2/(b^2-4*a*c))^p,x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && RationalQ[p] && 3<=Denominator[p]<=4 *) + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{d=Denominator[p]}, + d*Sqrt[(b+2*c*x)^2]/(b+2*c*x)*Subst[Int[x^(d*(p+1)-1)/Sqrt[b^2-4*a*c+4*c*x^d],x],x,(a+b*x+c*x^2)^(1/d)] /; + 3<=d<=4] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && RationalQ[p] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -(a+b*x+c*x^2)^(p+1)/(q*(p+1)*((q-b-2*c*x)/(2*q))^(p+1))*Hypergeometric2F1[-p,p+1,p+2,(b+q+2*c*x)/(2*q)]] /; +FreeQ[{a,b,c,p},x] && NeQ[b^2-4*a*c,0] && Not[IntegerQ[4*p]] + + +Int[(a_.+b_.*u_+c_.*u_^2)^p_,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x+c*x^2)^p,x],x,u] /; +FreeQ[{a,b,c,p},x] && LinearQ[u,x] && NeQ[u,x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.2 (d+e x)^m (a+b x+c x^2)^p*) + + +Int[(d_+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(p+1)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d^2-b*d*e+a*e^2,0] && IGtQ[p,0] + + +Int[(d_.+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e},x] && IGtQ[p,0] + + +Int[(d_.+e_.*x_)/(a_+b_.*x_+c_.*x_^2),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (c*d-e*(b/2-q/2))/q*Int[1/(b/2-q/2+c*x),x] - (c*d-e*(b/2+q/2))/q*Int[1/(b/2+q/2+c*x),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NiceSqrtQ[b^2-4*a*c] + + +Int[(d_+e_.*x_)/(a_+c_.*x_^2),x_Symbol] := + With[{q=Rt[-a*c,2]}, + (e/2+c*d/(2*q))*Int[1/(-q+c*x),x] + (e/2-c*d/(2*q))*Int[1/(q+c*x),x]] /; +FreeQ[{a,c,d,e},x] && NiceSqrtQ[-a*c] + + +Int[(d_.+e_.*x_)/(a_+b_.*x_+c_.*x_^2),x_Symbol] := +(* (d-b*e/(2*c))*Int[1/(a+b*x+c*x^2),x] + *) + (2*c*d-b*e)/(2*c)*Int[1/(a+b*x+c*x^2),x] + e/(2*c)*Int[(b+2*c*x)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && Not[NiceSqrtQ[b^2-4*a*c]] + + +Int[(d_+e_.*x_)/(a_+c_.*x_^2),x_Symbol] := + d*Int[1/(a+c*x^2),x] + e*Int[x/(a+c*x^2),x] /; +FreeQ[{a,c,d,e},x] && Not[NiceSqrtQ[-a*c]] + + +Int[(d_.+e_.*x_)/(a_.+b_.*x_+c_.*x_^2)^(3/2),x_Symbol] := + -2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*Sqrt[a+b*x+c*x^2]) /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] + + +Int[(d_+e_.*x_)/(a_+c_.*x_^2)^(3/2),x_Symbol] := + (-a*e+c*d*x)/(a*c*Sqrt[a+c*x^2]) /; +FreeQ[{a,c,d,e},x] + + +Int[(d_.+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (b*d-2*a*e+(2*c*d-b*e)*x)/((p+1)*(b^2-4*a*c))*(a+b*x+c*x^2)^(p+1) - + (2*p+3)*(2*c*d-b*e)/((p+1)*(b^2-4*a*c))*Int[(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] + + +Int[(d_+e_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (a*e-c*d*x)/(2*a*c*(p+1))*(a+c*x^2)^(p+1) + + d*(2*p+3)/(2*a*(p+1))*Int[(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && LtQ[p,-1] + + +Int[(d_.+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(a+b*x+c*x^2)^(p+1)/(2*c*(p+1)) + (2*c*d-b*e)/(2*c)*Int[(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p,-1] + + +Int[(d_.+e_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + e*(a+c*x^2)^(p+1)/(2*c*(p+1)) + d*Int[(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,p},x] && NeQ[p,-1] + + +Int[(d_+e_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^p/(d+e*x)^(2*p)*Int[(d+e*x)^(m+2*p),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2+c*x)^(2*FracPart[p]))*Int[(d+e*x)^m*(b/2+c*x)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[b^2-4*a*c,0] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && EqQ[c*d^2+a*e^2,0] && (IntegerQ[p] || PositiveQ[a] && PositiveQ[d] && IntegerQ[m+p]) + + +(* Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(a+b*x+c*x^2)^(m+p),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c*d^2-b*d*e+a*e^2,0] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] *) + + +(* Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(a+c*x^2)^(m+p),x] /; +FreeQ[{a,c,d,e,p},x] && EqQ[c*d^2+a*e^2,0] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] *) + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)/(c*(p+1)) /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && EqQ[m+p,0] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+c*x^2)^(p+1)/(c*(p+1)) /; +FreeQ[{a,c,d,e,m,p},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] && EqQ[m+p,0] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/((p+1)*(2*c*d-b*e)) /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && EqQ[m+2*p+2,0] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*c*d*(p+1)) /; +FreeQ[{a,c,d,e,m,p},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] && EqQ[m+2*p+2,0] + + +Int[(d_.+e_.*x_)^2*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)*(a+b*x+c*x^2)^(p+1)/(c*(p+1)) - e^2*(p+2)/(c*(p+1))*Int[(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && RationalQ[p] && p<-1 + + +Int[(d_+e_.*x_)^2*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)*(a+c*x^2)^(p+1)/(c*(p+1)) - e^2*(p+2)/(c*(p+1))*Int[(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,p},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] && RationalQ[p] && p<-1 + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[(a+b*x+c*x^2)^(m+p)/(a/d+c*x/e)^m,x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && IntegerQ[m] && + RationalQ[p] && (0<-m

0 && + (m<-2 || EqQ[m+2*p+1]) && NeQ[m+p+1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+c*x^2)^p/(e*(m+p+1)) - + c*p/(e^2*(m+p+1))*Int[(d+e*x)^(m+2)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m,p] && p>0 && + (m<-2 || EqQ[m+2*p+1]) && NeQ[m+p+1] && IntegerQ[2*p] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e*(m+2*p+1)) - + p*(2*c*d-b*e)/(e^2*(m+2*p+1))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m,p] && p>0 && + (-2<=m<0 || m+p+1==0) && NeQ[m+2*p+1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+c*x^2)^p/(e*(m+2*p+1)) - + 2*c*d*p/(e^2*(m+2*p+1))*Int[(d+e*x)^(m+1)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m,p] && p>0 && + (-2<=m<0 || m+p+1==0) && NeQ[m+2*p+1] && IntegerQ[2*p] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (2*c*d-b*e)*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(e*(p+1)*(b^2-4*a*c)) - + (2*c*d-b*e)*(m+2*p+2)/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m,p] && p<-1 && 01 && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+c*x^2)^(p+1)/(c*(p+1)) - + e^2*(m+p)/(c*(p+1))*Int[(d+e*x)^(m-2)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m,p] && p<-1 && m>1 && IntegerQ[2*p] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+1)) + + (m+p)*(2*c*d-b*e)/(c*(m+2*p+1))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m] && m>=1 && + NeQ[m+2*p+1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+c*x^2)^(p+1)/(c*(m+2*p+1)) + + 2*c*d*(m+p)/(c*(m+2*p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,p},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m] && m>=1 && NeQ[m+2*p+1] && IntegerQ[2*p] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -e*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/((m+p+1)*(2*c*d-b*e)) + + c*(m+2*p+2)/((m+p+1)*(2*c*d-b*e))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m] && m<0 && NeQ[m+p+1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + -e*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*c*d*(m+p+1)) + + (m+2*p+2)/(2*d*(m+p+1))*Int[(d+e*x)^(m+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,p},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m] && m<0 && NeQ[m+p+1] && IntegerQ[2*p] + + +Int[(e_.*x_)^m_*(b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (e*x)^m*(b*x+c*x^2)^p/(x^(m+p)*(b+c*x)^p)*Int[x^(m+p)*(b+c*x)^p,x] /; +FreeQ[{b,c,e,m},x] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] && PositiveQ[a] && PositiveQ[d] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (a+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,m},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] + + +Int[1/((d_+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + -4*b*c/(d*(b^2-4*a*c))*Int[1/(b+2*c*x),x] + + b^2/(d^2*(b^2-4*a*c))*Int[(d+e*x)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + 2*c*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(e*(p+1)*(b^2-4*a*c)) /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && EqQ[m+2*p+3] && NeQ[p+1] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && PositiveIntegerQ[p] && Not[EqQ[m-3] && p!=1] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e*(m+1)) - + b*p/(d*e*(m+1))*Int[(d+e*x)^(m+2)*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[m,p] && p>0 && m<-1 && + Not[EvenQ[m] && m+2*p+3<0] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e*(m+2*p+1)) - + d*p*(b^2-4*a*c)/(b*e*(m+2*p+1))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[p] && p>0 && + Not[RationalQ[m] && m<-1] && Not[PositiveIntegerQ[(m-1)/2] && (Not[IntegerQ[p]] || m<2*p)] && RationalQ[m] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + d*(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)/(b*(p+1)) - + d*e*(m-1)/(b*(p+1))*Int[(d+e*x)^(m-2)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[m,p] && p<-1 && m>1 && + IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + 2*c*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(e*(p+1)*(b^2-4*a*c)) - + 2*c*e*(m+2*p+3)/(e*(p+1)*(b^2-4*a*c))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[p] && p<-1 && + Not[RationalQ[m] && m>1] && RationalQ[m] && IntegerQ[2*p] + + +Int[1/((d_+e_.*x_)*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + 4*c*Subst[Int[1/(b^2*e-4*a*c*e+4*c*e*x^2),x],x,Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] + + +Int[1/(Sqrt[d_+e_.*x_]*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + 4/e*Sqrt[-c/(b^2-4*a*c)]*Subst[Int[1/Sqrt[Simp[1-b^2*x^4/(d^2*(b^2-4*a*c)),x]],x],x,Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NegativeQ[c/(b^2-4*a*c)] + + +Int[Sqrt[d_+e_.*x_]/Sqrt[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + 4/e*Sqrt[-c/(b^2-4*a*c)]*Subst[Int[x^2/Sqrt[Simp[1-b^2*x^4/(d^2*(b^2-4*a*c)),x]],x],x,Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NegativeQ[c/(b^2-4*a*c)] + + +Int[(d_+e_.*x_)^m_/Sqrt[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + Sqrt[-c*(a+b*x+c*x^2)/(b^2-4*a*c)]/Sqrt[a+b*x+c*x^2]* + Int[(d+e*x)^m/Sqrt[-a*c/(b^2-4*a*c)-b*c*x/(b^2-4*a*c)-c^2*x^2/(b^2-4*a*c)],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && EqQ[m^2,1/4] + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + 2*d*(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)/(b*(m+2*p+1)) + + d^2*(m-1)*(b^2-4*a*c)/(b^2*(m+2*p+1))*Int[(d+e*x)^(m-2)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[m] && m>1 && + NeQ[m+2*p+1] && (IntegerQ[2*p] || IntegerQ[m] && RationalQ[p] || OddQ[m]) + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -2*b*d*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(d^2*(m+1)*(b^2-4*a*c)) + + b^2*(m+2*p+3)/(d^2*(m+1)*(b^2-4*a*c))*Int[(d+e*x)^(m+2)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] && NeQ[m+2*p+3] && RationalQ[m] && m<-1 && + (IntegerQ[2*p] || IntegerQ[m] && RationalQ[p] || IntegerQ[(m+2*p+3)/2]) + + +Int[(d_+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + 1/e*Subst[Int[x^m*(a-b^2/(4*c)+(c*x^2)/e^2)^p,x],x,d+e*x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + PositiveIntegerQ[p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[p] && Not[EqQ[m-1] && p>1] + + +(* Int[Sqrt[d_.+e_.*x_]/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + With[{q=Rt[(c*d^2-b*d*e+a*e^2)/c,2]}, + 1/2*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x] + + 1/2*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + NegativeQ[b^2-4*a*c] *) + + +(* Int[Sqrt[d_+e_.*x_]/(a_+c_.*x_^2),x_Symbol] := + With[{q=Rt[(c*d^2+a*e^2)/c,2]}, + 1/2*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x] + + 1/2*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && NegativeQ[-a*c] *) + + +(* Int[Sqrt[d_.+e_.*x_]/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*d-b*e+e*q)/q*Int[1/(Sqrt[d+e*x]*(b-q+2*c*x)),x] - + (2*c*d-b*e-e*q)/q*Int[1/(Sqrt[d+e*x]*(b+q+2*c*x)),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] (* && + Not[NegativeQ[b^2-4*a*c]] *) *) + + +(* Int[Sqrt[d_+e_.*x_]/(a_+c_.*x_^2),x_Symbol] := + With[{q=Rt[-a*c,2]}, + (c*d+e*q)/(2*q)*Int[1/(Sqrt[d+e*x]*(-q+c*x)),x] - + (c*d-e*q)/(2*q)*Int[1/(Sqrt[d+e*x]*(+q+c*x)),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] (* && Not[NegativeQ[-a*c]] *) *) + + +Int[Sqrt[d_.+e_.*x_]/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + 2*e*Subst[Int[x^2/(c*d^2-b*d*e+a*e^2-(2*c*d-b*e)*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] + + +Int[Sqrt[d_+e_.*x_]/(a_+c_.*x_^2),x_Symbol] := + 2*e*Subst[Int[x^2/(c*d^2+a*e^2-2*c*d*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[(d_.+e_.*x_)^m_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[PolynomialDivide[(d+e*x)^m,a+b*x+c*x^2,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + IntegerQ[m] && m>1 && (NeQ[d] || m>2) + + +Int[(d_+e_.*x_)^m_/(a_+c_.*x_^2),x_Symbol] := + Int[PolynomialDivide[(d+e*x)^m,a+c*x^2,x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && IntegerQ[m] && m>1 && (NeQ[d] || m>2) + + +Int[(d_.+e_.*x_)^m_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m-1)/(c*(m-1)) + + 1/c*Int[(d+e*x)^(m-2)*Simp[c*d^2-a*e^2+e*(2*c*d-b*e)*x,x]/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && RationalQ[m] && m>1 + + +Int[(d_+e_.*x_)^m_/(a_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m-1)/(c*(m-1)) + + 1/c*Int[(d+e*x)^(m-2)*Simp[c*d^2-a*e^2+2*c*d*e*x,x]/(a+c*x^2),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[m] && m>1 + + +Int[1/((d_.+e_.*x_)*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + e^2/(c*d^2-b*d*e+a*e^2)*Int[1/(d+e*x),x] + + 1/(c*d^2-b*d*e+a*e^2)*Int[(c*d-b*e-c*e*x)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] + + +Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)),x_Symbol] := + e^2/(c*d^2+a*e^2)*Int[1/(d+e*x),x] + + 1/(c*d^2+a*e^2)*Int[(c*d-c*e*x)/(a+c*x^2),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +(* Int[1/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + With[{q=Rt[(c*d^2-b*d*e+a*e^2)/c,2]}, + 1/(2*q)*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x] - + 1/(2*q)*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + NegativeQ[b^2-4*a*c] *) + + +(* Int[1/(Sqrt[d_+e_.*x_]*(a_+c_.*x_^2)),x_Symbol] := + With[{q=Rt[(c*d^2+a*e^2)/c,2]}, + 1/(2*q)*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x] - + 1/(2*q)*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && NegativeQ[-a*c] *) + + +(* Int[1/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/(Sqrt[d+e*x]*(b-q+2*c*x)),x] - + 2*c/q*Int[1/(Sqrt[d+e*x]*(b+q+2*c*x)),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] (* && + Not[NegativeQ[b^2-4*a*c]] *) *) + + +(* Int[1/(Sqrt[d_+e_.*x_]*(a_+c_.*x_^2)),x_Symbol] := + With[{q=Rt[-a*c,2]}, + c/(2*q)*Int[1/(Sqrt[d+e*x]*(-q+c*x)),x] - + c/(2*q)*Int[1/(Sqrt[d+e*x]*(q+c*x)),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] (* && Not[NegativeQ[-a*c]] *) *) + + +Int[1/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + 2*e*Subst[Int[1/(c*d^2-b*d*e+a*e^2-(2*c*d-b*e)*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] + + +Int[1/(Sqrt[d_+e_.*x_]*(a_+c_.*x_^2)),x_Symbol] := + 2*e*Subst[Int[1/(c*d^2+a*e^2-2*c*d*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[(d_.+e_.*x_)^m_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x)^(m+1)*Simp[c*d-b*e-c*e*x,x]/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && RationalQ[m] && m<-1 + + +Int[(d_+e_.*x_)^m_/(a_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m+1)/((m+1)*(c*d^2+a*e^2)) + + c/(c*d^2+a*e^2)*Int[(d+e*x)^(m+1)*(d-e*x)/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && RationalQ[m] && m<-1 + + +Int[(d_.+e_.*x_)^m_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m,1/(a+b*x+c*x^2),x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && Not[IntegerQ[m]] + + +Int[(d_+e_.*x_)^m_/(a_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m,1/(a+c*x^2),x],x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[m]] + + +Int[(d_.+e_.*x_)^m_*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^FracPart[p]*(a+b*x+c*x^2)^FracPart[p]/(a*d+c*e*x^3)^FracPart[p]*Int[(d+e*x)^(m-p)*(a*d+c*e*x^3)^p,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[b*d+a*e] && EqQ[c*d+b*e] && PositiveIntegerQ[m-p+1] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_/Sqrt[b_.*x_+c_.*x_^2],x_Symbol] := + Int[(d+e*x)^m/(Sqrt[b*x]*Sqrt[1+c/b*x]),x] /; +FreeQ[{b,c,d,e},x] && NeQ[c*d-b*e] && NeQ[2*c*d-b*e] && RationalQ[m] && m^2==1/4 && NegativeQ[c] && RationalQ[b] + + +Int[(d_.+e_.*x_)^m_/Sqrt[b_.*x_+c_.*x_^2],x_Symbol] := + Sqrt[x]*Sqrt[b+c*x]/Sqrt[b*x+c*x^2]*Int[(d+e*x)^m/(Sqrt[x]*Sqrt[b+c*x]),x] /; +FreeQ[{b,c,d,e},x] && NeQ[c*d-b*e] && NeQ[2*c*d-b*e] && RationalQ[m] && m^2==1/4 + + +Int[x_^m_/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] := + 2*Subst[Int[x^(2*m+1)/Sqrt[a+b*x^2+c*x^4],x],x,Sqrt[x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c,0] && EqQ[m^2-1/4] + + +Int[(e_*x_)^m_/Sqrt[a_+b_.*x_+c_.*x_^2],x_Symbol] := + (e*x)^m/x^m*Int[x^m/Sqrt[a+b*x+c*x^2], x] /; +FreeQ[{a,b,c,e},x] && NeQ[b^2-4*a*c,0] && EqQ[m^2-1/4] + + +Int[(d_.+e_.*x_)^m_/Sqrt[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + 2*Rt[b^2-4*a*c,2]*(d+e*x)^m*Sqrt[-c*(a+b*x+c*x^2)/(b^2-4*a*c)]/ + (c*Sqrt[a+b*x+c*x^2]*(2*c*(d+e*x)/(2*c*d-b*e-e*Rt[b^2-4*a*c,2]))^m)* + Subst[Int[(1+2*e*Rt[b^2-4*a*c,2]*x^2/(2*c*d-b*e-e*Rt[b^2-4*a*c,2]))^m/Sqrt[1-x^2],x],x, + Sqrt[(b+Rt[b^2-4*a*c,2]+2*c*x)/(2*Rt[b^2-4*a*c,2])]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && EqQ[m^2-1/4] + + +Int[(d_+e_.*x_)^m_/Sqrt[a_+c_.*x_^2],x_Symbol] := + 2*a*Rt[-c/a,2]*(d+e*x)^m*Sqrt[1+c*x^2/a]/(c*Sqrt[a+c*x^2]*(c*(d+e*x)/(c*d-a*e*Rt[-c/a,2]))^m)* + Subst[Int[(1+2*a*e*Rt[-c/a,2]*x^2/(c*d-a*e*Rt[-c/a,2]))^m/Sqrt[1-x^2],x],x,Sqrt[(1-Rt[-c/a,2]*x)/2]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && EqQ[m^2-1/4] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^(m+1)*(d*b-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^p/(2*(m+1)*(c*d^2-b*d*e+a*e^2)) + + p*(b^2-4*a*c)/(2*(m+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+2)*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + RationalQ[m,p] && m+2*p+2==0 && p>0 + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^(m+1)*(-2*a*e+(2*c*d)*x)*(a+c*x^2)^p/(2*(m+1)*(c*d^2+a*e^2)) - + 4*a*c*p/(2*(m+1)*(c*d^2+a*e^2))*Int[(d+e*x)^(m+2)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[m,p] && m+2*p+2==0 && p>0 + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m-1)*(d*b-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) - + 2*(2*p+3)*(c*d^2-b*d*e+a*e^2)/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-2)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + RationalQ[m,p] && m+2*p+2==0 && p<-1 + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m-1)*(a*e-c*d*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) + + (2*p+3)*(c*d^2+a*e^2)/(2*a*c*(p+1))*Int[(d+e*x)^(m-2)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[m,p] && m+2*p+2==0 && p<-1 + + +Int[1/((d_.+e_.*x_)*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + -2*Subst[Int[1/(4*c*d^2-4*b*d*e+4*a*e^2-x^2),x],x,(2*a*e-b*d-(2*c*d-b*e)*x)/Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[2*c*d-b*e] + + +Int[1/((d_+e_.*x_)*Sqrt[a_+c_.*x_^2]),x_Symbol] := + -Subst[Int[1/(c*d^2+a*e^2-x^2),x],x,(a*e-c*d*x)/Sqrt[a+c*x^2]] /; +FreeQ[{a,c,d,e},x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(b-Rt[b^2-4*a*c,2]+2*c*x)*(d+e*x)^(m+1)*(a+b*x+c*x^2)^p/ + ((m+1)*(2*c*d-b*e+e*Rt[b^2-4*a*c,2])* + ((2*c*d-b*e+e*Rt[b^2-4*a*c,2])*(b+Rt[b^2-4*a*c,2]+2*c*x)/((2*c*d-b*e-e*Rt[b^2-4*a*c,2])*(b-Rt[b^2-4*a*c,2]+2*c*x)))^p)* + Hypergeometric2F1[m+1,-p,m+2,-4*c*Rt[b^2-4*a*c,2]*(d+e*x)/((2*c*d-b*e-e*Rt[b^2-4*a*c,2])*(b-Rt[b^2-4*a*c,2]+2*c*x))] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && Not[IntegerQ[p]] && + EqQ[m+2*p+2] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (Rt[-a*c,2]-c*x)*(d+e*x)^(m+1)*(a+c*x^2)^p/ + ((m+1)*(c*d+e*Rt[-a*c,2])*((c*d+e*Rt[-a*c,2])*(Rt[-a*c,2]+c*x)/((c*d-e*Rt[-a*c,2])*(-Rt[-a*c,2]+c*x)))^p)* + Hypergeometric2F1[m+1,-p,m+2,2*c*Rt[-a*c,2]*(d+e*x)/((c*d-e*Rt[-a*c,2])*(Rt[-a*c,2]-c*x))] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && EqQ[m+2*p+2] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(b+2*c*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) + + m*(2*c*d-b*e)/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && EqQ[m+2*p+3] && + RationalQ[p] && p<-1 + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^m*(2*c*x)*(a+c*x^2)^(p+1)/(4*a*c*(p+1)) - + m*(2*c*d)/(4*a*c*(p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && EqQ[m+2*p+3] && RationalQ[p] && p<-1 + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + (2*c*d-b*e)/(2*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && EqQ[m+2*p+3] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/((m+1)*(c*d^2+a*e^2)) + + c*d/(c*d^2+a*e^2)*Int[(d+e*x)^(m+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && EqQ[m+2*p+3] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e*(m+1)) - + p/(e*(m+1))*Int[(d+e*x)^(m+1)*(b+2*c*x)*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && RationalQ[p] && p>0 && + (IntegerQ[p] || RationalQ[m] && m<-1) && NeQ[m+1] && Not[NegativeIntegerQ[m+2*p+1]] && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+c*x^2)^p/(e*(m+1)) - + 2*c*p/(e*(m+1))*Int[x*(d+e*x)^(m+1)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && RationalQ[p] && p>0 && + (IntegerQ[p] || RationalQ[m] && m<-1) && NeQ[m+1] && Not[NegativeIntegerQ[m+2*p+1]] && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e*(m+2*p+1)) - + p/(e*(m+2*p+1))*Int[(d+e*x)^m*Simp[b*d-2*a*e+(2*c*d-b*e)*x,x]*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && RationalQ[p] && p>0 && + NeQ[m+2*p+1] && (Not[RationalQ[m]] || m<1) && Not[NegativeIntegerQ[m+2*p]] && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(a+c*x^2)^p/(e*(m+2*p+1)) + + 2*p/(e*(m+2*p+1))*Int[(d+e*x)^m*Simp[a*e-c*d*x,x]*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && RationalQ[p] && p>0 && + NeQ[m+2*p+1] && (Not[RationalQ[m]] || m<1) && Not[NegativeIntegerQ[m+2*p]] && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(b+2*c*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) - + 1/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(b*e*m+2*c*d*(2*p+3)+2*c*e*(m+2*p+3)*x)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + RationalQ[m,p] && p<-1 && m>0 && (m<1 || NegativeIntegerQ[m+2*p+3] && m!=2) && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + -x*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*a*(p+1)) + + 1/(2*a*(p+1))*Int[(d+e*x)^(m-1)*(d*(2*p+3)+e*(m+2*p+3)*x)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && + RationalQ[m,p] && p<-1 && m>0 && (m<1 || NegativeIntegerQ[m+2*p+3] && m!=2) && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m-1)*(d*b-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) + + 1/((p+1)*(b^2-4*a*c))* + Int[(d+e*x)^(m-2)* + Simp[e*(2*a*e*(m-1)+b*d*(2*p-m+4))-2*c*d^2*(2*p+3)+e*(b*e-2*d*c)*(m+2*p+2)*x,x]* + (a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + RationalQ[m,p] && p<-1 && m>1 && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m-1)*(a*e-c*d*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) + + 1/((p+1)*(-2*a*c))* + Int[(d+e*x)^(m-2)*Simp[a*e^2*(m-1)-c*d^2*(2*p+3)-d*c*e*(m+2*p+2)*x,x]*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[m,p] && p<-1 && m>1 && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) + + 1/((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2))* + Int[(d+e*x)^m* + Simp[b*c*d*e*(2*p-m+2)+b^2*e^2*(m+p+2)-2*c^2*d^2*(2*p+3)-2*a*c*e^2*(m+2*p+3)-c*e*(2*c*d-b*e)*(m+2*p+4)*x,x]* + (a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + RationalQ[p] && p<-1 && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^(m+1)*(a*e+c*d*x)*(a+c*x^2)^(p+1)/(2*a*(p+1)*(c*d^2+a*e^2)) + + 1/(2*a*(p+1)*(c*d^2+a*e^2))* + Int[(d+e*x)^m*Simp[c*d^2*(2*p+3)+a*e^2*(m+2*p+3)+c*e*d*(m+2*p+4)*x,x]*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,m},x] && NeQ[c*d^2+a*e^2] && RationalQ[p] && p<-1 && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+1)) + + 1/(c*(m+2*p+1))* + Int[(d+e*x)^(m-2)* + Simp[c*d^2*(m+2*p+1)-e*(a*e*(m-1)+b*d*(p+1))+e*(2*c*d-b*e)*(m+p)*x,x]* + (a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + If[RationalQ[m], m>1, SumSimplerQ[m,-2]] && NeQ[m+2*p+1] && IntQuadraticQ[a,b,c,d,e,m,p,x] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(a+c*x^2)^(p+1)/(c*(m+2*p+1)) + + 1/(c*(m+2*p+1))* + Int[(d+e*x)^(m-2)*Simp[c*d^2*(m+2*p+1)-a*e^2*(m-1)+2*c*d*e*(m+p)*x,x]*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && + If[RationalQ[m], m>1, SumSimplerQ[m,-2]] && NeQ[m+2*p+1] && IntQuadraticQ[a,0,c,d,e,m,p,x] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/((m+1)*(c*d^2-b*d*e+a*e^2))* + Int[(d+e*x)^(m+1)*Simp[c*d*(m+1)-b*e*(m+p+2)-c*e*(m+2*p+3)*x,x]*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && + (RationalQ[m] && m<-1 && IntQuadraticQ[a,b,c,d,e,m,p,x] || + SumSimplerQ[m,1] && IntegerQ[p] && NeQ[m+1] || + NegativeIntegerQ[Simplify[m+2*p+3]] && NeQ[m+1]) + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/((m+1)*(c*d^2+a*e^2)) + + c/((m+1)*(c*d^2+a*e^2))* + Int[(d+e*x)^(m+1)*Simp[d*(m+1)-e*(m+2*p+3)*x,x]*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && + (RationalQ[m] && m<-1 && IntQuadraticQ[a,0,c,d,e,m,p,x] || + SumSimplerQ[m,1] && IntegerQ[p] && NeQ[m+1] || + NegativeIntegerQ[Simplify[m+2*p+3]] && NeQ[m+1]) + + +Int[1/((d_.+e_.*x_)*(a_+b_.*x_+c_.*x_^2)^(1/3)),x_Symbol] := + With[{q=Rt[3*c*e^2*(2*c*d-b*e),3]}, + -Sqrt[3]*c*e*ArcTan[1/Sqrt[3]+2*(c*d-b*e-c*e*x)/(Sqrt[3]*q*(a+b*x+c*x^2)^(1/3))]/q^2 - + 3*c*e*Log[d+e*x]/(2*q^2) + + 3*c*e*Log[c*d-b*e-c*e*x-q*(a+b*x+c*x^2)^(1/3)]/(2*q^2)] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e] && EqQ[c^2*d^2-b*c*d*e+b^2*e^2-3*a*c*e^2] && PosQ[c*e^2*(2*c*d-b*e)] + + +Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)^(1/3)),x_Symbol] := + With[{q=Rt[6*c^2*e^2/d^2,3]}, + -Sqrt[3]*c*e*ArcTan[1/Sqrt[3]+2*c*(d-e*x)/(Sqrt[3]*d*q*(a+c*x^2)^(1/3))]/(d^2*q^2) - + 3*c*e*Log[d+e*x]/(2*d^2*q^2) + + 3*c*e*Log[c*d-c*e*x-d*q*(a+c*x^2)^(1/3)]/(2*d^2*q^2)] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2-3*a*e^2] + + +Int[1/((d_.+e_.*x_)*(a_+b_.*x_+c_.*x_^2)^(1/3)),x_Symbol] := + With[{q=Rt[-3*c*e^2*(2*c*d-b*e),3]}, + -Sqrt[3]*c*e*ArcTan[1/Sqrt[3]-2*(c*d-b*e-c*e*x)/(Sqrt[3]*q*(a+b*x+c*x^2)^(1/3))]/q^2 - + 3*c*e*Log[d+e*x]/(2*q^2) + + 3*c*e*Log[c*d-b*e-c*e*x+q*(a+b*x+c*x^2)^(1/3)]/(2*q^2)] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e] && EqQ[c^2*d^2-b*c*d*e+b^2*e^2-3*a*c*e^2] && NegQ[c*e^2*(2*c*d-b*e)] + + +(* Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)^(1/3)),x_Symbol] := + With[{q=Rt[-6*c^2*d*e^2,3]}, + -Sqrt[3]*c*e*ArcTan[1/Sqrt[3]-2*(c*d-c*e*x)/(Sqrt[3]*q*(a+c*x^2)^(1/3))]/q^2 - + 3*c*e*Log[d+e*x]/(2*q^2) + + 3*c*e*Log[c*d-c*e*x+q*(a+c*x^2)^(1/3)]/(2*q^2)] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2-3*a*e^2] && NegQ[c^2*d*e^2] *) + + +Int[1/((d_.+e_.*x_)*(a_+b_.*x_+c_.*x_^2)^(1/3)),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (b+q+2*c*x)^(1/3)*(b-q+2*c*x)^(1/3)/(a+b*x+c*x^2)^(1/3)*Int[1/((d+e*x)*(b+q+2*c*x)^(1/3)*(b-q+2*c*x)^(1/3)),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && EqQ[c^2*d^2-b*c*d*e-2*b^2*e^2+9*a*c*e^2] + + +Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)^(1/4)),x_Symbol] := + d*Int[1/((d^2-e^2*x^2)*(a+c*x^2)^(1/4)),x] - e*Int[x/((d^2-e^2*x^2)*(a+c*x^2)^(1/4)),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)^(3/4)),x_Symbol] := + d*Int[1/((d^2-e^2*x^2)*(a+c*x^2)^(3/4)),x] - e*Int[x/((d^2-e^2*x^2)*(a+c*x^2)^(3/4)),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + 1/(-4*c/(b^2-4*a*c))^p*Subst[Int[Simp[1-x^2/(b^2-4*a*c),x]^p/Simp[2*c*d-b*e+e*x,x],x],x,b+2*c*x] /; +FreeQ[{a,b,c,d,e,p},x] && PositiveQ[4*a-b^2/c] && IntegerQ[4*p] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + (a+b*x+c*x^2)^p/(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^p* + Int[(-a*c/(b^2-4*a*c)-b*c*x/(b^2-4*a*c)-c^2*x^2/(b^2-4*a*c))^p/(d+e*x),x] /; +FreeQ[{a,b,c,d,e,p},x] && Not[PositiveQ[4*a-b^2/c]] && IntegerQ[4*p] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[(d+e*x)^m*(Rt[a,2]+Rt[-c,2]*x)^p*(Rt[a,2]-Rt[-c,2]*x)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && PositiveQ[a] && NegativeQ[c] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -(1/(d+e*x))^(2*p)*(a+b*x+c*x^2)^p/(e*(e*(b-q+2*c*x)/(2*c*(d+e*x)))^p*(e*(b+q+2*c*x)/(2*c*(d+e*x)))^p)* + Subst[Int[x^(-m-2*(p+1))*Simp[1-(d-e*(b-q)/(2*c))*x,x]^p*Simp[1-(d-e*(b+q)/(2*c))*x,x]^p,x],x,1/(d+e*x)]] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && Not[IntegerQ[p]] && + NegativeIntegerQ[m] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[-a*c,2]}, + -(1/(d+e*x))^(2*p)*(a+c*x^2)^p/(e*(-e*(q-c*x)/(c*(d+e*x)))^p*(e*(q+c*x)/(c*(d+e*x)))^p)* + Subst[Int[x^(-m-2*(p+1))*Simp[1-(d+e*q/c)*x,x]^p*Simp[1-(d-e*q/c)*x,x]^p,x],x,1/(d+e*x)]] /; +FreeQ[{a,c,d,e,p},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && NegativeIntegerQ[m] + + +Int[(d_.+e_.*x_)^m_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (a+b*x+c*x^2)^p/(e*(1-(d+e*x)/(d-e*(b-q)/(2*c)))^p*(1-(d+e*x)/(d-e*(b+q)/(2*c)))^p)* + Subst[Int[x^m*Simp[1-x/(d-e*(b-q)/(2*c)),x]^p*Simp[1-x/(d-e*(b+q)/(2*c)),x]^p,x],x,d+e*x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2] && NeQ[2*c*d-b*e] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Rt[-a*c,2]}, + (a+c*x^2)^p/(e*(1-(d+e*x)/(d+e*q/c))^p*(1-(d+e*x)/(d-e*q/c))^p)* + Subst[Int[x^m*Simp[1-x/(d+e*q/c),x]^p*Simp[1-x/(d-e*q/c),x]^p,x],x,d+e*x]] /; +FreeQ[{a,c,d,e,m,p},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*u_)^m_.*(a_+b_.*u_+c_.*u_^2)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x)^m*(a+b*x+c*x^2)^p,x],x,u] /; +FreeQ[{a,b,c,d,e,m,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(d_.+e_.*u_)^m_.*(a_+c_.*u_^2)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x)^m*(a+c*x^2)^p,x],x,u] /; +FreeQ[{a,c,d,e,m,p},x] && LinearQ[u,x] && NeQ[u-x] + + +(* IntQuadraticQ[a,b,c,d,e,m,p,x] returns True iff (d+e*x)^m*(a+b*x+c*x^2)^p is integrable wrt x in terms of non-Appell functions. *) +IntQuadraticQ[a_,b_,c_,d_,e_,m_,p_,x_] := + IntegerQ[p] || PositiveIntegerQ[m] || IntegersQ[2*m,2*p] || IntegersQ[m,4*p] || + IntegersQ[m,p+1/3] && (EqQ[c^2*d^2-b*c*d*e+b^2*e^2-3*a*c*e^2] || EqQ[c^2*d^2-b*c*d*e-2*b^2*e^2+9*a*c*e^2]) + + +(* ::Subsection::Closed:: *) +(*1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p*) + + +Int[x_^n_.*(f_+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + f*Int[x^n*(a+c*x^2)^p,x] + g*Int[x^(n+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,f,g,p},x] && IntegerQ[n] && Not[IntegerQ[2*p]] + + +Int[(d_.+e_.*x_)^m_.*(f_+g_.*x_)*(a_+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -f*g*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(b*(p+1)*(e*f-d*g)) /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[b^2-4*a*c,0] && EqQ[m+2*p+3,0] && EqQ[2*c*f-b*g,0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (2*c*f-b*g)/(2*c*d-b*e)*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] - + (e*f-d*g)/(2*c*d-b*e)*Int[(d+e*x)^m*(b+2*c*x)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[b^2-4*a*c,0] && EqQ[m+2*p+3,0] && NeQ[2*c*f-b*g,0] && NeQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(2*c*(p+1)) - + e*g*m/(2*c*(p+1))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && EqQ[2*c*f-b*g,0] && LtQ[p,-1] && GtQ[m,0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2+c*x)^(2*FracPart[p]))*Int[(d+e*x)^m*(f+g*x)*(b/2+c*x)^(2*p),x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && EqQ[b^2-4*a*c,0] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(f+g*x)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(f+g*x)*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,f,g,m},x] && EqQ[c*d^2+a*e^2,0] && + (IntegerQ[p] || PositiveQ[a] && PositiveQ[d] && EqQ[m+p,0]) + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_.*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(f+g*x)^n*(a+b*x+c*x^2)^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[c*d^2-b*d*e+a*e^2,0] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_.*(a_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(f+g*x)^n*(a+c*x^2)^(m+p),x] /; +FreeQ[{a,c,d,e,f,g,n,p},x] && EqQ[c*d^2+a*e^2,0] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+2)) /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && + EqQ[m*(g*(c*d-b*e)+c*e*f)+e*(p+1)*(2*c*f-b*g),0] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(a+c*x^2)^(p+1)/(c*(m+2*p+2)) /; +FreeQ[{a,c,d,e,f,g,m,p},x] && EqQ[c*d^2+a*e^2,0] && EqQ[m*(d*g+e*f)+2*e*f*(p+1),0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (g*(c*d-b*e)+c*e*f)*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(c*(p+1)*(2*c*d-b*e)) - + e*(m*(g*(c*d-b*e)+c*e*f)+e*(p+1)*(2*c*f-b*g))/(c*(p+1)*(2*c*d-b*e))* + Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m,p] && p<-1 && m>0 + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (d*g+e*f)*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*c*d*(p+1)) - + e*(m*(d*g+e*f)+2*e*f*(p+1))/(2*c*d*(p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,f,g},x] && EqQ[c*d^2+a*e^2,0] && RationalQ[m,p] && p<-1 && m>0 + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (g*(c*d-b*e)+c*e*f)*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(c*(p+1)*(2*c*d-b*e)) - + e*(m*(g*(c*d-b*e)+c*e*f)+e*(p+1)*(2*c*f-b*g))/(c*(p+1)*(2*c*d-b*e))* + Int[(d+e*x)^Simplify[m-1]*(a+b*x+c*x^2)^Simplify[p+1],x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && + SumSimplerQ[p,1] && SumSimplerQ[m,-1] && NeQ[p+1] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (d*g+e*f)*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*c*d*(p+1)) - + e*(m*(d*g+e*f)+2*e*f*(p+1))/(2*c*d*(p+1))*Int[(d+e*x)^Simplify[m-1]*(a+c*x^2)^Simplify[p+1],x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && EqQ[c*d^2+a*e^2,0] && SumSimplerQ[p,1] && SumSimplerQ[m,-1] && NeQ[p+1] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d*g-e*f)*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/((2*c*d-b*e)*(m+p+1)) + + (m*(g*(c*d-b*e)+c*e*f)+e*(p+1)*(2*c*f-b*g))/(e*(2*c*d-b*e)*(m+p+1))* + Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && + (RationalQ[m] && m<-1 && Not[PositiveIntegerQ[m+p+1]] || RationalQ[m,p] && m<0 && p<-1 || EqQ[m+2*p+2]) && NeQ[m+p+1] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (d*g-e*f)*(d+e*x)^m*(a+c*x^2)^(p+1)/(2*c*d*(m+p+1)) + + (m*(g*c*d+c*e*f)+2*e*c*f*(p+1))/(e*(2*c*d)*(m+p+1))* + Int[(d+e*x)^(m+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && EqQ[c*d^2+a*e^2,0] && + (RationalQ[m] && m<-1 && Not[PositiveIntegerQ[m+p+1]] || RationalQ[m,p] && m<0 && p<-1 || EqQ[m+2*p+2]) && NeQ[m+p+1] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+2)) + + (m*(g*(c*d-b*e)+c*e*f)+e*(p+1)*(2*c*f-b*g))/(c*e*(m+2*p+2))*Int[(d+e*x)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && NeQ[m+2*p+2] + + +Int[(d_+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(a+c*x^2)^(p+1)/(c*(m+2*p+2)) + + (m*(d*g+e*f)+2*e*f*(p+1))/(e*(m+2*p+2))*Int[(d+e*x)^m*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && EqQ[c*d^2+a*e^2,0] && NeQ[m+2*p+2] + + +Int[x_^2*(f_+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + x^2*(a*g-c*f*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) - + 1/(2*a*c*(p+1))*Int[x*Simp[2*a*g-c*f*(2*p+5)*x,x]*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,f,g},x] && EqQ[a*g^2+f^2*c] && RationalQ[p] && p<-2 + + +Int[x_^2*(f_+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + 1/c*Int[(f+g*x)*(a+c*x^2)^(p+1),x] - f^2/c*Int[(a+c*x^2)^(p+1)/(f-g*x),x] /; +FreeQ[{a,c,f,g,p},x] && EqQ[a*g^2+f^2*c] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && IGtQ[p,0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,f,g,m},x] && NeQ[c*d^2+a*e^2,0] && IGtQ[p,0] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)/(a+b*x+c*x^2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[m] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)/(a_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)/(a+c*x^2),x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && IntegerQ[m] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -(e*f-d*g)*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(2*(p+1)*(c*d^2-b*d*e+a*e^2)) /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + EqQ[Simplify[m+2*p+3]] && EqQ[b*(e*f+d*g)-2*(c*d*f+a*e*g)] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + -(e*f-d*g)*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/(2*(p+1)*(c*d^2+a*e^2)) /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[c*d^2+a*e^2,0] && EqQ[Simplify[m+2*p+3]] && EqQ[c*d*f+a*e*g] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(a+b*x+c*x^2)^(p+1)*(b*f-2*a*g+(2*c*f-b*g)*x)/((p+1)*(b^2-4*a*c)) - + m*(b*(e*f+d*g)-2*(c*d*f+a*e*g))/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + EqQ[Simplify[m+2*p+3]] && RationalQ[p] && p<-1 && Not[m==1 && (EqQ[d] || EqQ[2*c*d-b*e])] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(a+c*x^2)^(p+1)*(a*g-c*f*x)/(2*a*c*(p+1)) - + m*(c*d*f+a*e*g)/(2*a*c*(p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && + EqQ[Simplify[m+2*p+3]] && RationalQ[p] && p<-1 && Not[m==1 && EqQ[d]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -(e*f-d*g)*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(2*(p+1)*(c*d^2-b*d*e+a*e^2)) - + (b*(e*f+d*g)-2*(c*d*f+a*e*g))/(2*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + EqQ[Simplify[m+2*p+3]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + -(e*f-d*g)*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/(2*(p+1)*(c*d^2+a*e^2)) + + (c*d*f+a*e*g)/(c*d^2+a*e^2)*Int[(d+e*x)^(m+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[c*d^2+a*e^2,0] && EqQ[Simplify[m+2*p+3]] + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(b*e*g*(p+2)-c*(e*f+d*g)*(2*p+3)-2*c*e*g*(p+1)*x)*(a+b*x+c*x^2)^(p+1)/(2*c^2*(p+1)*(2*p+3)) /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + EqQ[b^2*e*g*(p+2)-2*a*c*e*g+c*(2*c*d*f-b*(e*f+d*g))*(2*p+3)] + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + ((e*f+d*g)*(2*p+3)+2*e*g*(p+1)*x)*(a+c*x^2)^(p+1)/(2*c*(p+1)*(2*p+3)) /; +FreeQ[{a,c,d,e,f,g,p},x] && NeQ[c*d^2+a*e^2,0] && EqQ[a*e*g-c*d*f*(2*p+3)] + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(2*a*c*(e*f+d*g)-b*(c*d*f+a*e*g)-(b^2*e*g-b*c*(e*f+d*g)+2*c*(c*d*f-a*e*g))*x)*(a+b*x+c*x^2)^(p+1)/(c*(p+1)*(b^2-4*a*c)) - + (b^2*e*g*(p+2)-2*a*c*e*g+c*(2*c*d*f-b*(e*f+d*g))*(2*p+3))/(c*(p+1)*(b^2-4*a*c))*Int[(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[p] && p<-1 + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + -(c*d*f*x-a*(d*g+e*(f+g*x)))*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) - + (a*e*g-c*d*f*(2*p+3))/(2*a*c*(p+1))*Int[(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && RationalQ[p] && p<-1 + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(b*e*g*(p+2)-c*(e*f+d*g)*(2*p+3)-2*c*e*g*(p+1)*x)*(a+b*x+c*x^2)^(p+1)/(2*c^2*(p+1)*(2*p+3)) + + (b^2*e*g*(p+2)-2*a*c*e*g+c*(2*c*d*f-b*(e*f+d*g))*(2*p+3))/(2*c^2*(2*p+3))*Int[(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && Not[RationalQ[p] && p<=-1] + + +Int[(d_.+e_.*x_)*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + ((e*f+d*g)*(2*p+3)+2*e*g*(p+1)*x)*(a+c*x^2)^(p+1)/(2*c*(p+1)*(2*p+3)) - + (a*e*g-c*d*f*(2*p+3))/(c*(2*p+3))*Int[(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,p},x] && NeQ[c*d^2+a*e^2,0] && Not[RationalQ[p] && p<=-1] + + +Int[(e_.*x_)^m_*(f_+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + f*Int[(e*x)^m*(a+c*x^2)^p,x] + g/e*Int[(e*x)^(m+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,e,f,g,p},x] && Not[RationalQ[m]] && Not[PositiveIntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^FracPart[p]*(a+b*x+c*x^2)^FracPart[p]/(a*d+c*e*x^3)^FracPart[p]*Int[(f+g*x)*(a*d+c*e*x^3)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[m-p] && EqQ[b*d+a*e] && EqQ[c*d+b*e] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -(d+e*x)^(m+1)*(a+b*x+c*x^2)^p/(e^2*(m+1)*(m+2)*(c*d^2-b*d*e+a*e^2))* + ((d*g-e*f*(m+2))*(c*d^2-b*d*e+a*e^2)-d*p*(2*c*d-b*e)*(e*f-d*g)-e*(g*(m+1)*(c*d^2-b*d*e+a*e^2)+p*(2*c*d-b*e)*(e*f-d*g))*x) - + p/(e^2*(m+1)*(m+2)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+2)*(a+b*x+c*x^2)^(p-1)* + Simp[2*a*c*e*(e*f-d*g)*(m+2)+b^2*e*(d*g*(p+1)-e*f*(m+p+2))+b*(a*e^2*g*(m+1)-c*d*(d*g*(2*p+1)-e*f*(m+2*p+2)))- + c*(2*c*d*(d*g*(2*p+1)-e*f*(m+2*p+2))-e*(2*a*e*g*(m+1)-b*(d*g*(m-2*p)+e*f*(m+2*p+2))))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + RationalQ[m,p] && p>0 && m<-2 && m+2*p<0 && Not[NegativeIntegerQ[m+2*p+3]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + -(d+e*x)^(m+1)*(a+c*x^2)^p/(e^2*(m+1)*(m+2)*(c*d^2+a*e^2))* + ((d*g-e*f*(m+2))*(c*d^2+a*e^2)-2*c*d^2*p*(e*f-d*g)-e*(g*(m+1)*(c*d^2+a*e^2)+2*c*d*p*(e*f-d*g))*x) - + p/(e^2*(m+1)*(m+2)*(c*d^2+a*e^2))*Int[(d+e*x)^(m+2)*(a+c*x^2)^(p-1)* + Simp[2*a*c*e*(e*f-d*g)*(m+2)-c*(2*c*d*(d*g*(2*p+1)-e*f*(m+2*p+2))-2*a*e^2*g*(m+1))*x,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && + RationalQ[m,p] && p>0 && m<-2 && m+2*p<0 && Not[NegativeIntegerQ[m+2*p+3]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(e*f*(m+2*p+2)-d*g*(2*p+1)+e*g*(m+1)*x)*(a+b*x+c*x^2)^p/(e^2*(m+1)*(m+2*p+2)) + + p/(e^2*(m+1)*(m+2*p+2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p-1)* + Simp[g*(b*d+2*a*e+2*a*e*m+2*b*d*p)-f*b*e*(m+2*p+2)+(g*(2*c*d+b*e+b*e*m+4*c*d*p)-2*c*e*f*(m+2*p+2))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[p] && p>0 && + (RationalQ[m] && m<-1 || p==1 || IntegerQ[p] && Not[RationalQ[m]]) && NeQ[m+1] && Not[NegativeIntegerQ[m+2*p+1]] && + (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(e*f*(m+2*p+2)-d*g*(2*p+1)+e*g*(m+1)*x)*(a+c*x^2)^p/(e^2*(m+1)*(m+2*p+2)) + + p/(e^2*(m+1)*(m+2*p+2))*Int[(d+e*x)^(m+1)*(a+c*x^2)^(p-1)* + Simp[g*(2*a*e+2*a*e*m)+(g*(2*c*d+4*c*d*p)-2*c*e*f*(m+2*p+2))*x,x],x] /; +FreeQ[{a,c,d,e,f,g,m},x] && NeQ[c*d^2+a*e^2,0] && RationalQ[p] && p>0 && + (RationalQ[m] && m<-1 || p==1 || IntegerQ[p] && Not[RationalQ[m]]) && NeQ[m+1] && Not[NegativeIntegerQ[m+2*p+1]] && + (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(c*e*f*(m+2*p+2)-g*(c*d+2*c*d*p-b*e*p)+g*c*e*(m+2*p+1)*x)*(a+b*x+c*x^2)^p/ + (c*e^2*(m+2*p+1)*(m+2*p+2)) - + p/(c*e^2*(m+2*p+1)*(m+2*p+2))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p-1)* + Simp[c*e*f*(b*d-2*a*e)*(m+2*p+2)+g*(a*e*(b*e-2*c*d*m+b*e*m)+b*d*(b*e*p-c*d-2*c*d*p))+ + (c*e*f*(2*c*d-b*e)*(m+2*p+2)+g*(b^2*e^2*(p+m+1)-2*c^2*d^2*(1+2*p)-c*e*(b*d*(m-2*p)+2*a*e*(m+2*p+1))))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + RationalQ[p] && p>0 && (IntegerQ[p] || Not[RationalQ[m]] || -1<=m<0) && Not[NegativeIntegerQ[m+2*p]] && + (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m+1)*(c*e*f*(m+2*p+2)-g*c*d*(2*p+1)+g*c*e*(m+2*p+1)*x)*(a+c*x^2)^p/ + (c*e^2*(m+2*p+1)*(m+2*p+2)) + + 2*p/(c*e^2*(m+2*p+1)*(m+2*p+2))*Int[(d+e*x)^m*(a+c*x^2)^(p-1)* + Simp[f*a*c*e^2*(m+2*p+2)+a*c*d*e*g*m-(c^2*f*d*e*(m+2*p+2)-g*(c^2*d^2*(2*p+1)+a*c*e^2*(m+2*p+1)))*x,x],x] /; +FreeQ[{a,c,d,e,f,g,m},x] && NeQ[c*d^2+a*e^2,0] && + RationalQ[p] && p>0 && (IntegerQ[p] || Not[RationalQ[m]] || -1<=m<0) && Not[NegativeIntegerQ[m+2*p]] && + (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[(a+b*x+c*x^2)^p*ExpandIntegrand[(d+e*x)^m*(f+g*x),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && ILtQ[p,-1] && IGtQ[m,0] && RationalQ[a,b,c,d,e,f,g] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[(a+c*x^2)^p*ExpandIntegrand[(d+e*x)^m*(f+g*x),x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && ILtQ[p,-1] && IGtQ[m,0] && RationalQ[a,c,d,e,f,g] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + -(d+e*x)^(m-1)*(2*a*c*(e*f+d*g)-b*(c*d*f+a*e*g)-(2*c^2*d*f+b^2*e*g-c*(b*e*f+b*d*g+2*a*e*g))*x)* + (a+b*x+c*x^2)^(p+1)/(c*(p+1)*(b^2-4*a*c)) - + 1/(c*(p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-2)*(a+b*x+c*x^2)^(p+1)* + Simp[2*c^2*d^2*f*(2*p+3)+b*e*g*(a*e*(m-1)+b*d*(p+2))-c*(2*a*e*(e*f*(m-1)+d*g*m)+b*d*(d*g*(2*p+3)-e*f*(m-2*p-4))) + + e*(b^2*e*g*(m+p+1)+2*c^2*d*f*(m+2*p+2)-c*(2*a*e*g*m+b*(e*f+d*g)*(m+2*p+2)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] && GtQ[m,1] && + (EqQ[m,2] && EqQ[p,-3] && RationalQ[a,b,c,d,e,f,g] || Not[ILtQ[m+2*p+3,0]]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^(m-1)*(2*a*(e*f+d*g)-(2*c*d*f-2*a*e*g)*x)*(a+c*x^2)^(p+1)/(4*a*c*(p+1)) - + 1/(4*a*c*(p+1))*Int[(d+e*x)^(m-2)*(a+c*x^2)^(p+1)* + Simp[2*a*e*(e*f*(m-1)+d*g*m)-2*c*d^2*f*(2*p+3)+e*(2*a*e*g*m-2*c*d*f*(m+2*p+2))*x,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && LtQ[p,-1] && GtQ[m,1] && + (EqQ[m,2] && EqQ[p,-3] && RationalQ[a,c,d,e,f,g] || Not[ILtQ[m+2*p+3,0]]) + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(a+b*x+c*x^2)^(p+1)*(f*b-2*a*g+(2*c*f-b*g)*x)/((p+1)*(b^2-4*a*c)) + + 1/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)* + Simp[g*(2*a*e*m+b*d*(2*p+3))-f*(b*e*m+2*c*d*(2*p+3))-e*(2*c*f-b*g)*(m+2*p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + RationalQ[m,p] && p<-1 && m>0 && Not[m==1 && SimplerQ[d+e*x,f+g*x]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(a+c*x^2)^(p+1)*(a*g-c*f*x)/(2*a*c*(p+1)) - + 1/(2*a*c*(p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^(p+1)*Simp[a*e*g*m-c*d*f*(2*p+3)-c*e*f*(m+2*p+3)*x,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && + RationalQ[m,p] && p<-1 && m>0 && Not[m==1 && SimplerQ[d+e*x,f+g*x]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^(m+1)*(f*(b*c*d-b^2*e+2*a*c*e)-a*g*(2*c*d-b*e)+c*(f*(2*c*d-b*e)-g*(b*d-2*a*e))*x)*(a+b*x+c*x^2)^(p+1)/ + ((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) + + 1/((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p+1)* + Simp[f*(b*c*d*e*(2*p-m+2)+b^2*e^2*(p+m+2)-2*c^2*d^2*(2*p+3)-2*a*c*e^2*(m+2*p+3))- + g*(a*e*(b*e-2*c*d*m+b*e*m)-b*d*(3*c*d-b*e+2*c*d*p-b*e*p))+ + c*e*(g*(b*d-2*a*e)-f*(2*c*d-b*e))*(m+2*p+4)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + RationalQ[p] && p<-1 && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^(m+1)*(f*a*c*e-a*g*c*d+c*(c*d*f+a*e*g)*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)*(c*d^2+a*e^2)) + + 1/(2*a*c*(p+1)*(c*d^2+a*e^2))*Int[(d+e*x)^m*(a+c*x^2)^(p+1)* + Simp[f*(c^2*d^2*(2*p+3)+a*c*e^2*(m+2*p+3))-a*c*d*e*g*m+c*e*(c*d*f+a*e*g)*(m+2*p+4)*x,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && + RationalQ[p] && p<-1 && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + g*(d+e*x)^m/(c*m) + + 1/c*Int[(d+e*x)^(m-1)*Simp[c*d*f-a*e*g+(g*c*d-b*e*g+c*e*f)*x,x]/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && FractionQ[m] && m>0 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_+c_.*x_^2),x_Symbol] := + g*(d+e*x)^m/(c*m) + + 1/c*Int[(d+e*x)^(m-1)*Simp[c*d*f-a*e*g+(g*c*d+c*e*f)*x,x]/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && FractionQ[m] && m>0 + + +Int[(f_.+g_.*x_)/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + 2*Subst[Int[(e*f-d*g+g*x^2)/(c*d^2-b*d*e+a*e^2-(2*c*d-b*e)*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] + + +Int[(f_.+g_.*x_)/(Sqrt[d_.+e_.*x_]*(a_+c_.*x_^2)),x_Symbol] := + 2*Subst[Int[(e*f-d*g+g*x^2)/(c*d^2+a*e^2-2*c*d*x^2+c*x^4),x],x,Sqrt[d+e*x]] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x)^(m+1)*Simp[c*d*f-f*b*e+a*e*g-c*(e*f-d*g)*x,x]/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && FractionQ[m] && m<-1 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_+c_.*x_^2),x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)/((m+1)*(c*d^2+a*e^2)) + + 1/(c*d^2+a*e^2)*Int[(d+e*x)^(m+1)*Simp[c*d*f+a*e*g-c*(e*f-d*g)*x,x]/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,f,g,m},x] && NeQ[c*d^2+a*e^2,0] && FractionQ[m] && m<-1 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m,(f+g*x)/(a+b*x+c*x^2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && Not[RationalQ[m]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)/(a_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m,(f+g*x)/(a+c*x^2),x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2,0] && Not[RationalQ[m]] + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + g*(d+e*x)^m*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+2)) + + 1/(c*(m+2*p+2))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^p* + Simp[m*(c*d*f-a*e*g)+d*(2*c*f-b*g)*(p+1)+(m*(c*e*f+c*d*g-b*e*g)+e*(p+1)*(2*c*f-b*g))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && RationalQ[m] && m>0 && + NeQ[m+2*p+2] && Not[m==1 && SimplerQ[f+g*x,d+e*x]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + g*(d+e*x)^m*(a+c*x^2)^(p+1)/(c*(m+2*p+2)) + + 1/(c*(m+2*p+2))*Int[(d+e*x)^(m-1)*(a+c*x^2)^p* + Simp[c*d*f*(m+2*p+2)-a*e*g*m+c*(e*f*(m+2*p+2)+d*g*m)*x,x],x] /; +FreeQ[{a,c,d,e,f,g,p},x] && NeQ[c*d^2+a*e^2,0] && RationalQ[m] && m>0 && + NeQ[m+2*p+2] && Not[m==1 && SimplerQ[f+g*x,d+e*x]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/((m+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p* + Simp[(c*d*f-f*b*e+a*e*g)*(m+1)+b*(d*g-e*f)*(p+1)-c*(e*f-d*g)*(m+2*p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + RationalQ[m] && m<-1 && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/((m+1)*(c*d^2+a*e^2)) + + 1/((m+1)*(c*d^2+a*e^2))*Int[(d+e*x)^(m+1)*(a+c*x^2)^p*Simp[(c*d*f+a*e*g)*(m+1)-c*(e*f-d*g)*(m+2*p+3)*x,x],x] /; +FreeQ[{a,c,d,e,f,g,p},x] && NeQ[c*d^2+a*e^2,0] && + RationalQ[m] && m<-1 && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m,2*p]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/((m+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p* + Simp[(c*d*f-f*b*e+a*e*g)*(m+1)+b*(d*g-e*f)*(p+1)-c*(e*f-d*g)*(m+2*p+3)*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && + NegativeIntegerQ[Simplify[m+2*p+3]] && NeQ[m+1] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + (e*f-d*g)*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/((m+1)*(c*d^2+a*e^2)) + + 1/((m+1)*(c*d^2+a*e^2))*Int[(d+e*x)^(m+1)*(a+c*x^2)^p*Simp[(c*d*f+a*e*g)*(m+1)-c*(e*f-d*g)*(m+2*p+3)*x,x],x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[c*d^2+a*e^2,0] && NegativeIntegerQ[m+2*p+3] && NeQ[m+1] + + +Int[(f_+g_.*x_)/((d_.+e_.*x_)*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + 4*f*(a-d)/(b*d-a*e)*Subst[Int[1/(4*(a-d)-x^2),x],x,(2*(a-d)+(b-e)*x)/Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[4*c*(a-d)-(b-e)^2] && EqQ[e*f*(b-e)-2*g*(b*d-a*e)] && NeQ[b*d-a*e] + + +Int[(f_+g_.*x_)/(Sqrt[x_]*Sqrt[a_+b_.*x_+c_.*x_^2]),x_Symbol] := + 2*Subst[Int[(f+g*x^2)/Sqrt[a+b*x^2+c*x^4],x],x,Sqrt[x]] /; +FreeQ[{a,b,c,f,g},x] && NeQ[b^2-4*a*c,0] + + +Int[(f_+g_.*x_)/(Sqrt[x_]*Sqrt[a_+c_.*x_^2]),x_Symbol] := + 2*Subst[Int[(f+g*x^2)/Sqrt[a+c*x^4],x],x,Sqrt[x]] /; +FreeQ[{a,c,f,g},x] + + +Int[(f_+g_.*x_)/(Sqrt[e_*x_]*Sqrt[a_+b_.*x_+c_.*x_^2]),x_Symbol] := + Sqrt[x]/Sqrt[e*x]*Int[(f+g*x)/(Sqrt[x]*Sqrt[a+b*x+c*x^2]),x] /; +FreeQ[{a,b,c,e,f,g},x] && NeQ[b^2-4*a*c,0] + + +Int[(f_+g_.*x_)/(Sqrt[e_*x_]*Sqrt[a_+c_.*x_^2]),x_Symbol] := + Sqrt[x]/Sqrt[e*x]*Int[(f+g*x)/(Sqrt[x]*Sqrt[a+c*x^2]),x] /; +FreeQ[{a,c,e,f,g},x] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + g/e*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p,x] + (e*f-d*g)/e*Int[(d+e*x)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)*(a_+c_.*x_^2)^p_.,x_Symbol] := + g/e*Int[(d+e*x)^(m+1)*(a+c*x^2)^p,x] + (e*f-d*g)/e*Int[(d+e*x)^m*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[c*d^2+a*e^2,0] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p*) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2+c*x)^(2*FracPart[p]))*Int[(d+e*x)^m*(f+g*x)^n*(b/2+c*x)^(2*p),x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(f+g*x)^n*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(f+g*x)^n*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + (IntegerQ[p] || PositiveQ[a] && PositiveQ[d] && EqQ[m+p]) + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(f+g*x)^n*(a+b*x+c*x^2)^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[c*d^2-b*d*e+a*e^2] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + d^m*e^m*Int[(a*e+c*d*x)^(-m)*(f+g*x)^n*(a+c*x^2)^(m+p),x] /; +FreeQ[{a,c,d,e,f,g,n,p},x] && EqQ[c*d^2+a*e^2] && NegativeIntegerQ[m] && Not[IntegerQ[2*p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[(f+g*x)^n*(a+b*x+c*x^2)^(m+p)/(a/d+c*x/e)^m,x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] && + IntegersQ[m,n] && RationalQ[p] && (0<-m0 && n<-1 && Not[IntegerQ[n+p] && n+p+2<=0] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^m*(f+g*x)^(n+1)*(a+c*x^2)^p/(g*(n+1)) + + c*m/(e*g*(n+1))*Int[(d+e*x)^(m+1)*(f+g*x)^(n+1)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p>0 && n<-1 && Not[IntegerQ[n+p] && n+p+2<=0] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^m*(f+g*x)^(n+1)*(a+b*x+c*x^2)^p/(g*(m-n-1)) - + m*(c*e*f+c*d*g-b*e*g)/(e^2*g*(m-n-1))*Int[(d+e*x)^(m+1)*(f+g*x)^n*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p>0 && NeQ[m-n-1] && Not[PositiveIntegerQ[n]] && + Not[IntegerQ[n+p] && n+p+2<0] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + -(d+e*x)^m*(f+g*x)^(n+1)*(a+c*x^2)^p/(g*(m-n-1)) - + c*m*(e*f+d*g)/(e^2*g*(m-n-1))*Int[(d+e*x)^(m+1)*(f+g*x)^n*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e,f,g,n},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p>0 && NeQ[m-n-1] && Not[PositiveIntegerQ[n]] && + Not[IntegerQ[n+p] && n+p+2<0] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(f+g*x)^n*(a+b*x+c*x^2)^(p+1)/(c*(p+1)) - + e*g*n/(c*(p+1))*Int[(d+e*x)^(m-1)*(f+g*x)^(n-1)*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p<-1 && n>0 + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + e*(d+e*x)^(m-1)*(f+g*x)^n*(a+c*x^2)^(p+1)/(c*(p+1)) - + e*g*n/(c*(p+1))*Int[(d+e*x)^(m-1)*(f+g*x)^(n-1)*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p<-1 && n>0 + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p+1)/((p+1)*(c*e*f+c*d*g-b*e*g)) + + e^2*g*(m-n-2)/((p+1)*(c*e*f+c*d*g-b*e*g))*Int[(d+e*x)^(m-1)*(f+g*x)^n*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p<-1 + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+c*x^2)^(p+1)/(c*(p+1)*(e*f+d*g)) + + e^2*g*(m-n-2)/(c*(p+1)*(e*f+d*g))*Int[(d+e*x)^(m-1)*(f+g*x)^n*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,f,g,n},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n,p] && p<-1 + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -e*(d+e*x)^(m-1)*(f+g*x)^n*(a+b*x+c*x^2)^(p+1)/(c*(m-n-1)) - + n*(c*e*f+c*d*g-b*e*g)/(c*e*(m-n-1))*Int[(d+e*x)^m*(f+g*x)^(n-1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n] && n>0 && NeQ[m-n-1] && (IntegerQ[2*p] || IntegerQ[n]) + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + -e*(d+e*x)^(m-1)*(f+g*x)^n*(a+c*x^2)^(p+1)/(c*(m-n-1)) - + n*(e*f+d*g)/(e*(m-n-1))*Int[(d+e*x)^m*(f+g*x)^(n-1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n] && n>0 && NeQ[m-n-1] && (IntegerQ[2*p] || IntegerQ[n]) + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + -e^2*(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p+1)/((n+1)*(c*e*f+c*d*g-b*e*g)) - + c*e*(m-n-2)/((n+1)*(c*e*f+c*d*g-b*e*g))*Int[(d+e*x)^m*(f+g*x)^(n+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n] && n<-1 && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + -e^2*(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+c*x^2)^(p+1)/((n+1)*(c*e*f+c*d*g)) - + e*(m-n-2)/((n+1)*(e*f+d*g))*Int[(d+e*x)^m*(f+g*x)^(n+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p] && RationalQ[n] && n<-1 && IntegerQ[2*p] + + +Int[Sqrt[d_+e_.*x_]/((f_.+g_.*x_)*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + 2*e^2*Subst[Int[1/(c*(e*f+d*g)-b*e*g+e^2*g*x^2),x],x,Sqrt[a+b*x+c*x^2]/Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] + + +Int[Sqrt[d_+e_.*x_]/((f_.+g_.*x_)*Sqrt[a_+c_.*x_^2]),x_Symbol] := + 2*e^2*Subst[Int[1/(c*(e*f+d*g)+e^2*g*x^2),x],x,Sqrt[a+c*x^2]/Sqrt[d+e*x]] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p+1)/(c*g*(n+p+2)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && EqQ[b*e*g*(n+1)+c*e*f*(p+1)-c*d*g*(2*n+p+3)] && NeQ[n+p+2] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+c*x^2)^(p+1)/(c*g*(n+p+2)) /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && EqQ[e*f*(p+1)-d*g*(2*n+p+3)] && NeQ[n+p+2] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e^2*(e*f-d*g)*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p+1)/(g*(n+1)*(c*e*f+c*d*g-b*e*g)) - + e*(b*e*g*(n+1)+c*e*f*(p+1)-c*d*g*(2*n+p+3))/(g*(n+1)*(c*e*f+c*d*g-b*e*g))* + Int[(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && RationalQ[n] && n<-1 && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + e^2*(e*f-d*g)*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+c*x^2)^(p+1)/(c*g*(n+1)*(e*f+d*g)) - + e*(e*f*(p+1)-d*g*(2*n+p+3))/(g*(n+1)*(e*f+d*g))*Int[(d+e*x)^(m-1)*(f+g*x)^(n+1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && RationalQ[n] && n<-1 && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p+1)/(c*g*(n+p+2)) - + (b*e*g*(n+1)+c*e*f*(p+1)-c*d*g*(2*n+p+3))/(c*g*(n+p+2))*Int[(d+e*x)^(m-1)*(f+g*x)^n*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && Not[RationalQ[n] && n<-1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + e^2*(d+e*x)^(m-2)*(f+g*x)^(n+1)*(a+c*x^2)^(p+1)/(c*g*(n+p+2)) - + (e*f*(p+1)-d*g*(2*n+p+3))/(g*(n+p+2))*Int[(d+e*x)^(m-1)*(f+g*x)^n*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && + Not[IntegerQ[p]] && EqQ[m+p-1] && Not[RationalQ[n] && n<-1] && IntegerQ[2*p] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[p]] && (PositiveIntegerQ[m] || IntegersQ[m,n]) + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[ExpandIntegrand[1/Sqrt[a+c*x^2],(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^(p+1/2),x],x] /; +FreeQ[{a,c,d,e,f,g,n,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && IntegerQ[p-1/2] && IntegersQ[m,n] && Not[m<0 && p<0] && p!=1/2 + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,f,g,n,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && (PositiveIntegerQ[m] || IntegersQ[m,n]) + + +Int[x_^2*(a_.+b_.*x_+c_.*x_^2)^p_/(d_+e_.*x_),x_Symbol] := + d^2/e^2*Int[(a+b*x+c*x^2)^p/(d+e*x),x] - 1/e^2*Int[(d-e*x)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] + + +Int[x_^2*(a_+c_.*x_^2)^p_/(d_+e_.*x_),x_Symbol] := + d^2/e^2*Int[(a+c*x^2)^p/(d+e*x),x] - 1/e^2*Int[(d-e*x)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,p},x] && EqQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^2*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2)^(p+1)/(c*(m+2*p+3)) - + 1/(c*e^2*(m+2*p+3))*Int[(d+e*x)^m*(a+b*x+c*x^2)^p* + Simp[b*e*g*(d*g+e*f*(m+p+1))-c*(d^2*g^2+d*e*f*g*m+e^2*f^2*(m+2*p+3))+ + e*g*(b*e*g*(m+p+2)-c*(d*g*m+e*f*(m+2*p+4)))*x,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] && + NeQ[m+2*p+3] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^2*(a_+c_.*x_^2)^p_,x_Symbol] := + g*(d+e*x)^m*(f+g*x)*(a+c*x^2)^(p+1)/(c*(m+2*p+3)) - + 1/(c*e^2*(m+2*p+3))*Int[(d+e*x)^m*(a+c*x^2)^p* + Simp[-c*(d^2*g^2+d*e*f*g*m+e^2*f^2*(m+2*p+3))-c*e*g*(d*g*m+e*f*(m+2*p+4))*x,x],x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && NeQ[m+2*p+3] + + +Int[(e_.*x_)^m_*(f_.+g_.*x_)^n_*(b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (e*x)^m*(b*x+c*x^2)^p/(x^(m+p)*(b+c*x)^p)*Int[x^(m+p)*(f+g*x)^n*(b+c*x)^p,x] /; +FreeQ[{b,c,e,f,g,m,n},x] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[(d+e*x)^(m+p)*(f+g*x)^n*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] && PositiveQ[a] && PositiveQ[d] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := +(*(a+b*x+c*x^2)^p/((d+e*x)^p*(a*e+c*d*x)^p)*Int[(d+e*x)^(m+p)*(f+g*x)^n*(a*e+c*d*x)^p,x] /; *) + (a+b*x+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(f+g*x)^n*(a/d+c/e*x)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol] := + (a+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(f+g*x)^n*(a/d+c/e*x)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n},x] && NeQ[e*f-d*g] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && IntegersQ[m,n,p] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && IntegersQ[m,n,p] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_/((d_.+e_.*x_)*(f_.+g_.*x_)),x_Symbol] := + (c*d^2-b*d*e+a*e^2)/(e*(e*f-d*g))*Int[(a+b*x+c*x^2)^(p-1)/(d+e*x),x] - + 1/(e*(e*f-d*g))*Int[Simp[c*d*f-b*e*f+a*e*g-c*(e*f-d*g)*x,x]*(a+b*x+c*x^2)^(p-1)/(f+g*x),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && FractionQ[p] && p>0 + + +Int[(a_+c_.*x_^2)^p_/((d_.+e_.*x_)*(f_.+g_.*x_)),x_Symbol] := + (c*d^2+a*e^2)/(e*(e*f-d*g))*Int[(a+c*x^2)^(p-1)/(d+e*x),x] - + 1/(e*(e*f-d*g))*Int[Simp[c*d*f+a*e*g-c*(e*f-d*g)*x,x]*(a+c*x^2)^(p-1)/(f+g*x),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && FractionQ[p] && p>0 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + With[{q=Denominator[m]}, + q/e*Subst[Int[x^(q*(m+1)-1)*((e*f-d*g)/e+g*x^q/e)^n* + ((c*d^2-b*d*e+a*e^2)/e^2-(2*c*d-b*e)*x^q/e^2+c*x^(2*q)/e^2)^p,x],x,(d+e*x)^(1/q)]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && IntegersQ[n,p] && FractionQ[m] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + With[{q=Denominator[m]}, + q/e*Subst[Int[x^(q*(m+1)-1)*((e*f-d*g)/e+g*x^q/e)^n*((c*d^2+a*e^2)/e^2-2*c*d*x^q/e^2+c*x^(2*q)/e^2)^p,x],x,(d+e*x)^(1/q)]] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && IntegersQ[n,p] && FractionQ[m] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d*f+e*g*x^2)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[m-n] && EqQ[e*f+d*g] && (IntegerQ[m] || PositiveQ[d] && PositiveQ[f]) + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^n_*(a_.+c_.*x_^2)^p_.,x_Symbol] := + Int[(d*f+e*g*x^2)^m*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] && EqQ[m-n] && EqQ[e*f+d*g] && (IntegerQ[m] || PositiveQ[d] && PositiveQ[f]) + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^FracPart[m]*(f+g*x)^FracPart[m]/(d*f+e*g*x^2)^FracPart[m]*Int[(d*f+e*g*x^2)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[m-n] && EqQ[e*f+d*g] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^n_*(a_.+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^FracPart[m]*(f+g*x)^FracPart[m]/(d*f+e*g*x^2)^FracPart[m]*Int[(d*f+e*g*x^2)^m*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] && EqQ[m-n] && EqQ[e*f+d*g] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[(a+b*x)*(d+e*x)^m*(f+g*x)^n,x] + c*Int[x^2*(d+e*x)^m*(f+g*x)^n,x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && RationalQ[m,n] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2),x_Symbol] := + a*Int[(d+e*x)^m*(f+g*x)^n,x] + c*Int[x^2*(d+e*x)^m*(f+g*x)^n,x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && RationalQ[m,n] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + g/c^2*Int[Simp[2*c*e*f+c*d*g-b*e*g+c*e*g*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n-2),x] + + 1/c^2*Int[Simp[c^2*d*f^2-2*a*c*e*f*g-a*c*d*g^2+a*b*e*g^2+(c^2*e*f^2+2*c^2*d*f*g-2*b*c*e*f*g-b*c*d*g^2+b^2*e*g^2-a*c*e*g^2)*x,x]* + (d+e*x)^(m-1)*(f+g*x)^(n-2)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n>1 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_+c_.*x_^2),x_Symbol] := + g/c*Int[Simp[2*e*f+d*g+e*g*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n-2),x] + + 1/c*Int[Simp[c*d*f^2-2*a*e*f*g-a*d*g^2+(c*e*f^2+2*c*d*f*g-a*e*g^2)*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n-2)/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n>1 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*g/c*Int[(d+e*x)^(m-1)*(f+g*x)^(n-1),x] + + 1/c*Int[Simp[c*d*f-a*e*g+(c*e*f+c*d*g-b*e*g)*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n-1)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n>0 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_+c_.*x_^2),x_Symbol] := + e*g/c*Int[(d+e*x)^(m-1)*(f+g*x)^(n-1),x] + + 1/c*Int[Simp[c*d*f-a*e*g+(c*e*f+c*d*g)*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n-1)/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n>0 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + -g*(e*f-d*g)/(c*f^2-b*f*g+a*g^2)*Int[(d+e*x)^(m-1)*(f+g*x)^n,x] + + 1/(c*f^2-b*f*g+a*g^2)* + Int[Simp[c*d*f-b*d*g+a*e*g+c*(e*f-d*g)*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n+1)/(a+b*x+c*x^2),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n<-1 + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_+c_.*x_^2),x_Symbol] := + -g*(e*f-d*g)/(c*f^2+a*g^2)*Int[(d+e*x)^(m-1)*(f+g*x)^n,x] + + 1/(c*f^2+a*g^2)* + Int[Simp[c*d*f+a*e*g+c*(e*f-d*g)*x,x]*(d+e*x)^(m-1)*(f+g*x)^(n+1)/(a+c*x^2),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && RationalQ[m,n] && m>0 && n<-1 + + +Int[(d_.+e_.*x_)^m_/(Sqrt[f_.+g_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol] := + Int[ExpandIntegrand[1/(Sqrt[d+e*x]*Sqrt[f+g*x]),(d+e*x)^(m+1/2)/(a+b*x+c*x^2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[m+1/2] + + +Int[(d_.+e_.*x_)^m_/(Sqrt[f_.+g_.*x_]*(a_.+c_.*x_^2)),x_Symbol] := + Int[ExpandIntegrand[1/(Sqrt[d+e*x]*Sqrt[f+g*x]),(d+e*x)^(m+1/2)/(a+c*x^2),x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[m+1/2] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n,1/(a+b*x+c*x^2),x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[m]] && Not[IntegerQ[n]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_/(a_+c_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n,1/(a+c*x^2),x],x] /; +FreeQ[{a,c,d,e,f,g,m,n},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + (IntegerQ[p] || IntegersQ[m,n]) + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && (IntegerQ[p] || IntegersQ[m,n]) + + +Int[(g_.*x_)^n_*(d_.+e_.*x_)^m_*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (d+e*x)^FracPart[p]*(a+b*x+c*x^2)^FracPart[p]/(a*d+c*e*x^3)^FracPart[p]*Int[(g*x)^n*(a*d+c*e*x^3)^p,x] /; +FreeQ[{a,b,c,d,e,g,m,n,p},x] && EqQ[m-p] && EqQ[b*d+a*e] && EqQ[c*d+b*e] + + +Int[(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + (c*d^2-b*d*e+a*e^2)/(e*(e*f-d*g))*Int[(f+g*x)^(n+1)*(a+b*x+c*x^2)^(p-1)/(d+e*x),x] - + 1/(e*(e*f-d*g))*Int[(f+g*x)^n*(c*d*f-b*e*f+a*e*g-c*(e*f-d*g)*x)*(a+b*x+c*x^2)^(p-1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[n]] && Not[IntegerQ[p]] && RationalQ[n,p] && p>0 && n<-1 + + +Int[(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + (c*d^2+a*e^2)/(e*(e*f-d*g))*Int[(f+g*x)^(n+1)*(a+c*x^2)^(p-1)/(d+e*x),x] - + 1/(e*(e*f-d*g))*Int[(f+g*x)^n*(c*d*f+a*e*g-c*(e*f-d*g)*x)*(a+c*x^2)^(p-1),x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && + Not[IntegerQ[n]] && Not[IntegerQ[p]] && RationalQ[n,p] && p>0 && n<-1 + + +Int[(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + e*(e*f-d*g)/(c*d^2-b*d*e+a*e^2)*Int[(f+g*x)^(n-1)*(a+b*x+c*x^2)^(p+1)/(d+e*x),x] + + 1/(c*d^2-b*d*e+a*e^2)*Int[(f+g*x)^(n-1)*(c*d*f-b*e*f+a*e*g-c*(e*f-d*g)*x)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + Not[IntegerQ[n]] && Not[IntegerQ[p]] && RationalQ[n,p] && p<-1 && n>0 + + +Int[(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_/(d_.+e_.*x_),x_Symbol] := + e*(e*f-d*g)/(c*d^2+a*e^2)*Int[(f+g*x)^(n-1)*(a+c*x^2)^(p+1)/(d+e*x),x] + + 1/(c*d^2+a*e^2)*Int[(f+g*x)^(n-1)*(c*d*f+a*e*g-c*(e*f-d*g)*x)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && + Not[IntegerQ[n]] && Not[IntegerQ[p]] && RationalQ[n,p] && p<-1 && n>0 + + +Int[1/((d_.+e_.*x_)*Sqrt[f_.+g_.*x_]*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -2*Sqrt[-g*(b-q+2*c*x)/(2*c*f-b*g+g*q)]*Sqrt[-g*(b+q+2*c*x)/(2*c*f-b*g-g*q)]/ + ((e*f-d*g)*Sqrt[2*c/(2*c*f-b*g+g*q)]*Sqrt[a+b*x+c*x^2])* + EllipticPi[e*(2*c*f-b*g+g*q)/(2*c*(e*f-d*g)),ArcSin[Sqrt[2*c/(2*c*f-b*g+g*q)]*Sqrt[f+g*x]],(2*c*f-b*g+g*q)/(2*c*f-b*g-g*q)]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[1/((d_.+e_.*x_)*Sqrt[f_.+g_.*x_]*Sqrt[a_+c_.*x_^2]),x_Symbol] := + With[{q=Rt[-a*c,2]}, + -2*Sqrt[g*(q-c*x)/(c*f+g*q)]*Sqrt[-g*(q+c*x)/(c*f-g*q)]/((e*f-d*g)*Sqrt[c/(c*f+g*q)]*Sqrt[a+c*x^2])* + EllipticPi[e*(c*f+g*q)/(c*(e*f-d*g)),ArcSin[Sqrt[c/(c*f+g*q)]*Sqrt[f+g*x]],(c*f+g*q)/(c*f-g*q)]] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] + + +Int[(f_.+g_.*x_)^n_/((d_.+e_.*x_)*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + Int[ExpandIntegrand[1/(Sqrt[f+g*x]*Sqrt[a+b*x+c*x^2]),(f+g*x)^(n+1/2)/(d+e*x),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && IntegerQ[n+1/2] + + +Int[(f_.+g_.*x_)^n_/((d_.+e_.*x_)*Sqrt[a_+c_.*x_^2]),x_Symbol] := + Int[ExpandIntegrand[1/(Sqrt[f+g*x]*Sqrt[a+c*x^2]),(f+g*x)^(n+1/2)/(d+e*x),x],x] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && IntegerQ[n+1/2] + + +Int[1/(Sqrt[d_.+e_.*x_]*Sqrt[f_.+g_.*x_]*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + -2*(d+e*x)*Sqrt[(e*f-d*g)^2*(a+b*x+c*x^2)/((c*f^2-b*f*g+a*g^2)*(d+e*x)^2)]/((e*f-d*g)*Sqrt[a+b*x+c*x^2])* + Subst[ + Int[1/Sqrt[1-(2*c*d*f-b*e*f-b*d*g+2*a*e*g)*x^2/(c*f^2-b*f*g+a*g^2)+(c*d^2-b*d*e+a*e^2)*x^4/(c*f^2-b*f*g+a*g^2)],x], + x, + Sqrt[f+g*x]/Sqrt[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[1/(Sqrt[d_.+e_.*x_]*Sqrt[f_.+g_.*x_]*Sqrt[a_+c_.*x_^2]),x_Symbol] := + -2*(d+e*x)*Sqrt[(e*f-d*g)^2*(a+c*x^2)/((c*f^2+a*g^2)*(d+e*x)^2)]/((e*f-d*g)*Sqrt[a+c*x^2])* + Subst[ + Int[1/Sqrt[1-(2*c*d*f+2*a*e*g)*x^2/(c*f^2+a*g^2)+(c*d^2+a*e^2)*x^4/(c*f^2+a*g^2)],x],x,Sqrt[f+g*x]/Sqrt[d+e*x]] /; +FreeQ[{a,c,d,e,f,g},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] + + +Int[(e_.*x_)^m_*(f_+g_.*x_)^2*(a_+c_.*x_^2)^p_.,x_Symbol] := + 2*f*g/e*Int[(e*x)^(m+1)*(a+c*x^2)^p,x] + Int[(e*x)^m*(f^2+g^2*x^2)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,e,f,g,m,p},x] + + +Int[(e_.*x_)^m_*(f_+g_.*x_)^3*(a_+c_.*x_^2)^p_.,x_Symbol] := + f*Int[(e*x)^m*(f^2+3*g^2*x^2)*(a+c*x^2)^p,x] + g/e*Int[(e*x)^(m+1)*(3*f^2+g^2*x^2)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,e,f,g,m,p},x] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + g/e*Int[(d+e*x)^(m+1)*(f+g*x)^(n-1)*(a+b*x+c*x^2)^p,x] + + (e*f-d*g)/e*Int[(d+e*x)^m*(f+g*x)^(n-1)*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + g/e*Int[(d+e*x)^(m+1)*(f+g*x)^(n-1)*(a+c*x^2)^p,x] + + (e*f-d*g)/e*Int[(d+e*x)^m*(f+g*x)^(n-1)*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Defer[Int][(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +Int[(d_.+e_.*x_)^m_*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Defer[Int][(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] + + +Int[(d_.+e_.*u_)^m_.*(f_.+g_.*u_)^n_.*(a_+b_.*u_+c_.*u_^2)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p,x],x,u] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(d_.+e_.*u_)^m_.*(f_.+g_.*u_)^n_.*(a_+c_.*u_^2)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^p,x],x,u] /; +FreeQ[{a,c,d,e,f,g,m,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q*) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.,x_Symbol] := + (c/f)^p*Int[(d+e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && (IntegerQ[p] || PositiveQ[c/f]) && + (Not[IntegerQ[q]] || LeafCount[d+e*x+f*x^2]<=LeafCount[a+b*x+c*x^2]) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.,x_Symbol] := + a^IntPart[p]*(a+b*x+c*x^2)^FracPart[p]/(d^IntPart[p]*(d+e*x+f*x^2)^FracPart[p])*Int[(d+e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[PositiveQ[c/f]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_.,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_.,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,f,p,q},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +(* Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + 1/(2^(2*p+2*q+1)*c*(-c/(b^2-4*a*c))^p*(-f/(e^2-4*d*f))^q)* + Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x,b+2*c*x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] && + (IntegerQ[p] || PositiveQ[-c/(b^2-4*a*c)]) && (IntegerQ[q] || PositiveQ[-f/(e^2-4*d*f)]) *) + + +(* Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (d+e*x+f*x^2)^q/(2^(2*p+2*q+1)*c*(-c/(b^2-4*a*c))^p*(-f*(d+e*x+f*x^2)/(e^2-4*d*f))^q)* + Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x,b+2*c*x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] && + (IntegerQ[p] || PositiveQ[-c/(b^2-4*a*c)]) && Not[IntegerQ[q] || PositiveQ[-f/(e^2-4*d*f)]] *) + + +(* Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q/(2^(2*p+2*q+1)*c*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^p*(-f*(d+e*x+f*x^2)/(e^2-4*d*f))^q)* + Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x,b+2*c*x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] *) + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (b+2*c*x)*(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/((b^2-4*a*c)*(p+1)) - + (1/((b^2-4*a*c)*(p+1)))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[2*c*d*(2*p+3)+b*e*q+(2*b*f*q+2*c*e*(2*p+q+3))*x+2*c*f*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + (b+2*c*x)*(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q/((b^2-4*a*c)*(p+1)) - + (1/((b^2-4*a*c)*(p+1)))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q-1)* + Simp[2*c*d*(2*p+3)+(2*b*f*q)*x+2*c*f*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_.+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (2*c*x)*(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/((-4*a*c)*(p+1)) - + (1/((-4*a*c)*(p+1)))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[2*c*d*(2*p+3)+(2*c*e*(2*p+q+3))*x+2*c*f*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,c,d,e,f},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (2*a*c^2*e-b^2*c*e+b^3*f+b*c*(c*d-3*a*f)+c*(2*c^2*d+b^2*f-c*(b*e+2*a*f))*x)*(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/ + ((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)) - + (1/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[2*c*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)- + (2*c^2*d+b^2*f-c*(b*e+2*a*f))*(a*f*(p+1)-c*d*(p+2))- + e*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f))*(p+q+2)+ + (2*f*(2*a*c^2*e-b^2*c*e+b^3*f+b*c*(c*d-3*a*f))*(p+q+2)-(2*c^2*d+b^2*f-c*(b*e+2*a*f))*(b*f*(p+1)-c*e*(2*p+q+4)))*x+ + c*f*(2*c^2*d+b^2*f-c*(b*e+2*a*f))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,e,f,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && + NeQ[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f)] && Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + (b^3*f+b*c*(c*d-3*a*f)+c*(2*c^2*d+b^2*f-c*(2*a*f))*x)*(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q+1)/ + ((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1)) - + (1/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1)))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q* + Simp[2*c*(b^2*d*f+(c*d-a*f)^2)*(p+1)- + (2*c^2*d+b^2*f-c*(2*a*f))*(a*f*(p+1)-c*d*(p+2))+ + (2*f*(b^3*f+b*c*(c*d-3*a*f))*(p+q+2)-(2*c^2*d+b^2*f-c*(2*a*f))*(b*f*(p+1)))*x+ + c*f*(2*c^2*d+b^2*f-c*(2*a*f))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,f,q},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && NeQ[b^2*d*f+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_.+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (2*a*c^2*e+c*(2*c^2*d-c*(2*a*f))*x)*(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/ + ((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1)) - + (1/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1)))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[2*c*((c*d-a*f)^2-(-a*e)*(c*e))*(p+1)-(2*c^2*d-c*(2*a*f))*(a*f*(p+1)-c*d*(p+2))-e*(-2*a*c^2*e)*(p+q+2)+ + (2*f*(2*a*c^2*e)*(p+q+2)-(2*c^2*d-c*(2*a*f))*(-c*e*(2*p+q+4)))*x+ + c*f*(2*c^2*d-c*(2*a*f))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,c,d,e,f,q},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && NeQ[a*c*e^2+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (b*f*(3*p+2*q)-c*e*(2*p+q)+2*c*f*(p+q)*x)*(a+b*x+c*x^2)^(p-1)*(d+e*x+f*x^2)^(q+1)/(2*f^2*(p+q)*(2*p+2*q+1)) - + 1/(2*f^2*(p+q)*(2*p+2*q+1))* + Int[(a+b*x+c*x^2)^(p-2)*(d+e*x+f*x^2)^q* + Simp[(b*d-a*e)*(c*e-b*f)*(1-p)*(2*p+q)- + (p+q)*(b^2*d*f*(1-p)-a*(f*(b*e-2*a*f)*(2*p+2*q+1)+c*(2*d*f-e^2*(2*p+q))))+ + (2*(c*d-a*f)*(c*e-b*f)*(1-p)*(2*p+q)- + (p+q)*((b^2-4*a*c)*e*f*(1-p)+b*(c*(e^2-4*d*f)*(2*p+q)+f*(2*c*d-b*e+2*a*f)*(2*p+2*q+1))))*x+ + ((c*e-b*f)^2*(1-p)*p+c*(p+q)*(f*(b*e-2*a*f)*(4*p+2*q-1)-c*(2*d*f*(1-2*p)+e^2*(3*p+q-1))))*x^2,x],x]/; +FreeQ[{a,b,c,d,e,f,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && GtQ[p,1] && + NeQ[p+q] && NeQ[2*p+2*q+1] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + (b*(3*p+2*q)+2*c*(p+q)*x)*(a+b*x+c*x^2)^(p-1)*(d+f*x^2)^(q+1)/(2*f*(p+q)*(2*p+2*q+1)) - + 1/(2*f*(p+q)*(2*p+2*q+1))* + Int[(a+b*x+c*x^2)^(p-2)*(d+f*x^2)^q* + Simp[b^2*d*(p-1)*(2*p+q)-(p+q)*(b^2*d*(1-p)-2*a*(c*d-a*f*(2*p+2*q+1)))- + (2*b*(c*d-a*f)*(1-p)*(2*p+q)-2*(p+q)*b*(2*c*d*(2*p+q)-(c*d+a*f)*(2*p+2*q+1)))*x+ + (b^2*f*p*(1-p)+2*c*(p+q)*(c*d*(2*p-1)-a*f*(4*p+2*q-1)))*x^2,x],x]/; +FreeQ[{a,b,c,d,f,q},x] && NeQ[b^2-4*a*c] && GtQ[p,1] && NeQ[p+q] && NeQ[2*p+2*q+1] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_.+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + -c*(e*(2*p+q)-2*f*(p+q)*x)*(a+c*x^2)^(p-1)*(d+e*x+f*x^2)^(q+1)/(2*f^2*(p+q)*(2*p+2*q+1)) - + 1/(2*f^2*(p+q)*(2*p+2*q+1))* + Int[(a+c*x^2)^(p-2)*(d+e*x+f*x^2)^q* + Simp[-a*c*e^2*(1-p)*(2*p+q)+a*(p+q)*(-2*a*f^2*(2*p+2*q+1)+c*(2*d*f-e^2*(2*p+q)))+ + (2*(c*d-a*f)*(c*e)*(1-p)*(2*p+q)+4*a*c*e*f*(1-p)*(p+q))*x+ + (p*c^2*e^2*(1-p)-c*(p+q)*(2*a*f^2*(4*p+2*q-1)+c*(2*d*f*(1-2*p)+e^2*(3*p+q-1))))*x^2,x],x]/; +FreeQ[{a,c,d,e,f,q},x] && NeQ[e^2-4*d*f] && GtQ[p,1] && NeQ[p+q] && NeQ[2*p+2*q+1] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[1/((a_+b_.*x_+c_.*x_^2)*(d_+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2-b*c*d*e+a*c*e^2+b^2*d*f-2*a*c*d*f-a*b*e*f+a^2*f^2}, + 1/q*Int[(c^2*d-b*c*e+b^2*f-a*c*f-(c^2*e-b*c*f)*x)/(a+b*x+c*x^2),x] + + 1/q*Int[(c*e^2-c*d*f-b*e*f+a*f^2+(c*e*f-b*f^2)*x)/(d+e*x+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[1/((a_+b_.*x_+c_.*x_^2)*(d_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2+b^2*d*f-2*a*c*d*f+a^2*f^2}, + 1/q*Int[(c^2*d+b^2*f-a*c*f+b*c*f*x)/(a+b*x+c*x^2),x] - + 1/q*Int[(c*d*f-a*f^2+b*f^2*x)/(d+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] + + +Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + -2*e*Subst[Int[1/(e*(b*e-4*a*f)-(b*d-a*e)*x^2),x],x,(e+2*f*x)/Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] + + +(* Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{k=Rt[(a/c-d/f)^2+(b/c-e/f)*(b*d/(c*f)-a*e/(c*f)),2]}, + -2*(c*d-a*f+c*f*k+(c*e-b*f)*x)*Sqrt[(d+e*x+f*x^2)*((c*f*k)/(c*d-a*f+c*f*k+(c*e-b*f)*x))^2]/(c*Sqrt[d+e*x+f*x^2])* + Subst[Int[(1-x)/( + (b*d-a*e-b*f*k-(c*d-a*f-c*f*k)^2/(c*e-b*f)+(b*d-a*e+b*f*k-(a*f-c*d-c*f*k)^2/(c*e-b*f))*x^2)* + Sqrt[-f*((b*d-a*e-c*e*k)/(c*e-b*f)-(c*d-a*f-c*f*k)^2/(c*e-b*f)^2)-f*((b*d-a*e+c*e*k)/(c*e-b*f)-(a*f-c*d-c*f*k)^2/(c*e-b*f)^2)*x^2]),x],x, + (c*d-a*f-c*f*k+(c*e-b*f)*x)/(c*d-a*f+c*f*k+(c*e-b*f)*x)]] /; +FreeQ[{a,b,c,d,e,f},x] && RationalQ[a,b,c,d,e,f] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && NeQ[c*e-b*f] && NegativeQ[b^2-4*a*c] *) + + +(* Int[1/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{k=Rt[(a/c-d/f)^2+a*e^2/(c*f^2),2]}, + -2*(c*d-a*f+c*f*k+c*e*x)*Sqrt[(d+e*x+f*x^2)*((c*f*k)/(c*d-a*f+c*f*k+c*e*x))^2]/(c*Sqrt[d+e*x+f*x^2])* + Subst[Int[(1-x)/( + (-a*e-(c*d-a*f-c*f*k)^2/(c*e)+(-a*e-(a*f-c*d-c*f*k)^2/(c*e))*x^2)* + Sqrt[-f*((-a*e-c*e*k)/(c*e)-(c*d-a*f-c*f*k)^2/(c*e)^2)-f*((-a*e+c*e*k)/(c*e)-(a*f-c*d-c*f*k)^2/(c*e)^2)*x^2]),x],x, + (c*d-a*f-c*f*k+(c*e)*x)/(c*d-a*f+c*f*k+(c*e)*x)]] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[a,c,d,e,f] && NeQ[e^2-4*d*f] && NegativeQ[-a*c] *) + + +(* Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + With[{k=Rt[(a/c-d/f)^2+b^2*d/(c^2*f),2]}, + -2*(c*d-a*f+c*f*k-b*f*x)*Sqrt[(d+f*x^2)*((c*f*k)/(c*d-a*f+c*f*k-b*f*x))^2]/(c*Sqrt[d+f*x^2])* + Subst[Int[(1-x)/( + (b*d-b*f*k+(c*d-a*f-c*f*k)^2/(b*f)+(b*d+b*f*k+(a*f-c*d-c*f*k)^2/(b*f))*x^2)* + Sqrt[-f*(-d/f-(c*d-a*f-c*f*k)^2/(b*f)^2)-f*(-d/f-(a*f-c*d-c*f*k)^2/(b*f)^2)*x^2]),x],x, + (c*d-a*f-c*f*k+(-b*f)*x)/(c*d-a*f+c*f*k+(-b*f)*x)]] /; +FreeQ[{a,b,c,d,f},x] && RationalQ[a,b,c,d,f] && NeQ[b^2-4*a*c] && NegativeQ[b^2-4*a*c] *) + + +Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/((b-q+2*c*x)*Sqrt[d+e*x+f*x^2]),x] - + 2*c/q*Int[1/((b+q+2*c*x)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && NeQ[c*e-b*f] && PosQ[b^2-4*a*c] + + +Int[1/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + 1/2*Int[1/((a-Rt[-a*c,2]*x)*Sqrt[d+e*x+f*x^2]),x] + + 1/2*Int[1/((a+Rt[-a*c,2]*x)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,c,d,e,f},x] && NeQ[e^2-4*d*f] && PosQ[-a*c] + + +Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/((b-q+2*c*x)*Sqrt[d+f*x^2]),x] - + 2*c/q*Int[1/((b+q+2*c*x)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] && PosQ[b^2-4*a*c] + + +Int[1/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f),2]}, + 1/(2*q)*Int[(c*d-a*f+q+(c*e-b*f)*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] - + 1/(2*q)*Int[(c*d-a*f-q+(c*e-b*f)*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && NeQ[c*e-b*f] && NegQ[b^2-4*a*c] + + +Int[1/((a_.+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2+a*c*e^2,2]}, + 1/(2*q)*Int[(c*d-a*f+q+c*e*x)/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x] - + 1/(2*q)*Int[(c*d-a*f-q+c*e*x)/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,c,d,e,f},x] && NeQ[e^2-4*d*f] && NegQ[-a*c] + + +Int[1/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2+b^2*d*f,2]}, + 1/(2*q)*Int[(c*d-a*f+q+(-b*f)*x)/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x] - + 1/(2*q)*Int[(c*d-a*f-q+(-b*f)*x)/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] && NegQ[b^2-4*a*c] + + +Int[Sqrt[a_+b_.*x_+c_.*x_^2]/(d_+e_.*x_+f_.*x_^2),x_Symbol] := + c/f*Int[1/Sqrt[a+b*x+c*x^2],x] - + 1/f*Int[(c*d-a*f+(c*e-b*f)*x)/(Sqrt[a+b*x+c*x^2]*(d+e*x+f*x^2)),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[Sqrt[a_+b_.*x_+c_.*x_^2]/(d_+f_.*x_^2),x_Symbol] := + c/f*Int[1/Sqrt[a+b*x+c*x^2],x] - + 1/f*Int[(c*d-a*f-b*f*x)/(Sqrt[a+b*x+c*x^2]*(d+f*x^2)),x] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] + + +Int[Sqrt[a_+c_.*x_^2]/(d_+e_.*x_+f_.*x_^2),x_Symbol] := + c/f*Int[1/Sqrt[a+c*x^2],x] - + 1/f*Int[(c*d-a*f+c*e*x)/(Sqrt[a+c*x^2]*(d+e*x+f*x^2)),x] /; +FreeQ[{a,c,d,e,f},x] && NeQ[e^2-4*d*f] + + +Int[1/(Sqrt[a_+b_.*x_+c_.*x_^2]*Sqrt[d_+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{r=Rt[b^2-4*a*c,2]}, + Sqrt[b+r+2*c*x]*Sqrt[2*a+(b+r)*x]/Sqrt[a+b*x+c*x^2]*Int[1/(Sqrt[b+r+2*c*x]*Sqrt[2*a+(b+r)*x]*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[1/(Sqrt[a_+b_.*x_+c_.*x_^2]*Sqrt[d_+f_.*x_^2]),x_Symbol] := + With[{r=Rt[b^2-4*a*c,2]}, + Sqrt[b+r+2*c*x]*Sqrt[2*a+(b+r)*x]/Sqrt[a+b*x+c*x^2]*Int[1/(Sqrt[b+r+2*c*x]*Sqrt[2*a+(b+r)*x]*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,c,d,f},x] && NeQ[b^2-4*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Defer[Int][(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Defer[Int][(a+c*x^2)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,c,d,e,f,p,q},x] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_.+b_.*u_+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q,x],x,u] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+c*x^2)^p*(d+e*x+f*x^2)^q,x],x,u] /; +FreeQ[{a,c,d,e,f,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q*) + + +Int[(g_.+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (c/f)^p*Int[(g+h*x)^m*(d+e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && (IntegerQ[p] || PositiveQ[c/f]) && + (Not[IntegerQ[q]] || LeafCount[d+e*x+f*x^2]<=LeafCount[a+b*x+c*x^2]) + + +Int[(g_.+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + a^IntPart[p]*(a+b*x+c*x^2)^FracPart[p]/(d^IntPart[p]*(d+e*x+f*x^2)^FracPart[p])*Int[(g+h*x)^m*(d+e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[PositiveQ[c/f]] + + +Int[(g_.+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(g+h*x)^m*(b+2*c*x)^(2*p)*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(g+h*x)^m*(b+2*c*x)^(2*p)*(d+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,f,g,h,m,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(g_+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^m_.,x_Symbol] := + Int[(d*g/a+f*h*x/c)^m*(a+b*x+c*x^2)^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p},x] && EqQ[c*g^2-b*g*h+a*h^2] && EqQ[c^2*d*g^2-a*c*e*g*h+a^2*f*h^2] && IntegerQ[m] + + +Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^m_.,x_Symbol] := + Int[(d*g/a+f*h*x/c)^m*(a+c*x^2)^(m+p),x] /; +FreeQ[{a,c,d,e,f,g,h,p},x] && EqQ[c*g^2+a*h^2] && EqQ[c^2*d*g^2-a*c*e*g*h+a^2*f*h^2] && IntegerQ[m] + + +Int[(g_+h_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^m_.,x_Symbol] := + Int[(d*g/a+f*h*x/c)^m*(a+b*x+c*x^2)^(m+p),x] /; +FreeQ[{a,b,c,d,f,g,h,p},x] && EqQ[c*g^2-b*g*h+a*h^2] && EqQ[c^2*d*g^2+a^2*f*h^2] && IntegerQ[m] + + +Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^m_.,x_Symbol] := + Int[(d*g/a+f*h*x/c)^m*(a+c*x^2)^(m+p),x] /; +FreeQ[{a,c,d,f,g,h,p},x] && EqQ[c*g^2+a*h^2] && EqQ[c^2*d*g^2+a^2*f*h^2] && IntegerQ[m] + + +(* Int[(g_+h_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,q},x] && EqQ[c*g^2-b*g*h+a*h^2] && IntegerQ[p] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,c,d,e,f,g,h,m,q},x] && NeQ[e^2-4*d*f] && EqQ[c*g^2+a*h^2] && IntegerQ[p] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,f,g,h,m,q},x] && NeQ[b^2-4*a*c] && EqQ[c*g^2-b*g*h+a*h^2] && IntegerQ[p] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x] /; +FreeQ[{a,c,d,f,g,h,m,q},x] && EqQ[c*g^2+a*h^2] && IntegerQ[p] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[p])*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,q},x] && EqQ[c*g^2-b*g*h+a*h^2] && Not[IntegerQ[p]] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol] := + (a+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[p])*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+e*x+f*x^2)^q,x] /; +FreeQ[{a,c,d,e,f,g,h,m,q},x] && NeQ[e^2-4*d*f] && EqQ[c*g^2+a*h^2] && Not[IntegerQ[p]] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[p])*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x] /; +FreeQ[{a,b,c,d,f,g,h,m,q},x] && NeQ[b^2-4*a*c] && EqQ[c*g^2-b*g*h+a*h^2] && Not[IntegerQ[p]] *) + + +(* Int[(g_+h_.*x_)^m_.*(a_+c_.*x_^2)^p_*(d_.+f_.*x_^2)^q_,x_Symbol] := + (a+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[p])*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x] /; +FreeQ[{a,c,d,f,g,h,m,q},x] && EqQ[c*g^2+a*h^2] && Not[IntegerQ[p]] *) + + +Int[x_^p_*(a_.+b_.*x_+c_.*x_^2)^p_*(e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Int[(a/e+c/f*x)^p*(e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,b,c,e,f,q},x] && NeQ[b^2-4*a*c] && EqQ[c*e^2-b*e*f+a*f^2] && IntegerQ[p] + + +Int[x_^p_*(a_+c_.*x_^2)^p_*(e_.*x_+f_.*x_^2)^q_,x_Symbol] := + Int[(a/e+c/f*x)^p*(e*x+f*x^2)^(p+q),x] /; +FreeQ[{a,c,e,f,q},x] && EqQ[c*e^2+a*f^2] && IntegerQ[p] + + +Int[(g_+h_.*x_)/((a_+c_.*x_^2)^(1/3)*(d_+f_.*x_^2)),x_Symbol] := + Sqrt[3]*h*ArcTan[1/Sqrt[3]-2^(2/3)*(1-3*h*x/g)^(2/3)/(Sqrt[3]*(1+3*h*x/g)^(1/3))]/(2^(2/3)*a^(1/3)*f) + + h*Log[d+f*x^2]/(2^(5/3)*a^(1/3)*f) - + 3*h*Log[(1-3*h*x/g)^(2/3)+2^(1/3)*(1+3*h*x/g)^(1/3)]/(2^(5/3)*a^(1/3)*f) /; +FreeQ[{a,c,d,f,g,h},x] && EqQ[c*d+3*a*f] && EqQ[c*g^2+9*a*h^2] && PositiveQ[a] + + +Int[(g_+h_.*x_)/((a_+c_.*x_^2)^(1/3)*(d_+f_.*x_^2)),x_Symbol] := + (1+c*x^2/a)^(1/3)/(a+c*x^2)^(1/3)*Int[(g+h*x)/((1+c*x^2/a)^(1/3)*(d+f*x^2)),x] /; +FreeQ[{a,c,d,f,g,h},x] && EqQ[c*d+3*a*f] && EqQ[c*g^2+9*a*h^2] && Not[PositiveQ[a]] + + +Int[(g_+h_.*x_)*(a_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_,x_Symbol] := + g*Int[(a+c*x^2)^p*(d+f*x^2)^q,x] + h*Int[x*(a+c*x^2)^p*(d+f*x^2)^q,x] /; +FreeQ[{a,c,d,f,g,h,p,q},x] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + Int[ExpandIntegrand[(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*(g+h*x),x],x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && IntegersQ[p,q] && p>0 + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + Int[ExpandIntegrand[(a+c*x^2)^p*(d+e*x+f*x^2)^q*(g+h*x),x],x] /; +FreeQ[{a,c,d,e,f,g,h},x] && NeQ[e^2-4*d*f] && IntegersQ[p,q] && (p>0 || q>0) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (g*b-2*a*h-(b*h-2*g*c)*x)*(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/((b^2-4*a*c)*(p+1)) - + 1/((b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[e*q*(g*b-2*a*h)-d*(b*h-2*g*c)*(2*p+3)+ + (2*f*q*(g*b-2*a*h)-e*(b*h-2*g*c)*(2*p+q+3))*x- + f*(b*h-2*g*c)*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && RationalQ[p,q] && p<-1 && q>0 + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (a*h-g*c*x)*(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/(2*a*c*(p+1)) + + 2/(4*a*c*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[g*c*d*(2*p+3)-a*(h*e*q)+(g*c*e*(2*p+q+3)-a*(2*h*f*q))*x+g*c*f*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,c,d,e,f,g,h},x] && NeQ[e^2-4*d*f] && RationalQ[p,q] && p<-1 && q>0 + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (g*b-2*a*h-(b*h-2*g*c)*x)*(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q/((b^2-4*a*c)*(p+1)) - + 1/((b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q-1)* + Simp[-d*(b*h-2*g*c)*(2*p+3)+(2*f*q*(g*b-2*a*h))*x-f*(b*h-2*g*c)*(2*p+2*q+3)*x^2,x],x] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] && RationalQ[p,q] && p<-1 && q>0 + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + (g*c*(2*a*c*e-b*(c*d+a*f))+(g*b-a*h)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+ + c*(g*(2*c^2*d+b^2*f-c*(b*e+2*a*f))-h*(b*c*d-2*a*c*e+a*b*f))*x) + + 1/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(b*h-2*g*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)+ + (b^2*(g*f)-b*(h*c*d+g*c*e+a*h*f)+2*(g*c*(c*d-a*f)-a*(-h*c*e)))*(a*f*(p+1)-c*d*(p+2))- + e*((g*c)*(2*a*c*e-b*(c*d+a*f))+(g*b-a*h)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (2*f*((g*c)*(2*a*c*e-b*(c*d+a*f))+(g*b-a*h)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (b^2*g*f-b*(h*c*d+g*c*e+a*h*f)+2*(g*c*(c*d-a*f)-a*(-h*c*e)))* + (b*f*(p+1)-c*e*(2*p+q+4)))*x- + c*f*(b^2*(g*f)-b*(h*c*d+g*c*e+a*h*f)+2*(g*c*(c*d-a*f)+a*h*c*e))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,e,f,g,h,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && RationalQ[p] && p<-1 && + NeQ[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f)] && Not[Not[IntegerQ[p]] && IntegerQ[q] && q<-1] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + (g*c*(2*a*c*e)+(-a*h)*(2*c^2*d-c*(2*a*f))+ + c*(g*(2*c^2*d-c*(2*a*f))-h*(-2*a*c*e))*x) + + 1/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(-2*g*c)*((c*d-a*f)^2-(-a*e)*(c*e))*(p+1)+ + (2*(g*c*(c*d-a*f)-a*(-h*c*e)))*(a*f*(p+1)-c*d*(p+2))- + e*((g*c)*(2*a*c*e)+(-a*h)*(2*c^2*d-c*(+2*a*f)))*(p+q+2)- + (2*f*((g*c)*(2*a*c*e)+(-a*h)*(2*c^2*d+-c*(+2*a*f)))*(p+q+2)-(2*(g*c*(c*d-a*f)-a*(-h*c*e)))*(-c*e*(2*p+q+4)))*x- + c*f*(2*(g*c*(c*d-a*f)-a*(-h*c*e)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,c,d,e,f,g,h,q},x] && NeQ[e^2-4*d*f] && RationalQ[p] && p<-1 && NeQ[a*c*e^2+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && IntegerQ[q] && q<-1] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q+1)/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + ((g*c)*(-b*(c*d+a*f))+(g*b-a*h)*(2*c^2*d+b^2*f-c*(2*a*f))+ + c*(g*(2*c^2*d+b^2*f-c*(2*a*f))-h*(b*c*d+a*b*f))*x) + + 1/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q* + Simp[(b*h-2*g*c)*((c*d-a*f)^2-(b*d)*(-b*f))*(p+1)+ + (b^2*(g*f)-b*(h*c*d+a*h*f)+2*(g*c*(c*d-a*f)))*(a*f*(p+1)-c*d*(p+2))- + (2*f*((g*c)*(-b*(c*d+a*f))+(g*b-a*h)*(2*c^2*d+b^2*f-c*(2*a*f)))*(p+q+2)- + (b^2*(g*f)-b*(h*c*d+a*h*f)+2*(g*c*(c*d-a*f)))* + (b*f*(p+1)))*x- + c*f*(b^2*(g*f)-b*(h*c*d+a*h*f)+2*(g*c*(c*d-a*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,f,g,h,q},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && NeQ[b^2*d*f+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && IntegerQ[q] && q<-1] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + h*(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^(q+1)/(2*f*(p+q+1)) - + (1/(2*f*(p+q+1)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[h*p*(b*d-a*e)+a*(h*e-2*g*f)*(p+q+1)+ + (2*h*p*(c*d-a*f)+b*(h*e-2*g*f)*(p+q+1))*x+ + (h*p*(c*e-b*f)+c*(h*e-2*g*f)*(p+q+1))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && RationalQ[p] && p>0 && NeQ[p+q+1] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + h*(a+c*x^2)^p*(d+e*x+f*x^2)^(q+1)/(2*f*(p+q+1)) + + (1/(2*f*(p+q+1)))* + Int[(a+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[a*h*e*p-a*(h*e-2*g*f)*(p+q+1)-2*h*p*(c*d-a*f)*x-(h*c*e*p+c*(h*e-2*g*f)*(p+q+1))*x^2,x],x] /; +FreeQ[{a,c,d,e,f,g,h,q},x] && NeQ[e^2-4*d*f] && RationalQ[p] && p>0 && NeQ[p+q+1] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + h*(a+b*x+c*x^2)^p*(d+f*x^2)^(q+1)/(2*f*(p+q+1)) - + (1/(2*f*(p+q+1)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+f*x^2)^q* + Simp[h*p*(b*d)+a*(-2*g*f)*(p+q+1)+ + (2*h*p*(c*d-a*f)+b*(-2*g*f)*(p+q+1))*x+ + (h*p*(-b*f)+c*(-2*g*f)*(p+q+1))*x^2,x],x] /; +FreeQ[{a,b,c,d,f,g,h,q},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p>0 && NeQ[p+q+1] + + +Int[(g_.+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*(d_+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=Simplify[c^2*d^2-b*c*d*e+a*c*e^2+b^2*d*f-2*a*c*d*f-a*b*e*f+a^2*f^2]}, + 1/q*Int[Simp[g*c^2*d-g*b*c*e+a*h*c*e+g*b^2*f-a*b*h*f-a*g*c*f+c*(h*c*d-g*c*e+g*b*f-a*h*f)*x,x]/(a+b*x+c*x^2),x] + + 1/q*Int[Simp[-h*c*d*e+g*c*e^2+b*h*d*f-g*c*d*f-g*b*e*f+a*g*f^2-f*(h*c*d-g*c*e+g*b*f-a*h*f)*x,x]/(d+e*x+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(g_.+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*(d_+f_.*x_^2)),x_Symbol] := + With[{q=Simplify[c^2*d^2+b^2*d*f-2*a*c*d*f+a^2*f^2]}, + 1/q*Int[Simp[g*c^2*d+g*b^2*f-a*b*h*f-a*g*c*f+c*(h*c*d+g*b*f-a*h*f)*x,x]/(a+b*x+c*x^2),x] + + 1/q*Int[Simp[b*h*d*f-g*c*d*f+a*g*f^2-f*(h*c*d+g*b*f-a*h*f)*x,x]/(d+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] + + +Int[(g_+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + -2*g*Subst[Int[1/(b*d-a*e-b*x^2),x],x,Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] && EqQ[h*e-2*g*f] + + +Int[(g_.+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + -(h*e-2*g*f)/(2*f)*Int[1/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] + + h/(2*f)*Int[(e+2*f*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[c*e-b*f] && NeQ[h*e-2*g*f] + + +Int[x_/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_+e_.*x_+f_.*x_^2]),x_Symbol] := + -2*e*Subst[Int[(1-d*x^2)/(c*e-b*f-e*(2*c*d-b*e+2*a*f)*x^2+d^2*(c*e-b*f)*x^4),x],x, + (1+(e+Sqrt[e^2-4*d*f])*x/(2*d))/Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[b*d-a*e] + + +Int[(g_+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_+e_.*x_+f_.*x_^2]),x_Symbol] := + g*Subst[Int[1/(a+(c*d-a*f)*x^2),x],x,x/Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[b*d-a*e] && EqQ[2*h*d-g*e] + + +Int[(g_+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_+e_.*x_+f_.*x_^2]),x_Symbol] := + -(2*h*d-g*e)/e*Int[1/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] + + h/e*Int[(2*d+e*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && EqQ[b*d-a*e] && NeQ[2*h*d-g*e] + + +Int[(g_.+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + -2*g*(g*b-2*a*h)* + Subst[Int[1/Simp[g*(g*b-2*a*h)*(b^2-4*a*c)-(b*d-a*e)*x^2,x],x],x,Simp[g*b-2*a*h-(b*h-2*g*c)*x,x]/Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && NeQ[b*d-a*e] && + EqQ[h^2*(b*d-a*e)-2*g*h*(c*d-a*f)+g^2*(c*e-b*f)] + + +Int[(g_+h_.*x_)/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + -2*a*g*h*Subst[Int[1/Simp[2*a^2*g*h*c+a*e*x^2,x],x],x,Simp[a*h-g*c*x,x]/Sqrt[d+e*x+f*x^2]] /; +FreeQ[{a,c,d,e,f,g,h},x] && EqQ[a*h^2*e+2*g*h*(c*d-a*f)-g^2*c*e] + + +Int[(g_+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_+f_.*x_^2]),x_Symbol] := + -2*g*(g*b-2*a*h)*Subst[Int[1/Simp[g*(g*b-2*a*h)*(b^2-4*a*c)-b*d*x^2,x],x],x,Simp[g*b-2*a*h-(b*h-2*g*c)*x,x]/Sqrt[d+f*x^2]] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] && EqQ[b*h^2*d-2*g*h*(c*d-a*f)-g^2*b*f] + + +Int[(g_.+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*g-h*(b-q))/q*Int[1/((b-q+2*c*x)*Sqrt[d+e*x+f*x^2]),x] - + (2*c*g-h*(b+q))/q*Int[1/((b+q+2*c*x)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && PosQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[-a*c,2]}, + (h/2+c*g/(2*q))*Int[1/((-q+c*x)*Sqrt[d+e*x+f*x^2]),x] + + (h/2-c*g/(2*q))*Int[1/((q+c*x)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,c,d,e,f,g,h},x] && NeQ[e^2-4*d*f] && PosQ[-a*c] + + +Int[(g_.+h_.*x_)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*g-h*(b-q))/q*Int[1/((b-q+2*c*x)*Sqrt[d+f*x^2]),x] - + (2*c*g-h*(b+q))/q*Int[1/((b+q+2*c*x)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] && PosQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f),2]}, + 1/(2*q)*Int[Simp[h*(b*d-a*e)-g*(c*d-a*f-q)-(g*(c*e-b*f)-h*(c*d-a*f+q))*x,x]/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] - + 1/(2*q)*Int[Simp[h*(b*d-a*e)-g*(c*d-a*f+q)-(g*(c*e-b*f)-h*(c*d-a*f-q))*x,x]/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && NeQ[b*d-a*e] && NegQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2+a*c*e^2,2]}, + 1/(2*q)*Int[Simp[-a*h*e-g*(c*d-a*f-q)+(h*(c*d-a*f+q)-g*c*e)*x,x]/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x] - + 1/(2*q)*Int[Simp[-a*h*e-g*(c*d-a*f+q)+(h*(c*d-a*f-q)-g*c*e)*x,x]/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,c,d,e,f,g,h},x] && NeQ[e^2-4*d*f] && NegQ[-a*c] + + +Int[(g_.+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)*Sqrt[d_+f_.*x_^2]),x_Symbol] := + With[{q=Rt[(c*d-a*f)^2+b^2*d*f,2]}, + 1/(2*q)*Int[Simp[h*b*d-g*(c*d-a*f-q)+(h*(c*d-a*f+q)+g*b*f)*x,x]/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x] - + 1/(2*q)*Int[Simp[h*b*d-g*(c*d-a*f+q)+(h*(c*d-a*f-q)+g*b*f)*x,x]/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] && NegQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)/(Sqrt[a_+b_.*x_+c_.*x_^2]*Sqrt[d_+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{s=Rt[b^2-4*a*c,2],t=Rt[e^2-4*d*f,2]}, + Sqrt[b+s+2*c*x]*Sqrt[2*a+(b+s)*x]*Sqrt[e+t+2*f*x]*Sqrt[2*d+(e+t)*x]/(Sqrt[a+b*x+c*x^2]*Sqrt[d+e*x+f*x^2])* + Int[(g+h*x)/(Sqrt[b+s+2*c*x]*Sqrt[2*a+(b+s)*x]*Sqrt[e+t+2*f*x]*Sqrt[2*d+(e+t)*x]),x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(g_.+h_.*x_)/(Sqrt[a_+b_.*x_+c_.*x_^2]*Sqrt[d_+f_.*x_^2]),x_Symbol] := + With[{s=Rt[b^2-4*a*c,2],t=Rt[-4*d*f,2]}, + Sqrt[b+s+2*c*x]*Sqrt[2*a+(b+s)*x]*Sqrt[t+2*f*x]*Sqrt[2*d+t*x]/(Sqrt[a+b*x+c*x^2]*Sqrt[d+f*x^2])* + Int[(g+h*x)/(Sqrt[b+s+2*c*x]*Sqrt[2*a+(b+s)*x]*Sqrt[t+2*f*x]*Sqrt[2*d+t*x]),x]] /; +FreeQ[{a,b,c,d,f,g,h},x] && NeQ[b^2-4*a*c] + + +Int[(g_.+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)^(1/3)*(d_.+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=(-9*c*h^2/(2*c*g-b*h)^2)^(1/3)}, + Sqrt[3]*h*q*ArcTan[1/Sqrt[3]-2^(2/3)*(1-(3*h*(b+2*c*x))/(2*c*g-b*h))^(2/3)/(Sqrt[3]*(1+(3*h*(b+2*c*x))/(2*c*g-b*h))^(1/3))]/f + + h*q*Log[d+e*x+f*x^2]/(2*f) - + 3*h*q*Log[(1-3*h*(b+2*c*x)/(2*c*g-b*h))^(2/3)+2^(1/3)*(1+3*h*(b+2*c*x)/(2*c*g-b*h))^(1/3)]/(2*f)] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[c*e-b*f] && EqQ[c^2*d-f*(b^2-3*a*c)] && EqQ[c^2*g^2-b*c*g*h-2*b^2*h^2+9*a*c*h^2] && + PositiveQ[-9*c*h^2/(2*c*g-b*h)^2] + + +Int[(g_.+h_.*x_)/((a_.+b_.*x_+c_.*x_^2)^(1/3)*(d_.+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=-c/(b^2-4*a*c)}, + (q*(a+b*x+c*x^2))^(1/3)/(a+b*x+c*x^2)^(1/3)*Int[(g+h*x)/((q*a+b*q*x+c*q*x^2)^(1/3)*(d+e*x+f*x^2)),x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[c*e-b*f] && EqQ[c^2*d-f*(b^2-3*a*c)] && EqQ[c^2*g^2-b*c*g*h-2*b^2*h^2+9*a*c*h^2] && + Not[PositiveQ[4*a-b^2/c]] + + +Int[(a_.+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + Defer[Int][(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*(g+h*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] + + +Int[(a_.+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_*(g_.+h_.*x_),x_Symbol] := + Defer[Int][(a+c*x^2)^p*(d+e*x+f*x^2)^q*(g+h*x),x] /; +FreeQ[{a,c,d,e,f,g,h,p,q},x] + + +Int[(g_.+h_.*u_)^m_.*(a_.+b_.*u_+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(g+h*x)^m*(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q,x],x,u] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(g_.+h_.*u_)^m_.*(a_.+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(g+h*x)^m*(a+c*x^2)^p*(d+e*x+f*x^2)^q,x],x,u] /; +FreeQ[{a,c,d,e,f,g,h,m,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)^n_.*(h_.+i_.*x_)^q_.*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(h+i*x)^q*(d*f+e*g*x^2)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,m,n,p,q},x] && EqQ[e*f+d*g] && EqQ[m-n] && (IntegerQ[m] || PositiveQ[d] && PositiveQ[f]) + + +Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)^n_.*(h_.+i_.*x_)^q_.*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^n*(h+i*x)^q*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,m,n,p,q},x] && PositiveIntegerQ[p] && NegativeIntegerQ[m] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^n_*(h_.+i_.*x_)^q_.*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + (d+e*x)^FracPart[m]*(f+g*x)^FracPart[m]/(d*f+e*g*x^2)^FracPart[m]*Int[(h+i*x)^q*(d*f+e*g*x^2)^m*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,m,n,p,q},x] && EqQ[e*f+d*g] && EqQ[m-n] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2)*) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (c/f)^p*Int[(d+e*x+f*x^2)^(p+q)*(A+B*x+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && (IntegerQ[p] || PositiveQ[c/f]) && + (Not[IntegerQ[q]] || LeafCount[d+e*x+f*x^2]<=LeafCount[a+b*x+c*x^2]) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.*(A_.+C_.*x_^2),x_Symbol] := + (c/f)^p*Int[(d+e*x+f*x^2)^(p+q)*(A+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && (IntegerQ[p] || PositiveQ[c/f]) && + (Not[IntegerQ[q]] || LeafCount[d+e*x+f*x^2]<=LeafCount[a+b*x+c*x^2]) + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + a^IntPart[p]*(a+b*x+c*x^2)^FracPart[p]/(d^IntPart[p]*(d+e*x+f*x^2)^FracPart[p])*Int[(d+e*x+f*x^2)^(p+q)*(A+B*x+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[PositiveQ[c/f]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_+e_.*x_+f_.*x_^2)^q_.*(A_.+C_.*x_^2),x_Symbol] := + a^IntPart[p]*(a+b*x+c*x^2)^FracPart[p]/(d^IntPart[p]*(d+e*x+f*x^2)^FracPart[p])*Int[(d+e*x+f*x^2)^(p+q)*(A+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,p,q},x] && EqQ[c*d-a*f] && EqQ[b*d-a*e] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[PositiveQ[c/f]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_.+e_.*x_+f_.*x_^2)^q_.*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+e*x+f*x^2)^q*(A+B*x+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_.+e_.*x_+f_.*x_^2)^q_.*(A_.+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+e*x+f*x^2)^q*(A+C*x^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_.+f_.*x_^2)^q_.*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+f*x^2)^q*(A+B*x+C*x^2),x] /; +FreeQ[{a,b,c,d,f,A,B,C,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_.*(d_.+f_.*x_^2)^q_.*(A_.+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(b+2*c*x)^(2*p)*(d+f*x^2)^q*(A+C*x^2),x] /; +FreeQ[{a,b,c,d,f,A,C,p,q},x] && EqQ[b^2-4*a*c] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (A*b*c-2*a*B*c+a*b*C-(c*(b*B-2*A*c)-C*(b^2-2*a*c))*x)*(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/(c*(b^2-4*a*c)*(p+1)) - + 1/(c*(b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[e*q*(A*b*c-2*a*B*c+a*b*C)-d*(c*(b*B-2*A*c)*(2*p+3)+C*(2*a*c-b^2*(p+2)))+ + (2*f*q*(A*b*c-2*a*B*c+a*b*C)-e*(c*(b*B-2*A*c)*(2*p+q+3)+C*(2*a*c*(q+1)-b^2*(p+q+2))))*x- + f*(c*(b*B-2*A*c)*(2*p+2*q+3)+C*(2*a*c*(2*q+1)-b^2*(p+2*q+2)))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (A*b*c+a*b*C+(2*A*c^2+C*(b^2-2*a*c))*x)*(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/(c*(b^2-4*a*c)*(p+1)) - + 1/(c*(b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[A*c*(2*c*d*(2*p+3)+b*e*q)-C*(2*a*c*d-b^2*d*(p+2)-a*b*e*q)+ + (C*(2*a*b*f*q-2*a*c*e*(q+1)+b^2*e*(p+q+2))+2*A*c*(b*f*q+c*e*(2*p+q+3)))*x- + f*(-2*A*c^2*(2*p+2*q+3)+C*(2*a*c*(2*q+1)-b^2*(p+2*q+2)))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a*B-(A*c-a*C)*x)*(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/(2*a*c*(p+1)) - + 2/((-4*a*c)*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[A*c*d*(2*p+3)-a*(C*d+B*e*q)+(A*c*e*(2*p+q+3)-a*(2*B*f*q+C*e*(q+1)))*x-f*(a*C*(2*q+1)-A*c*(2*p+2*q+3))*x^2,x],x] /; +FreeQ[{a,c,d,e,f,A,B,C},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + -(A*c-a*C)*x*(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q/(2*a*c*(p+1)) + + 2/(4*a*c*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q-1)* + Simp[A*c*d*(2*p+3)-a*C*d+(A*c*e*(2*p+q+3)-a*C*e*(q+1))*x-f*(a*C*(2*q+1)-A*c*(2*p+2*q+3))*x^2,x],x] /; +FreeQ[{a,c,d,e,f,A,C},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (A*b*c-2*a*B*c+a*b*C-(c*(b*B-2*A*c)-C*(b^2-2*a*c))*x)*(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q/(c*(b^2-4*a*c)*(p+1)) - + 1/(c*(b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q-1)* + Simp[-d*(c*(b*B-2*A*c)*(2*p+3)+C*(2*a*c-b^2*(p+2)))+ + (2*f*q*(A*b*c-2*a*B*c+a*b*C))*x- + f*(c*(b*B-2*A*c)*(2*p+2*q+3)+C*(2*a*c*(2*q+1)-b^2*(p+2*q+2)))*x^2,x],x] /; +FreeQ[{a,b,c,d,f,A,B,C},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (A*b*c+a*b*C+(2*A*c^2+C*(b^2-2*a*c))*x)*(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q/(c*(b^2-4*a*c)*(p+1)) - + 1/(c*(b^2-4*a*c)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q-1)* + Simp[A*c*(2*c*d*(2*p+3))-C*(2*a*c*d-b^2*d*(p+2))+ + (C*(2*a*b*f*q)+2*A*c*(b*f*q))*x- + f*(-2*A*c^2*(2*p+2*q+3)+C*(2*a*c*(2*q+1)-b^2*(p+2*q+2)))*x^2,x],x] /; +FreeQ[{a,b,c,d,f,A,C},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && GtQ[q,0] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + ((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b-a*B)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+ + c*(A*(2*c^2*d+b^2*f-c*(b*e+2*a*f))-B*(b*c*d-2*a*c*e+a*b*f)+C*(b^2*d-a*b*e-2*a*(c*d-a*f)))*x) + + 1/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(b*B-2*A*c-2*a*C)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)+ + (b^2*(C*d+A*f)-b*(B*c*d+A*c*e+a*C*e+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + e*((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b-a*B)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (2*f*((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b-a*B)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (b^2*(C*d+A*f)-b*(B*c*d+A*c*e+a*C*e+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))* + (b*f*(p+1)-c*e*(2*p+q+4)))*x- + c*f*(b^2*(C*d+A*f)-b*(B*c*d+A*c*e+a*C*e+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,e,f,A,B,C,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && + NeQ[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f)] && Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + ((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b)*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+ + c*(A*(2*c^2*d+b^2*f-c*(b*e+2*a*f))+C*(b^2*d-a*b*e-2*a*(c*d-a*f)))*x) + + 1/((b^2-4*a*c)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(-2*A*c-2*a*C)*((c*d-a*f)^2-(b*d-a*e)*(c*e-b*f))*(p+1)+ + (b^2*(C*d+A*f)-b*(+A*c*e+a*C*e)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + e*((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (2*f*((A*c-a*C)*(2*a*c*e-b*(c*d+a*f))+(A*b)*(2*c^2*d+b^2*f-c*(b*e+2*a*f)))*(p+q+2)- + (b^2*(C*d+A*f)-b*(A*c*e+a*C*e)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))* + (b*f*(p+1)-c*e*(2*p+q+4)))*x- + c*f*(b^2*(C*d+A*f)-b*(A*c*e+a*C*e)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,e,f,A,C,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && LtQ[p,-1] && + NeQ[(c*d-a*f)^2-(b*d-a*e)*(c*e-b*f)] && Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + ((A*c-a*C)*(2*a*c*e)+(-a*B)*(2*c^2*d-c*(2*a*f))+ + c*(A*(2*c^2*d-c*(2*a*f))-B*(-2*a*c*e)+C*(-2*a*(c*d-a*f)))*x) + + 1/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(-2*A*c-2*a*C)*((c*d-a*f)^2-(-a*e)*(c*e))*(p+1)+ + (2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + e*((A*c-a*C)*(2*a*c*e)+(-a*B)*(2*c^2*d-c*(+2*a*f)))*(p+q+2)- + (2*f*((A*c-a*C)*(2*a*c*e)+(-a*B)*(2*c^2*d+-c*(+2*a*f)))*(p+q+2)- + (2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))* + (-c*e*(2*p+q+4)))*x- + c*f*(2*(A*c*(c*d-a*f)-a*(c*C*d-B*c*e-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,c,d,e,f,A,B,C,q},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && NeQ[a*c*e^2+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (a+c*x^2)^(p+1)*(d+e*x+f*x^2)^(q+1)/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + ((A*c-a*C)*(2*a*c*e)+c*(A*(2*c^2*d-c*(2*a*f))+C*(-2*a*(c*d-a*f)))*x) + + 1/((-4*a*c)*(a*c*e^2+(c*d-a*f)^2)*(p+1))* + Int[(a+c*x^2)^(p+1)*(d+e*x+f*x^2)^q* + Simp[(-2*A*c-2*a*C)*((c*d-a*f)^2-(-a*e)*(c*e))*(p+1)+ + (2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + e*((A*c-a*C)*(2*a*c*e))*(p+q+2)- + (2*f*((A*c-a*C)*(2*a*c*e))*(p+q+2)-(2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(-c*e*(2*p+q+4)))*x- + c*f*(2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,c,d,e,f,A,C,q},x] && NeQ[e^2-4*d*f] && LtQ[p,-1] && NeQ[a*c*e^2+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q+1)/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + ((A*c-a*C)*(-b*(c*d+a*f))+(A*b-a*B)*(2*c^2*d+b^2*f-c*(2*a*f))+ + c*(A*(2*c^2*d+b^2*f-c*(2*a*f))-B*(b*c*d+a*b*f)+C*(b^2*d-2*a*(c*d-a*f)))*x) + + 1/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q* + Simp[(b*B-2*A*c-2*a*C)*((c*d-a*f)^2-(b*d)*(-b*f))*(p+1)+ + (b^2*(C*d+A*f)-b*(B*c*d+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + (2*f*((A*c-a*C)*(-b*(c*d+a*f))+(A*b-a*B)*(2*c^2*d+b^2*f-c*(2*a*f)))*(p+q+2)- + (b^2*(C*d+A*f)-b*(B*c*d+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))* + (b*f*(p+1)))*x- + c*f*(b^2*(C*d+A*f)-b*(B*c*d+a*B*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,f,A,B,C,q},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && NeQ[b^2*d*f+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (a+b*x+c*x^2)^(p+1)*(d+f*x^2)^(q+1)/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + ((A*c-a*C)*(-b*(c*d+a*f))+(A*b)*(2*c^2*d+b^2*f-c*(2*a*f))+ + c*(A*(2*c^2*d+b^2*f-c*(2*a*f))+C*(b^2*d-2*a*(c*d-a*f)))*x) + + 1/((b^2-4*a*c)*(b^2*d*f+(c*d-a*f)^2)*(p+1))* + Int[(a+b*x+c*x^2)^(p+1)*(d+f*x^2)^q* + Simp[(-2*A*c-2*a*C)*((c*d-a*f)^2-(b*d)*(-b*f))*(p+1)+ + (b^2*(C*d+A*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(a*f*(p+1)-c*d*(p+2))- + (2*f*((A*c-a*C)*(-b*(c*d+a*f))+(A*b)*(2*c^2*d+b^2*f-c*(2*a*f)))*(p+q+2)- + (b^2*(C*d+A*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))* + (b*f*(p+1)))*x- + c*f*(b^2*(C*d+A*f)+2*(A*c*(c*d-a*f)-a*(c*C*d-a*C*f)))*(2*p+2*q+5)*x^2,x],x]/; +FreeQ[{a,b,c,d,f,A,C,q},x] && NeQ[b^2-4*a*c] && LtQ[p,-1] && NeQ[b^2*d*f+(c*d-a*f)^2] && + Not[Not[IntegerQ[p]] && ILtQ[q,-1]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (B*c*f*(2*p+2*q+3)+C*(b*f*p-c*e*(2*p+q+2))+2*c*C*f*(p+q+1)*x)*(a+b*x+c*x^2)^p* + (d+e*x+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[p*(b*d-a*e)*(C*(c*e-b*f)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(b^2*C*d*f*p+a*c*(C*(2*d*f-e^2*(2*p+q+2))+f*(B*e-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(c*e-b*f)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(C*e*f*p*(b^2-4*a*c)-b*c*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d-B*e+2*A*f)*(2*p+2*q+3))))*x+ + (p*(c*e-b*f)*(C*(c*e-b*f)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(C*f^2*p*(b^2-4*a*c)-c^2*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d-B*e+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && GtQ[p,0] && + NeQ[p+q+1] && NeQ[2*p+2*q+3] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (C*(b*f*p-c*e*(2*p+q+2))+2*c*C*f*(p+q+1)*x)*(a+b*x+c*x^2)^p* + (d+e*x+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[p*(b*d-a*e)*(C*(c*e-b*f)*(q+1)-c*(C*e)*(2*p+2*q+3))+ + (p+q+1)*(b^2*C*d*f*p+a*c*(C*(2*d*f-e^2*(2*p+q+2))+f*(-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(c*e-b*f)*(q+1)-c*(C*e)*(2*p+2*q+3))+ + (p+q+1)*(C*e*f*p*(b^2-4*a*c)-b*c*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x+ + (p*(c*e-b*f)*(C*(c*e-b*f)*(q+1)-c*(C*e)*(2*p+2*q+3))+ + (p+q+1)*(C*f^2*p*(b^2-4*a*c)-c^2*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C,q},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] && GtQ[p,0] && + NeQ[p+q+1] && NeQ[2*p+2*q+3] && Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (B*c*f*(2*p+2*q+3)+C*(-c*e*(2*p+q+2))+2*c*C*f*(p+q+1)*x)*(a+c*x^2)^p* + (d+e*x+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[p*(-a*e)*(C*(c*e)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(a*c*(C*(2*d*f-e^2*(2*p+q+2))+f*(B*e-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(c*e)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(C*e*f*p*(-4*a*c)))*x+ + (p*(c*e)*(C*(c*e)*(q+1)-c*(C*e-B*f)*(2*p+2*q+3))+ + (p+q+1)*(C*f^2*p*(-4*a*c)-c^2*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d-B*e+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,c,d,e,f,A,B,C,q},x] && NeQ[e^2-4*d*f] && GtQ[p,0] && NeQ[p+q+1] && NeQ[2*p+2*q+3] && + Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+c_.*x_^2)^p_*(d_+e_.*x_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (C*(-c*e*(2*p+q+2))+2*c*C*f*(p+q+1)*x)*(a+c*x^2)^p*(d+e*x+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+c*x^2)^(p-1)*(d+e*x+f*x^2)^q* + Simp[p*(-a*e)*(C*(c*e)*(q+1)-c*(C*e)*(2*p+2*q+3))+(p+q+1)*(a*c*(C*(2*d*f-e^2*(2*p+q+2))+f*(-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(c*e)*(q+1)-c*(C*e)*(2*p+2*q+3))+(p+q+1)*(C*e*f*p*(-4*a*c)))*x+ + (p*(c*e)*(C*(c*e)*(q+1)-c*(C*e)*(2*p+2*q+3))+ + (p+q+1)*(C*f^2*p*(-4*a*c)-c^2*(C*(e^2-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,c,d,e,f,A,C,q},x] && NeQ[e^2-4*d*f] && GtQ[p,0] && NeQ[p+q+1] && NeQ[2*p+2*q+3] && + Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + (B*c*f*(2*p+2*q+3)+C*(b*f*p)+2*c*C*f*(p+q+1)*x)*(a+b*x+c*x^2)^p* + (d+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+f*x^2)^q* + Simp[p*(b*d)*(C*(-b*f)*(q+1)-c*(-B*f)*(2*p+2*q+3))+ + (p+q+1)*(b^2*C*d*f*p+a*c*(C*(2*d*f)+f*(-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(-b*f)*(q+1)-c*(-B*f)*(2*p+2*q+3))+ + (p+q+1)*(-b*c*(C*(-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x+ + (p*(-b*f)*(C*(-b*f)*(q+1)-c*(-B*f)*(2*p+2*q+3))+ + (p+q+1)*(C*f^2*p*(b^2-4*a*c)-c^2*(C*(-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,b,c,d,f,A,B,C,q},x] && NeQ[b^2-4*a*c] && GtQ[p,0] && NeQ[p+q+1] && NeQ[2*p+2*q+3] && + Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_+f_.*x_^2)^q_*(A_.+C_.*x_^2),x_Symbol] := + (C*(b*f*p)+2*c*C*f*(p+q+1)*x)*(a+b*x+c*x^2)^p* + (d+f*x^2)^(q+1)/(2*c*f^2*(p+q+1)*(2*p+2*q+3)) - + (1/(2*c*f^2*(p+q+1)*(2*p+2*q+3)))* + Int[(a+b*x+c*x^2)^(p-1)*(d+f*x^2)^q* + Simp[p*(b*d)*(C*(-b*f)*(q+1))+ + (p+q+1)*(b^2*C*d*f*p+a*c*(C*(2*d*f)+f*(-2*A*f)*(2*p+2*q+3)))+ + (2*p*(c*d-a*f)*(C*(-b*f)*(q+1))+ + (p+q+1)*(-b*c*(C*(-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x+ + (p*(-b*f)*(C*(-b*f)*(q+1))+ + (p+q+1)*(C*f^2*p*(b^2-4*a*c)-c^2*(C*(-4*d*f)*(2*p+q+2)+f*(2*C*d+2*A*f)*(2*p+2*q+3))))*x^2,x],x] /; +FreeQ[{a,b,c,d,f,A,C,q},x] && NeQ[b^2-4*a*c] && GtQ[p,0] && NeQ[p+q+1] && NeQ[2*p+2*q+3] && + Not[IGtQ[p,0]] && Not[IGtQ[q,0]] + + +Int[(A_.+B_.*x_+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*(d_+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2-b*c*d*e+a*c*e^2+b^2*d*f-2*a*c*d*f-a*b*e*f+a^2*f^2}, + 1/q*Int[(A*c^2*d-a*c*C*d-A*b*c*e+a*B*c*e+A*b^2*f-a*b*B*f-a*A*c*f+a^2*C*f+ + c*(B*c*d-b*C*d-A*c*e+a*C*e+A*b*f-a*B*f)*x)/(a+b*x+c*x^2),x] + + 1/q*Int[(c*C*d^2-B*c*d*e+A*c*e^2+b*B*d*f-A*c*d*f-a*C*d*f-A*b*e*f+a*A*f^2- + f*(B*c*d-b*C*d-A*c*e+a*C*e+A*b*f-a*B*f)*x)/(d+e*x+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(A_.+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*(d_+e_.*x_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2-b*c*d*e+a*c*e^2+b^2*d*f-2*a*c*d*f-a*b*e*f+a^2*f^2}, + 1/q*Int[(A*c^2*d-a*c*C*d-A*b*c*e+A*b^2*f-a*A*c*f+a^2*C*f+ + c*(-b*C*d-A*c*e+a*C*e+A*b*f)*x)/(a+b*x+c*x^2),x] + + 1/q*Int[(c*C*d^2+A*c*e^2-A*c*d*f-a*C*d*f-A*b*e*f+a*A*f^2- + f*(-b*C*d-A*c*e+a*C*e+A*b*f)*x)/(d+e*x+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(A_.+B_.*x_+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*(d_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2+b^2*d*f-2*a*c*d*f+a^2*f^2}, + 1/q*Int[(A*c^2*d-a*c*C*d+A*b^2*f-a*b*B*f-a*A*c*f+a^2*C*f+c*(B*c*d-b*C*d+A*b*f-a*B*f)*x)/(a+b*x+c*x^2),x] + + 1/q*Int[(c*C*d^2+b*B*d*f-A*c*d*f-a*C*d*f+a*A*f^2-f*(B*c*d-b*C*d+A*b*f-a*B*f)*x)/(d+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,f,A,B,C},x] && NeQ[b^2-4*a*c] + + +Int[(A_.+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*(d_+f_.*x_^2)),x_Symbol] := + With[{q=c^2*d^2+b^2*d*f-2*a*c*d*f+a^2*f^2}, + 1/q*Int[(A*c^2*d-a*c*C*d+A*b^2*f-a*A*c*f+a^2*C*f+c*(-b*C*d+A*b*f)*x)/(a+b*x+c*x^2),x] + + 1/q*Int[(c*C*d^2-A*c*d*f-a*C*d*f+a*A*f^2-f*(-b*C*d+A*b*f)*x)/(d+f*x^2),x] /; + NeQ[q]] /; +FreeQ[{a,b,c,d,f,A,C},x] && NeQ[b^2-4*a*c] + + +Int[(A_.+B_.*x_+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+e*x+f*x^2],x] + + 1/c*Int[(A*c-a*C+(B*c-b*C)*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(A_.+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+e*x+f*x^2],x] + 1/c*Int[(A*c-a*C-b*C*x)/((a+b*x+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b^2-4*a*c] && NeQ[e^2-4*d*f] + + +Int[(A_.+B_.*x_+C_.*x_^2)/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+e*x+f*x^2],x] + 1/c*Int[(A*c-a*C+B*c*x)/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,c,d,e,f,A,B,C},x] && NeQ[e^2-4*d*f] + + +Int[(A_.+C_.*x_^2)/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+e*x+f*x^2],x] + (A*c-a*C)/c*Int[1/((a+c*x^2)*Sqrt[d+e*x+f*x^2]),x] /; +FreeQ[{a,c,d,e,f,A,C},x] && NeQ[e^2-4*d*f] + + +Int[(A_.+B_.*x_+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+f*x^2],x] + 1/c*Int[(A*c-a*C+(B*c-b*C)*x)/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x] /; +FreeQ[{a,b,c,d,f,A,B,C},x] && NeQ[b^2-4*a*c] + + +Int[(A_.+C_.*x_^2)/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + C/c*Int[1/Sqrt[d+f*x^2],x] + 1/c*Int[(A*c-a*C-b*C*x)/((a+b*x+c*x^2)*Sqrt[d+f*x^2]),x] /; +FreeQ[{a,b,c,d,f,A,C},x] && NeQ[b^2-4*a*c] + + +Int[(a_.+b_.*u_+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+B_.*u_+C_.*u_^2),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*(A+B*x+C*x^2),x],x,u] /; +FreeQ[{a,b,c,d,e,f,A,B,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*u_+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+B_.*u_),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*(A+B*x),x],x,u] /; +FreeQ[{a,b,c,d,e,f,A,B,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*u_+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+C_.*u_^2),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*(A+C*x^2),x],x,u] /; +FreeQ[{a,b,c,d,e,f,A,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+B_.*u_+C_.*u_^2),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+c*x^2)^p*(d+e*x+f*x^2)^q*(A+B*x+C*x^2),x],x,u] /; +FreeQ[{a,c,d,e,f,A,B,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+B_.*u_),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+c*x^2)^p*(d+e*x+f*x^2)^q*(A+B*x),x],x,u] /; +FreeQ[{a,c,d,e,f,A,B,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+c_.*u_^2)^p_.*(d_.+e_.*u_+f_.*u_^2)^q_.*(A_.+C_.*u_^2),x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+c*x^2)^p*(d+e*x+f*x^2)^q*(A+C*x^2),x],x,u] /; +FreeQ[{a,c,d,e,f,A,C,p,q},x] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.9 (d+e x)^m Pq(x) (a+b x+c x^2)^p*) + + +Int[x_^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + 1/2*Subst[Int[x^((m-1)/2)*SubstFor[x^2,Pq,x]*(a+c*x)^p,x],x,x^2] /; +FreeQ[{a,c,p},x] && PolyQ[Pq,x^2] && IntegerQ[(m-1)/2] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+1)*PolynomialQuotient[Pq,d+e*x,x]*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[PolynomialRemainder[Pq,d+e*x,x]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+1)*PolynomialQuotient[Pq,d+e*x,x]*(a+c*x^2)^p,x] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[PolynomialRemainder[Pq,d+e*x,x]] + + +Int[(d_.+e_.*x_)^m_.*P2_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + With[{f=Coeff[P2,x,0],g=Coeff[P2,x,1],h=Coeff[P2,x,2]}, + h*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)/(c*e*(m+2*p+3)) /; + EqQ[b*e*h*(m+p+2)+2*c*d*h*(p+1)-c*e*g*(m+2*p+3)] && EqQ[b*d*h*(p+1)+a*e*h*(m+1)-c*e*f*(m+2*p+3)]] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[P2,x,2] && NeQ[m+2*p+3] + + +Int[(d_.+e_.*x_)^m_.*P2_*(a_+c_.*x_^2)^p_.,x_Symbol] := + With[{f=Coeff[P2,x,0],g=Coeff[P2,x,1],h=Coeff[P2,x,2]}, + h*(d+e*x)^(m+1)*(a+c*x^2)^(p+1)/(c*e*(m+2*p+3)) /; + EqQ[2*d*h*(p+1)-e*g*(m+2*p+3)] && EqQ[a*h*(m+1)-c*f*(m+2*p+3)]] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[P2,x,2] && NeQ[m+2*p+3] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*Pq*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && PolyQ[Pq,x] && IGtQ[p,-2] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*Pq*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c,d,e,m},x] && PolyQ[Pq,x] && IGtQ[p,-2] + + +Int[(d_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + 1/d*Int[(d*x)^(m+1)*ExpandToSum[Pq/x,x]*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[(d+e*x)^m*Pq*(b+2*c*x)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[b^2-4*a*c,0] + + +Int[(e_.*x_)^m_.*Pq_*(b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(e*x)^(m-1)*ExpandToSum[e*Pq/(b+c*x),x]*(b*x+c*x^2)^(p+1),x] /; +FreeQ[{b,c,e,m,p},x] && PolyQ[Pq,x] && EqQ[PolynomialRemainder[Pq,b+c*x,x]] + + +Int[(d_+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m-1)*ExpandToSum[d*e*Pq/(a*e+c*d*x),x]*(a+b*x+c*x^2)^(p+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && EqQ[PolynomialRemainder[Pq,a*e+c*d*x,x]] + + +Int[(d_+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m-1)*ExpandToSum[d*e*Pq/(a*e+c*d*x),x]*(a+c*x^2)^(p+1),x] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[c*d^2+a*e^2,0] && EqQ[PolynomialRemainder[Pq,a*e+c*d*x,x]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],f=Coeff[Pq,x,Expon[Pq,x]]}, + f*(d+e*x)^(m+q-1)*(a+b*x+c*x^2)^(p+1)/(c*e^(q-1)*(m+q+2*p+1)) + + 1/(c*e^q*(m+q+2*p+1))*Int[(d+e*x)^m*(a+b*x+c*x^2)^p* + ExpandToSum[c*e^q*(m+q+2*p+1)*Pq-c*f*(m+q+2*p+1)*(d+e*x)^q+e*f*(m+p+q)*(d+e*x)^(q-2)*(b*d-2*a*e+(2*c*d-b*e)*x),x],x] /; + NeQ[m+q+2*p+1]] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] + + +Int[(d_+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],f=Coeff[Pq,x,Expon[Pq,x]]}, + f*(d+e*x)^(m+q-1)*(a+c*x^2)^(p+1)/(c*e^(q-1)*(m+q+2*p+1)) + + 1/(c*e^q*(m+q+2*p+1))*Int[(d+e*x)^m*(a+c*x^2)^p* + ExpandToSum[c*e^q*(m+q+2*p+1)*Pq-c*f*(m+q+2*p+1)*(d+e*x)^q-2*e*f*(m+p+q)*(d+e*x)^(q-2)*(a*e-c*d*x),x],x] /; + NeQ[m+q+2*p+1]] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[c*d^2+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p*Pq,x] /; +FreeQ[{a,b,c,d,e,m},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p] + + +Int[(d_+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p*Pq,x] /; +FreeQ[{a,c,d,e,m},x] && PolyQ[Pq,x] && EqQ[c*d^2+a*e^2,0] && IntegerQ[p] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p*Pq,x] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + (a+c*x^2)^FracPart[p]/((d+e*x)^FracPart[p]*(a/d+(c*x)/e)^FracPart[p])*Int[(d+e*x)^(m+p)*(a/d+c/e*x)^p*Pq,x] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[(a+b*x+c*x^2)^p*ExpandIntegrand[(d+e*x)^m*Pq,x],x] /; +FreeQ[{a,b,c,d,e},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] && IGtQ[m,0] && RationalQ[a,b,c,d,e] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[(a+c*x^2)^p*ExpandIntegrand[(d+e*x)^m*Pq,x],x] /; +FreeQ[{a,c,d,e},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] && LtQ[p,-1] && IGtQ[m,0] && RationalQ[a,c,d,e] && Not[IntegerQ[p]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+b*x+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,1]}, + (d+e*x)^m*(a+b*x+c*x^2)^(p+1)*(f*b-2*a*g+(2*c*f-b*g)*x)/((p+1)*(b^2-4*a*c)) + + 1/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^(m-1)*(a+b*x+c*x^2)^(p+1)* + ExpandToSum[(p+1)*(b^2-4*a*c)*(d+e*x)*Q+g*(2*a*e*m+b*d*(2*p+3))-f*(b*e*m+2*c*d*(2*p+3))-e*(2*c*f-b*g)*(m+2*p+3)*x,x],x]] /; +FreeQ[{a,b,c,d,e},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] && GtQ[m,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,1]}, + (d+e*x)^m*(a+c*x^2)^(p+1)*(a*g-c*f*x)/(2*a*c*(p+1)) + + 1/(2*a*c*(p+1))*Int[(d+e*x)^(m-1)*(a+c*x^2)^(p+1)* + ExpandToSum[2*a*c*(p+1)*(d+e*x)*Q-a*e*g*m+c*d*f*(2*p+3)+c*e*f*(m+2*p+3)*x,x],x]] /; +FreeQ[{a,c,d,e},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] && LtQ[p,-1] && GtQ[m,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[(d+e*x)^m*Pq,a+b*x+c*x^2,x], + f=Coeff[PolynomialRemainder[(d+e*x)^m*Pq,a+b*x+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[(d+e*x)^m*Pq,a+b*x+c*x^2,x],x,1]}, + (b*f-2*a*g+(2*c*f-b*g)*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) + + 1/((p+1)*(b^2-4*a*c))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p+1)* + ExpandToSum[(p+1)*(b^2-4*a*c)*(d+e*x)^(-m)*Q-(2*p+3)*(2*c*f-b*g)*(d+e*x)^(-m),x],x]] /; +FreeQ[{a,b,c,d,e},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] && ILtQ[m,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[(d+e*x)^m*Pq,a+c*x^2,x], + f=Coeff[PolynomialRemainder[(d+e*x)^m*Pq,a+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[(d+e*x)^m*Pq,a+c*x^2,x],x,1]}, + (a*g-c*f*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) + + 1/(2*a*c*(p+1))*Int[(d+e*x)^m*(a+c*x^2)^(p+1)* + ExpandToSum[2*a*c*(p+1)*(d+e*x)^(-m)*Q+c*f*(2*p+3)*(d+e*x)^(-m),x],x]] /; +FreeQ[{a,c,d,e},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] && LtQ[p,-1] && ILtQ[m,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+b*x+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,1]}, + (d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1)*(f*(b*c*d-b^2*e+2*a*c*e)-a*g*(2*c*d-b*e)+c*(f*(2*c*d-b*e)-g*(b*d-2*a*e))*x)/ + ((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) + + 1/((p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^m*(a+b*x+c*x^2)^(p+1)* + ExpandToSum[(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*Q+ + f*(b*c*d*e*(2*p-m+2)+b^2*e^2*(p+m+2)-2*c^2*d^2*(2*p+3)-2*a*c*e^2*(m+2*p+3))- + g*(a*e*(b*e-2*c*d*m+b*e*m)-b*d*(3*c*d-b*e+2*c*d*p-b*e*p))+ + c*e*(g*(b*d-2*a*e)-f*(2*c*d-b*e))*(m+2*p+4)*x,x],x]] /; +FreeQ[{a,b,c,d,e,m},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] && Not[GtQ[m,0]] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,1]}, + -(d+e*x)^(m+1)*(a+c*x^2)^(p+1)*(a*(e*f-d*g)+(c*d*f+a*e*g)*x)/(2*a*(p+1)*(c*d^2+a*e^2)) + + 1/(2*a*(p+1)*(c*d^2+a*e^2))*Int[(d+e*x)^m*(a+c*x^2)^(p+1)* + ExpandToSum[2*a*(p+1)*(c*d^2+a*e^2)*Q+c*d^2*f*(2*p+3)-a*e*(d*g*m-e*f*(m+2*p+3))+e*(c*d*f+a*e*g)*(m+2*p+4)*x,x],x]] /; +FreeQ[{a,c,d,e,m},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] && LtQ[p,-1] && Not[GtQ[m,0]] + + +Int[(d_.+e_.*x_)^m_*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,d+e*x,x], R=PolynomialRemainder[Pq,d+e*x,x]}, + (e*R*(d+e*x)^(m+1)*(a+b*x+c*x^2)^(p+1))/((m+1)*(c*d^2-b*d*e+a*e^2)) + + 1/((m+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x)^(m+1)*(a+b*x+c*x^2)^p* + ExpandToSum[(m+1)*(c*d^2-b*d*e+a*e^2)*Q+c*d*R*(m+1)-b*e*R*(m+p+2)-c*e*R*(m+2*p+3)*x,x],x]] /; +FreeQ[{a,b,c,d,e,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)^m_*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,d+e*x,x], R=PolynomialRemainder[Pq,d+e*x,x]}, + (e*R*(d+e*x)^(m+1)*(a+c*x^2)^(p+1))/((m+1)*(c*d^2+a*e^2)) + + 1/((m+1)*(c*d^2+a*e^2))*Int[(d+e*x)^(m+1)*(a+c*x^2)^p* + ExpandToSum[(m+1)*(c*d^2+a*e^2)*Q+c*d*R*(m+1)-c*e*R*(m+2*p+3)*x,x],x]] /; +FreeQ[{a,c,d,e,p},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + With[{q=Expon[Pq,x]}, + Coeff[Pq,x,q]/e^q*Int[(d+e*x)^(m+q)*(a+b*x+c*x^2)^p,x] + + 1/e^q*Int[(d+e*x)^m*(a+b*x+c*x^2)^p*ExpandToSum[e^q*Pq-Coeff[Pq,x,q]*(d+e*x)^q,x],x] /; + EqQ[m+q+2*p+1,0]] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + With[{q=Expon[Pq,x]}, + Coeff[Pq,x,q]/e^q*Int[(d+e*x)^(m+q)*(a+c*x^2)^p,x] + + 1/e^q*Int[(d+e*x)^m*(a+c*x^2)^p*ExpandToSum[e^q*Pq-Coeff[Pq,x,q]*(d+e*x)^q,x],x] /; + EqQ[m+q+2*p+1,0]] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],f=Coeff[Pq,x,Expon[Pq,x]]}, + f*(d+e*x)^(m+q-1)*(a+b*x+c*x^2)^(p+1)/(c*e^(q-1)*(m+q+2*p+1)) + + 1/(c*e^q*(m+q+2*p+1))*Int[(d+e*x)^m*(a+b*x+c*x^2)^p*ExpandToSum[c*e^q*(m+q+2*p+1)*Pq-c*f*(m+q+2*p+1)*(d+e*x)^q- + f*(d+e*x)^(q-2)*(b*d*e*(p+1)+a*e^2*(m+q-1)-c*d^2*(m+q+2*p+1)-e*(2*c*d-b*e)*(m+q+p)*x),x],x] /; + NeQ[m+q+2*p+1,0]] /; +FreeQ[{a,b,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] + + +Int[(d_.+e_.*x_)^m_.*Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],f=Coeff[Pq,x,Expon[Pq,x]]}, + f*(d+e*x)^(m+q-1)*(a+c*x^2)^(p+1)/(c*e^(q-1)*(m+q+2*p+1)) + + 1/(c*e^q*(m+q+2*p+1))*Int[(d+e*x)^m*(a+c*x^2)^p*ExpandToSum[c*e^q*(m+q+2*p+1)*Pq-c*f*(m+q+2*p+1)*(d+e*x)^q- + f*(d+e*x)^(q-2)*(a*e^2*(m+q-1)-c*d^2*(m+q+2*p+1)-2*c*d*e*(m+q+p)*x),x],x] /; + NeQ[m+q+2*p+1,0]] /; +FreeQ[{a,c,d,e,m,p},x] && PolyQ[Pq,x] && NeQ[c*d^2+a*e^2,0] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.1.8 Pq(x) (a+b x+c x^2)^p*) + + +Int[Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],k}, + Int[Sum[Coeff[Pq,x,2*k]*x^(2*k),{k,0,q/2}]*(a+c*x^2)^p,x] + + Int[x*Sum[Coeff[Pq,x,2*k+1]*x^(2*k),{k,0,(q-1)/2}]*(a+c*x^2)^p,x]] /; +FreeQ[{a,c,p},x] && PolyQ[Pq,x] && Not[PolyQ[Pq,x^2]] && NeQ[p,-1] + + +Int[Pq_*(a_+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x+c*x^2)^p,x],x] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x] && PositiveIntegerQ[p+2] + + +Int[Pq_*(a_+c_.*x_^2)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+c*x^2)^p,x],x] /; +FreeQ[{a,c},x] && PolyQ[Pq,x] && PositiveIntegerQ[p+2] + + +Int[Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + Int[x*ExpandToSum[Pq/x,x]*(a+b*x+c*x^2)^p,x] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] + + +Int[Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + Int[x*ExpandToSum[Pq/x,x]*(a+c*x^2)^p,x] /; +FreeQ[{a,c,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] + + +Int[Pq_*(a_+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + (a+b*x+c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x)^(2*FracPart[p]))*Int[Pq*(b+2*c*x)^(2*p),x] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && EqQ[b^2-4*a*c] + + +Int[Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+b*x+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+b*x+c*x^2,x],x,1]}, + (b*f-2*a*g+(2*c*f-b*g)*x)*(a+b*x+c*x^2)^(p+1)/((p+1)*(b^2-4*a*c)) + + 1/((p+1)*(b^2-4*a*c))*Int[(a+b*x+c*x^2)^(p+1)*ExpandToSum[(p+1)*(b^2-4*a*c)*Q-(2*p+3)*(2*c*f-b*g),x],x]] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && LtQ[p,-1] + + +Int[Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{Q=PolynomialQuotient[Pq,a+c*x^2,x], + f=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,0], + g=Coeff[PolynomialRemainder[Pq,a+c*x^2,x],x,1]}, + (a*g-c*f*x)*(a+c*x^2)^(p+1)/(2*a*c*(p+1)) + + 1/(2*a*c*(p+1))*Int[(a+c*x^2)^(p+1)*ExpandToSum[2*a*c*(p+1)*Q+c*f*(2*p+3),x],x]] /; +FreeQ[{a,c},x] && PolyQ[Pq,x] && LtQ[p,-1] + + +Int[Pq_*(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],e=Coeff[Pq,x,Expon[Pq,x]]}, + e*x^(q-1)*(a+b*x+c*x^2)^(p+1)/(c*(q+2*p+1)) + + 1/(c*(q+2*p+1))*Int[(a+b*x+c*x^2)^p* + ExpandToSum[c*(q+2*p+1)*Pq-a*e*(q-1)*x^(q-2)-b*e*(q+p)*x^(q-1)-c*e*(q+2*p+1)*x^q,x],x]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && Not[LeQ[p,-1]] + + +Int[Pq_*(a_+c_.*x_^2)^p_,x_Symbol] := + With[{q=Expon[Pq,x],e=Coeff[Pq,x,Expon[Pq,x]]}, + e*x^(q-1)*(a+c*x^2)^(p+1)/(c*(q+2*p+1)) + + 1/(c*(q+2*p+1))*Int[(a+c*x^2)^p* + ExpandToSum[c*(q+2*p+1)*Pq-a*e*(q-1)*x^(q-2)-c*e*(q+2*p+1)*x^q,x],x]] /; +FreeQ[{a,c,p},x] && PolyQ[Pq,x] && Not[LeQ[p,-1]] + + + +`) + +func resourcesRubi121QuadraticTrinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi121QuadraticTrinomialProductsM, nil +} + +func resourcesRubi121QuadraticTrinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi121QuadraticTrinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.2.1 Quadratic trinomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi122QuarticTrinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.2.2 Quartic Trinomial Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*1.2.2.1 (a+b x^2+c x^4)^p*) + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^p/(b+2*c*x^2)^(2*p)*Int[(b+2*c*x^2)^(2*p),x] /; +FreeQ[{a,b,c,p},x] && EqQ[b^2-4*a*c,0] + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + x*(a+b*x^2+c*x^4)^p/(4*p+1) + + 2*p/(4*p+1)*Int[(2*a+b*x^2)*(a+b*x^2+c*x^4)^(p-1),x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p>0 && IntegerQ[2*p] + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + -x*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(b^2-2*a*c+2*(p+1)*(b^2-4*a*c)+b*c*(4*p+7)*x^2)*(a+b*x^2+c*x^4)^(p+1),x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[1/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + c/q*Int[1/(b/2-q/2+c*x^2),x] - c/q*Int[1/(b/2+q/2+c*x^2),x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && PosQ[b^2-4*a*c] + + +Int[1/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*q*r)*Int[(r-x)/(q-r*x+x^2),x] + 1/(2*c*q*r)*Int[(r+x)/(q+r*x+x^2),x]]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && NegQ[b^2-4*a*c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*Sqrt[-c]*Int[1/(Sqrt[b+q+2*c*x^2]*Sqrt[-b+q-2*c*x^2]),x]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && NegativeQ[c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,4]}, + (1+q^2*x^2)*Sqrt[(a+b*x^2+c*x^4)/(a*(1+q^2*x^2)^2)]/(2*q*Sqrt[a+b*x^2+c*x^4])*EllipticF[2*ArcTan[q*x],1/2-b*q^2/(4*c)]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && PositiveQ[c/a] && NegativeQ[b/a] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[-2*a-(b-q)*x^2]*Sqrt[(2*a+(b+q)*x^2)/q]/(2*Sqrt[-a]*Sqrt[a+b*x^2+c*x^4])* + EllipticF[ArcSin[x/Sqrt[(2*a+(b+q)*x^2)/(2*q)]],(b+q)/(2*q)] /; + IntegerQ[q]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && NegativeQ[a] && PositiveQ[c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[(2*a+(b-q)*x^2)/(2*a+(b+q)*x^2)]*Sqrt[(2*a+(b+q)*x^2)/q]/(2*Sqrt[a+b*x^2+c*x^4]*Sqrt[a/(2*a+(b+q)*x^2)])* + EllipticF[ArcSin[x/Sqrt[(2*a+(b+q)*x^2)/(2*q)]],(b+q)/(2*q)]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && NegativeQ[a] && PositiveQ[c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*a+(b+q)*x^2)*Sqrt[(2*a+(b-q)*x^2)/(2*a+(b+q)*x^2)]/(2*a*Rt[(b+q)/(2*a),2]*Sqrt[a+b*x^2+c*x^4])* + EllipticF[ArcTan[Rt[(b+q)/(2*a),2]*x],2*q/(b+q)] /; + PosQ[(b+q)/a] && Not[PosQ[(b-q)/a] && SimplerSqrtQ[(b-q)/(2*a),(b+q)/(2*a)]]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*a+(b-q)*x^2)*Sqrt[(2*a+(b+q)*x^2)/(2*a+(b-q)*x^2)]/(2*a*Rt[(b-q)/(2*a),2]*Sqrt[a+b*x^2+c*x^4])* + EllipticF[ArcTan[Rt[(b-q)/(2*a),2]*x],-2*q/(b-q)] /; + PosQ[(b-q)/a]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+(b+q)*x^2/(2*a)]*Sqrt[1+(b-q)*x^2/(2*a)]/(Rt[-(b+q)/(2*a),2]*Sqrt[a+b*x^2+c*x^4])* + EllipticF[ArcSin[Rt[-(b+q)/(2*a),2]*x],(b-q)/(b+q)] /; + NegQ[(b+q)/a] && Not[NegQ[(b-q)/a] && SimplerSqrtQ[-(b-q)/(2*a),-(b+q)/(2*a)]]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+(b-q)*x^2/(2*a)]*Sqrt[1+(b+q)*x^2/(2*a)]/(Rt[-(b-q)/(2*a),2]*Sqrt[a+b*x^2+c*x^4])* + EllipticF[ArcSin[Rt[-(b-q)/(2*a),2]*x],(b+q)/(b-q)] /; + NegQ[(b-q)/a]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,4]}, + (1+q^2*x^2)*Sqrt[(a+b*x^2+c*x^4)/(a*(1+q^2*x^2)^2)]/(2*q*Sqrt[a+b*x^2+c*x^4])*EllipticF[2*ArcTan[q*x],1/2-b*q^2/(4*c)]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && PosQ[c/a] + + +Int[1/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]/Sqrt[a+b*x^2+c*x^4]* + Int[1/(Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]),x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && NegQ[c/a] + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + a^IntPart[p]*(a+b*x^2+c*x^4)^FracPart[p]/ + ((1+2*c*x^2/(b+Rt[b^2-4*a*c,2]))^FracPart[p]*(1+2*c*x^2/(b-Rt[b^2-4*a*c,2]))^FracPart[p])* + Int[(1+2*c*x^2/(b+Sqrt[b^2-4*a*c]))^p*(1+2*c*x^2/(b-Sqrt[b^2-4*a*c]))^p,x] /; +FreeQ[{a,b,c,p},x] && NeQ[b^2-4*a*c] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.2.2 (d x)^m (a+b x^2+c x^4)^p*) + + +Int[x_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,p},x] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[p] && Not[IntegerQ[Simplify[(m+1)/2]]] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(2*a*d*n*(p+1)*(2*p+1)) - + (d*x)^(m+1)*(2*a+b*x^n)*(a+b*x^n+c*x^(2*n))^p/(2*a*d*n*(2*p+1)) /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] && EqQ[m+2*n(p+1)+1,0] && NeQ[2*p+1,0] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2+c*x^n)^(2*FracPart[p]))*Int[(d*x)^m*(b/2+c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] + + +Int[x_^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[x^(Simplify[(m+1)/2]-1)*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,p},x] && NeQ[b^2-4*a*c] && IntegerQ[(m+1)/2] + + +Int[(d_.*x_)^m_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/d*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(2*k)/d^2+c*x^(4*k)/d^4)^p,x],x,(d*x)^(1/k)]] /; +FreeQ[{a,b,c,d,p},x] && NeQ[b^2-4*a*c] && FractionQ[m] && IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + d*(d*x)^(m-1)*(2*b*p+c*(m+4*p-1)*x^2)*(a+b*x^2+c*x^4)^p/(c*(m+4*p+1)*(m+4*p-1)) - + 2*p*d^2/(c*(m+4*p+1)*(m+4*p-1))* + Int[(d*x)^(m-2)*(a+b*x^2+c*x^4)^(p-1)*Simp[a*b*(m-1)-(2*a*c*(m+4*p-1)-b^2*(m+2*p-1))*x^2,x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m,p] && p>0 && m>1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^2+c*x^4)^p/(d*(m+1)) - + 2*p/(d^2*(m+1))*Int[(d*x)^(m+2)*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(p-1),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m,p] && p>0 && m<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^2+c*x^4)^p/(d*(m+4*p+1)) + + 2*p/(m+4*p+1)*Int[(d*x)^m*(2*a+b*x^2)*(a+b*x^2+c*x^4)^(p-1),x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p>0 && NeQ[m+4*p+1] && IntegerQ[2*p] && + (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + d*(d*x)^(m-1)*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*(p+1)*(b^2-4*a*c)) - + d^2/(2*(p+1)*(b^2-4*a*c))* + Int[(d*x)^(m-2)*(b*(m-1)+2*c*(m+4*(p+1)+1)*x^2)*(a+b*x^2+c*x^4)^(p+1),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m,p] && p<-1 && 13 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + -(d*x)^(m+1)*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*d*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))* + Int[(d*x)^m*(a+b*x^2+c*x^4)^(p+1)*Simp[b^2*(m+2*p+3)-2*a*c*(m+4*(p+1)+1)+b*c*(m+4*p+7)*x^2,x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + d^3*(d*x)^(m-3)*(a+b*x^2+c*x^4)^(p+1)/(c*(m+4*p+1)) - + d^4/(c*(m+4*p+1))* + Int[(d*x)^(m-4)*Simp[a*(m-3)+b*(m+2*p-1)*x^2,x]*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,p},x] && NeQ[b^2-4*a*c] && RationalQ[m] && m>3 && NeQ[m+4*p+1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^2+c*x^4)^(p+1)/(a*d*(m+1)) - + 1/(a*d^2*(m+1))*Int[(d*x)^(m+2)*(b*(m+2*p+3)+c*(m+4*p+5)*x^2)*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,p},x] && NeQ[b^2-4*a*c] && RationalQ[m] && m<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(d_.*x_)^m_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + (d*x)^(m+1)/(a*d*(m+1)) - + 1/(a*d^2)*Int[(d*x)^(m+2)*(b+c*x^2)/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m] && m<-1 + + +Int[x_^m_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + Int[PolynomialDivide[x^m,(a+b*x^2+c*x^4),x],x] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && IntegerQ[m] && m>5 + + +Int[(d_.*x_)^m_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + d^3*(d*x)^(m-3)/(c*(m-3)) - d^4/c*Int[(d*x)^(m-4)*(a+b*x^2)/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m] && m>3 + + +Int[x_^2/(a_+b_.*x_^2+c_.*x_^4), x_Symbol] := + With[{q=Rt[a/c,2]}, + 1/2*Int[(q+x^2)/(a+b*x^2+c*x^4),x] - 1/2*Int[(q-x^2)/(a+b*x^2+c*x^4),x]] /; +FreeQ[{a,b,c},x] && NegativeQ[b^2-4*a*c] && PosQ[a*c] + + +Int[x_^m_./(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*r)*Int[x^(m-3)*(q+r*x)/(q+r*x+x^2),x] - + 1/(2*c*r)*Int[x^(m-3)*(q-r*x)/(q-r*x+x^2),x]]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && 3<=m<4 && NegQ[b^2-4*a*c] + + +Int[x_^m_./(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*r)*Int[x^(m-1)/(q-r*x+x^2),x] - 1/(2*c*r)*Int[x^(m-1)/(q+r*x+x^2),x]]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && 1<=m<3 && NegQ[b^2-4*a*c] + + +Int[(d_.*x_)^m_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + d^2/2*(b/q+1)*Int[(d*x)^(m-2)/(b/2+q/2+c*x^2),x] - + d^2/2*(b/q-1)*Int[(d*x)^(m-2)/(b/2-q/2+c*x^2),x]] /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-4*a*c] && RationalQ[m] && m>=2 + + +Int[(d_.*x_)^m_./(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + c/q*Int[(d*x)^m/(b/2-q/2+c*x^2),x] - c/q*Int[(d*x)^m/(b/2+q/2+c*x^2),x]] /; +FreeQ[{a,b,c,d,m},x] && NeQ[b^2-4*a*c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*Sqrt[-c]*Int[x^2/(Sqrt[b+q+2*c*x^2]*Sqrt[-b+q-2*c*x^2]),x]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && NegativeQ[c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,2]}, + 1/q*Int[1/Sqrt[a+b*x^2+c*x^4],x] - 1/q*Int[(1-q*x^2)/Sqrt[a+b*x^2+c*x^4],x]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && PositiveQ[c/a] && NegativeQ[b/a] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -(b-q)/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + 1/(2*c)*Int[(b-q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] && NegativeQ[a] && PositiveQ[c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + x*(b+q+2*c*x^2)/(2*c*Sqrt[a+b*x^2+c*x^4]) - + Rt[(b+q)/(2*a),2]*(2*a+(b+q)*x^2)*Sqrt[(2*a+(b-q)*x^2)/(2*a+(b+q)*x^2)]/(2*c*Sqrt[a+b*x^2+c*x^4])* + EllipticE[ArcTan[Rt[(b+q)/(2*a),2]*x],2*q/(b+q)] /; + PosQ[(b+q)/a] && Not[PosQ[(b-q)/a] && SimplerSqrtQ[(b-q)/(2*a),(b+q)/(2*a)]]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + x*(b-q+2*c*x^2)/(2*c*Sqrt[a+b*x^2+c*x^4]) - + Rt[(b-q)/(2*a),2]*(2*a+(b-q)*x^2)*Sqrt[(2*a+(b+q)*x^2)/(2*a+(b-q)*x^2)]/(2*c*Sqrt[a+b*x^2+c*x^4])* + EllipticE[ArcTan[Rt[(b-q)/(2*a),2]*x],-2*q/(b-q)] /; + PosQ[(b-q)/a]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -(b+q)/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + 1/(2*c)*Int[(b+q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NegQ[(b+q)/a] && Not[NegQ[(b-q)/a] && SimplerSqrtQ[-(b-q)/(2*a),-(b+q)/(2*a)]]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -(b-q)/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + 1/(2*c)*Int[(b-q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NegQ[(b-q)/a]] /; +FreeQ[{a,b,c},x] && PositiveQ[b^2-4*a*c] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,2]}, + 1/q*Int[1/Sqrt[a+b*x^2+c*x^4],x] - 1/q*Int[(1-q*x^2)/Sqrt[a+b*x^2+c*x^4],x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && PosQ[c/a] + + +Int[x_^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]/Sqrt[a+b*x^2+c*x^4]* + Int[x^2/(Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]),x]] /; +FreeQ[{a,b,c},x] && NeQ[b^2-4*a*c] && NegQ[c/a] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + a^IntPart[p]*(a+b*x^2+c*x^4)^FracPart[p]/ + ((1+2*c*x^2/(b+Rt[b^2-4*a*c,2]))^FracPart[p]*(1+2*c*x^2/(b-Rt[b^2-4*a*c,2]))^FracPart[p])* + Int[(d*x)^m*(1+2*c*x^2/(b+Sqrt[b^2-4*a*c]))^p*(1+2*c*x^2/(b-Sqrt[b^2-4*a*c]))^p,x] /; +FreeQ[{a,b,c,d,m,p},x] + + +Int[u_^m_.*(a_.+b_.*v_^2+c_.*v_^4)^p_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(a+b*x^2+c*x^(2*2))^p,x],x,v] /; +FreeQ[{a,b,c,m,p},x] && LinearPairQ[u,v,x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p*) + + +Int[(d_+e_.*x_^2)/(b_.*x_^2+c_.*x_^4)^(3/4),x_Symbol] := + -2*(c*d-b*e)*(b*x^2+c*x^4)^(1/4)/(b*c*x) + e/c*Int[(b*x^2+c*x^4)^(1/4)/x^2,x] /; +FreeQ[{b,c,d,e},x] + + +Int[(d_+e_.*x_^2)*(b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + e*(b*x^2+c*x^4)^(p+1)/(c*(4*p+3)*x) /; +FreeQ[{b,c,d,e,p},x] && Not[IntegerQ[p]] && NeQ[4*p+3] && EqQ[b*e*(2*p+1)-c*d*(4*p+3)] + + +Int[(d_+e_.*x_^2)*(b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + e*(b*x^2+c*x^4)^(p+1)/(c*(4*p+3)*x) - ((b*e*(2*p+1)-c*d*(4*p+3))/(c*(4*p+3)))*Int[(b*x^2+c*x^4)^p,x] /; +FreeQ[{b,c,d,e,p},x] && Not[IntegerQ[p]] && NeQ[4*p+3] && NeQ[b*e*(2*p+1)-c*d*(4*p+3)] + + +Int[(d_+e_.*x_^2)^q_.*(b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (b*x^2+c*x^4)^FracPart[p]/(x^(2*FracPart[p])*(b+c*x^2)^FracPart[p])*Int[x^(2*p)*(d+e*x^2)^q*(b+c*x^2)^p,x] /; +FreeQ[{b,c,d,e,p,q},x] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_.)^q_.*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^p/(d+e*x^n)^(2*p)*Int[(d+e*x^n)^(q+2*p),x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] && EqQ[2*c*d-b*e,0] + + +Int[(d_+e_.*x_^n_.)^q_.*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2+c*x^n)^(2*FracPart[p]))*Int[(d+e*x^n)^q*(b/2+c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] + + +Int[(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,q},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Int[(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,c,d,e,q},x] && EqQ[c*d^2+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+(c*x^2)/e)^FracPart[p])*Int[(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + (a+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+(c*x^2)/e)^FracPart[p])*Int[(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,c,d,e,p,q},x] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[p,q] + + +Int[(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q*(a+c*x^4)^p,x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[p,q] + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + -(c*d^2-b*d*e+a*e^2)*x*(d+e*x^2)^(q+1)/(2*d*e^2*(q+1)) + + 1/(2*(q+1)*d*e^2)*Int[(d+e*x^2)^(q+1)*Simp[c*d^2-b*d*e+a*e^2*(2*(q+1)+1)+2*c*d*e*(q+1)*x^2,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4),x_Symbol] := + -(c*d^2+a*e^2)*x*(d+e*x^2)^(q+1)/(2*d*e^2*(q+1)) + + 1/(2*(q+1)*d*e^2)*Int[(d+e*x^2)^(q+1)*Simp[c*d^2+a*e^2*(2*(q+1)+1)+2*c*d*e*(q+1)*x^2,x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + c*x^(2+1)*(d+e*x^2)^(q+1)/(e*(2*(q+2)+1)) + + 1/(e*(2*(q+2)+1))*Int[(d+e*x^2)^q*(a*e*(2*(q+2)+1)-(3*c*d-b*e*(2*(q+2)+1))*x^2),x] /; +FreeQ[{a,b,c,d,e,q},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4),x_Symbol] := + c*x^(2+1)*(d+e*x^2)^(q+1)/(e*(2*(q+2)+1)) + + 1/(e*(2*(q+2)+1))*Int[(d+e*x^2)^q*(a*e*(2*(q+2)+1)-3*c*d*x^2),x] /; +FreeQ[{a,c,d,e,q},x] && NeQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[2*d/e-b/c,2]}, + e/(2*c)*Int[1/Simp[d/e+q*x+x^2,x],x] + e/(2*c)*Int[1/Simp[d/e-q*x+x^2,x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && + (PositiveQ[2*d/e-b/c] || Not[NegativeQ[2*d/e-b/c]] && EqQ[d-e*Rt[a/c,2]]) + + +Int[(d_+e_.*x_^2)/(a_+c_.*x_^4),x_Symbol] := + With[{q=Rt[2*d/e,2]}, + e/(2*c)*Int[1/Simp[d/e+q*x+x^2,x],x] + e/(2*c)*Int[1/Simp[d/e-q*x+x^2,x],x]] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2-a*e^2] && PosQ[d*e] + + +Int[(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[1/(b/2-q/2+c*x^2),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[1/(b/2+q/2+c*x^2),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[-2*d/e-b/c,2]}, + e/(2*c*q)*Int[(q-2*x)/Simp[d/e+q*x-x^2,x],x] + + e/(2*c*q)*Int[(q+2*x)/Simp[d/e-q*x-x^2,x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && Not[PositiveQ[b^2-4*a*c]] + + +Int[(d_+e_.*x_^2)/(a_+c_.*x_^4),x_Symbol] := + With[{q=Rt[-2*d/e,2]}, + e/(2*c*q)*Int[(q-2*x)/Simp[d/e+q*x-x^2,x],x] + + e/(2*c*q)*Int[(q+2*x)/Simp[d/e-q*x-x^2,x],x]] /; +FreeQ[{a,c,d,e},x] && EqQ[c*d^2-a*e^2] && NegQ[d*e] + + +Int[(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[1/(b/2-q/2+c*x^2),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[1/(b/2+q/2+c*x^2),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-a*e^2] && PosQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/(a_+c_.*x_^4),x_Symbol] := + With[{q=Rt[-a*c,2]}, + (e/2+c*d/(2*q))*Int[1/(-q+c*x^2),x] + (e/2-c*d/(2*q))*Int[1/(q+c*x^2),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2-a*e^2] && PosQ[-a*c] + + +Int[(d_+e_.*x_^2)/(a_+c_.*x_^4),x_Symbol] := + With[{q=Rt[a*c,2]}, + (d*q+a*e)/(2*a*c)*Int[(q+c*x^2)/(a+c*x^4),x] + + (d*q-a*e)/(2*a*c)*Int[(q-c*x^2)/(a+c*x^4),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && NeQ[c*d^2-a*e^2] && NegQ[-a*c] + + +Int[(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*q*r)*Int[(d*r-(d-e*q)*x)/(q-r*x+x^2),x] + + 1/(2*c*q*r)*Int[(d*r+(d-e*q)*x)/(q+r*x+x^2),x]]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && NegQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q/(a+b*x^2+c*x^4),x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && IntegerQ[q] + + +Int[(d_+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q/(a+c*x^4),x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && IntegerQ[q] + + +Int[(d_+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + e^2/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x^2)^q,x] + + 1/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x^2)^(q+1)*(c*d-b*e-c*e*x^2)/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[q]] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + e^2/(c*d^2+a*e^2)*Int[(d+e*x^2)^q,x] + + c/(c*d^2+a*e^2)*Int[(d+e*x^2)^(q+1)*(d-e*x^2)/(a+c*x^4),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[q]] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{r=Rt[b^2-4*a*c,2]}, + 2*c/r*Int[(d+e*x^2)^q/(b-r+2*c*x^2),x] - 2*c/r*Int[(d+e*x^2)^q/(b+r+2*c*x^2),x]] /; +FreeQ[{a,b,c,d,e,q},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[q]] + + +Int[(d_+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + With[{r=Rt[-a*c,2]}, + -c/(2*r)*Int[(d+e*x^2)^q/(r-c*x^2),x] - c/(2*r)*Int[(d+e*x^2)^q/(r+c*x^2),x]] /; +FreeQ[{a,c,d,e,q},x] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[q]] + + +Int[(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + x*(2*b*e*p+c*d*(4*p+3)+c*e*(4*p+1)*x^2)*(a+b*x^2+c*x^4)^p/(c*(4*p+1)*(4*p+3)) + + 2*p/(c*(4*p+1)*(4*p+3))*Int[Simp[2*a*c*d*(4*p+3)-a*b*e+(2*a*c*e*(4*p+1)+b*c*d*(4*p+3)-b^2*e*(2*p+1))*x^2,x]* + (a+b*x^2+c*x^4)^(p-1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && FractionQ[p] && p>0 && IntegerQ[2*p] + + +Int[(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + x*(d*(4*p+3)+e*(4*p+1)*x^2)*(a+c*x^4)^p/((4*p+1)*(4*p+3)) + + 2*p/((4*p+1)*(4*p+3))*Int[Simp[2*a*d*(4*p+3)+(2*a*e*(4*p+1))*x^2,x]*(a+c*x^4)^(p-1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && FractionQ[p] && p>0 && IntegerQ[2*p] + + +Int[(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + -x*(d*b^2-a*b*e-2*a*c*d+(b*d-2*a*e)*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[Simp[(2*p+3)*d*b^2-a*b*e-2*a*c*d*(4*p+5)+(4*p+7)*(d*b-2*a*e)*c*x^2,x]* + (a+b*x^2+c*x^4)^(p+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + -x*(d+e*x^2)*(a+c*x^4)^(p+1)/(4*a*(p+1)) + + 1/(4*a*(p+1))*Int[Simp[d*(4*p+5)+e*(4*p+7)*x^2,x]*(a+c*x^4)^(p+1),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*Sqrt[-c]*Int[(d+e*x^2)/(Sqrt[b+q+2*c*x^2]*Sqrt[-b+q-2*c*x^2]),x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && NegativeQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[-a*c,2]}, + Sqrt[-c]*Int[(d+e*x^2)/(Sqrt[q+c*x^2]*Sqrt[q-c*x^2]),x]] /; +FreeQ[{a,c,d,e},x] && PositiveQ[a] && NegativeQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,4]}, + -d*x*Sqrt[a+b*x^2+c*x^4]/(a*(1+q^2*x^2)) + + d*(1+q^2*x^2)*Sqrt[(a+b*x^2+c*x^4)/(a*(1+q^2*x^2)^2)]/(q*Sqrt[a+b*x^2+c*x^4])*EllipticE[2*ArcTan[q*x],1/2-b*q^2/(4*c)] /; + EqQ[e+d*q^2]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && PositiveQ[c/a] && NegativeQ[b/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,2]}, + (e+d*q)/q*Int[1/Sqrt[a+b*x^2+c*x^4],x] - e/q*Int[(1-q*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NeQ[e+d*q]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && PositiveQ[c/a] && NegativeQ[b/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + e*x*(b+q+2*c*x^2)/(2*c*Sqrt[a+b*x^2+c*x^4]) - + e*q*Sqrt[(2*a+(b-q)*x^2)/(2*a+(b+q)*x^2)]*Sqrt[(2*a+(b+q)*x^2)/q]/(2*c*Sqrt[a+b*x^2+c*x^4]*Sqrt[a/(2*a+(b+q)*x^2)])* + EllipticE[ArcSin[x/Sqrt[(2*a+(b+q)*x^2)/(2*q)]],(b+q)/(2*q)] /; + EqQ[2*c*d-e*(b-q)]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && NegativeQ[a] && PositiveQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[-a*c,2]}, + e*x*(q+c*x^2)/(c*Sqrt[a+c*x^4]) - + Sqrt[2]*e*q*Sqrt[-a+q*x^2]*Sqrt[(a+q*x^2)/q]/(Sqrt[-a]*c*Sqrt[a+c*x^4])* + EllipticE[ArcSin[x/Sqrt[(a+q*x^2)/(2*q)]],1/2] /; + EqQ[c*d+e*q] && IntegerQ[q]] /; +FreeQ[{a,c,d,e},x] && NegativeQ[a] && PositiveQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[-a*c,2]}, + e*x*(q+c*x^2)/(c*Sqrt[a+c*x^4]) - + Sqrt[2]*e*q*Sqrt[(a-q*x^2)/(a+q*x^2)]*Sqrt[(a+q*x^2)/q]/(c*Sqrt[a+c*x^4]*Sqrt[a/(a+q*x^2)])* + EllipticE[ArcSin[x/Sqrt[(a+q*x^2)/(2*q)]],1/2] /; + EqQ[c*d+e*q]] /; +FreeQ[{a,c,d,e},x] && NegativeQ[a] && PositiveQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*d-e*(b-q))/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + e/(2*c)*Int[(b-q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NeQ[2*c*d-e*(b-q)]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && NegativeQ[a] && PositiveQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[-a*c,2]}, + (c*d+e*q)/c*Int[1/Sqrt[a+c*x^4],x] - e/c*Int[(q-c*x^2)/Sqrt[a+c*x^4],x] /; + NeQ[c*d+e*q]] /; +FreeQ[{a,c,d,e},x] && NegativeQ[a] && PositiveQ[c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + d*Int[1/Sqrt[a+b*x^2+c*x^4],x] + e*Int[x^2/Sqrt[a+b*x^2+c*x^4],x] /; + PosQ[(b+q)/a] || PosQ[(b-q)/a]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + d*Int[1/Sqrt[a+c*x^4],x] + e*Int[x^2/Sqrt[a+c*x^4],x] /; +FreeQ[{a,c,d,e},x] && PositiveQ[-a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -a*e*Rt[-(b+q)/(2*a),2]*Sqrt[1+(b+q)*x^2/(2*a)]*Sqrt[1+(b-q)*x^2/(2*a)]/(c*Sqrt[a+b*x^2+c*x^4])* + EllipticE[ArcSin[Rt[-(b+q)/(2*a),2]*x],(b-q)/(b+q)] /; + NegQ[(b+q)/a] && EqQ[2*c*d-e*(b+q)] && Not[SimplerSqrtQ[-(b-q)/(2*a),-(b+q)/(2*a)]]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*d-e*(b+q))/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + e/(2*c)*Int[(b+q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NegQ[(b+q)/a] && NeQ[2*c*d-e*(b+q)] && Not[SimplerSqrtQ[-(b-q)/(2*a),-(b+q)/(2*a)]]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + -a*e*Rt[-(b-q)/(2*a),2]*Sqrt[1+(b-q)*x^2/(2*a)]*Sqrt[1+(b+q)*x^2/(2*a)]/(c*Sqrt[a+b*x^2+c*x^4])* + EllipticE[ArcSin[Rt[-(b-q)/(2*a),2]*x],(b+q)/(b-q)] /; + NegQ[(b-q)/a] && EqQ[2*c*d-e*(b-q)]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*d-e*(b-q))/(2*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + e/(2*c)*Int[(b-q+2*c*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NegQ[(b-q)/a] && NeQ[2*c*d-e*(b-q)]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,4]}, + -d*x*Sqrt[a+b*x^2+c*x^4]/(a*(1+q^2*x^2)) + + d*(1+q^2*x^2)*Sqrt[(a+b*x^2+c*x^4)/(a*(1+q^2*x^2)^2)]/(q*Sqrt[a+b*x^2+c*x^4])*EllipticE[2*ArcTan[q*x],1/2-b*q^2/(4*c)] /; + EqQ[e+d*q^2]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && PosQ[c/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,4]}, + -d*x*Sqrt[a+c*x^4]/(a*(1+q^2*x^2)) + + d*(1+q^2*x^2)*Sqrt[(a+c*x^4)/(a*(1+q^2*x^2)^2)]/(q*Sqrt[a+c*x^4])*EllipticE[2*ArcTan[q*x],1/2] /; + EqQ[e+d*q^2]] /; +FreeQ[{a,c,d,e},x] && PosQ[c/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,2]}, + (e+d*q)/q*Int[1/Sqrt[a+b*x^2+c*x^4],x] - e/q*Int[(1-q*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; + NeQ[e+d*q]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && PosQ[c/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[c/a,2]}, + (e+d*q)/q*Int[1/Sqrt[a+c*x^4],x] - e/q*Int[(1-q*x^2)/Sqrt[a+c*x^4],x] /; + NeQ[e+d*q]] /; +FreeQ[{a,c,d,e},x] && PosQ[c/a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + d/Sqrt[a]*Int[Sqrt[1+e*x^2/d]/Sqrt[1-e*x^2/d],x] /; +FreeQ[{a,c,d,e},x] && NegQ[c/a] && EqQ[c*d^2+a*e^2] && PositiveQ[a] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + Sqrt[1+c*x^4/a]/Sqrt[a+c*x^4]*Int[(d+e*x^2)/Sqrt[1+c*x^4/a],x] /; +FreeQ[{a,c,d,e},x] && NegQ[c/a] && EqQ[c*d^2+a*e^2] && Not[PositiveQ[a]] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+c_.*x_^4],x_Symbol] := + With[{q=Rt[-c/a,2]}, + (d*q-e)/q*Int[1/Sqrt[a+c*x^4],x] + e/q*Int[(1+q*x^2)/Sqrt[a+c*x^4],x]] /; +FreeQ[{a,c,d,e},x] && NegQ[c/a] && NeQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_^2)/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]/Sqrt[a+b*x^2+c*x^4]* + Int[(d+e*x^2)/(Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NegQ[c/a] + + +Int[(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)*(a+c*x^4)^p,x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +(* Int[(d_+e_.*x_^2)^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + e^2*x*Sqrt[a+b*x^2+c*x^4]/(3*c) + + 2*(3*c*d-b*e)/(3*c)*Int[(d+e*x^2)/Sqrt[a+b*x^2+c*x^4],x] - + (3*c*d^2-2*b*d*e+a*e^2)/(3*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] *) + + +(* Int[(d_+e_.*x_^2)^2/Sqrt[a_+c_.*x_^4],x_Symbol] := + e^2*x*Sqrt[a+c*x^4]/(3*c) + + 2*d*Int[(d+e*x^2)/Sqrt[a+c*x^4],x] - + (3*c*d^2+a*e^2)/(3*c)*Int[1/Sqrt[a+c*x^4],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] *) + + +(* Int[(d_+e_.*x_^2)^q_/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + e^2*x*(d+e*x^2)^(q-2)*Sqrt[a+b*x^2+c*x^4]/(c*(2*q-1)) + + 2*(q-1)*(3*c*d-b*e)/(c*(2*q-1))*Int[(d+e*x^2)^(q-1)/Sqrt[a+b*x^2+c*x^4],x] - + (2*q-3)*(3*c*d^2-2*b*d*e+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-2)/Sqrt[a+b*x^2+c*x^4],x] + + 2*d*(q-2)*(c*d^2-b*d*e+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-3)/Sqrt[a+b*x^2+c*x^4],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[q-2] *) + + +(* Int[(d_+e_.*x_^2)^q_/Sqrt[a_+c_.*x_^4],x_Symbol] := + e^2*x*(d+e*x^2)^(q-2)*Sqrt[a+c*x^4]/(c*(2*q-1)) + + 6*d*(q-1)/(2*q-1)*Int[(d+e*x^2)^(q-1)/Sqrt[a+c*x^4],x] - + (2*q-3)*(3*c*d^2+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-2)/Sqrt[a+c*x^4],x] + + 2*d*(q-2)*(c*d^2+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-3)/Sqrt[a+c*x^4],x] /; +FreeQ[{a,c,d,e},x] && PositiveIntegerQ[q-2] *) + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{f=Coeff[PolynomialRemainder[(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,0], + g=Coeff[PolynomialRemainder[(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*g-f*(b^2-2*a*c)-c*(b*f-2*a*g)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(a+b*x^2+c*x^4)^(p+1)* + ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*PolynomialQuotient[(d+e*x^2)^q,a+b*x^2+c*x^4,x]+ + b^2*f*(2*p+3)-2*a*c*f*(4*p+5)-a*b*g+c*(4*p+7)*(b*f-2*a*g)*x^2,x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[q-1] && + RationalQ[p] && p<-1 + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + e^q*x^(2*q-3)*(a+b*x^2+c*x^4)^(p+1)/(c*(4*p+2*q+1)) + + 1/(c*(4*p+2*q+1))*Int[(a+b*x^2+c*x^4)^p* + ExpandToSum[c*(4*p+2*q+1)*(d+e*x^2)^q-a*(2*q-3)*e^q*x^(2*q-4)-b*(2*p+2*q-1)*e^q*x^(2*q-2)-c*(4*p+2*q+1)*e^q*x^(2*q),x],x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[q-1] + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + e^q*x^(2*q-3)*(a+c*x^4)^(p+1)/(c*(4*p+2*q+1)) + + 1/(c*(4*p+2*q+1))*Int[(a+c*x^4)^p* + ExpandToSum[c*(4*p+2*q+1)*(d+e*x^2)^q-a*(2*q-3)*e^q*x^(2*q-4)-c*(4*p+2*q+1)*e^q*x^(2*q),x],x] /; +FreeQ[{a,c,d,e,p},x] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[q-1] + + +Int[(a_+b_.*x_^2+c_.*x_^4)^p_/(d_+e_.*x_^2),x_Symbol] := + -x*(b^2*c*d-b^3*e-2*a*c^2*d+3*a*b*c*e+c*(b*c*d-b^2*e+2*a*c*e)*x^2)*(a+b*x^2+c*x^4)^(p+1)/ + (2*a*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) - + 1/(2*a*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2))*Int[((a+b*x^2+c*x^4)^(p+1)/(d+e*x^2))* + Simp[b^3*d*e*(3+2*p)-a*b*c*d*e*(11+8*p)-b^2*(2*a*e^2*(p+1)+c*d^2*(3+2*p))+ + 2*a*c*(4*a*e^2*(p+1)+c*d^2*(5+4*p))- + (4*a*c^2*d*e-2*b^2*c*d*e*(2+p)-b^3*e^2*(3+2*p)+b*c*(c*d^2*(7+4*p)+a*e^2*(11+8*p)))*x^2- + c*e*(b*c*d-b^2*e+2*a*c*e)*(7+4*p)*x^4,x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && NegativeIntegerQ[p+1/2] + + +Int[(a_+c_.*x_^4)^p_/(d_+e_.*x_^2),x_Symbol] := + -x*(-2*a*c^2*d+c*(2*a*c*e)*x^2)*(a+c*x^4)^(p+1)/(2*a*(p+1)*(-4*a*c)*(c*d^2+a*e^2)) - + 1/(2*a*(p+1)*(-4*a*c)*(c*d^2+a*e^2))*Int[((a+c*x^4)^(p+1)/(d+e*x^2))* + Simp[2*a*c*(4*a*e^2*(p+1)+c*d^2*(5+4*p))-(4*a*c^2*d*e)*x^2-c*e*(2*a*c*e)*(7+4*p)*x^4,x],x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && NegativeIntegerQ[p+1/2] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/(d_+e_.*x_^2),x_Symbol] := + -1/e^2*Int[(c*d-b*e-c*e*x^2)/Sqrt[a+b*x^2+c*x^4],x] + + (c*d^2-b*d*e+a*e^2)/e^2*Int[1/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[Sqrt[a_+c_.*x_^4]/(d_+e_.*x_^2),x_Symbol] := + -c/e^2*Int[(d-e*x^2)/Sqrt[a+c*x^4],x] + + (c*d^2+a*e^2)/e^2*Int[1/((d+e*x^2)*Sqrt[a+c*x^4]),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*Sqrt[-c]*Int[1/((d+e*x^2)*Sqrt[b+q+2*c*x^2]*Sqrt[-b+q-2*c*x^2]),x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && NegativeQ[c] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + With[{q=Rt[-a*c,2]}, + Sqrt[-c]*Int[1/((d+e*x^2)*Sqrt[q+c*x^2]*Sqrt[q-c*x^2]),x]] /; +FreeQ[{a,c,d,e},x] && PositiveQ[a] && NegativeQ[c] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/(2*c*d-e*(b-q))*Int[1/Sqrt[a+b*x^2+c*x^4],x] - e/(2*c*d-e*(b-q))*Int[(b-q+2*c*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[b^2-4*a*c] && Not[NegativeQ[c]] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + With[{q=Rt[-a*c,2]}, + c/(c*d+e*q)*Int[1/Sqrt[a+c*x^4],x] + e/(c*d+e*q)*Int[(q-c*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x]] /; +FreeQ[{a,c,d,e},x] && PositiveQ[-a*c] && Not[NegativeQ[c]] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + With[{q=Rt[c/a,4]}, + ArcTan[Sqrt[(c*d^2-b*d*e+a*e^2)/(d*e)]*x/Sqrt[a+b*x^2+c*x^4]]/(2*d*Sqrt[(c*d^2-b*d*e+a*e^2)/(d*e)]) + + (e+d*q^2)*(1+q^2*x^2)*Sqrt[(a+b*x^2+c*x^4)/(a*(1+q^2*x^2)^2)]/(4*d*q*(e-d*q^2)*Sqrt[a+b*x^2+c*x^4])* + EllipticPi[-(e-d*q^2)^2/(4*d*e*q^2),2*ArcTan[q*x],1/2-b*q^2/(4*c)] - + q^2/(e-d*q^2)*Int[1/Sqrt[a+b*x^2+c*x^4],x] /; + NeQ[e-d*q^2]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PosQ[c/a] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + With[{q=Rt[c/a,4]}, + ArcTan[Sqrt[(c*d^2+a*e^2)/(d*e)]*x/Sqrt[a+c*x^4]]/(2*d*Sqrt[(c*d^2+a*e^2)/(d*e)]) + + (e+d*q^2)*(1+q^2*x^2)*Sqrt[(a+c*x^4)/(a*(1+q^2*x^2)^2)]/(4*d*q*(e-d*q^2)*Sqrt[a+c*x^4])* + EllipticPi[-(e-d*q^2)^2/(4*d*e*q^2),2*ArcTan[q*x],1/2] - + q^2/(e-d*q^2)*Int[1/Sqrt[a+c*x^4],x] /; + NeQ[e-d*q^2]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && PosQ[c/a] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + With[{q=Rt[-c/a,4]}, + 1/(d*Sqrt[a]*q)*EllipticPi[-e/(d*q^2),ArcSin[q*x],-1]] /; +FreeQ[{a,c,d,e},x] && NegQ[c/a] && PositiveQ[a] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + Sqrt[1+c*x^4/a]/Sqrt[a+c*x^4]*Int[1/((d+e*x^2)*Sqrt[1+c*x^4/a]),x] /; +FreeQ[{a,c,d,e},x] && NegQ[c/a] && Not[PositiveQ[a]] + + +Int[1/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]/Sqrt[a+b*x^2+c*x^4]* + Int[1/((d+e*x^2)*Sqrt[1+2*c*x^2/(b-q)]*Sqrt[1+2*c*x^2/(b+q)]),x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NegQ[c/a] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/(d_+e_.*x_^2)^2,x_Symbol] := + x*Sqrt[a+b*x^2+c*x^4]/(2*d*(d+e*x^2)) + + c/(2*d*e^2)*Int[(d-e*x^2)/Sqrt[a+b*x^2+c*x^4],x] - + (c*d^2-a*e^2)/(2*d*e^2)*Int[1/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[Sqrt[a_+c_.*x_^4]/(d_+e_.*x_^2)^2,x_Symbol] := + x*Sqrt[a+c*x^4]/(2*d*(d+e*x^2)) + + c/(2*d*e^2)*Int[(d-e*x^2)/Sqrt[a+c*x^4],x] - + (c*d^2-a*e^2)/(2*d*e^2)*Int[1/((d+e*x^2)*Sqrt[a+c*x^4]),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[1/((d_+e_.*x_^2)^2*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + e^2*x*Sqrt[a+b*x^2+c*x^4]/(2*d*(c*d^2-b*d*e+a*e^2)*(d+e*x^2)) - + c/(2*d*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x^2)/Sqrt[a+b*x^2+c*x^4],x] + + (3*c*d^2-2*b*d*e+a*e^2)/(2*d*(c*d^2-b*d*e+a*e^2))*Int[1/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[1/((d_+e_.*x_^2)^2*Sqrt[a_+c_.*x_^4]),x_Symbol] := + e^2*x*Sqrt[a+c*x^4]/(2*d*(c*d^2+a*e^2)*(d+e*x^2)) - + c/(2*d*(c*d^2+a*e^2))*Int[(d+e*x^2)/Sqrt[a+c*x^4],x] + + (3*c*d^2+a*e^2)/(2*d*(c*d^2+a*e^2))*Int[1/((d+e*x^2)*Sqrt[a+c*x^4]),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Module[{aa,bb,cc}, + Int[ReplaceAll[ExpandIntegrand[1/Sqrt[aa+bb*x^2+cc*x^4],(d+e*x^2)^q*(aa+bb*x^2+cc*x^4)^(p+1/2),x],{aa->a,bb->b,cc->c}],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && NegativeIntegerQ[q] && IntegerQ[p+1/2] && p!=-1/2 + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + Module[{aa,cc}, + Int[ReplaceAll[ExpandIntegrand[1/Sqrt[aa+cc*x^4],(d+e*x^2)^q*(aa+cc*x^4)^(p+1/2),x],{aa->a,cc->c}],x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && NegativeIntegerQ[q] && IntegerQ[p+1/2] && p!=-1/2 + + +Int[(d_+e_.*x_^2)^q_/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] := + -e^2*x*(d+e*x^2)^(q+1)*Sqrt[a+b*x^2+c*x^4]/(2*d*(q+1)*(c*d^2-b*d*e+a*e^2)) + + (2*q+3)*(3*c*d^2-2*b*d*e+a*e^2)/(2*d*(q+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x^2)^(q+1)/Sqrt[a+b*x^2+c*x^4],x] - + (q+2)*(3*c*d-b*e)/(d*(q+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x^2)^(q+2)/Sqrt[a+b*x^2+c*x^4],x] + + c*(2*q+5)/(2*d*(q+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x^2)^(q+3)/Sqrt[a+b*x^2+c*x^4],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c] && NegativeIntegerQ[q+2] + + +Int[(d_+e_.*x_^2)^q_/Sqrt[a_+c_.*x_^4],x_Symbol] := + -e^2*x*(d+e*x^2)^(q+1)*Sqrt[a+c*x^4]/(2*d*(q+1)*(c*d^2+a*e^2)) + + (2*q+3)*(3*c*d^2+a*e^2)/(2*d*(q+1)*(c*d^2+a*e^2))*Int[(d+e*x^2)^(q+1)/Sqrt[a+c*x^4],x] - + 3*c*(q+2)/((q+1)*(c*d^2+a*e^2))*Int[(d+e*x^2)^(q+2)/Sqrt[a+c*x^4],x] + + c*(2*q+5)/(2*d*(q+1)*(c*d^2+a*e^2))*Int[(d+e*x^2)^(q+3)/Sqrt[a+c*x^4],x] /; +FreeQ[{a,c,d,e},x] && NegativeIntegerQ[q+2] + + +Int[1/(Sqrt[d_+e_.*x_^2]*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + 1/(2*Sqrt[a]*Sqrt[d]*Rt[-e/d,2])*EllipticF[2*ArcSin[Rt[-e/d,2]*x],b*d/(4*a*e)] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d-b*e] && PositiveQ[a] && PositiveQ[d] + + +Int[1/(Sqrt[d_+e_.*x_^2]*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + Sqrt[(d+e*x^2)/d]*Sqrt[(a+b*x^2+c*x^4)/a]/(Sqrt[d+e*x^2]*Sqrt[a+b*x^2+c*x^4])* + Int[1/(Sqrt[1+e/d*x^2]*Sqrt[1+b/a*x^2+c/a*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d-b*e] && Not[PositiveQ[a] && PositiveQ[d]] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[a]/(2*Sqrt[d]*Rt[-e/d,2])*EllipticE[2*ArcSin[Rt[-e/d,2]*x],b*d/(4*a*e)] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d-b*e] && PositiveQ[a] && PositiveQ[d] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2+c*x^4]*Sqrt[(d+e*x^2)/d]/(Sqrt[d+e*x^2]*Sqrt[(a+b*x^2+c*x^4)/a])* + Int[Sqrt[1+b/a*x^2+c/a*x^4]/Sqrt[1+e/d*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d-b*e] && Not[PositiveQ[a] && PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + c^p*x^(4*p-1)*(d+e*x^2)^(q+1)/(e*(4*p+2*q+1)) + + Int[(d+e*x^2)^q*ExpandToSum[(a+b*x^2+c*x^4)^p-c^p*x^(4*p)-d*c^p*(4*p-1)*x^(4*p-2)/(e*(4*p+2*q+1)),x],x] /; +FreeQ[{a,b,c,d,e,q},x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[p] && + NeQ[4*p+2*q+1] && Not[PositiveIntegerQ[q]] + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + c^p*x^(4*p-1)*(d+e*x^2)^(q+1)/(e*(4*p+2*q+1)) + + Int[(d+e*x^2)^q*ExpandToSum[(a+c*x^4)^p-c^p*x^(4*p)-d*c^p*(4*p-1)*x^(4*p-2)/(e*(4*p+2*q+1)),x],x] /; +FreeQ[{a,c,d,e,q},x] && PositiveIntegerQ[p] && + NeQ[4*p+2*q+1] && Not[PositiveIntegerQ[q]] + + +Int[(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b^2-4*a*c] && (IntegersQ[p,q] || PositiveIntegerQ[p] || PositiveIntegerQ[q]) + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^q*(a+c*x^4)^p,x],x] /; +FreeQ[{a,c,d,e,p,q},x] && (IntegersQ[p,q] || PositiveIntegerQ[p]) + + +Int[(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + Int[ExpandIntegrand[(a+c*x^4)^p,(d/(d^2-e^2*x^4)-e*x^2/(d^2-e^2*x^4))^(-q),x],x] /; +FreeQ[{a,c,d,e,p},x] && NeQ[c*d^2+a*e^2] && NegativeIntegerQ[q] && Not[IntegersQ[2*p]] + + +Int[(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Defer[Int][(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,e,p,q},x] + + +Int[(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Defer[Int][(d+e*x^2)^q*(a+c*x^4)^p,x] /; +FreeQ[{a,c,d,e,p,q},x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p*) + + +Int[x_^m_.*(e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + 1/(2*e^((m+1)/2-1))*Subst[Int[(e*x)^(q+(m+1)/2-1)*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,e,p,q},x] && IntegerQ[(m+1)/2] + + +Int[x_^m_.*(e_.*x_^2)^q_*(a_+c_.*x_^4)^p_.,x_Symbol] := + 1/(2*e^((m+1)/2-1))*Subst[Int[(e*x)^(q+(m+1)/2-1)*(a+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,c,e,p,q},x] && IntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_.*(e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + f^m*e^IntPart[q]*(e*x^2)^FracPart[q]/x^(2*FracPart[q])*Int[x^(m+2*q)*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,e,f,m,p,q},x] && (IntegerQ[m] || PositiveQ[f]) && Not[IntegerQ[(m+1)/2]] + + +Int[(f_.*x_)^m_.*(e_.*x_^2)^q_*(a_+c_.*x_^4)^p_.,x_Symbol] := + f^m*e^IntPart[q]*(e*x^2)^FracPart[q]/x^(2*FracPart[q])*Int[x^(m+2*q)*(a+c*x^4)^p,x] /; +FreeQ[{a,c,e,f,m,p,q},x] && (IntegerQ[m] || PositiveQ[f]) && Not[IntegerQ[(m+1)/2]] + + +Int[(f_*x_)^m_.*(e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(e*x^2)^q*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,e,f,m,p,q},x] && Not[IntegerQ[m]] + + +Int[(f_*x_)^m_.*(e_.*x_^2)^q_*(a_+c_.*x_^4)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(e*x^2)^q*(a+c*x^4)^p,x] /; +FreeQ[{a,c,e,f,m,p,q},x] && Not[IntegerQ[m]] + + +Int[x_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,d,e,p,q},x] + + +Int[x_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[(d+e*x)^q*(a+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,c,d,e,p,q},x] + + +Int[x_^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + 1/2*Subst[Int[x^((m+1)/2-1)*(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[b^2-4*a*c,0] && Not[IntegerQ[p]] && PositiveIntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^FracPart[p]/(c^IntPart[p]*(b/2+c*x^2)^(2*FracPart[p]))* + Int[(f*x)^m*(d+e*x^2)^q*(b/2+c*x^2)^(2*p),x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && EqQ[b^2-4*a*c,0] && Not[IntegerQ[p]] + + +Int[x_^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[x^((m+1)/2-1)*(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,d,e,m,p,q},x] && IntegerQ[(m+1)/2] + + +Int[x_^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + 1/2*Subst[Int[x^((m+1)/2-1)*(d+e*x)^q*(a+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,c,d,e,m,p,q},x] && IntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[(f*x)^m*(d+e*x^2)^(q+p)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Int[(f*x)^m*(d+e*x^2)^(q+p)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,q,m,q},x] && EqQ[c*d^2+a*e^2] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+(c*x^2)/e)^FracPart[p])* + Int[(f*x)^m*(d+e*x^2)^(q+p)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + (a+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+(c*x^2)/e)^FracPart[p])*Int[(f*x)^m*(d+e*x^2)^(q+p)*(a/d+c/e*x^2)^p,x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[x_^m_.*(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + (-d)^(m/2-1)*(c*d^2-b*d*e+a*e^2)^p*x*(d+e*x^2)^(q+1)/(2*e^(2*p+m/2)*(q+1)) + + 1/(2*e^(2*p+m/2)*(q+1))*Int[(d+e*x^2)^(q+1)* + ExpandToSum[Together[1/(d+e*x^2)*(2*e^(2*p+m/2)*(q+1)*x^m*(a+b*x^2+c*x^4)^p- + (-d)^(m/2-1)*(c*d^2-b*d*e+a*e^2)^p*(d+e*(2*q+3)*x^2))],x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && PositiveIntegerQ[p] && IntegersQ[m/2,q] && q<-1 && m>0 + + +Int[x_^m_.*(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_.,x_Symbol] := + (-d)^(m/2-1)*(c*d^2+a*e^2)^p*x*(d+e*x^2)^(q+1)/(2*e^(2*p+m/2)*(q+1)) + + 1/(2*e^(2*p+m/2)*(q+1))*Int[(d+e*x^2)^(q+1)* + ExpandToSum[Together[1/(d+e*x^2)*(2*e^(2*p+m/2)*(q+1)*x^m*(a+c*x^4)^p- + (-d)^(m/2-1)*(c*d^2+a*e^2)^p*(d+e*(2*q+3)*x^2))],x],x] /; +FreeQ[{a,c,d,e},x] && PositiveIntegerQ[p] && IntegersQ[m/2,q] && q<-1 && m>0 + + +Int[x_^m_*(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + (-d)^(m/2-1)*(c*d^2-b*d*e+a*e^2)^p*x*(d+e*x^2)^(q+1)/(2*e^(2*p+m/2)*(q+1)) + + (-d)^(m/2-1)/(2*e^(2*p)*(q+1))*Int[x^m*(d+e*x^2)^(q+1)* + ExpandToSum[Together[1/(d+e*x^2)*(2*(-d)^(-m/2+1)*e^(2*p)*(q+1)*(a+b*x^2+c*x^4)^p - + (e^(-m/2)*(c*d^2-b*d*e+a*e^2)^p*x^(-m))*(d+e*(2*q+3)*x^2))],x],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && PositiveIntegerQ[p] && IntegersQ[m/2,q] && q<-1 && m<0 + + +Int[x_^m_*(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_.,x_Symbol] := + (-d)^(m/2-1)*(c*d^2+a*e^2)^p*x*(d+e*x^2)^(q+1)/(2*e^(2*p+m/2)*(q+1)) + + (-d)^(m/2-1)/(2*e^(2*p)*(q+1))*Int[x^m*(d+e*x^2)^(q+1)* + ExpandToSum[Together[1/(d+e*x^2)*(2*(-d)^(-m/2+1)*e^(2*p)*(q+1)*(a+c*x^4)^p - + (e^(-m/2)*(c*d^2+a*e^2)^p*x^(-m))*(d+e*(2*q+3)*x^2))],x],x] /; +FreeQ[{a,c,d,e},x] && PositiveIntegerQ[p] && IntegersQ[m/2,q] && q<-1 && m<0 + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + c^p*(f*x)^(m+4*p-1)*(d+e*x^2)^(q+1)/(e*f^(4*p-1)*(m+4*p+2*q+1)) + + 1/(e*(m+4*p+2*q+1))*Int[(f*x)^m*(d+e*x^2)^q* + ExpandToSum[e*(m+4*p+2*q+1)*((a+b*x^2+c*x^4)^p-c^p*x^(4*p))-d*c^p*(m+4*p-1)*x^(4*p-2),x],x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && NeQ[b^2-4*a*c,0] && PositiveIntegerQ[p] && Not[IntegerQ[q]] && NeQ[m+4*p+2*q+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + c^p*(f*x)^(m+4*p-1)*(d+e*x^2)^(q+1)/(e*f^(4*p-1)*(m+4*p+2*q+1)) + + 1/(e*(m+4*p+2*q+1))*Int[(f*x)^m*(d+e*x^2)^q* + ExpandToSum[e*(m+4*p+2*q+1)*((a+c*x^4)^p-c^p*x^(4*p))-d*c^p*(m+4*p-1)*x^(4*p-2),x],x] /; +FreeQ[{a,c,d,e,f,m,q},x] && PositiveIntegerQ[p] && Not[IntegerQ[q]] && NeQ[m+4*p+2*q+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m(d+e*x^2)^q*(a+c*x^4)^p,x],x] /; +FreeQ[{a,c,d,e,f,m,q},x] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/f*Subst[Int[x^(k*(m+1)-1)*(d+e*x^(2*k)/f^2)^q*(a+b*x^(2*k)/f^k+c*x^(4*k)/f^4)^p,x],x,(f*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c,0] && FractionQ[m] && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/f*Subst[Int[x^(k*(m+1)-1)*(d+e*x^(2*k)/f)^q*(a+c*x^(4*k)/f)^p,x],x,(f*x)^(1/k)]] /; +FreeQ[{a,c,d,e,f,p,q},x] && FractionQ[m] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+b*x^2+c*x^4)^p*(d*(m+4*p+3)+e*(m+1)*x^2)/(f*(m+1)*(m+4*p+3)) + + 2*p/(f^2*(m+1)*(m+4*p+3))*Int[(f*x)^(m+2)*(a+b*x^2+c*x^4)^(p-1)* + Simp[2*a*e*(m+1)-b*d*(m+4*p+3)+(b*e*(m+1)-2*c*d*(m+4*p+3))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && RationalQ[m,p] && p>0 && m<-1 && + m+4*p+3!=0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+c*x^4)^p*(d*(m+4*p+3)+e*(m+1)*x^2)/(f*(m+1)*(m+4*p+3)) + + 4*p/(f^2*(m+1)*(m+4*p+3))*Int[(f*x)^(m+2)*(a+c*x^4)^(p-1)*(a*e*(m+1)-c*d*(m+4*p+3)*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p>0 && m<-1 && + m+4*p+3!=0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+b*x^2+c*x^4)^p*(b*e*2*p+c*d*(m+4*p+3)+c*e*(4*p+m+1)*x^2)/ + (c*f*(4*p+m+1)*(m+4*p+3)) + + 2*p/(c*(4*p+m+1)*(m+4*p+3))*Int[(f*x)^m*(a+b*x^2+c*x^4)^(p-1)* + Simp[2*a*c*d*(m+4*p+3)-a*b*e*(m+1)+(2*a*c*e*(4*p+m+1)+b*c*d*(m+4*p+3)-b^2*e*(m+2*p+1))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] && RationalQ[p] && p>0 && + NeQ[4*p+m+1] && NeQ[m+4*p+3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+c*x^4)^p*(c*d*(m+4*p+3)+c*e*(4*p+m+1)*x^2)/(c*f*(4*p+m+1)*(m+4*p+3)) + + 4*a*p/((4*p+m+1)*(m+4*p+3))*Int[(f*x)^m*(a+c*x^4)^(p-1)*Simp[d*(m+4*p+3)+e*(4*p+m+1)*x^2,x],x] /; +FreeQ[{a,c,d,e,f,m},x] && RationalQ[p] && p>0 && + NeQ[4*p+m+1] && NeQ[m+4*p+3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + f*(f*x)^(m-1)*(a+b*x^2+c*x^4)^(p+1)*(b*d-2*a*e-(b*e-2*c*d)*x^2)/(2*(p+1)*(b^2-4*a*c)) - + f^2/(2*(p+1)*(b^2-4*a*c))*Int[(f*x)^(m-2)*(a+b*x^2+c*x^4)^(p+1)* + Simp[(m-1)*(b*d-2*a*e)-(4*p+4+m+1)*(b*e-2*c*d)*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && RationalQ[m,p] && p<-1 && m>1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_.,x_Symbol] := + f*(f*x)^(m-1)*(a+c*x^4)^(p+1)*(a*e-c*d*x^2)/(4*a*c*(p+1)) - + f^2/(4*a*c*(p+1))*Int[(f*x)^(m-2)*(a+c*x^4)^(p+1)*(a*e*(m-1)-c*d*(4*p+4+m+1)*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p<-1 && m>1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+b*x^2+c*x^4)^(p+1)*(d*(b^2-2*a*c)-a*b*e+(b*d-2*a*e)*c*x^2)/(2*a*f*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(f*x)^m*(a+b*x^2+c*x^4)^(p+1)* + Simp[d*(b^2*(m+2*(p+1)+1)-2*a*c*(m+4*(p+1)+1))-a*b*e*(m+1)+c*(m+2*(2*p+3)+1)*(b*d-2*a*e)*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] && RationalQ[p] && p<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+c*x^4)^(p+1)*(d+e*x^2)/(4*a*f*(p+1)) + + 1/(4*a*(p+1))*Int[(f*x)^m*(a+c*x^4)^(p+1)*Simp[d*(m+4*(p+1)+1)+e*(m+2*(2*p+3)+1)*x^2,x],x] /; +FreeQ[{a,c,d,e,f,m},x] && RationalQ[p] && p<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + e*f*(f*x)^(m-1)*(a+b*x^2+c*x^4)^(p+1)/(c*(m+4*p+3)) - + f^2/(c*(m+4*p+3))*Int[(f*x)^(m-2)*(a+b*x^2+c*x^4)^p*Simp[a*e*(m-1)+(b*e*(m+2*p+1)-c*d*(m+4*p+3))*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && NeQ[b^2-4*a*c,0] && RationalQ[m] && m>1 && + NeQ[m+4*p+3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + e*f*(f*x)^(m-1)*(a+c*x^4)^(p+1)/(c*(m+4*p+3)) - + f^2/(c*(m+4*p+3))*Int[(f*x)^(m-2)*(a+c*x^4)^p*(a*e*(m-1)-c*d*(m+4*p+3)*x^2),x] /; +FreeQ[{a,c,d,e,f,p},x] && RationalQ[m] && m>1 && NeQ[m+4*p+3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + d*(f*x)^(m+1)*(a+b*x^2+c*x^4)^(p+1)/(a*f*(m+1)) + + 1/(a*f^2*(m+1))*Int[(f*x)^(m+2)*(a+b*x^2+c*x^4)^p*Simp[a*e*(m+1)-b*d*(m+2*p+3)-c*d*(m+4*p+5)*x^2,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && NeQ[b^2-4*a*c,0] && RationalQ[m] && m<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)*(a_+c_.*x_^4)^p_,x_Symbol] := + d*(f*x)^(m+1)*(a+c*x^4)^(p+1)/(a*f*(m+1)) + + 1/(a*f^2*(m+1))*Int[(f*x)^(m+2)*(a+c*x^4)^p*(a*e*(m+1)-c*d*(m+4*p+5)*x^2),x] /; +FreeQ[{a,c,d,e,f,p},x] && RationalQ[m] && m<-1 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4), x_Symbol] := + With[{r=Rt[c/e*(2*c*d-b*e),2]}, + e/2*Int[(f*x)^m/(c*d/e-r*x+c*x^2),x] + + e/2*Int[(f*x)^m/(c*d/e+r*x+c*x^2),x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-a*e^2] && PositiveQ[d/e] && PosQ[c/e*(2*c*d-b*e)] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)/(a_+c_.*x_^4), x_Symbol] := + With[{r=Rt[2*c^2*d/e,2]}, + e/2*Int[(f*x)^m/(c*d/e-r*x+c*x^2),x] + + e/2*Int[(f*x)^m/(c*d/e+r*x+c*x^2),x]] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[c*d^2-a*e^2] && PositiveQ[d/e] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[(f*x)^m/(b/2-q/2+c*x^2),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[(f*x)^m/(b/2+q/2+c*x^2),x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)/(a_+c_.*x_^4),x_Symbol] := + With[{q=Rt[-a*c,2]}, + -(e/2+c*d/(2*q))*Int[(f*x)^m/(q-c*x^2),x] + (e/2-c*d/(2*q))*Int[(f*x)^m/(q+c*x^2),x]] /; +FreeQ[{a,c,d,e,f,m},x] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_./(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^2)^q/(a+b*x^2+c*x^4),x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] && IntegerQ[q] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_./(a_+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^2)^q/(a+c*x^4),x],x] /; +FreeQ[{a,c,d,e,f,m},x] && IntegerQ[q] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_./(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m,(d+e*x^2)^q/(a+b*x^2+c*x^4),x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b^2-4*a*c,0] && IntegerQ[q] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_./(a_+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m,(d+e*x^2)^q/(a+c*x^4),x],x] /; +FreeQ[{a,c,d,e,f,m},x] && IntegerQ[q] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + f^4/c^2*Int[(f*x)^(m-4)*(c*d-b*e+c*e*x^2)*(d+e*x^2)^(q-1),x] - + f^4/c^2*Int[(f*x)^(m-4)*(d+e*x^2)^(q-1)*Simp[a*(c*d-b*e)+(b*c*d-b^2*e+a*c*e)*x^2,x]/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && Not[IntegerQ[q]] && RationalQ[m,q] && q>0 && m>3 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + f^4/c*Int[(f*x)^(m-4)*(d+e*x^2)^q,x] - + a*f^4/c*Int[(f*x)^(m-4)*(d+e*x^2)^q/(a+c*x^4),x] /; +FreeQ[{a,c,d,e,f,q},x] && Not[IntegerQ[q]] && RationalQ[m] && m>3 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + e*f^2/c*Int[(f*x)^(m-2)*(d+e*x^2)^(q-1),x] - + f^2/c*Int[(f*x)^(m-2)*(d+e*x^2)^(q-1)*Simp[a*e-(c*d-b*e)*x^2,x]/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && Not[IntegerQ[q]] && RationalQ[m,q] && q>0 && 10 && 10 && m<0 + + +Int[(f_.*x_)^m_*(d_.+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + d/a*Int[(f*x)^m*(d+e*x^2)^(q-1),x] + + 1/(a*f^2)*Int[(f*x)^(m+2)*(d+e*x^2)^(q-1)*Simp[a*e-c*d*x^2,x]/(a+c*x^4),x] /; +FreeQ[{a,c,d,e,f},x] && Not[IntegerQ[q]] && RationalQ[m,q] && q>0 && m<0 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + d^2*f^4/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-4)*(d+e*x^2)^q,x] - + f^4/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-4)*(d+e*x^2)^(q+1)*Simp[a*d+(b*d-a*e)*x^2,x]/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && Not[IntegerQ[q]] && RationalQ[m,q] && q<-1 && m>3 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+c_.*x_^4),x_Symbol] := + d^2*f^4/(c*d^2+a*e^2)*Int[(f*x)^(m-4)*(d+e*x^2)^q,x] - + a*f^4/(c*d^2+a*e^2)*Int[(f*x)^(m-4)*(d+e*x^2)^(q+1)*(d-e*x^2)/(a+c*x^4),x] /; +FreeQ[{a,c,d,e,f},x] && Not[IntegerQ[q]] && RationalQ[m,q] && q<-1 && m>3 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^2)^q_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + -d*e*f^2/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2)*(d+e*x^2)^q,x] + + f^2/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2)*(d+e*x^2)^(q+1)*Simp[a*e+c*d*x^2,x]/(a+b*x^2+c*x^4),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && Not[IntegerQ[q]] && RationalQ[m,q] && q<-1 && 10 && m<-2 + + +Int[(f_.*x_)^m_*(a_+c_.*x_^4)^p_./(d_.+e_.*x_^2),x_Symbol] := + a/d^2*Int[(f*x)^m*(d-e*x^2)*(a+c*x^4)^(p-1),x] + + (c*d^2+a*e^2)/(d^2*f^4)*Int[(f*x)^(m+4)*(a+c*x^4)^(p-1)/(d+e*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p>0 && m<-2 + + +Int[(f_.*x_)^m_*(a_.+b_.*x_^2+c_.*x_^4)^p_./(d_.+e_.*x_^2),x_Symbol] := + 1/(d*e)*Int[(f*x)^m*(a*e+c*d*x^2)*(a+b*x^2+c*x^4)^(p-1),x] - + (c*d^2-b*d*e+a*e^2)/(d*e*f^2)*Int[(f*x)^(m+2)*(a+b*x^2+c*x^4)^(p-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && RationalQ[m,p] && p>0 && m<0 + + +Int[(f_.*x_)^m_*(a_+c_.*x_^4)^p_./(d_.+e_.*x_^2),x_Symbol] := + 1/(d*e)*Int[(f*x)^m*(a*e+c*d*x^2)*(a+c*x^4)^(p-1),x] - + (c*d^2+a*e^2)/(d*e*f^2)*Int[(f*x)^(m+2)*(a+c*x^4)^(p-1)/(d+e*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p>0 && m<0 + + +Int[(f_.*x_)^m_.*(a_.+b_.*x_^2+c_.*x_^4)^p_/(d_.+e_.*x_^2),x_Symbol] := + -f^4/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-4)*(a*d+(b*d-a*e)*x^2)*(a+b*x^2+c*x^4)^p,x] + + d^2*f^4/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-4)*(a+b*x^2+c*x^4)^(p+1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && RationalQ[m,p] && p<-1 && m>2 + + +Int[(f_.*x_)^m_.*(a_+c_.*x_^4)^p_/(d_.+e_.*x_^2),x_Symbol] := + -a*f^4/(c*d^2+a*e^2)*Int[(f*x)^(m-4)*(d-e*x^2)*(a+c*x^4)^p,x] + + d^2*f^4/(c*d^2+a*e^2)*Int[(f*x)^(m-4)*(a+c*x^4)^(p+1)/(d+e*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p<-1 && m>2 + + +Int[(f_.*x_)^m_.*(a_.+b_.*x_^2+c_.*x_^4)^p_/(d_.+e_.*x_^2),x_Symbol] := + f^2/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2)*(a*e+c*d*x^2)*(a+b*x^2+c*x^4)^p,x] - + d*e*f^2/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2)*(a+b*x^2+c*x^4)^(p+1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b^2-4*a*c,0] && RationalQ[m,p] && p<-1 && m>0 + + +Int[(f_.*x_)^m_.*(a_+c_.*x_^4)^p_/(d_.+e_.*x_^2),x_Symbol] := + f^2/(c*d^2+a*e^2)*Int[(f*x)^(m-2)*(a*e+c*d*x^2)*(a+c*x^4)^p,x] - + d*e*f^2/(c*d^2+a*e^2)*Int[(f*x)^(m-2)*(a+c*x^4)^(p+1)/(d+e*x^2),x] /; +FreeQ[{a,c,d,e,f},x] && RationalQ[m,p] && p<-1 && m>0 + + +Int[x_^m_*(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{f=Coeff[PolynomialRemainder[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,0], + g=Coeff[PolynomialRemainder[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*g-f*(b^2-2*a*c)-c*(b*f-2*a*g)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(a+b*x^2+c*x^4)^(p+1)* + Simp[ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*PolynomialQuotient[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x]+ + b^2*f*(2*p+3)-2*a*c*f*(4*p+5)-a*b*g+c*(4*p+7)*(b*f-2*a*g)*x^2,x],x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] && IGtQ[q,1] && IGtQ[m/2,0] + + +Int[x_^m_*(d_+e_.*x_^2)^q_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{f=Coeff[PolynomialRemainder[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,0], + g=Coeff[PolynomialRemainder[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*g-f*(b^2-2*a*c)-c*(b*f-2*a*g)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[x^m*(a+b*x^2+c*x^4)^(p+1)* + Simp[ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*x^(-m)*PolynomialQuotient[x^m*(d+e*x^2)^q,a+b*x^2+c*x^4,x]+ + (b^2*f*(2*p+3)-2*a*c*f*(4*p+5)-a*b*g)*x^(-m)+c*(4*p+7)*(b*f-2*a*g)*x^(2-m),x],x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] && IGtQ[q,1] && ILtQ[m/2,0] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && NeQ[b^2-4*a*c,0] && (PositiveIntegerQ[p] || PositiveIntegerQ[q] || IntegersQ[m,q]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^2)^q*(a+c*x^4)^p,x],x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && (PositiveIntegerQ[p] || PositiveIntegerQ[q] || IntegersQ[m,q]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + f^m*Int[ExpandIntegrand[x^m*(a+c*x^4)^p,(d/(d^2-e^2*x^4)-e*x^2/(d^2-e^2*x^4))^(-q),x],x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && NegativeIntegerQ[q] && (IntegerQ[m] || PositiveQ[f]) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^q_*(a_+c_.*x_^4)^p_,x_Symbol] := + (f*x)^m/x^m*Int[x^m*(d+e*x^2)^q*(a+c*x^4)^p,x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && NegativeIntegerQ[q] && Not[IntegerQ[m] || PositiveQ[f]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^q*(a+c*x^4)^p,x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.2.6 (d x)^m Pq(x) (a+b x^2+c x^4)^p*) + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],k}, + Int[(d*x)^m*Sum[Coeff[Pq,x,2*k]*x^(2*k),{k,0,q/2+1}]*(a+b*x^2+c*x^4)^p,x] + + 1/d*Int[(d*x)^(m+1)*Sum[Coeff[Pq,x,2*k+1]*x^(2*k),{k,0,(q-1)/2+1}]*(a+b*x^2+c*x^4)^p,x]] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x] && Not[PolyQ[Pq,x^2]] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + 1/2*Subst[Int[x^((m-1)/2)*SubstFor[x^2,Pq,x]*(a+b*x+c*x^2)^p,x],x,x^2] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x^2] && IntegerQ[(m-1)/2] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*Pq*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c,d,m},x] && PolyQ[Pq,x^2] && IGtQ[p,-2] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + 1/d^2*Int[(d*x)^(m+2)*ExpandToSum[Pq/x^2,x]*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x^2] && EqQ[Coeff[Pq,x,0]] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + With[{e=Coeff[Pq,x,0],f=Coeff[Pq,x,2],g=Coeff[Pq,x,4]}, + e*(d*x)^(m+1)*(a+b*x^2+c*x^4)^(p+1)/(a*d*(m+1)) /; + EqQ[a*f*(m+1)-b*e*(m+2*p+3)] && EqQ[a*g*(m+1)-c*e*(m+4*p+5)] && NeQ[m+1]] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x^2] && Expon[Pq,x]==4 + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x^2)^(2*FracPart[p]))*Int[(d*x)^m*Pq*(b+2*c*x^2)^(2*p),x] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && EqQ[b^2-4*a*c] + + +Int[x_^m_*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{d=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,0], + e=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*e-d*(b^2-2*a*c)-c*(b*d-2*a*e)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(a+b*x^2+c*x^4)^(p+1)* + ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*PolynomialQuotient[x^m*Pq,a+b*x^2+c*x^4,x]+ + b^2*d*(2*p+3)-2*a*c*d*(4*p+5)-a*b*e+c*(4*p+7)*(b*d-2*a*e)*x^2,x],x]] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && + PositiveIntegerQ[m/2] + + +Int[x_^m_*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{d=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,0], + e=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*e-d*(b^2-2*a*c)-c*(b*d-2*a*e)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[x^m*(a+b*x^2+c*x^4)^(p+1)* + ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*x^(-m)*PolynomialQuotient[x^m*Pq,a+b*x^2+c*x^4,x]+ + (b^2*d*(2*p+3)-2*a*c*d*(4*p+5)-a*b*e)*x^(-m)+c*(4*p+7)*(b*d-2*a*e)*x^(2-m),x],x]] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && + NegativeIntegerQ[m/2] + + +(* Int[x_^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{d=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,1], + e=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,3]}, + x^2*(a+b*x^2+c*x^4)^(p+1)*(a*b*e-d*(b^2-2*a*c)-c*(b*d-2*a*e)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(a*(p+1)*(b^2-4*a*c))*Int[x^m*(a+b*x^2+c*x^4)^(p+1)* + ExpandToSum[a*(p+1)*(b^2-4*a*c)*x^(-m)*PolynomialQuotient[x^m*Pq,a+b*x^2+c*x^4,x]+ + (b^2*d*(p+2)-2*a*c*d*(2*p+3)-a*b*e)*x^(1-m)+2*c*(p+2)*(b*d-2*a*e)*x^(3-m),x],x]] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && + IntegerQ[(m-1)/2] *) + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Defer[Int][(d*x)^m*Pq*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && PolyQ[Pq,x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.2.5 Pq(x) (a+b x^2+c x^4)^p*) + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x^2+c*x^4)^p,x],x] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x] && IGtQ[p,0] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Int[x*ExpandToSum[Pq/x,x]*(a+b*x^2+c*x^4)^p,x] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0],0] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],k}, + Int[Sum[Coeff[Pq,x,2*k]*x^(2*k),{k,0,q/2}]*(a+b*x^2+c*x^4)^p,x] + + Int[x*Sum[Coeff[Pq,x,2*k+1]*x^(2*k),{k,0,(q-1)/2}]*(a+b*x^2+c*x^4)^p,x]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && Not[PolyQ[Pq,x^2]] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + With[{d=Coeff[Pq,x,0],e=Coeff[Pq,x,2],f=Coeff[Pq,x,4]}, + d*x*(a+b*x^2+c*x^4)^(p+1)/a /; + EqQ[a*e-b*d*(2*p+3),0] && EqQ[a*f-c*d*(4*p+5),0]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x^2] && EqQ[Expon[Pq,x],4] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] := + With[{d=Coeff[Pq,x,0],e=Coeff[Pq,x,2],f=Coeff[Pq,x,4],g=Coeff[Pq,x,6]}, + x*(3*a*d+(a*e-b*d*(2*p+3))*x^2)*(a+b*x^2+c*x^4)^(p+1)/(3*a^2) /; + EqQ[3*a^2*g-c*(4*p+7)*(a*e-b*d*(2*p+3)),0] && EqQ[3*a^2*f-3*a*c*d*(4*p+5)-b*(2*p+5)*(a*e-b*d*(2*p+3)),0]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x^2] && EqQ[Expon[Pq,x],6] + + +Int[Pq_/(a_+b_.*x_^2+c_.*x_^4),x_Symbol] := + Int[ExpandIntegrand[Pq/(a+b*x^2+c*x^4),x],x] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + (a+b*x^2+c*x^4)^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x^2)^(2*FracPart[p]))*Int[Pq*(b+2*c*x^2)^(2*p),x] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && EqQ[b^2-4*a*c,0] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{d=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,0], + e=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,2]}, + x*(a+b*x^2+c*x^4)^(p+1)*(a*b*e-d*(b^2-2*a*c)-c*(b*d-2*a*e)*x^2)/(2*a*(p+1)*(b^2-4*a*c)) + + 1/(2*a*(p+1)*(b^2-4*a*c))*Int[(a+b*x^2+c*x^4)^(p+1)* + ExpandToSum[2*a*(p+1)*(b^2-4*a*c)*PolynomialQuotient[Pq,a+b*x^2+c*x^4,x]+ + b^2*d*(2*p+3)-2*a*c*d*(4*p+5)-a*b*e+c*(4*p+7)*(b*d-2*a*e)*x^2,x],x]] /; +FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && NeQ[b^2-4*a*c,0] && LtQ[p,-1] + + +Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] := + With[{q=Expon[Pq,x^2],e=Coeff[Pq,x^2,Expon[Pq,x^2]]}, + e*x^(2*q-3)*(a+b*x^2+c*x^4)^(p+1)/(c*(2*q+4*p+1)) + + 1/(c*(2*q+4*p+1))*Int[(a+b*x^2+c*x^4)^p* + ExpandToSum[c*(2*q+4*p+1)*Pq-a*e*(2*q-3)*x^(2*q-4)-b*e*(2*q+2*p-1)*x^(2*q-2)-c*e*(2*q+4*p+1)*x^(2*q),x],x]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>1 && NeQ[b^2-4*a*c,0] && Not[LtQ[p,-1]] + + + +`) + +func resourcesRubi122QuarticTrinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi122QuarticTrinomialProductsM, nil +} + +func resourcesRubi122QuarticTrinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi122QuarticTrinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.2.2 Quartic trinomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi123GeneralTrinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.2.3 General Trinomial Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*1.2.3.1 (a+b x^n+c x^(2 n))^p*) + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(2*n*p)*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c},x] && EqQ[n2,2*n] && LtQ[n,0] && IntegerQ[p] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[n]}, + k*Subst[Int[x^(k-1)*(a+b*x^(k*n)+c*x^(2*k*n))^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,p},x] && EqQ[n2,2*n] && FractionQ[n] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -Subst[Int[(a+b*x^(-n)+c*x^(-2*n))^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,p},x] && EqQ[n2,2*n] && ILtQ[n,0] + + +Int[(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^p/(b+2*c*x^n)^(2*p)*Int[(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c,0] && IGtQ[p,0] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + x*(a+b*x^n+c*x^(2*n))^p/(2*n*p+1) + + n*p/(2*n*p+1)*Int[(2*a+b*x^n)*(a+b*x^n+c*x^(2*n))^(p-1),x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c,0] && GtQ[p,0] && NeQ[2*n*p+1,0] && IntegerQ[p] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -x*(b^2-2*a*c+b*c*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))* + Int[(b^2-2*a*c+n*(p+1)*(b^2-4*a*c)+b*c*(n*(2*p+3)+1)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1),x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] && IntegerQ[p] + + +Int[1/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*q*r)*Int[(r-x^(n/2))/(q-r*x^(n/2)+x^n),x] + + 1/(2*c*q*r)*Int[(r+x^(n/2))/(q+r*x^(n/2)+x^n),x]]] /; +FreeQ[{a,b,c},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c,0] && IGtQ[n/2,0] && NegQ[b^2-4*a*c] + + +Int[1/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + c/q*Int[1/(b/2-q/2+c*x^n),x] - c/q*Int[1/(b/2+q/2+c*x^n),x]] /; +FreeQ[{a,b,c},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c,0] + + +Int[(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + a^IntPart[p]*(a+b*x^n+c*x^(2*n))^FracPart[p]/ + ((1+2*c*x^n/(b+Rt[b^2-4*a*c,2]))^FracPart[p]*(1+2*c*x^n/(b-Rt[b^2-4*a*c,2]))^FracPart[p])* + Int[(1+2*c*x^n/(b+Sqrt[b^2-4*a*c]))^p*(1+2*c*x^n/(b-Sqrt[b^2-4*a*c]))^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2,2*n] + + +Int[(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + Int[(b+a*x^n+c*x^(2*n))^p/x^(n*p),x] /; +FreeQ[{a,b,c,n},x] && EqQ[mn,-n] && IntegerQ[p] && PosQ[n] + + +Int[(a_+b_.*x_^mn_+c_.*x_^n_.)^p_,x_Symbol] := + x^(n*FracPart[p])*(a+b*x^(-n)+c*x^n)^FracPart[p]/(b+a*x^n+c*x^(2*n))^FracPart[p]*Int[(b+a*x^n+c*x^(2*n))^p/x^(n*p),x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[mn,-n] && Not[IntegerQ[p]] && PosQ[n] + + +Int[(a_+b_.*u_^n_+c_.*u_^n2_.)^p_,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*x^n+c*x^(2*n))^p,x],x,u] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2,2*n] && LinearQ[u,x] && NeQ[u,x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p*) + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2,2*n] && EqQ[Simplify[m-n+1]] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2,2*n] && PositiveIntegerQ[p] && Not[IntegerQ[Simplify[(m+1)/n]]] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + Int[x^(m+2*n*p)*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[n2,2*n] && NegativeIntegerQ[p] && NegQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(d_*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=GCD[m+1,n]}, + 1/k*Subst[Int[x^((m+1)/k-1)*(a+b*x^(n/k)+c*x^(2*n/k))^p,x],x,x^k] /; + k!=1] /; +FreeQ[{a,b,c,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(d_.*x_)^m_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/d*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n)/d^n+c*x^(2*k*n)/d^(2*n))^p,x],x,(d*x)^(1/k)]] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && FractionQ[m] && IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^(n-1)*(d*x)^(m-n+1)*(b*n*p+c*(m+n*(2*p-1)+1)*x^n)*(a+b*x^n+c*x^(2*n))^p/(c*(m+2*n*p+1)*(m+n*(2*p-1)+1)) - + n*p*d^n/(c*(m+2*n*p+1)*(m+n*(2*p-1)+1))* + Int[(d*x)^(m-n)*(a+b*x^n+c*x^(2*n))^(p-1)*Simp[a*b*(m-n+1)-(2*a*c*(m+n*(2*p-1)+1)-b^2*(m+n*(p-1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m>n-1 && + m+2*n*p+1!=0 && m+n*(2*p-1)+1!=0 && IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^n+c*x^(2*n))^p/(d*(m+1)) - + n*p/(d^n*(m+1))*Int[(d*x)^(m+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^(p-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<-1 && IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^n+c*x^(2*n))^p/(d*(m+2*n*p+1)) + + n*p/(m+2*n*p+1)*Int[(d*x)^m*(2*a+b*x^n)*(a+b*x^n+c*x^(2*n))^(p-1),x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p>0 && NeQ[m+2*n*p+1] && + IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^(n-1)*(d*x)^(m-n+1)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(n*(p+1)*(b^2-4*a*c)) - + d^n/(n*(p+1)*(b^2-4*a*c))* + Int[(d*x)^(m-n)*(b*(m-n+1)+2*c*(m+2*n*(p+1)+1)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && n-12*n-1 && + IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -(d*x)^(m+1)*(b^2-2*a*c+b*c*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*d*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))* + Int[(d*x)^m*(a+b*x^n+c*x^(2*n))^(p+1)*Simp[b^2*(n*(p+1)+m+1)-2*a*c*(m+2*n*(p+1)+1)+b*c*(2*n*p+3*n+m+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && IntegerQ[p] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^(2*n-1)*(d*x)^(m-2*n+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(c*(m+2*n*p+1)) - + d^(2*n)/(c*(m+2*n*p+1))* + Int[(d*x)^(m-2*n)*Simp[a*(m-2*n+1)+b*(m+n*(p-1)+1)*x^n,x]*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m>2*n-1 && + NeQ[m+2*n*p+1] && IntegerQ[p] + + +Int[(d_.*x_)^m_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*d*(m+1)) - + 1/(a*d^n*(m+1))*Int[(d*x)^(m+n)*(b*(m+n*(p+1)+1)+c*(m+2*n*(p+1)+1)*x^n)*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 && IntegerQ[p] + + +Int[(d_.*x_)^m_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + (d*x)^(m+1)/(a*d*(m+1)) - + 1/(a*d^n)*Int[(d*x)^(m+n)*(b+c*x^n)/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[x_^m_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + Int[PolynomialDivide[x^m,(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[m] && m>3*n-1 + + +Int[(d_.*x_)^m_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + d^(2*n-1)*(d*x)^(m-2*n+1)/(c*(m-2*n+1)) - + d^(2*n)/c*Int[(d*x)^(m-2*n)*(a+b*x^n)/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m>2*n-1 + + +Int[x_^m_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*r)*Int[x^(m-3*(n/2))*(q+r*x^(n/2))/(q+r*x^(n/2)+x^n),x] - + 1/(2*c*r)*Int[x^(m-3*(n/2))*(q-r*x^(n/2))/(q-r*x^(n/2)+x^n),x]]] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n/2,m] && 3*n/2<=m<2*n && NegQ[b^2-4*a*c] + + +Int[x_^m_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*r)*Int[x^(m-n/2)/(q-r*x^(n/2)+x^n),x] - + 1/(2*c*r)*Int[x^(m-n/2)/(q+r*x^(n/2)+x^n),x]]] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n/2,m] && n/2<=m<3*n/2 && NegQ[b^2-4*a*c] + + +Int[(d_.*x_)^m_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + d^n/2*(b/q+1)*Int[(d*x)^(m-n)/(b/2+q/2+c*x^n),x] - + d^n/2*(b/q-1)*Int[(d*x)^(m-n)/(b/2-q/2+c*x^n),x]] /; +FreeQ[{a,b,c,d},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m>=n + + +Int[(d_.*x_)^m_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + c/q*Int[(d*x)^m/(b/2-q/2+c*x^n),x] - c/q*Int[(d*x)^m/(b/2+q/2+c*x^n),x]] /; +FreeQ[{a,b,c,d,m},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -Subst[Int[(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[m]}, + -k/d*Subst[Int[(a+b*d^(-n)*x^(-k*n)+c*d^(-2*n)*x^(-2*k*n))^p/x^(k*(m+1)+1),x],x,1/(d*x)^(1/k)]] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(d_.*x_)^m_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -d^IntPart[m]*(d*x)^FracPart[m]*(x^(-1))^FracPart[m]*Subst[Int[(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[n]}, + k*Subst[Int[x^(k*(m+1)-1)*(a+b*x^(k*n)+c*x^(2*k*n))^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,m,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[(d_*x_)^m_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/(m+1)*Subst[Int[(a+b*x^Simplify[n/(m+1)]+c*x^Simplify[2*n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(d_*x_)^m_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(d_.*x_)^m_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[(d*x)^m/(b-q+2*c*x^n),x] - + 2*c/q*Int[(d*x)^m/(b+q+2*c*x^n),x]] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -(d*x)^(m+1)*(b^2-2*a*c+b*c*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*d*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))* + Int[(d*x)^m*(a+b*x^n+c*x^(2*n))^(p+1)*Simp[b^2*(n*(p+1)+m+1)-2*a*c*(m+2*n*(p+1)+1)+b*c*(2*n*p+3*n+m+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2,2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + a^IntPart[p]*(a+b*x^n+c*x^(2*n))^FracPart[p]/ + ((1+2*c*x^n/(b+Rt[b^2-4*a*c,2]))^FracPart[p]*(1+2*c*x^n/(b-Rt[b^2-4*a*c,2]))^FracPart[p])* + Int[(d*x)^m*(1+2*c*x^n/(b+Sqrt[b^2-4*a*c]))^p*(1+2*c*x^n/(b-Sqrt[b^2-4*a*c]))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] + + +Int[x_^m_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + Int[x^(m-n*p)*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[mn,-n] && IntegerQ[p] && PosQ[n] + + +Int[x_^m_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + x^(n*FracPart[p])*(a+b/x^n+c*x^n)^FracPart[p]/(b+a*x^n+c*x^(2*n))^FracPart[p]*Int[x^(m-n*p)*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[mn,-n] && Not[IntegerQ[p]] && PosQ[n] + + +Int[(d_*x_)^m_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*x^(-n)+c*x^n)^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[mn,-n] + + +Int[x_^m_.*(a_.+b_.*v_^n_+c_.*v_^n2_.)^p_.,x_Symbol] := + 1/Coefficient[v,x,1]^(m+1)*Subst[Int[SimplifyIntegrand[(x-Coefficient[v,x,0])^m*(a+b*x^n+c*x^(2*n))^p,x],x],x,v] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2,2*n] && LinearQ[v,x] && IntegerQ[m] && NeQ[v-x] + + +Int[u_^m_.*(a_.+b_.*v_^n_+c_.*v_^n2_.)^p_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(a+b*x^n+c*x^(2*n))^p,x],x,v] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2,2*n] && LinearPairQ[u,v,x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p*) + + +Int[(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(n*(2*p+q))*(e+d*x^(-n))^q*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && IntegersQ[p,q] && NegQ[n] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(n*(2*p+q))*(e+d*x^(-n))^q*(c+a*x^(-2*n))^p,x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && IntegersQ[p,q] && NegQ[n] + + +Int[(d_+e_.*x_^n_)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -Subst[Int[(d+e*x^(-n))^q*(a+b*x^(-n)+c*x^(-2*n))^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[n] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -Subst[Int[(d+e*x^(-n))^q*(a+c*x^(-2*n))^p/x^2,x],x,1/x] /; +FreeQ[{a,c,d,e,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[n] + + +Int[(d_+e_.*x_^n_)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g-1)*(d+e*x^(g*n))^q*(a+b*x^(g*n)+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[n2-2*n] && FractionQ[n] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g-1)*(d+e*x^(g*n))^q*(a+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,c,d,e,p,q},x] && EqQ[n2-2*n] && FractionQ[n] + + +Int[(d_+e_.*x_^n_)*(b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (b*e-d*c)*(b*x^n+c*x^(2*n))^(p+1)/(b*c*n*(p+1)*x^(2*n*(p+1))) + + e/c*Int[x^(-n)*(b*x^n+c*x^(2*n))^(p+1),x] /; +FreeQ[{b,c,d,e,n,p},x] && EqQ[n2-2*n] && Not[IntegerQ[p]] && EqQ[n*(2*p+1)+1] + + +Int[(d_+e_.*x_^n_)*(b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + e*x^(-n+1)*(b*x^n+c*x^(2*n))^(p+1)/(c*(n*(2*p+1)+1)) /; +FreeQ[{b,c,d,e,n,p},x] && EqQ[n2-2*n] && Not[IntegerQ[p]] && NeQ[n*(2*p+1)+1] && EqQ[b*e*(n*p+1)-c*d*(n*(2*p+1)+1)] + + +Int[(d_+e_.*x_^n_)*(b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + e*x^(-n+1)*(b*x^n+c*x^(2*n))^(p+1)/(c*(n*(2*p+1)+1)) - + (b*e*(n*p+1)-c*d*(n*(2*p+1)+1))/(c*(n*(2*p+1)+1))*Int[(b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{b,c,d,e,n,p},x] && EqQ[n2-2*n] && Not[IntegerQ[p]] && NeQ[n*(2*p+1)+1] && NeQ[b*e*(n*p+1)-c*d*(n*(2*p+1)+1)] + + +Int[(d_+e_.*x_^n_)^q_.*(b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (b*x^n+c*x^(2*n))^FracPart[p]/(x^(n*FracPart[p])*(b+c*x^n)^FracPart[p])*Int[x^(n*p)*(d+e*x^n)^q*(b+c*x^n)^p,x] /; +FreeQ[{b,c,d,e,n,p,q},x] && EqQ[n2-2*n] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(d+e*x^n)^(p+q)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(d+e*x^n)^(p+q)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,n,q},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])*Int[(d+e*x^n)^(p+q)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + (a+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])*Int[(d+e*x^n)^(p+q)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,n,p,q},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q*(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[q] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q*(a+c*x^(2*n)),x],x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && PositiveIntegerQ[q] + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + -(c*d^2-b*d*e+a*e^2)*x*(d+e*x^n)^(q+1)/(d*e^2*n*(q+1)) + + 1/(n*(q+1)*d*e^2)*Int[(d+e*x^n)^(q+1)*Simp[c*d^2-b*d*e+a*e^2*(n*(q+1)+1)+c*d*e*n*(q+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_),x_Symbol] := + -(c*d^2+a*e^2)*x*(d+e*x^n)^(q+1)/(d*e^2*n*(q+1)) + + 1/(n*(q+1)*d*e^2)*Int[(d+e*x^n)^(q+1)*Simp[c*d^2+a*e^2*(n*(q+1)+1)+c*d*e*n*(q+1)*x^n,x],x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + c*x^(n+1)*(d+e*x^n)^(q+1)/(e*(n*(q+2)+1)) + + 1/(e*(n*(q+2)+1))*Int[(d+e*x^n)^q*(a*e*(n*(q+2)+1)-(c*d*(n+1)-b*e*(n*(q+2)+1))*x^n),x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_),x_Symbol] := + c*x^(n+1)*(d+e*x^n)^(q+1)/(e*(n*(q+2)+1)) + + 1/(e*(n*(q+2)+1))*Int[(d+e*x^n)^q*(a*e*(n*(q+2)+1)-c*d*(n+1)*x^n),x] /; +FreeQ[{a,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] + + +Int[(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[2*d*e,2]}, + e^2/(2*c)*Int[1/(d+q*x^(n/2)+e*x^n),x] + e^2/(2*c)*Int[1/(d-q*x^(n/2)+e*x^n),x]] /; +FreeQ[{a,c,d,e},x] && EqQ[n2-2*n] && EqQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && PosQ[d*e] + + +Int[(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[-2*d*e,2]}, + d/(2*a)*Int[(d-q*x^(n/2))/(d-q*x^(n/2)-e*x^n),x] + + d/(2*a)*Int[(d+q*x^(n/2))/(d+q*x^(n/2)-e*x^n),x]] /; +FreeQ[{a,c,d,e},x] && EqQ[n2-2*n] && EqQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && NegQ[d*e] + + +Int[(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[a/c,4]}, + 1/(2*Sqrt[2]*c*q^3)*Int[(Sqrt[2]*d*q-(d-e*q^2)*x^(n/2))/(q^2-Sqrt[2]*q*x^(n/2)+x^n),x] + + 1/(2*Sqrt[2]*c*q^3)*Int[(Sqrt[2]*d*q+(d-e*q^2)*x^(n/2))/(q^2+Sqrt[2]*q*x^(n/2)+x^n),x]] /; +FreeQ[{a,c,d,e},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && NeQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && PosQ[a*c] + + +Int[(d_+e_.*x_^3)/(a_+c_.*x_^6),x_Symbol] := + With[{q=Rt[c/a,6]}, + 1/(3*a*q^2)*Int[(q^2*d-e*x)/(1+q^2*x^2),x] + + 1/(6*a*q^2)*Int[(2*q^2*d-(Sqrt[3]*q^3*d-e)*x)/(1-Sqrt[3]*q*x+q^2*x^2),x] + + 1/(6*a*q^2)*Int[(2*q^2*d+(Sqrt[3]*q^3*d+e)*x)/(1+Sqrt[3]*q*x+q^2*x^2),x]] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2] && PosQ[c/a] + + +Int[(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[-a/c,2]}, + (d+e*q)/2*Int[1/(a+c*q*x^n),x] + (d-e*q)/2*Int[1/(a-c*q*x^n),x]] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && NegQ[a*c] && IntegerQ[n] + + +Int[(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + d*Int[1/(a+c*x^(2*n)),x] + e*Int[x^n/(a+c*x^(2*n)),x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && (PosQ[a*c] || Not[IntegerQ[n]]) + + +Int[(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[2*d/e-b/c,2]}, + e/(2*c)*Int[1/Simp[d/e+q*x^(n/2)+x^n,x],x] + + e/(2*c)*Int[1/Simp[d/e-q*x^(n/2)+x^n,x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && + (PositiveQ[2*d/e-b/c] || Not[NegativeQ[2*d/e-b/c]] && EqQ[d-e*Rt[a/c,2]]) + + +Int[(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[1/(b/2-q/2+c*x^n),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[1/(b/2+q/2+c*x^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && + PositiveQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[-2*d/e-b/c,2]}, + e/(2*c*q)*Int[(q-2*x^(n/2))/Simp[d/e+q*x^(n/2)-x^n,x],x] + + e/(2*c*q)*Int[(q+2*x^(n/2))/Simp[d/e-q*x^(n/2)-x^n,x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-a*e^2] && PositiveIntegerQ[n/2] && + Not[PositiveQ[b^2-4*a*c]] + + +Int[(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[1/(b/2-q/2+c*x^n),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[1/(b/2+q/2+c*x^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + (PosQ[b^2-4*a*c] || Not[PositiveIntegerQ[n/2]]) + + +Int[(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[a/c,2]}, + With[{r=Rt[2*q-b/c,2]}, + 1/(2*c*q*r)*Int[(d*r-(d-e*q)*x^(n/2))/(q-r*x^(n/2)+x^n),x] + + 1/(2*c*q*r)*Int[(d*r+(d-e*q)*x^(n/2))/(q+r*x^(n/2)+x^n),x]]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && PositiveIntegerQ[n/2] && + NegQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && IntegerQ[q] + + +Int[(d_+e_.*x_^n_)^q_/(a_+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q/(a+c*x^(2*n)),x],x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && IntegerQ[q] + + +Int[(d_+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + e^2/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x^n)^q,x] + + 1/(c*d^2-b*d*e+a*e^2)*Int[(d+e*x^n)^(q+1)*(c*d-b*e-c*e*x^n)/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[q]] && + RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^n_)^q_/(a_+c_.*x_^n2_),x_Symbol] := + e^2/(c*d^2+a*e^2)*Int[(d+e*x^n)^q,x] + + c/(c*d^2+a*e^2)*Int[(d+e*x^n)^(q+1)*(d-e*x^n)/(a+c*x^(2*n)),x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[q]] && RationalQ[q] && q<-1 + + +Int[(d_+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{r=Rt[b^2-4*a*c,2]}, + 2*c/r*Int[(d+e*x^n)^q/(b-r+2*c*x^n),x] - 2*c/r*Int[(d+e*x^n)^q/(b+r+2*c*x^n),x]] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[q]] + + +Int[(d_+e_.*x_^n_)^q_/(a_+c_.*x_^n2_),x_Symbol] := + With[{r=Rt[-a*c,2]}, + -c/(2*r)*Int[(d+e*x^n)^q/(r-c*x^n),x] - c/(2*r)*Int[(d+e*x^n)^q/(r+c*x^n),x]] /; +FreeQ[{a,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && Not[IntegerQ[q]] + + +Int[(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + x*(b*e*n*p+c*d*(2*n*p+n+1)+c*e*(2*n*p+1)*x^n)*(a+b*x^n+c*x^(2*n))^p/(c*(2*n*p+1)*(2*n*p+n+1)) + + n*p/(c*(2*n*p+1)*(2*n*p+n+1))* + Int[Simp[2*a*c*d*(2*n*p+n+1)-a*b*e+(2*a*c*e*(2*n*p+1)+b*d*c*(2*n*p+n+1)-b^2*e*(n*p+1))*x^n,x]* + (a+b*x^n+c*x^(2*n))^(p-1),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && FractionQ[p] && p>0 && NeQ[2*n*p+1] && NeQ[2*n*p+n+1] && + IntegerQ[p] + + +Int[(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + x*(d*(2*n*p+n+1)+e*(2*n*p+1)*x^n)*(a+c*x^(2*n))^p/((2*n*p+1)*(2*n*p+n+1)) + + 2*a*n*p/((2*n*p+1)*(2*n*p+n+1))*Int[(d*(2*n*p+n+1)+e*(2*n*p+1)*x^n)*(a+c*x^(2*n))^(p-1),x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && FractionQ[p] && p>0 && NeQ[2*n*p+1] && NeQ[2*n*p+n+1] && + IntegerQ[p] + + +Int[(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(d*b^2-a*b*e-2*a*c*d+(b*d-2*a*e)*c*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))* + Int[Simp[(n*p+n+1)*d*b^2-a*b*e-2*a*c*d*(2*n*p+2*n+1)+(2*n*p+3*n+1)*(d*b-2*a*e)*c*x^n,x]* + (a+b*x^n+c*x^(2*n))^(p+1),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(d+e*x^n)*(a+c*x^(2*n))^(p+1)/(2*a*n*(p+1)) + + 1/(2*a*n*(p+1))*Int[(d*(2*n*p+2*n+1)+e*(2*n*p+3*n+1)*x^n)*(a+c*x^(2*n))^(p+1),x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] && RationalQ[p] && p<-1 && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)*(a+c*x^(2*n))^p,x],x] /; +FreeQ[{a,c,d,e,n},x] && EqQ[n2-2*n] + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + c^p*x^(2*n*p-n+1)*(d+e*x^n)^(q+1)/(e*(2*n*p+n*q+1)) + + Int[(d+e*x^n)^q*ExpandToSum[(a+b*x^n+c*x^(2*n))^p-c^p*x^(2*n*p)-d*c^p*(2*n*p-n+1)*x^(2*n*p-n)/(e*(2*n*p+n*q+1)),x],x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[p] && + NeQ[2*n*p+n*q+1] && PositiveIntegerQ[n] && Not[PositiveIntegerQ[q]] + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + c^p*x^(2*n*p-n+1)*(d+e*x^n)^(q+1)/(e*(2*n*p+n*q+1)) + + Int[(d+e*x^n)^q*ExpandToSum[(a+c*x^(2*n))^p-c^p*x^(2*n*p)-d*c^p*(2*n*p-n+1)*x^(2*n*p-n)/(e*(2*n*p+n*q+1)),x],x] /; +FreeQ[{a,c,d,e,n,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[p] && + NeQ[2*n*p+n*q+1] && PositiveIntegerQ[n] && Not[PositiveIntegerQ[q]] + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && + (IntegersQ[p,q] && Not[IntegerQ[n]] || PositiveIntegerQ[p] || PositiveIntegerQ[q] && Not[IntegerQ[n]]) + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^n)^q*(a+c*x^(2*n))^p,x],x] /; +FreeQ[{a,c,d,e,n,p,q},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && + (IntegersQ[p,q] && Not[IntegerQ[n]] || PositiveIntegerQ[p] || PositiveIntegerQ[q] && Not[IntegerQ[n]]) + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + Int[ExpandIntegrand[(a+c*x^(2*n))^p,(d/(d^2-e^2*x^(2*n))-e*x^n/(d^2-e^2*x^(2*n)))^(-q),x],x] /; +FreeQ[{a,c,d,e,n,p},x] && EqQ[n2-2*n] && NeQ[c*d^2+a*e^2] && NegativeIntegerQ[q] && Not[IntegersQ[n,2*p]] + + +Int[(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Defer[Int][(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2-2*n] && Not[IntegersQ[n,q]] + + +Int[(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + Defer[Int][(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,n,p,q},x] && EqQ[n2-2*n] && Not[IntegersQ[n,q]] + + +Int[(d_+e_.*u_^n_)^q_.*(a_+b_.*u_^n_+c_.*u_^n2_)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x],x,u] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2-2*n] && LinearQ[u,x] && NeQ[u-x] + + +Int[(d_+e_.*u_^n_)^q_.*(a_+c_.*u_^n2_)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(d+e*x^n)^q*(a+c*x^(2*n))^p,x],x,u] /; +FreeQ[{a,c,d,e,n,p,q},x] && EqQ[n2-2*n] && LinearQ[u,x] && NeQ[u-x] + + +Int[(d_+e_.*x_^mn_.)^q_.*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[n2,2*n] && EqQ[mn,-n] && IntegerQ[q] + + +Int[(d_+e_.*x_^mn_.)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x] /; +FreeQ[{a,c,d,e,mn,p},x] && EqQ[n2,-2*mn] && IntegerQ[q] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(2*n*p)*(d+e*x^(-n))^q*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] && Not[IntegerQ[q]] && IntegerQ[p] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(-2*mn*p)*(d+e*x^mn)^q*(c+a*x^(2*mn))^p,x] /; +FreeQ[{a,c,d,e,mn,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[q]] && IntegerQ[p] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + x^(n*FracPart[q])*(d+e*x^(-n))^FracPart[q]/(e+d*x^n)^FracPart[q]*Int[x^(-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] && Not[IntegerQ[q]] && Not[IntegerQ[p]] && PosQ[n] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + x^(-mn*FracPart[q])*(d+e*x^mn)^FracPart[q]/(e+d*x^(-mn))^FracPart[q]*Int[x^(mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x] /; +FreeQ[{a,c,d,e,mn,p,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[q]] && Not[IntegerQ[p]] && PosQ[n2] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/(x^(2*n*FracPart[p])*(c+b*x^(-n)+a*x^(-2*n))^FracPart[p])* + Int[x^(2*n*p)*(d+e*x^(-n))^q*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] && Not[IntegerQ[q]] && Not[IntegerQ[p]] && NegQ[n] + + +Int[(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + (a+c*x^n2)^FracPart[p]/(x^(n2*FracPart[p])*(c+a*x^(2*mn))^FracPart[p])* + Int[x^(n2*p)*(d+e*x^mn)^q*(c+a*x^(2*mn))^p,x] /; +FreeQ[{a,c,d,e,mn,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[q]] && Not[IntegerQ[p]] && NegQ[n2] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + Int[x^(-n*p)*(d+e*x^n)^q*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,q},x] && EqQ[mn,-n] && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + x^(n*FracPart[p])*(a+b/x^n+c*x^n)^FracPart[p]/(b+a*x^n+c*x^(2*n))^FracPart[p]* + Int[x^(-n*p)*(d+e*x^n)^q*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[mn,-n] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_.*(f_+g_.*x_^n_)^r_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x^n)^(2*FracPart[p]))* + Int[(d+e*x^n)^q*(f+g*x^n)^r*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p,q,r},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_.*(f_+g_.*x_^n_)^r_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(d+e*x^n)^(p+q)*(f+g*x^n)^r*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,q,r},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)^q_.*(f_+g_.*x_^n_)^r_.*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(d+e*x^n)^(p+q)*(f+g*x^n)^r*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,f,g,n,q,r},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && IntegerQ[p] + + +Int[(d_+e_.*x_^n_)^q_.*(f_+g_.*x_^n_)^r_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])* + Int[(d+e*x^n)^(p+q)*(f+g*x^n)^r*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p,q,r},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^n_)^q_.*(f_+g_.*x_^n_)^r_.*(a_+c_.*x_^n2_)^p_,x_Symbol] := + (a+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])* + Int[(d+e*x^n)^(p+q)*(f+g*x^n)^r*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,f,g,n,p,q,r},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[(f_.+g_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (2*c*f-g*(b-q))/(2*c*d-e*(b-q))*Int[1/Sqrt[a+b*x^2+c*x^4],x] - + (e*f-d*g)/(2*c*d-e*(b-q))*Int[(b-q+2*c*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /; + NeQ[2*c*f-g*(b-q)]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveQ[b^2-4*a*c] && NeQ[c*d^2-b*d*e+a*e^2] && Not[NegativeQ[c]] + + +Int[(f_+g_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + With[{q=Rt[-a*c,2]}, + (c*f+g*q)/(c*d+e*q)*Int[1/Sqrt[a+c*x^4],x] + (e*f-d*g)/(c*d+e*q)*Int[(q-c*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x] /; + NeQ[c*f+g*q]] /; +FreeQ[{a,c,d,e,f,g},x] && PositiveQ[-a*c] && NeQ[c*d^2+a*e^2] && Not[NegativeQ[c]] + + +Int[(d1_+e1_.*x_^non2_.)^q_.*(d2_+e2_.*x_^non2_.)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(d1*d2+e1*e2*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n,p,q},x] && EqQ[n2-2*n] && EqQ[non2-n/2] && EqQ[d2*e1+d1*e2] && + (IntegerQ[q] || PositiveQ[d1] && PositiveQ[d2]) + + +Int[(d1_+e1_.*x_^non2_.)^q_.*(d2_+e2_.*x_^non2_.)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + (d1+e1*x^(n/2))^FracPart[q]*(d2+e2*x^(n/2))^FracPart[q]/(d1*d2+e1*e2*x^n)^FracPart[q]* + Int[(d1*d2+e1*e2*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n,p,q},x] && EqQ[n2-2*n] && EqQ[non2-n/2] && EqQ[d2*e1+d1*e2] + + +Int[(A_+B_.*x_^m_.)*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + A*Int[(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] + B*Int[x^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,A,B,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[m-n+1] + + +Int[(A_+B_.*x_^m_.)*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + A*Int[(d+e*x^n)^q*(a+c*x^(2*n))^p,x] + B*Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,A,B,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[m-n+1] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p*) + + +Int[(f_.*x_)^m_.*(e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^m/(n*e^((m+1)/n-1))*Subst[Int[(e*x)^(q+(m+1)/n-1)*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && (IntegerQ[m] || PositiveQ[f]) && IntegerQ[Simplify[(m+1)/n]] + + +Int[(f_.*x_)^m_.*(e_.*x_^n_)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^m/(n*e^((m+1)/n-1))*Subst[Int[(e*x)^(q+(m+1)/n-1)*(a+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && (IntegerQ[m] || PositiveQ[f]) && IntegerQ[Simplify[(m+1)/n]] + + +Int[(f_.*x_)^m_.*(e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^m*e^IntPart[q]*(e*x^n)^FracPart[q]/x^(n*FracPart[q])*Int[x^(m+n*q)*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && (IntegerQ[m] || PositiveQ[f]) && Not[IntegerQ[Simplify[(m+1)/n]]] + + +Int[(f_.*x_)^m_.*(e_.*x_^n_)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^m*e^IntPart[q]*(e*x^n)^FracPart[q]/x^(n*FracPart[q])*Int[x^(m+n*q)*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && (IntegerQ[m] || PositiveQ[f]) && Not[IntegerQ[Simplify[(m+1)/n]]] + + +Int[(f_*x_)^m_.*(e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && Not[IntegerQ[m]] + + +Int[(f_*x_)^m_.*(e_.*x_^n_)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,e,f,m,n,p,q},x] && EqQ[n2-2*n] && Not[IntegerQ[m]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[Simplify[m-n+1]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[(d+e*x)^q*(a+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[Simplify[m-n+1]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m+n*(2*p+q))*(e+d*x^(-n))^q*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && IntegersQ[p,q] && NegQ[n] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m+n*(2*p+q))*(e+d*x^(-n))^q*(c+a*x^(-2*n))^p,x] /; +FreeQ[{a,c,d,e,m,n},x] && EqQ[n2-2*n] && IntegersQ[p,q] && NegQ[n] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/n*Subst[Int[x^((m+1)/n-1)*(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] && PositiveIntegerQ[m,n,(m+1)/n] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2+c*x^n)^(2*FracPart[p]))* + Int[(f*x)^m*(d+e*x^n)^q*(b/2+c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(d+e*x)^q*(a+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(f_*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(f_*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(f*x)^m*(d+e*x^n)^(q+p)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(f*x)^m*(d+e*x^n)^(q+p)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,f,q,m,n,q},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])* + Int[(f*x)^m*(d+e*x^n)^(q+p)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EqQ[c*d^2-b*d*e+a*e^2] && Not[IntegerQ[p]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + (a+c*x^(2*n))^FracPart[p]/((d+e*x^n)^FracPart[p]*(a/d+(c*x^n)/e)^FracPart[p])*Int[(f*x)^m*(d+e*x^n)^(q+p)*(a/d+c/e*x^n)^p,x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && EqQ[c*d^2+a*e^2] && Not[IntegerQ[p]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + (-d)^((m-Mod[m,n])/n-1)*(c*d^2-b*d*e+a*e^2)^p*x^(Mod[m,n]+1)*(d+e*x^n)^(q+1)/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)) + + 1/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1))*Int[x^Mod[m,n]*(d+e*x^n)^(q+1)* + ExpandToSum[Together[1/(d+e*x^n)*(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)*x^(m-Mod[m,n])*(a+b*x^n+c*x^(2*n))^p- + (-d)^((m-Mod[m,n])/n-1)*(c*d^2-b*d*e+a*e^2)^p*(d*(Mod[m,n]+1)+e*(Mod[m,n]+n*(q+1)+1)*x^n))],x],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n,p] && IntegersQ[m,q] && q<-1 && m>0 + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + (-d)^((m-Mod[m,n])/n-1)*(c*d^2+a*e^2)^p*x^(Mod[m,n]+1)*(d+e*x^n)^(q+1)/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)) + + 1/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1))*Int[x^Mod[m,n]*(d+e*x^n)^(q+1)* + ExpandToSum[Together[1/(d+e*x^n)*(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)*x^(m-Mod[m,n])*(a+c*x^(2*n))^p- + (-d)^((m-Mod[m,n])/n-1)*(c*d^2+a*e^2)^p*(d*(Mod[m,n]+1)+e*(Mod[m,n]+n*(q+1)+1)*x^n))],x],x] /; +FreeQ[{a,c,d,e},x] && EqQ[n2-2*n] && PositiveIntegerQ[n,p] && IntegersQ[m,q] && q<-1 && m>0 + + +Int[x_^m_*(d_+e_.*x_^n_)^q_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + (-d)^((m-Mod[m,n])/n-1)*(c*d^2-b*d*e+a*e^2)^p*x^(Mod[m,n]+1)*(d+e*x^n)^(q+1)/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)) + + (-d)^((m-Mod[m,n])/n-1)/(n*e^(2*p)*(q+1))*Int[x^m*(d+e*x^n)^(q+1)* + ExpandToSum[Together[1/(d+e*x^n)*(n*(-d)^(-(m-Mod[m,n])/n+1)*e^(2*p)*(q+1)*(a+b*x^n+c*x^(2*n))^p - + (e^(-(m-Mod[m,n])/n)*(c*d^2-b*d*e+a*e^2)^p*x^(-(m-Mod[m,n])))*(d*(Mod[m,n]+1)+e*(Mod[m,n]+n*(q+1)+1)*x^n))],x],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n,p] && IntegersQ[m,q] && q<-1 && m<0 + + +Int[x_^m_*(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + (-d)^((m-Mod[m,n])/n-1)*(c*d^2+a*e^2)^p*x^(Mod[m,n]+1)*(d+e*x^n)^(q+1)/(n*e^(2*p+(m-Mod[m,n])/n)*(q+1)) + + (-d)^((m-Mod[m,n])/n-1)/(n*e^(2*p)*(q+1))*Int[x^m*(d+e*x^n)^(q+1)* + ExpandToSum[Together[1/(d+e*x^n)*(n*(-d)^(-(m-Mod[m,n])/n+1)*e^(2*p)*(q+1)*(a+c*x^(2*n))^p - + (e^(-(m-Mod[m,n])/n)*(c*d^2+a*e^2)^p*x^(-(m-Mod[m,n])))*(d*(Mod[m,n]+1)+e*(Mod[m,n]+n*(q+1)+1)*x^n))],x],x] /; +FreeQ[{a,c,d,e},x] && EqQ[n2-2*n] && PositiveIntegerQ[n,p] && IntegersQ[m,q] && q<-1 && m<0 + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + c^p*(f*x)^(m+2*n*p-n+1)*(d+e*x^n)^(q+1)/(e*f^(2*n*p-n+1)*(m+2*n*p+n*q+1)) + + 1/(e*(m+2*n*p+n*q+1))*Int[(f*x)^m*(d+e*x^n)^q* + ExpandToSum[e*(m+2*n*p+n*q+1)*((a+b*x^n+c*x^(2*n))^p-c^p*x^(2*n*p))-d*c^p*(m+2*n*p-n+1)*x^(2*n*p-n),x],x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n,p] && 2*n*p>n-1 && + Not[IntegerQ[q]] && NeQ[m+2*n*p+n*q+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + c^p*(f*x)^(m+2*n*p-n+1)*(d+e*x^n)^(q+1)/(e*f^(2*n*p-n+1)*(m+2*n*p+n*q+1)) + + 1/(e*(m+2*n*p+n*q+1))*Int[(f*x)^m*(d+e*x^n)^q* + ExpandToSum[e*(m+2*n*p+n*q+1)*((a+c*x^(2*n))^p-c^p*x^(2*n*p))-d*c^p*(m+2*n*p-n+1)*x^(2*n*p-n),x],x] /; +FreeQ[{a,c,d,e,f,m,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n,p] && 2*n*p>n-1 && + Not[IntegerQ[q]] && NeQ[m+2*n*p+n*q+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n,p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m(d+e*x^n)^q*(a+c*x^(2*n))^p,x],x] /; +FreeQ[{a,c,d,e,f,m,q},x] && EqQ[n2-2*n] && EqQ[n2-2*n] && PositiveIntegerQ[n,p] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=GCD[m+1,n]}, + 1/k*Subst[Int[x^((m+1)/k-1)*(d+e*x^(n/k))^q*(a+b*x^(n/k)+c*x^(2*n/k))^p,x],x,x^k] /; + k!=1] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=GCD[m+1,n]}, + 1/k*Subst[Int[x^((m+1)/k-1)*(d+e*x^(n/k))^q*(a+c*x^(2*n/k))^p,x],x,x^k] /; + k!=1] /; +FreeQ[{a,c,d,e,p,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/f*Subst[Int[x^(k*(m+1)-1)*(d+e*x^(k*n)/f^n)^q*(a+b*x^(k*n)/f^n+c*x^(2*k*n)/f^(2*n))^p,x],x,(f*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && FractionQ[m] && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{k=Denominator[m]}, + k/f*Subst[Int[x^(k*(m+1)-1)*(d+e*x^(k*n)/f)^q*(a+c*x^(2*k*n)/f)^p,x],x,(f*x)^(1/k)]] /; +FreeQ[{a,c,d,e,f,p,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && FractionQ[m] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+b*x^n+c*x^(2*n))^p*(d*(m+n*(2*p+1)+1)+e*(m+1)*x^n)/(f*(m+1)*(m+n*(2*p+1)+1)) + + n*p/(f^n*(m+1)*(m+n*(2*p+1)+1))*Int[(f*x)^(m+n)*(a+b*x^n+c*x^(2*n))^(p-1)* + Simp[2*a*e*(m+1)-b*d*(m+n*(2*p+1)+1)+(b*e*(m+1)-2*c*d*(m+n*(2*p+1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<-1 && + m+n*(2*p+1)+1!=0 && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+c*x^(2*n))^p*(d*(m+n*(2*p+1)+1)+e*(m+1)*x^n)/(f*(m+1)*(m+n*(2*p+1)+1)) + + 2*n*p/(f^n*(m+1)*(m+n*(2*p+1)+1))*Int[(f*x)^(m+n)*(a+c*x^(2*n))^(p-1)*(a*e*(m+1)-c*d*(m+n*(2*p+1)+1)*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<-1 && + m+n*(2*p+1)+1!=0 && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+b*x^n+c*x^(2*n))^p*(b*e*n*p+c*d*(m+n*(2*p+1)+1)+c*e*(2*n*p+m+1)*x^n)/ + (c*f*(2*n*p+m+1)*(m+n*(2*p+1)+1)) + + n*p/(c*(2*n*p+m+1)*(m+n*(2*p+1)+1))*Int[(f*x)^m*(a+b*x^n+c*x^(2*n))^(p-1)* + Simp[2*a*c*d*(m+n*(2*p+1)+1)-a*b*e*(m+1)+(2*a*c*e*(2*n*p+m+1)+b*c*d*(m+n*(2*p+1)+1)-b^2*e*(m+n*p+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p>0 && + NeQ[2*n*p+m+1] && NeQ[m+n*(2*p+1)+1] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + (f*x)^(m+1)*(a+c*x^(2*n))^p*(c*d*(m+n*(2*p+1)+1)+c*e*(2*n*p+m+1)*x^n)/(c*f*(2*n*p+m+1)*(m+n*(2*p+1)+1)) + + 2*a*n*p/((2*n*p+m+1)*(m+n*(2*p+1)+1))*Int[(f*x)^m*(a+c*x^(2*n))^(p-1)*Simp[d*(m+n*(2*p+1)+1)+e*(2*n*p+m+1)*x^n,x],x] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[p] && p>0 && + NeQ[2*n*p+m+1] && NeQ[m+n*(2*p+1)+1] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + f^(n-1)*(f*x)^(m-n+1)*(a+b*x^n+c*x^(2*n))^(p+1)*(b*d-2*a*e-(b*e-2*c*d)*x^n)/(n*(p+1)*(b^2-4*a*c)) + + f^n/(n*(p+1)*(b^2-4*a*c))*Int[(f*x)^(m-n)*(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[(n-m-1)*(b*d-2*a*e)+(2*n*p+2*n+m+1)*(b*e-2*c*d)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p] && p<-1 && m>n-1 && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_.,x_Symbol] := + f^(n-1)*(f*x)^(m-n+1)*(a+c*x^(2*n))^(p+1)*(a*e-c*d*x^n)/(2*a*c*n*(p+1)) + + f^n/(2*a*c*n*(p+1))*Int[(f*x)^(m-n)*(a+c*x^(2*n))^(p+1)*(a*e*(n-m-1)+c*d*(2*n*p+2*n+m+1)*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m>n-1 && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)*(d*(b^2-2*a*c)-a*b*e+(b*d-2*a*e)*c*x^n)/(a*f*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))*Int[(f*x)^m*(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[d*(b^2*(m+n*(p+1)+1)-2*a*c*(m+2*n*(p+1)+1))-a*b*e*(m+1)+c*(m+n*(2*p+3)+1)*(b*d-2*a*e)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && + IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+c*x^(2*n))^(p+1)*(d+e*x^n)/(2*a*f*n*(p+1)) + + 1/(2*a*n*(p+1))*Int[(f*x)^m*(a+c*x^(2*n))^(p+1)*Simp[d*(m+2*n*(p+1)+1)+e*(m+n*(2*p+3)+1)*x^n,x],x] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + e*f^(n-1)*(f*x)^(m-n+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(c*(m+n(2*p+1)+1)) - + f^n/(c*(m+n(2*p+1)+1))* + Int[(f*x)^(m-n)*(a+b*x^n+c*x^(2*n))^p*Simp[a*e*(m-n+1)+(b*e*(m+n*p+1)-c*d*(m+n(2*p+1)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m>n-1 && + NeQ[m+n(2*p+1)+1] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + e*f^(n-1)*(f*x)^(m-n+1)*(a+c*x^(2*n))^(p+1)/(c*(m+n(2*p+1)+1)) - + f^n/(c*(m+n(2*p+1)+1))*Int[(f*x)^(m-n)*(a+c*x^(2*n))^p*(a*e*(m-n+1)-c*d*(m+n(2*p+1)+1)*x^n),x] /; +FreeQ[{a,c,d,e,f,p},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m] && m>n-1 && NeQ[m+n(2*p+1)+1] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + d*(f*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*f*(m+1)) + + 1/(a*f^n*(m+1))*Int[(f*x)^(m+n)*(a+b*x^n+c*x^(2*n))^p*Simp[a*e*(m+1)-b*d*(m+n*(p+1)+1)-c*d*(m+2*n(p+1)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 && + IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + d*(f*x)^(m+1)*(a+c*x^(2*n))^(p+1)/(a*f*(m+1)) + + 1/(a*f^n*(m+1))*Int[(f*x)^(m+n)*(a+c*x^(2*n))^p*(a*e*(m+1)-c*d*(m+2*n(p+1)+1)*x^n),x] /; +FreeQ[{a,c,d,e,f,p},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[a*c,2]}, + With[{r=Rt[2*c*q-b*c,2]}, + c/(2*q*r)*Int[(f*x)^m*Simp[d*r-(c*d-e*q)*x^(n/2),x]/(q-r*x^(n/2)+c*x^n),x] + + c/(2*q*r)*Int[(f*x)^m*Simp[d*r+(c*d-e*q)*x^(n/2),x]/(q+r*x^(n/2)+c*x^n),x]] /; + Not[NegativeQ[2*c*q-b*c]]] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NegativeQ[b^2-4*a*c] && IntegersQ[m,n/2] && 02 && PosQ[a*c] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[a*c,2]}, + With[{r=Rt[2*c*q,2]}, + c/(2*q*r)*Int[(f*x)^m*(d*r-(c*d-e*q)*x^(n/2))/(q-r*x^(n/2)+c*x^n),x] + + c/(2*q*r)*Int[(f*x)^m*(d*r+(c*d-e*q)*x^(n/2))/(q+r*x^(n/2)+c*x^n),x]] /; + Not[NegativeQ[2*c*q]]] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && IntegerQ[n/2] && n>2 && PositiveQ[a*c] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)/(a_+b_.*x_^n_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (e/2+(2*c*d-b*e)/(2*q))*Int[(f*x)^m/(b/2-q/2+c*x^n),x] + (e/2-(2*c*d-b*e)/(2*q))*Int[(f*x)^m/(b/2+q/2+c*x^n),x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)/(a_+c_.*x_^n2_),x_Symbol] := + With[{q=Rt[-a*c,2]}, + -(e/2+c*d/(2*q))*Int[(f*x)^m/(q-c*x^n),x] + (e/2-c*d/(2*q))*Int[(f*x)^m/(q+c*x^n),x]] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[q] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_./(a_+c_.*x_^n2_.),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^n)^q/(a+c*x^(2*n)),x],x] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && IntegerQ[q] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_./(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m,(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[q] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_./(a_+c_.*x_^n2_.),x_Symbol] := + Int[ExpandIntegrand[(f*x)^m,(d+e*x^n)^q/(a+c*x^(2*n)),x],x] /; +FreeQ[{a,c,d,e,f,m},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && IntegerQ[q] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + f^(2*n)/c^2*Int[(f*x)^(m-2*n)*(c*d-b*e+c*e*x^n)*(d+e*x^n)^(q-1),x] - + f^(2*n)/c^2*Int[(f*x)^(m-2*n)*(d+e*x^n)^(q-1)*Simp[a*(c*d-b*e)+(b*c*d-b^2*e+a*c*e)*x^n,x]/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && + RationalQ[m,q] && q>0 && m>2*n-1 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+c_.*x_^n2_.),x_Symbol] := + f^(2*n)/c*Int[(f*x)^(m-2*n)*(d+e*x^n)^q,x] - + a*f^(2*n)/c*Int[(f*x)^(m-2*n)*(d+e*x^n)^q/(a+c*x^(2*n)),x] /; +FreeQ[{a,c,d,e,f,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && RationalQ[m] && m>2*n-1 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + e*f^n/c*Int[(f*x)^(m-n)*(d+e*x^n)^(q-1),x] - + f^n/c*Int[(f*x)^(m-n)*(d+e*x^n)^(q-1)*Simp[a*e-(c*d-b*e)*x^n,x]/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && + RationalQ[m,q] && q>0 && n-10 && n-10 && m<0 + + +Int[(f_.*x_)^m_*(d_.+e_.*x_^n_)^q_/(a_+c_.*x_^n2_.),x_Symbol] := + d/a*Int[(f*x)^m*(d+e*x^n)^(q-1),x] + + 1/(a*f^n)*Int[(f*x)^(m+n)*(d+e*x^n)^(q-1)*Simp[a*e-c*d*x^n,x]/(a+c*x^(2*n)),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && RationalQ[m,q] && q>0 && m<0 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + d^2*f^(2*n)/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2*n)*(d+e*x^n)^q,x] - + f^(2*n)/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2*n)*(d+e*x^n)^(q+1)*Simp[a*d+(b*d-a*e)*x^n,x]/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && + RationalQ[m,q] && q<-1 && m>2*n-1 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+c_.*x_^n2_.),x_Symbol] := + d^2*f^(2*n)/(c*d^2+a*e^2)*Int[(f*x)^(m-2*n)*(d+e*x^n)^q,x] - + a*f^(2*n)/(c*d^2+a*e^2)*Int[(f*x)^(m-2*n)*(d+e*x^n)^(q+1)*(d-e*x^n)/(a+c*x^(2*n)),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && RationalQ[m,q] && q<-1 && m>2*n-1 + + +Int[(f_.*x_)^m_.*(d_.+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + -d*e*f^n/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-n)*(d+e*x^n)^q,x] + + f^n/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-n)*(d+e*x^n)^(q+1)*Simp[a*e+c*d*x^n,x]/(a+b*x^n+c*x^(2*n)),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[IntegerQ[q]] && + RationalQ[m,q] && q<-1 && n-10 && m<-n + + +Int[(f_.*x_)^m_*(a_+c_.*x_^n2_.)^p_./(d_.+e_.*x_^n_),x_Symbol] := + a/d^2*Int[(f*x)^m*(d-e*x^n)*(a+c*x^(2*n))^(p-1),x] + + (c*d^2+a*e^2)/(d^2*f^(2*n))*Int[(f*x)^(m+2*n)*(a+c*x^(2*n))^(p-1)/(d+e*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<-n + + +Int[(f_.*x_)^m_*(a_.+b_.*x_^n_+c_.*x_^n2_.)^p_./(d_.+e_.*x_^n_),x_Symbol] := + 1/(d*e)*Int[(f*x)^m*(a*e+c*d*x^n)*(a+b*x^n+c*x^(2*n))^(p-1),x] - + (c*d^2-b*d*e+a*e^2)/(d*e*f^n)*Int[(f*x)^(m+n)*(a+b*x^n+c*x^(2*n))^(p-1)/(d+e*x^n),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<0 + + +Int[(f_.*x_)^m_*(a_+c_.*x_^n2_.)^p_./(d_.+e_.*x_^n_),x_Symbol] := + 1/(d*e)*Int[(f*x)^m*(a*e+c*d*x^n)*(a+c*x^(2*n))^(p-1),x] - + (c*d^2+a*e^2)/(d*e*f^n)*Int[(f*x)^(m+n)*(a+c*x^(2*n))^(p-1)/(d+e*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m<0 + + +Int[(f_.*x_)^m_.*(a_.+b_.*x_^n_+c_.*x_^n2_.)^p_/(d_.+e_.*x_^n_),x_Symbol] := + -f^(2*n)/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2*n)*(a*d+(b*d-a*e)*x^n)*(a+b*x^n+c*x^(2*n))^p,x] + + d^2*f^(2*n)/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-2*n)*(a+b*x^n+c*x^(2*n))^(p+1)/(d+e*x^n),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m>n + + +Int[(f_.*x_)^m_.*(a_+c_.*x_^n2_.)^p_/(d_.+e_.*x_^n_),x_Symbol] := + -a*f^(2*n)/(c*d^2+a*e^2)*Int[(f*x)^(m-2*n)*(d-e*x^n)*(a+c*x^(2*n))^p,x] + + d^2*f^(2*n)/(c*d^2+a*e^2)*Int[(f*x)^(m-2*n)*(a+c*x^(2*n))^(p+1)/(d+e*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m>n + + +Int[(f_.*x_)^m_.*(a_.+b_.*x_^n_+c_.*x_^n2_.)^p_/(d_.+e_.*x_^n_),x_Symbol] := + f^n/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-n)*(a*e+c*d*x^n)*(a+b*x^n+c*x^(2*n))^p,x] - + d*e*f^n/(c*d^2-b*d*e+a*e^2)*Int[(f*x)^(m-n)*(a+b*x^n+c*x^(2*n))^(p+1)/(d+e*x^n),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m>0 + + +Int[(f_.*x_)^m_.*(a_+c_.*x_^n2_.)^p_/(d_.+e_.*x_^n_),x_Symbol] := + f^n/(c*d^2+a*e^2)*Int[(f*x)^(m-n)*(a*e+c*d*x^n)*(a+c*x^(2*n))^p,x] - + d*e*f^n/(c*d^2+a*e^2)*Int[(f*x)^(m-n)*(a+c*x^(2*n))^(p+1)/(d+e*x^n),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && RationalQ[m,p] && p<-1 && m>0 + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*x^n+c*x^(2*n))^p,(f*x)^m(d+e*x^n)^q,x],x] /; +FreeQ[{a,b,c,d,e,f,m,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && (PositiveIntegerQ[q] || IntegersQ[m,q]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(a+c*x^(2*n))^p,(f*x)^m(d+e*x^n)^q,x],x] /; +FreeQ[{a,c,d,e,f,m,q},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] && (PositiveIntegerQ[q] || IntegersQ[m,q]) + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -Subst[Int[(d+e*x^(-n))^q*(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,d,e,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + -Subst[Int[(d+e*x^(-n))^q*(a+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,c,d,e,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[m]}, + -g/f*Subst[Int[(d+e*f^(-n)*x^(-g*n))^q*(a+b*f^(-n)*x^(-g*n)+c*f^(-2*n)*x^(-2*g*n))^p/x^(g*(m+1)+1),x],x,1/(f*x)^(1/g)]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[m]}, + -g/f*Subst[Int[(d+e*f^(-n)*x^(-g*n))^q*(a+c*f^(-2*n)*x^(-2*g*n))^p/x^(g*(m+1)+1),x],x,1/(f*x)^(1/g)]] /; +FreeQ[{a,c,d,e,f,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + -f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^FracPart[m]*Subst[Int[(d+e*x^(-n))^q*(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + -f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^FracPart[m]*Subst[Int[(d+e*x^(-n))^q*(a+c*x^(-2*n))^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g*(m+1)-1)*(d+e*x^(g*n))^q*(a+b*x^(g*n)+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,c,d,e,m,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g*(m+1)-1)*(d+e*x^(g*n))^q*(a+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,c,d,e,m,p,q},x] && EqQ[n2-2*n] && FractionQ[n] + + +Int[(f_*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[(f_*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && FractionQ[n] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/(m+1)*Subst[Int[(d+e*x^Simplify[n/(m+1)])^q*(a+b*x^Simplify[n/(m+1)]+c*x^Simplify[2*n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/(m+1)*Subst[Int[(d+e*x^Simplify[n/(m+1)])^q*(a+c*x^Simplify[2*n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(f_*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(f_*x_)^m_*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,p,q},x] && EqQ[n2-2*n] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_/(a_+b_.*x_^n_+c_.*x_^n2_.),x_Symbol] := + With[{r=Rt[b^2-4*a*c,2]}, + 2*c/r*Int[(f*x)^m*(d+e*x^n)^q/(b-r+2*c*x^n),x] - 2*c/r*Int[(f*x)^m*(d+e*x^n)^q/(b+r+2*c*x^n),x]] /; +FreeQ[{a,b,c,d,e,f,m,n,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_/(a_+c_.*x_^n2_.),x_Symbol] := + With[{r=Rt[-a*c,2]}, + -c/(2*r)*Int[(f*x)^m*(d+e*x^n)^q/(r-c*x^n),x] - c/(2*r)*Int[(f*x)^m*(d+e*x^n)^q/(r+c*x^n),x]] /; +FreeQ[{a,c,d,e,f,m,n,q},x] && EqQ[n2-2*n] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)*(d*(b^2-2*a*c)-a*b*e+(b*d-2*a*e)*c*x^n)/(a*f*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))*Int[(f*x)^m*(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[d*(b^2*(m+n*(p+1)+1)-2*a*c*(m+2*n*(p+1)+1))-a*b*e*(m+1)+(m+n*(2*p+3)+1)*(b*d-2*a*e)*c*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -(f*x)^(m+1)*(a+c*x^(2*n))^(p+1)*(d+e*x^n)/(2*a*f*n*(p+1)) + + 1/(2*a*n*(p+1))*Int[(f*x)^m*(a+c*x^(2*n))^(p+1)*Simp[d*(m+2*n*(p+1)+1)+e*(m+n*(2*p+3)+1)*x^n,x],x] /; +FreeQ[{a,c,d,e,f,m,n},x] && EqQ[n2-2*n] && NegativeIntegerQ[p+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && (PositiveIntegerQ[p] || PositiveIntegerQ[q]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(f*x)^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x],x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && (PositiveIntegerQ[p] || PositiveIntegerQ[q]) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + f^m*Int[ExpandIntegrand[x^m*(a+c*x^(2*n))^p,(d/(d^2-e^2*x^(2*n))-e*x^n/(d^2-e^2*x^(2*n)))^(-q),x],x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[q] && (IntegerQ[m] || PositiveQ[f]) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^n_)^q_*(a_+c_.*x_^n2_)^p_,x_Symbol] := + (f*x)^m/x^m*Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] && NegativeIntegerQ[q] && Not[IntegerQ[m] || PositiveQ[f]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,n,p,q},x] && EqQ[n2-2*n] + + +Int[u_^m_.*(d_+e_.*v_^n_)^q_.*(a_+b_.*v_^n_+c_.*v_^n2_.)^p_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(d+e*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x],x,v] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2-2*n] && LinearPairQ[u,v,x] + + +Int[u_^m_.*(d_+e_.*v_^n_)^q_.*(a_+c_.*v_^n2_.)^p_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x],x,v] /; +FreeQ[{a,c,d,e,m,n,p},x] && EqQ[n2-2*n] && LinearPairQ[u,v,x] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_.*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2,2*n] && EqQ[mn,-n] && IntegerQ[q] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_.*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m+mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x] /; +FreeQ[{a,c,d,e,m,mn,p},x] && EqQ[n2,-2*mn] && IntegerQ[q] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m+2*n*p)*(d+e*x^(-n))^q*(c+b*x^(-n)+a*x^(-2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] && Not[IntegerQ[q]] && IntegerQ[p] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[x^(m-2*mn*p)*(d+e*x^mn)^q*(c+a*x^(2*mn))^p,x] /; +FreeQ[{a,c,d,e,m,mn,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[q]] && IntegerQ[p] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + x^(n*FracPart[q])*(d+e*x^(-n))^FracPart[q]/(e+d*x^n)^FracPart[q]*Int[x^(m-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] && Not[IntegerQ[q]] && Not[IntegerQ[p]] + + +Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + x^(-mn*FracPart[q])*(d+e*x^mn)^FracPart[q]/(e+d*x^(-mn))^FracPart[q]*Int[x^(m+mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x] /; +FreeQ[{a,c,d,e,m,mn,p,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[q]] && Not[IntegerQ[p]] + + +Int[(f_*x_)^m_*(d_+e_.*x_^mn_.)^q_.*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^mn)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] + + +Int[(f_*x_)^m_*(d_+e_.*x_^mn_.)^q_.*(a_+c_.*x_^n2_.)^p_,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^mn)^q*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,c,d,e,f,m,mn,p,q},x] && EqQ[n2,-2*mn] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + Int[x^(m-n*p)*(d+e*x^n)^q*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n,q},x] && EqQ[mn,-n] && IntegerQ[p] + + +Int[x_^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + x^(n*FracPart[p])*(a+b/x^n+c*x^n)^FracPart[p]/(b+a*x^n+c*x^(2*n))^FracPart[p]* + Int[x^(m-n*p)*(d+e*x^n)^q*(b+a*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[mn,-n] && Not[IntegerQ[p]] + + +Int[(f_*x_)^m_.*(d_+e_.*x_^n_)^q_.*(a_+b_.*x_^mn_+c_.*x_^n_.)^p_.,x_Symbol] := + f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(d+e*x^n)^q*(a+b*x^(-n)+c*x^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[mn,-n] + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_^non2_.)^q_.*(d2_+e2_.*x_^non2_.)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + Int[(f*x)^m*(d1*d2+e1*e2*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,n,p,q},x] && EqQ[n2-2*n] && EqQ[non2-n/2] && EqQ[d2*e1+d1*e2] && + (IntegerQ[q] || PositiveQ[d1] && PositiveQ[d2]) + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_^non2_.)^q_.*(d2_+e2_.*x_^non2_.)^q_.*(a_.+b_.*x_^n_+c_.*x_^n2_)^p_.,x_Symbol] := + (d1+e1*x^(n/2))^FracPart[q]*(d2+e2*x^(n/2))^FracPart[q]/(d1*d2+e1*e2*x^n)^FracPart[q]* + Int[(f*x)^m*(d1*d2+e1*e2*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,n,p,q},x] && EqQ[n2-2*n] && EqQ[non2-n/2] && EqQ[d2*e1+d1*e2] + + + + + +(* ::Subsection::Closed:: *) +(*1.1.3.7 (c x)^m Pq(x) (a+b x^n)^p*) + + +Int[x_^m_.*(e_+f_.*x_^q_.+g_.*x_^r_.+h_.*x_^n_.)/(a_+c_.*x_^n_.)^(3/2),x_Symbol] := + -(2*a*g+4*a*h*x^(n/4)-2*c*f*x^(n/2))/(a*c*n*Sqrt[a+c*x^n]) /; +FreeQ[{a,c,e,f,g,h,m,n},x] && EqQ[q-n/4] && EqQ[r-3*n/4] && EqQ[4*m-n+4] && EqQ[c*e+a*h] + + +Int[(d_*x_)^m_.*(e_+f_.*x_^q_.+g_.*x_^r_.+h_.*x_^n_.)/(a_+c_.*x_^n_.)^(3/2),x_Symbol] := + (d*x)^m/x^m*Int[x^m*(e+f*x^(n/4)+g*x^((3*n)/4)+h*x^n)/(a+c*x^n)^(3/2),x] /; +FreeQ[{a,c,d,e,f,g,h,m,n},x] && EqQ[4*m-n+4] && EqQ[q-n/4] && EqQ[r-3*n/4] && EqQ[c*e+a*h] + + +Int[(c_.*x_)^m_*Pq_*(a_+b_.*x_)^p_,x_Symbol] := + With[{n=Denominator[p]}, + n/b*Subst[Int[x^(n*p+n-1)*(-a*c/b+c*x^n/b)^m*ReplaceAll[Pq,x->-a/b+x^n/b],x],x,(a+b*x)^(1/n)]] /; +FreeQ[{a,b,c,m},x] && PolyQ[Pq,x] && FractionQ[p] && NegativeIntegerQ[m+1] + + +(* Int[Pq_*(a_+b_.*x_)^p_,x_Symbol] := + With[{n=Denominator[p]}, + n/b*Subst[Int[x^(n*p+n-1)*ReplaceAll[Pq,x->-a/b+x^n/b],x],x,(a+b*x)^(1/n)]] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && FractionQ[p] *) + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_.,x_Symbol] := + 1/(m+1)*Subst[Int[SubstFor[x^(m+1),Pq,x]*(a+b*x^Simplify[n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,m,n,p},x] && NeQ[m+1] && PositiveIntegerQ[Simplify[n/(m+1)]] && PolyQ[Pq,x^(m+1)] + + +Int[Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + Coeff[Pq,x,n-1]*(a+b*x^n)^(p+1)/(b*n*(p+1)) + + Int[ExpandToSum[Pq-Coeff[Pq,x,n-1]*x^(n-1),x]*(a+b*x^n)^p,x] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n,p] && NeQ[Coeff[Pq,x,n-1]] + + +Int[(c_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(c*x)^m*Pq*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,m,n},x] && PolyQ[Pq,x] && (PositiveIntegerQ[p] || EqQ[n-1]) + + +Int[Pq_*(a_+b_.*x_^n_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,n},x] && PolyQ[Pq,x] && (PositiveIntegerQ[p] || EqQ[n-1]) + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*SubstFor[x^n,Pq,x]*(a+b*x)^p,x],x,x^n] /; +FreeQ[{a,b,m,n,p},x] && PolyQ[Pq,x^n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(c_*x_)^m_.*Pq_*(a_+b_.*x_^n_)^p_.,x_Symbol] := + c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]*Int[x^m*Pq*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,n,p},x] && PolyQ[Pq,x^n] && IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + Pq*(a+b*x^n)^(p+1)/(b*n*(p+1)) - + 1/(b*n*(p+1))*Int[D[Pq,x]*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b,m,n},x] && PolyQ[Pq,x] && EqQ[m-n+1] && RationalQ[p] && p<-1 + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + 1/d*Int[(d*x)^(m+1)*ExpandToSum[Pq/x,x]*(a+b*x^n)^p,x] /; +FreeQ[{a,b,d,m,n,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] + + +Int[Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + Int[x*ExpandToSum[Pq/x,x]*(a+b*x^n)^p,x] /; +FreeQ[{a,b,n,p},x] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] && SumQ[Pq] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + Module[{u=IntHide[x^m*Pq,x]}, + u*(a+b*x^n)^p - b*n*p*Int[x^(m+n)*(a+b*x^n)^(p-1)*ExpandToSum[u/x^(m+1),x],x]] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] && RationalQ[m,p] && p>0 && m+Expon[Pq,x]+1<0 + + +Int[(c_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],i}, + (c*x)^m*(a+b*x^n)^p*Sum[Coeff[Pq,x,i]*x^(i+1)/(m+n*p+i+1),{i,0,q}] + + a*n*p*Int[(c*x)^m*(a+b*x^n)^(p-1)*Sum[Coeff[Pq,x,i]*x^i/(m+n*p+i+1),{i,0,q}],x]] /; +FreeQ[{a,b,c,m},x] && PolyQ[Pq,x] && PositiveIntegerQ[(n-1)/2] && RationalQ[p] && p>0 + + +Int[Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],i}, + (a+b*x^n)^p*Sum[Coeff[Pq,x,i]*x^(i+1)/(n*p+i+1),{i,0,q}] + + a*n*p*Int[(a+b*x^n)^(p-1)*Sum[Coeff[Pq,x,i]*x^i/(n*p+i+1),{i,0,q}],x]] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[(n-1)/2] && RationalQ[p] && p>0 + + +Int[Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],i}, + (a*Coeff[Pq,x,q]-b*x*ExpandToSum[Pq-Coeff[Pq,x,q]*x^q,x])*(a+b*x^n)^(p+1)/(a*b*n*(p+1)) + + 1/(a*n*(p+1))*Int[Sum[(n*(p+1)+i+1)*Coeff[Pq,x,i]*x^i,{i,0,q-1}]*(a+b*x^n)^(p+1),x] /; + q==n-1] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + -x*Pq*(a+b*x^n)^(p+1)/(a*n*(p+1)) + + 1/(a*n*(p+1))*Int[ExpandToSum[n*(p+1)*Pq+D[x*Pq,x],x]*(a+b*x^n)^(p+1),x] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && Expon[Pq,x]=n] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[x_^m_*Pq_*(a_+b_.*x_^n_.)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + Module[{Q=PolynomialQuotient[a*b^(Floor[(q-1)/n]+1)*x^m*Pq,a+b*x^n,x], + R=PolynomialRemainder[a*b^(Floor[(q-1)/n]+1)*x^m*Pq,a+b*x^n,x],i}, + -x*R*(a+b*x^n)^(p+1)/(a^2*n*(p+1)*b^(Floor[(q-1)/n]+1)) + + 1/(a*n*(p+1)*b^(Floor[(q-1)/n]+1))*Int[x^m*(a+b*x^n)^(p+1)* + ExpandToSum[n*(p+1)*x^(-m)*Q+Sum[(n*(p+1)+i+1)/a*Coeff[R,x,i]*x^(i-m),{i,0,n-1}],x],x]]] /; +FreeQ[{a,b},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 && NegativeIntegerQ[m] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{g=GCD[m+1,n]}, + 1/g*Subst[Int[x^((m+1)/g-1)*ReplaceAll[Pq,x->x^(1/g)]*(a+b*x^(n/g))^p,x],x,x^g] /; + g!=1] /; +FreeQ[{a,b,p},x] && PolyQ[Pq,x^n] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(A_+B_.*x_)/(a_+b_.*x_^3),x_Symbol] := + B^3/b*Int[1/(A^2-A*B*x+B^2*x^2),x] /; +FreeQ[{a,b,A,B},x] && EqQ[a*B^3-b*A^3] + + +Int[(A_+B_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{r=Numerator[Rt[a/b,3]], s=Denominator[Rt[a/b,3]]}, + -r*(B*r-A*s)/(3*a*s)*Int[1/(r+s*x),x] + + r/(3*a*s)*Int[(r*(B*r+2*A*s)+s*(B*r-A*s)*x)/(r^2-r*s*x+s^2*x^2),x]] /; +FreeQ[{a,b,A,B},x] && NeQ[a*B^3-b*A^3] && PosQ[a/b] + + +Int[(A_+B_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{r=Numerator[Rt[-a/b,3]], s=Denominator[Rt[-a/b,3]]}, + r*(B*r+A*s)/(3*a*s)*Int[1/(r-s*x),x] - + r/(3*a*s)*Int[(r*(B*r-2*A*s)-s*(B*r+A*s)*x)/(r^2+r*s*x+s^2*x^2),x]] /; +FreeQ[{a,b,A,B},x] && NeQ[a*B^3-b*A^3] && NegQ[a/b] + + +Int[(A_+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + -C^2/b*Int[1/(B-C*x),x] /; +FreeQ[{a,b,A,B,C},x] && EqQ[B^2-A*C] && EqQ[b*B^3+a*C^3] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/b^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A*b^(2/3)-a^(1/3)*b^(1/3)*B-2*a^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/b^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[a^(1/3)*b^(1/3)*B+2*a^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/b^(1/3)}, + C/b*Int[1/(q+x),x] + C*q/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A*b^(2/3)-2*a^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/(-b)^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A*(-b)^(2/3)-(-a)^(1/3)*(-b)^(1/3)*B-2*(-a)^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/(-b)^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[(-a)^(1/3)*(-b)^(1/3)*B+2*(-a)^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/(-b)^(1/3)}, + C/b*Int[1/(q+x),x] + C*q/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A*(-b)^(2/3)-2*(-a)^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/b^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A*b^(2/3)+(-a)^(1/3)*b^(1/3)*B-2*(-a)^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/b^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[(-a)^(1/3)*b^(1/3)*B-2*(-a)^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a)^(1/3)/b^(1/3)}, + -C/b*Int[1/(q-x),x] - C*q/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A*b^(2/3)-2*(-a)^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/(-b)^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A*(-b)^(2/3)+a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/(-b)^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=a^(1/3)/(-b)^(1/3)}, + -C/b*Int[1/(q-x),x] - C*q/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A*(-b)^(2/3)-2*a^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A-(a/b)^(1/3)*B-2*(a/b)^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[(a/b)^(1/3)*B+2*(a/b)^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + C/b*Int[1/(q+x),x] + C*q/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A-2*(a/b)^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[a/b,3]}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A-Rt[a/b,3]*B-2*Rt[a/b,3]^2*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[a/b,3]}, + C/b*Int[1/(q+x),x] + (B+C*q)/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[Rt[a/b,3]*B+2*Rt[a/b,3]^2*C] + + +Int[(A_.+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[a/b,3]}, + C/b*Int[1/(q+x),x] + C*q/b*Int[1/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A-2*Rt[a/b,3]^2*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A+(-a/b)^(1/3)*B-2*(-a/b)^(2/3)*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[(-a/b)^(1/3)*B-2*(-a/b)^(2/3)*C] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + -C/b*Int[1/(q-x),x] - C*q/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A-2*(-a/b)^(2/3)*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[-a/b,3]}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A+Rt[-a/b,3]*B-2*Rt[-a/b,3]^2*C] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[-a/b,3]}, + -C/b*Int[1/(q-x),x] + (B-C*q)/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[Rt[-a/b,3]*B-2*Rt[-a/b,3]^2*C] + + +Int[(A_.+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=Rt[-a/b,3]}, + -C/b*Int[1/(q-x),x] - C*q/b*Int[1/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A-2*Rt[-a/b,3]^2*C] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + Int[(A+B*x)/(a+b*x^3),x] + C*Int[x^2/(a+b*x^3),x] /; +FreeQ[{a,b,A,B,C},x] && (EqQ[a*B^3-b*A^3] || Not[RationalQ[a/b]]) + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + B*Int[x/(a+b*x^3),x] + C*Int[x^2/(a+b*x^3),x] /; +FreeQ[{a,b,B,C},x] && Not[RationalQ[a/b]] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + A*Int[1/(a+b*x^3),x] + C*Int[x^2/(a+b*x^3),x] /; +FreeQ[{a,b,A,C},x] && Not[RationalQ[a,b,A,C]] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + q^2/a*Int[(A+C*q*x)/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A-B*(a/b)^(1/3)+C*(a/b)^(2/3)] + + +Int[x_*(B_.+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + C*q^3/a*Int[x/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[B*(a/b)^(1/3)-C*(a/b)^(2/3)] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + q^2/a*Int[(A+C*q*x)/(q^2-q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A+C*(a/b)^(2/3)] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + q/a*Int[(A*q+(A+B*q)*x)/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A+B*(-a/b)^(1/3)+C*(-a/b)^(2/3)] + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + B*q^2/a*Int[x/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,B,C},x] && EqQ[B*(-a/b)^(1/3)+C*(-a/b)^(2/3)] + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + A*q/a*Int[(q+x)/(q^2+q*x+x^2),x]] /; +FreeQ[{a,b,A,C},x] && EqQ[A+C*(-a/b)^(2/3)] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + q*(A-B*q+C*q^2)/(3*a)*Int[1/(q+x),x] + + q/(3*a)*Int[(q*(2*A+B*q-C*q^2)-(A-B*q-2*C*q^2)*x)/(q^2-q*x+x^2),x] /; + NeQ[A-B*q+C*q^2]] /; +FreeQ[{a,b,A,B,C},x] && NeQ[a*B^3-b*A^3] && RationalQ[a/b] && a/b>0 + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + -q*(B*q-C*q^2)/(3*a)*Int[1/(q+x),x] + + q/(3*a)*Int[(q*(B*q-C*q^2)+(B*q+2*C*q^2)*x)/(q^2-q*x+x^2),x] /; + NeQ[B*q-C*q^2]] /; +FreeQ[{a,b,B,C},x] && RationalQ[a/b] && a/b>0 + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(a/b)^(1/3)}, + q*(A+C*q^2)/(3*a)*Int[1/(q+x),x] + + q/(3*a)*Int[(q*(2*A-C*q^2)-(A-2*C*q^2)*x)/(q^2-q*x+x^2),x] /; + NeQ[A+C*q^2]] /; +FreeQ[{a,b,A,C},x] && RationalQ[a/b] && a/b>0 + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + q*(A+B*q+C*q^2)/(3*a)*Int[1/(q-x),x] + + q/(3*a)*Int[(q*(2*A-B*q-C*q^2)+(A+B*q-2*C*q^2)*x)/(q^2+q*x+x^2),x] /; + NeQ[A+B*q+C*q^2]] /; +FreeQ[{a,b,A,B,C},x] && NeQ[a*B^3-b*A^3] && RationalQ[a/b] && a/b<0 + + +Int[x_*(B_+C_.*x_)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + q*(B*q+C*q^2)/(3*a)*Int[1/(q-x),x] + + q/(3*a)*Int[(-q*(B*q+C*q^2)+(B*q-2*C*q^2)*x)/(q^2+q*x+x^2),x] /; + NeQ[B*q+C*q^2]] /; +FreeQ[{a,b,B,C},x] && RationalQ[a/b] && a/b<0 + + +Int[(A_+C_.*x_^2)/(a_+b_.*x_^3),x_Symbol] := + With[{q=(-a/b)^(1/3)}, + q*(A+C*q^2)/(3*a)*Int[1/(q-x),x] + + q/(3*a)*Int[(q*(2*A-C*q^2)+(A-2*C*q^2)*x)/(q^2+q*x+x^2),x] /; + NeQ[A+C*q^2]] /; +FreeQ[{a,b,A,C},x] && RationalQ[a/b] && a/b<0 + + +Int[(c_.*x_)^m_.*Pq_/(a_+b_.*x_^n_),x_Symbol] := + With[{v=Sum[(c*x)^(m+ii)*(Coeff[Pq,x,ii]+Coeff[Pq,x,n/2+ii]*x^(n/2))/(c^ii*(a+b*x^n)),{ii,0,n/2-1}]}, + Int[v,x] /; + SumQ[v]] /; +FreeQ[{a,b,c,m},x] && PolyQ[Pq,x] && PositiveIntegerQ[n/2] && Expon[Pq,x]=0 && (IntegerQ[2*p] || IntegerQ[p+(q+1)/(2*n)])] /; +FreeQ[{a,b,c,m,p},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] + + +Int[Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + With[{Pqq=Coeff[Pq,x,q]}, + Pqq*x^(q-n+1)*(a+b*x^n)^(p+1)/(b*(q+n*p+1)) + + 1/(b*(q+n*p+1))*Int[ExpandToSum[b*(q+n*p+1)*(Pq-Pqq*x^q)-a*Pqq*(q-n+1)*x^(q-n),x]*(a+b*x^n)^p,x]] /; + NeQ[q+n*p+1] && q-n>=0 && (IntegerQ[2*p] || IntegerQ[p+(q+1)/(2*n)])] /; +FreeQ[{a,b,p},x] && PolyQ[Pq,x] && PositiveIntegerQ[n] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + -Subst[Int[ExpandToSum[x^q*ReplaceAll[Pq,x->x^(-1)],x]*(a+b*x^(-n))^p/x^(m+q+2),x],x,1/x]] /; +FreeQ[{a,b,p},x] && PolyQ[Pq,x] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[(c_.*x_)^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{g=Denominator[m],q=Expon[Pq,x]}, + -g/c*Subst[Int[ExpandToSum[x^(g*q)*ReplaceAll[Pq,x->c^(-1)*x^(-g)],x]* + (a+b*c^(-n)*x^(-g*n))^p/x^(g*(m+q+1)+1),x],x,1/(c*x)^(1/g)]] /; +FreeQ[{a,b,c,p},x] && PolyQ[Pq,x] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(c_.*x_)^m_*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + -(c*x)^m*(x^(-1))^m*Subst[Int[ExpandToSum[x^q*ReplaceAll[Pq,x->x^(-1)],x]*(a+b*x^(-n))^p/x^(m+q+2),x],x,1/x]] /; +FreeQ[{a,b,c,m,p},x] && PolyQ[Pq,x] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g*(m+1)-1)*ReplaceAll[Pq,x->x^g]*(a+b*x^(g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,m,p},x] && PolyQ[Pq,x] && FractionQ[n] + + +Int[Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g-1)*ReplaceAll[Pq,x->x^g]*(a+b*x^(g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,p},x] && PolyQ[Pq,x] && FractionQ[n] + + +Int[(c_*x_)^m_*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]*Int[x^m*Pq*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,p},x] && PolyQ[Pq,x] && FractionQ[n] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + 1/(m+1)*Subst[Int[ReplaceAll[SubstFor[x^n,Pq,x],x->x^Simplify[n/(m+1)]]*(a+b*x^Simplify[n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,m,n,p},x] && PolyQ[Pq,x^n] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(c_*x_)^m_*Pq_*(a_+b_.*x_^n_)^p_,x_Symbol] := + c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]*Int[x^m*Pq*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,m,n,p},x] && PolyQ[Pq,x^n] && IntegerQ[Simplify[n/(m+1)]] && Not[IntegerQ[n]] + + +Int[(A_+B_.*x_^m_.)*(a_+b_.*x_^n_)^p_.,x_Symbol] := + A*Int[(a+b*x^n)^p,x] + B*Int[x^m*(a+b*x^n)^p,x] /; +FreeQ[{a,b,A,B,m,n,p},x] && EqQ[m-n+1] + + +Int[(c_.*x_)^m_.*Pq_*(a_+b_.*x_^n_)^p_.,x_Symbol] := + Int[ExpandIntegrand[(c*x)^m*Pq*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,m,n,p},x] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) + + +Int[Pq_*(a_+b_.*x_^n_)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,n,p},x] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) + + +Int[u_^m_.*Pq_*(a_+b_.*v_^n_.)^p_,x_Symbol] := + u^m/(Coeff[v,x,1]*v^m)*Subst[Int[x^m*SubstFor[v,Pq,x]*(a+b*x^n)^p,x],x,v] /; +FreeQ[{a,b,m,n,p},x] && LinearPairQ[u,v,x] && PolyQ[Pq,v^n] + + +Int[Pq_*(a_+b_.*v_^n_.)^p_,x_Symbol] := + 1/Coeff[v,x,1]*Subst[Int[SubstFor[v,Pq,x]*(a+b*x^n)^p,x],x,v] /; +FreeQ[{a,b,n,p},x] && LinearQ[v,x] && PolyQ[Pq,v^n] + + +Int[(c_.*x_)^m_.*Pq_*(a1_+b1_.*x_^n_.)^p_.*(a2_+b2_.*x_^n_.)^p_.,x_Symbol] := + Int[(c*x)^m*Pq*(a1*a2+b1*b2*x^(2*n))^p,x] /; +FreeQ[{a1,b1,a2,b2,c,m,n,p},x] && PolyQ[Pq,x] && EqQ[a2*b1+a1*b2] && (IntegerQ[p] || PositiveQ[a1] && PositiveQ[a2]) + + +Int[Pq_*(a1_+b1_.*x_^n_.)^p_.*(a2_+b2_.*x_^n_.)^p_.,x_Symbol] := + Int[Pq*(a1*a2+b1*b2*x^(2*n))^p,x] /; +FreeQ[{a1,b1,a2,b2,n,p},x] && PolyQ[Pq,x] && EqQ[a2*b1+a1*b2] && (IntegerQ[p] || PositiveQ[a1] && PositiveQ[a2]) + + +Int[(c_.*x_)^m_.*Pq_*(a1_+b1_.*x_^n_.)^p_.*(a2_+b2_.*x_^n_.)^p_.,x_Symbol] := + (a1+b1*x^n)^FracPart[p]*(a2+b2*x^n)^FracPart[p]/(a1*a2+b1*b2*x^(2*n))^FracPart[p]* + Int[(c*x)^m*Pq*(a1*a2+b1*b2*x^(2*n))^p,x] /; +FreeQ[{a1,b1,a2,b2,c,m,n,p},x] && PolyQ[Pq,x] && EqQ[a2*b1+a1*b2] && Not[EqQ[n,1] && LinearQ[Pq,x]] + + +Int[Pq_*(a1_+b1_.*x_^n_.)^p_.*(a2_+b2_.*x_^n_.)^p_.,x_Symbol] := + (a1+b1*x^n)^FracPart[p]*(a2+b2*x^n)^FracPart[p]/(a1*a2+b1*b2*x^(2*n))^FracPart[p]* + Int[Pq*(a1*a2+b1*b2*x^(2*n))^p,x] /; +FreeQ[{a1,b1,a2,b2,n,p},x] && PolyQ[Pq,x] && EqQ[a2*b1+a1*b2] && Not[EqQ[n,1] && LinearQ[Pq,x]] + + +Int[(e_+f_.*x_^n_.+g_.*x_^n2_.)*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^n_.)^p_.,x_Symbol] := + e*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(p+1)/(a*c) /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[n2-2*n] && EqQ[a*c*f-e*(b*c+a*d)*(n*(p+1)+1)] && EqQ[a*c*g-b*d*e*(2*n*(p+1)+1)] + + +Int[(e_+g_.*x_^n2_.)*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^n_.)^p_.,x_Symbol] := + e*x*(a+b*x^n)^(p+1)*(c+d*x^n)^(p+1)/(a*c) /; +FreeQ[{a,b,c,d,e,g,n,p},x] && EqQ[n2-2*n] && EqQ[n*(p+1)+1] && EqQ[a*c*g-b*d*e*(2*n*(p+1)+1)] + + +Int[(h_.*x_)^m_.*(e_+f_.*x_^n_.+g_.*x_^n2_.)*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^n_.)^p_.,x_Symbol] := + e*(h*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(p+1)/(a*c*h*(m+1)) /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[n2-2*n] && EqQ[a*c*f*(m+1)-e*(b*c+a*d)*(m+n*(p+1)+1)] && + EqQ[a*c*g*(m+1)-b*d*e*(m+2*n*(p+1)+1)] && NeQ[m+1] + + +Int[(h_.*x_)^m_.*(e_+g_.*x_^n2_.)*(a_+b_.*x_^n_.)^p_.*(c_+d_.*x_^n_.)^p_.,x_Symbol] := + e*(h*x)^(m+1)*(a+b*x^n)^(p+1)*(c+d*x^n)^(p+1)/(a*c*h*(m+1)) /; +FreeQ[{a,b,c,d,e,g,h,m,n,p},x] && EqQ[n2-2*n] && EqQ[m+n*(p+1)+1] && EqQ[a*c*g*(m+1)-b*d*e*(m+2*n*(p+1)+1)] && + NeQ[m+1] + + +Int[(A_+B_.*x_^m_.)*(a_.+b_.*x_^n_)^p_.*(c_+d_.*x_^n_)^q_.,x_Symbol] := + A*Int[(a+b*x^n)^p*(c+d*x^n)^q,x] + B*Int[x^m*(a+b*x^n)^p*(c+d*x^n)^q,x] /; +FreeQ[{a,b,c,d,A,B,m,n,p,q},x] && NeQ[b*c-a*d] && EqQ[m-n+1] + + +Int[Px_^q_.*(a_.+b_.*(c_+d_.*x_)^n_)^p_,x_Symbol] := + With[{k=Denominator[n]}, + k/d*Subst[Int[SimplifyIntegrand[x^(k-1)*ReplaceAll[Px,x->x^k/d-c/d]^q*(a+b*x^(k*n))^p,x],x],x,(c+d*x)^(1/k)]] /; +FreeQ[{a,b,c,d,p},x] && PolynomialQ[Px,x] && IntegerQ[q] && RationalQ[n] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.3.5 (d x)^m Pq(x) (a+b x^n+c x^(2 n))^p*) + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/n*Subst[Int[SubstFor[x^n,Pq,x]*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && EqQ[Simplify[m-n+1]] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*Pq*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && PositiveIntegerQ[p] + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && PositiveIntegerQ[p] + + +Int[(d_+e_.*x_^n_.+f_.*x_^n2_.)*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + d*x*(a+b*x^n+c*x^(2*n))^(p+1)/a /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[n2-2*n] && EqQ[a*e-b*d*(n*(p+1)+1)] && EqQ[a*f-c*d*(2*n*(p+1)+1)] + + +Int[(d_+f_.*x_^n2_.)*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + d*x*(a+b*x^n+c*x^(2*n))^(p+1)/a /; +FreeQ[{a,b,c,d,f,n,p},x] && EqQ[n2-2*n] && EqQ[n*(p+1)+1] && EqQ[c*d+a*f] + + +Int[(g_.*x_)^m_.*(d_+e_.*x_^n_.+f_.*x_^n2_.)*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + d*(g*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*g*(m+1)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[n2-2*n] && EqQ[a*e*(m+1)-b*d*(m+n*(p+1)+1)] && EqQ[a*f*(m+1)-c*d*(m+2*n*(p+1)+1)] && + NeQ[m+1] + + +Int[(g_.*x_)^m_.*(d_+f_.*x_^n2_.)*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + d*(g*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*g*(m+1)) /; +FreeQ[{a,b,c,d,f,g,m,n,p},x] && EqQ[n2-2*n] && EqQ[m+n*(p+1)+1] && EqQ[c*d+a*f] && NeQ[m+1] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x^n)^(2*FracPart[p]))*Int[(d*x)^m*Pq*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && EqQ[b^2-4*a*c] && Not[IntegerQ[2*p]] + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^FracPart[p]/((4*c)^IntPart[p]*(b+2*c*x^n)^(2*FracPart[p]))*Int[Pq*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && EqQ[b^2-4*a*c] && Not[IntegerQ[2*p]] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*SubstFor[x^n,Pq,x]*(a+b*x+c*x^2)^p,x],x,x^n] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(d_*x_)^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^m/x^m*Int[x^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + 1/d*Int[(d*x)^(m+1)*ExpandToSum[Pq/x,x]*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + Int[x*ExpandToSum[Pq/x,x]*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && EqQ[Coeff[Pq,x,0]] && SumQ[Pq] + + +Int[(d_+e_.*x_^n_+f_.*x_^n2_.+g_.*x_^n3_.)*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + x*(a*d*(n+1)+(a*e-b*d*(n*(p+1)+1))*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a^2*(n+1)) /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && + EqQ[a^2*g*(n+1)-c*(n*(2*p+3)+1)*(a*e-b*d*(n*(p+1)+1))] && + EqQ[a^2*f*(n+1)-a*c*d*(n+1)*(2*n*(p+1)+1)-b*(n*(p+2)+1)*(a*e-b*d*(n*(p+1)+1))] + + +Int[(d_+f_.*x_^n2_.+g_.*x_^n3_.)*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + d*x*(a*(n+1)-b*(n*(p+1)+1)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a^2*(n+1)) /; +FreeQ[{a,b,c,d,f,g,n,p},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && + EqQ[a^2*g*(n+1)+c*b*d*(n*(2*p+3)+1)*(n*(p+1)+1)] && + EqQ[a^2*f*(n+1)-a*c*d*(n+1)*(2*n*(p+1)+1)+b^2*d*(n*(p+2)+1)*(n*(p+1)+1)] + + +Int[(d_+e_.*x_^n_+g_.*x_^n3_.)*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + x*(a*d*(n+1)+(a*e-b*d*(n*(p+1)+1))*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a^2*(n+1)) /; +FreeQ[{a,b,c,d,e,g,n,p},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && + EqQ[a^2*g*(n+1)-c*(n*(2*p+3)+1)*(a*e-b*d*(n*(p+1)+1))] && + EqQ[a*c*d*(n+1)*(2*n*(p+1)+1)+b*(n*(p+2)+1)*(a*e-b*d*(n*(p+1)+1))] + + +Int[(d_+g_.*x_^n3_.)*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + d*x*(a*(n+1)-b*(n*(p+1)+1)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a^2*(n+1)) /; +FreeQ[{a,b,c,d,g,n,p},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && + EqQ[a^2*g*(n+1)+c*b*d*(n*(2*p+3)+1)*(n*(p+1)+1)] && + EqQ[a*c*d*(n+1)*(2*n*(p+1)+1)-b^2*d*(n*(p+2)+1)*(n*(p+1)+1)] + + +Int[x_^m_.*(e_+f_.*x_^q_.+g_.*x_^r_.+h_.*x_^s_.)/(a_+b_.*x_^n_.+c_.*x_^n2_.)^(3/2),x_Symbol] := + -(2*c*(b*f-2*a*g)+2*h*(b^2-4*a*c)*x^(n/2)+2*c*(2*c*f-b*g)*x^n)/(c*n*(b^2-4*a*c)*Sqrt[a+b*x^n+c*x^(2*n)]) /; +FreeQ[{a,b,c,e,f,g,h,m,n},x] && EqQ[n2-2*n] && EqQ[q-n/2] && EqQ[r-3*n/2] && EqQ[s-2*n] && + NeQ[b^2-4*a*c] && EqQ[2*m-n+2] && EqQ[c*e+a*h] + + +Int[(d_*x_)^m_.*(e_+f_.*x_^q_.+g_.*x_^r_.+h_.*x_^s_.)/(a_+b_.*x_^n_.+c_.*x_^n2_.)^(3/2),x_Symbol] := + (d*x)^m/x^m*Int[x^m*(e+f*x^(n/2)+g*x^((3*n)/2)+h*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[n2-2*n] && EqQ[q-n/2] && EqQ[r-3*n/2] && EqQ[s-2*n] && + NeQ[b^2-4*a*c] && EqQ[2*m-n+2] && EqQ[c*e+a*h] + + +Int[Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],i}, + -x*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c))* + Sum[((b^2-2*a*c)*Coeff[Pq,x,i]-a*b*Coeff[Pq,x,n+i])*x^i+ + c*(b*Coeff[Pq,x,i]-2*a*Coeff[Pq,x,n+i])*x^(n+i),{i,0,n-1}] + + 1/(a*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Sum[((b^2*(n*(p+1)+i+1)-2*a*c*(2*n*(p+1)+i+1))*Coeff[Pq,x,i]-a*b*(i+1)*Coeff[Pq,x,n+i])*x^i+ + c*(n*(2*p+3)+i+1)*(b*Coeff[Pq,x,i]-2*a*Coeff[Pq,x,n+i])*x^(n+i),{i,0,n-1}],x] /; + q<2*n] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + Module[{Q=PolynomialQuotient[(b*c)^(Floor[(q-1)/n]+1)*Pq,a+b*x^n+c*x^(2*n),x], + R=PolynomialRemainder[(b*c)^(Floor[(q-1)/n]+1)*Pq,a+b*x^n+c*x^(2*n),x],i}, + -x*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c)*(b*c)^(Floor[(q-1)/n]+1))* + Sum[((b^2-2*a*c)*Coeff[R,x,i]-a*b*Coeff[R,x,n+i])*x^i+ + c*(b*Coeff[R,x,i]-2*a*Coeff[R,x,n+i])*x^(n+i),{i,0,n-1}] + + 1/(a*n*(p+1)*(b^2-4*a*c)*(b*c)^(Floor[(q-1)/n]+1))*Int[(a+b*x^n+c*x^(2*n))^(p+1)*ExpandToSum[a*n*(p+1)*(b^2-4*a*c)*Q+ + Sum[((b^2*(n*(p+1)+i+1)-2*a*c*(2*n*(p+1)+i+1))*Coeff[R,x,i]-a*b*(i+1)*Coeff[R,x,n+i])*x^i+ + c*(n*(2*p+3)+i+1)*(b*Coeff[R,x,i]-2*a*Coeff[R,x,n+i])*x^(n+i),{i,0,n-1}],x],x]] /; + q>=2*n] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && RationalQ[p] && p<-1 + + +Int[x_^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Module[{q=Expon[Pq,x]}, + Module[{Q=PolynomialQuotient[a*(b*c)^(Floor[(q-1)/n]+1)*x^m*Pq,a+b*x^n+c*x^(2*n),x], + R=PolynomialRemainder[a*(b*c)^(Floor[(q-1)/n]+1)*x^m*Pq,a+b*x^n+c*x^(2*n),x],i}, + -x*(a+b*x^n+c*x^(2*n))^(p+1)/(a^2*n*(p+1)*(b^2-4*a*c)*(b*c)^(Floor[(q-1)/n]+1))* + Sum[((b^2-2*a*c)*Coeff[R,x,i]-a*b*Coeff[R,x,n+i])*x^i+ + c*(b*Coeff[R,x,i]-2*a*Coeff[R,x,n+i])*x^(n+i),{i,0,n-1}] + + 1/(a*n*(p+1)*(b^2-4*a*c)*(b*c)^(Floor[(q-1)/n]+1))*Int[x^m*(a+b*x^n+c*x^(2*n))^(p+1)* + ExpandToSum[n*(p+1)*(b^2-4*a*c)*x^(-m)*Q+ + Sum[((b^2*(n*(p+1)+i+1)/a-2*c*(2*n*(p+1)+i+1))*Coeff[R,x,i]-b*(i+1)*Coeff[R,x,n+i])*x^(i-m)+ + c*(n*(2*p+3)+i+1)*(b/a*Coeff[R,x,i]-2*Coeff[R,x,n+i])*x^(n+i-m),{i,0,n-1}],x],x]] /; + q>=2*n] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[p] && p<-1 && NegativeIntegerQ[m] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + With[{g=GCD[m+1,n]}, + 1/g*Subst[Int[x^((m+1)/g-1)*ReplaceAll[Pq,x->x^(1/g)]*(a+b*x^(n/g)+c*x^(2*n/g))^p,x],x,x^g] /; + g!=1] /; +FreeQ[{a,b,c,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(d_.*x_)^m_.*Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*Pq/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && NiceSqrtQ[b^2-4*a*c] + + +Int[Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_),x_Symbol] := + Int[ExpandIntegrand[Pq/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + (NiceSqrtQ[b^2-4*a*c] || Expon[Pq,x]=2*n && m+q+2*n*p+1!=0 && (IntegerQ[2*p] || n==1 && IntegerQ[4*p] || IntegerQ[p+(q+1)/(2*n)])] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + With[{Pqq=Coeff[Pq,x,q]}, + Pqq*x^(q-2*n+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(c*(q+2*n*p+1)) + + Int[ExpandToSum[Pq-Pqq*x^q-Pqq*(a*(q-2*n+1)*x^(q-2*n)+b*(q+n*(p-1)+1)*x^(q-n))/(c*(q+2*n*p+1)),x]*(a+b*x^n+c*x^(2*n))^p,x]] /; + q>=2*n && q+2*n*p+1!=0 && (IntegerQ[2*p] || n==1 && IntegerQ[4*p] || IntegerQ[p+(q+1)/(2*n)])] /; +FreeQ[{a,b,c,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],j,k}, + Int[Sum[1/d^j*(d*x)^(m+j)*Sum[Coeff[Pq,x,j+k*n]*x^(k*n),{k,0,(q-j)/n+1}]*(a+b*x^n+c*x^(2*n))^p,{j,0,n-1}],x]] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[PolyQ[Pq,x^n]] + + +Int[Pq_*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + Module[{q=Expon[Pq,x],j,k}, + Int[Sum[x^j*Sum[Coeff[Pq,x,j+k*n]*x^(k*n),{k,0,(q-j)/n+1}]*(a+b*x^n+c*x^(2*n))^p,{j,0,n-1}],x]] /; +FreeQ[{a,b,c,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && Not[PolyQ[Pq,x^n]] + + +Int[(d_.*x_)^m_.*Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_.),x_Symbol] := + Int[RationalFunctionExpand[(d*x)^m*Pq/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_.),x_Symbol] := + Int[RationalFunctionExpand[Pq/(a+b*x^n+c*x^(2*n)),x],x] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + -Subst[Int[ExpandToSum[x^q*ReplaceAll[Pq,x->x^(-1)],x]*(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+q+2),x],x,1/x]] /; +FreeQ[{a,b,c,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && IntegerQ[m] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[m],q=Expon[Pq,x]}, + -g/d*Subst[Int[ExpandToSum[x^(g*q)*ReplaceAll[Pq,x->d^(-1)*x^(-g)],x]* + (a+b*d^(-n)*x^(-g*n)+c*d^(-2*n)*x^(-2*g*n))^p/x^(g*(m+q+1)+1),x],x,1/(d*x)^(1/g)]] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && FractionQ[m] + + +Int[(d_.*x_)^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{q=Expon[Pq,x]}, + -(d*x)^m*(x^(-1))^m*Subst[Int[ExpandToSum[x^q*ReplaceAll[Pq,x->x^(-1)],x]*(a+b*x^(-n)+c*x^(-2*n))^p/x^(m+q+2),x],x,1/x]] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && NegativeIntegerQ[n] && Not[RationalQ[m]] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g*(m+1)-1)*ReplaceAll[Pq,x->x^g]*(a+b*x^(g*n)+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,c,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + With[{g=Denominator[n]}, + g*Subst[Int[x^(g-1)*ReplaceAll[Pq,x->x^g]*(a+b*x^(g*n)+c*x^(2*g*n))^p,x],x,x^(1/g)]] /; +FreeQ[{a,b,c,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[(d_*x_)^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^(m-1/2)*Sqrt[d*x]/Sqrt[x]*Int[x^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && FractionQ[n] && PositiveIntegerQ[m+1/2] + + +Int[(d_*x_)^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + d^(m+1/2)*Sqrt[x]/Sqrt[d*x]*Int[x^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && FractionQ[n] && NegativeIntegerQ[m-1/2] + + +Int[(d_*x_)^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^m/x^m*Int[x^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] && FractionQ[n] + + +Int[x_^m_.*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + 1/(m+1)*Subst[Int[ReplaceAll[SubstFor[x^n,Pq,x],x->x^Simplify[n/(m+1)]]*(a+b*x^Simplify[n/(m+1)]+c*x^Simplify[2*n/(m+1)])^p,x],x,x^(m+1)] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && + Not[IntegerQ[n]] + + +Int[(d_*x_)^m_*Pq_*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol] := + (d*x)^m/x^m*Int[x^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,p},x] && EqQ[n2-2*n] && PolyQ[Pq,x^n] && NeQ[b^2-4*a*c] && IntegerQ[Simplify[n/(m+1)]] && + Not[IntegerQ[n]] + + +Int[(d_.*x_)^m_.*Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[(d*x)^m*Pq/(b-q+2*c*x^n),x] - + 2*c/q*Int[(d*x)^m*Pq/(b+q+2*c*x^n),x]] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] + + +Int[Pq_/(a_+b_.*x_^n_.+c_.*x_^n2_.),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[Pq/(b-q+2*c*x^n),x] - + 2*c/q*Int[Pq/(b+q+2*c*x^n),x]] /; +FreeQ[{a,b,c,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NeQ[b^2-4*a*c] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + Int[ExpandIntegrand[(d*x)^m*Pq*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NegativeIntegerQ[p+1] + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + Int[ExpandIntegrand[Pq*(a+b*x^n+c*x^(2*n))^p,x],x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2-2*n] && PolyQ[Pq,x] && NegativeIntegerQ[p+1] + + +Int[(d_.*x_)^m_.*Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Defer[Int][(d*x)^m*Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2-2*n] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) + + +Int[Pq_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + Defer[Int][Pq*(a+b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && (PolyQ[Pq,x] || PolyQ[Pq,x^n]) + + +Int[u_^m_.*Pq_*(a_+b_.*v_^n_+c_.*v_^n2_.)^p_.,x_Symbol] := + u^m/(Coefficient[v,x,1]*v^m)*Subst[Int[x^m*SubstFor[v,Pq,x]*(a+b*x^n+c*x^(2*n))^p,x],x,v] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2-2*n] && LinearPairQ[u,v,x] && PolyQ[Pq,v^n] + + +Int[Pq_*(a_+b_.*v_^n_+c_.*v_^n2_.)^p_.,x_Symbol] := + 1/Coefficient[v,x,1]*Subst[Int[SubstFor[v,Pq,x]*(a+b*x^n+c*x^(2*n))^p,x],x,v] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && LinearQ[v,x] && PolyQ[Pq,v^n] + + + +`) + +func resourcesRubi123GeneralTrinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi123GeneralTrinomialProductsM, nil +} + +func resourcesRubi123GeneralTrinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi123GeneralTrinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.2.3 General trinomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi124ImproperTrinomialProductsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.2.4 Improper Trinomial Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p*) + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + Int[((a+b+c)*x^n)^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n-q] && EqQ[r-n] + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + Int[x^(p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && IntegerQ[p] + + +Int[Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))])* + Int[x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))],x] /; +FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] + + +Int[1/Sqrt[a_.*x_^2+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + -2/(n-2)*Subst[Int[1/(4*a-x^2),x],x,x*(2*a+b*x^(n-2))/Sqrt[a*x^2+b*x^n+c*x^r]] /; +FreeQ[{a,b,c,n,r},x] && EqQ[r,2*n-2] && PosQ[n-2] && NeQ[b^2-4*a*c,0] + + +Int[1/Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[1/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]),x] /; +FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x*(a*x^q+b*x^n+c*x^(2*n-q))^p/(p*(2*n-q)+1) + + (n-q)*p/(p*(2*n-q)+1)* + Int[x^q*(2*a+b*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && RationalQ[p] && p>0 && + NeQ[p*(2*n-q)+1] + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + -x^(-q+1)*(b^2-2*a*c+b*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(b^2-4*a*c)) + + 1/(a*(n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(-q)*((p*q+1)*(b^2-2*a*c)+(n-q)*(p+1)*(b^2-4*a*c)+b*c*(p*q+(n-q)*(2*p+3)+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + (a*x^q+b*x^n+c*x^(2*n-q))^p/(x^(p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p)* + Int[x^(p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,n,p,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] + + +Int[(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + Defer[Int][(a*x^q+b*x^n+c*x^(2*n-q))^p,x] /; +FreeQ[{a,b,c,n,p,q},x] && EqQ[r-(2*n-q)] + + +Int[(a_.*u_^q_.+b_.*u_^n_.+c_.*u_^r_.)^p_,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a*x^q+b*x^n+c*x^(2*n-q))^p,x],x,u] /; +FreeQ[{a,b,c,n,p,q},x] && EqQ[r-(2*n-q)] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p*) + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_.,x_Symbol] := + Int[x^m*((a+b+c)*x^n)^p,x] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[q-n] && EqQ[r-n] + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_.,x_Symbol] := + Int[x^(m+p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,m,n,q},x] && EqQ[r-(2*n-q)] && IntegerQ[p] && PosQ[n-q] + + +Int[x_^m_./Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + -2/(n-q)*Subst[Int[1/(4*a-x^2),x],x,x^(m+1)*(2*a+b*x^(n-q))/Sqrt[a*x^q+b*x^n+c*x^r]] /; +FreeQ[{a,b,c,m,n,q,r},x] && EqQ[r,2*n-q] && PosQ[n-q] && NeQ[b^2-4*a*c,0] && EqQ[m,q/2-1] + + +Int[x_^m_./Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[x^(m-q/2)/Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))],x] /; +FreeQ[{a,b,c,m,n,q},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && (EqQ[m-1] && EqQ[n-3] && EqQ[q-2] || + (EqQ[m+1/2] || EqQ[m-3/2] || EqQ[m-1/2] || EqQ[m-5/2]) && EqQ[n-3] && EqQ[q-1]) + + +Int[x_^m_./(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^(3/2),x_Symbol] := + -2*x^((n-1)/2)*(b+2*c*x)/((b^2-4*a*c)*Sqrt[a*x^(n-1)+b*x^n+c*x^(n+1)]) /; +FreeQ[{a,b,c,n},x] && EqQ[m-3*(n-1)/2] && EqQ[q-n+1] && EqQ[r-n-1] && NeQ[b^2-4*a*c] + + +Int[x_^m_./(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^(3/2),x_Symbol] := + x^((n-1)/2)*(4*a+2*b*x)/((b^2-4*a*c)*Sqrt[a*x^(n-1)+b*x^n+c*x^(n+1)]) /; +FreeQ[{a,b,c,n},x] && EqQ[m-(3*n-1)/2] && EqQ[q-n+1] && EqQ[r-n-1] && NeQ[b^2-4*a*c] + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m-n)*(a*x^(n-1)+b*x^n+c*x^(n+1))^(p+1)/(2*c*(p+1)) - + b/(2*c)*Int[x^(m-1)*(a*x^(n-1)+b*x^n+c*x^(n+1))^p,x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && m+p*(n-1)-1==0 + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m-n+q+1)*(b+2*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p/(2*c*(n-q)*(2*p+1)) - + p*(b^2-4*a*c)/(2*c*(2*p+1))*Int[x^(m+q)*(a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q+1==n-q + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m-n+q+1)*(b*(n-q)*p+c*(m+p*q+(n-q)*(2*p-1)+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p/(c*(m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p-1)+1)) + + (n-q)*p/(c*(m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p-1)+1))* + Int[x^(m-(n-2*q))* + Simp[-a*b*(m+p*q-n+q+1)+(2*a*c*(m+p*q+(n-q)*(2*p-1)+1)-b^2*(m+p*q+(n-q)*(p-1)+1))*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q+1>n-q && m+p*(2*n-q)+1!=0 && m+p*q+(n-q)*(2*p-1)+1!=0 + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m+1)*(a*x^q+b*x^n+c*x^(2*n-q))^p/(m+p*q+1) - + (n-q)*p/(m+p*q+1)*Int[x^(m+n)*(b+2*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q+1<=-(n-q)+1 && NeQ[m+p*q+1] + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m+1)*(a*x^q+b*x^n+c*x^(2*n-q))^p/(m+p*(2*n-q)+1) + + (n-q)*p/(m+p*(2*n-q)+1)*Int[x^(m+q)*(2*a+b*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q+1>-(n-q) && m+p*(2*n-q)+1!=0 + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + -x^(m-q+1)*(b^2-2*a*c+b*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(b^2-4*a*c)) + + (2*a*c-b^2*(p+2))/(a*(p+1)*(b^2-4*a*c))* + Int[x^(m-q)*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p<-1 && m+p*q+1==-(n-q)*(2*p+3) + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + -x^(m-2*n+q+1)*(2*a+b*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/((n-q)*(p+1)*(b^2-4*a*c)) + + 1/((n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(m-2*n+q)*(2*a*(m+p*q-2*(n-q)+1)+b*(m+p*q+(n-q)*(2*p+1)+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p<-1 && m+p*q+1>2*(n-q) + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + -x^(m-q+1)*(b^2-2*a*c+b*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(b^2-4*a*c)) + + 1/(a*(n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(m-q)* + (b^2*(m+p*q+(n-q)*(p+1)+1)-2*a*c*(m+p*q+2*(n-q)*(p+1)+1)+b*c*(m+p*q+(n-q)*(2*p+3)+1)*x^(n-q))* + (a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p<-1 && m+p*q+12*(n-q) + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(m-q+1)*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(m+p*q+1)) - + 1/(a*(m+p*q+1))* + Int[x^(m+n-q)*(b*(m+p*q+(n-q)*(p+1)+1)+c*(m+p*q+2*(n-q)*(p+1)+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p,x] /; +FreeQ[{a,b,c},x] && EqQ[r-(2*n-q)] && PosQ[n-q] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && -1<=p<0 && m+p*q+1<0 + + +Int[x_^m_.*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + (a*x^q+b*x^n+c*x^(2*n-q))^p/(x^(p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p)* + Int[x^(m+p*q)*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,m,n,p,q},x] && EqQ[r-(2*n-q)] && Not[IntegerQ[p]] && PosQ[n-q] + + +Int[u_^m_.*(a_.*u_^q_.+b_.*u_^n_.+c_.*u_^r_.)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[x^m*(a*x^q+b*x^n+c*x^(2*n-q))^p,x],x,u] /; +FreeQ[{a,b,c,m,n,p,q},x] && EqQ[r-(2*n-q)] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p*) + + +Int[(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + Int[x^(p*q)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,n,q},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && IntegerQ[p] && PosQ[n-q] + + +(* Int[(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))])* + Int[x^(q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PosQ[n-q] && PositiveIntegerQ[p+1/2] *) + + +(* Int[(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[x^(q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PosQ[n-q] && NegativeIntegerQ[p-1/2] *) + + +(* Int[(A_+B_.*x_^j_.)*Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))])* + Int[x^(q/2)*(A+B*x^(n-q))*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))],x] /; +FreeQ[{a,b,c,A,B,n,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PosQ[n-q] *) + + +Int[(A_+B_.*x_^j_.)/Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[(A+B*x^(n-q))/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]),x] /; +FreeQ[{a,b,c,A,B,n,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PosQ[n-q] && + EqQ[n-3] && EqQ[q-2] + + +Int[(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_,x_Symbol] := + x*(b*B*(n-q)*p+A*c*(p*q+(n-q)*(2*p+1)+1)+B*c*(p*(2*n-q)+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p/ + (c*(p*(2*n-q)+1)*(p*q+(n-q)*(2*p+1)+1)) + + (n-q)*p/(c*(p*(2*n-q)+1)*(p*q+(n-q)*(2*p+1)+1))* + Int[x^q* + (2*a*A*c*(p*q+(n-q)*(2*p+1)+1)-a*b*B*(p*q+1)+(2*a*B*c*(p*(2*n-q)+1)+A*b*c*(p*q+(n-q)*(2*p+1)+1)-b^2*B*(p*q+(n-q)*p+1))*x^(n-q))* + (a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c,A,B,n,q},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && RationalQ[p] && p>0 && + NeQ[p*(2*n-q)+1] && NeQ[p*q+(n-q)*(2*p+1)+1] + + +Int[(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_,x_Symbol] := + With[{n=q+r}, + x*(A*(p*q+(n-q)*(2*p+1)+1)+B*(p*(2*n-q)+1)*x^(n-q))*(a*x^q+c*x^(2*n-q))^p/((p*(2*n-q)+1)*(p*q+(n-q)*(2*p+1)+1)) + + (n-q)*p/((p*(2*n-q)+1)*(p*q+(n-q)*(2*p+1)+1))* + Int[x^q*(2*a*A*(p*q+(n-q)*(2*p+1)+1)+(2*a*B*(p*(2*n-q)+1))*x^(n-q))*(a*x^q+c*x^(2*n-q))^(p-1),x] /; + EqQ[j-(2*n-q)] && NeQ[p*(2*n-q)+1] && NeQ[p*q+(n-q)*(2*p+1)+1]] /; +FreeQ[{a,c,A,B,q},x] && Not[IntegerQ[p]] && RationalQ[p] && p>0 + + +Int[(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_,x_Symbol] := + -x^(-q+1)*(A*b^2-a*b*B-2*a*A*c+(A*b-2*a*B)*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(b^2-4*a*c)) + + 1/(a*(n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(-q)* + ((A*b^2*(p*q+(n-q)*(p+1)+1)-a*b*B*(p*q+1)-2*a*A*c*(p*q+2*(n-q)*(p+1)+1)+(p*q+(n-q)*(2*p+3)+1)*(A*b-2*a*B)*c*x^(n-q))* + (a*x^q+b*x^n+c*x^(2*n-q))^(p+1)),x] /; +FreeQ[{a,b,c,A,B,n,q},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && RationalQ[p] && p<-1 + + +Int[(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_,x_Symbol] := + With[{n=q+r}, + -x^(-q+1)*(a*A*c+a*B*c*x^(n-q))*(a*x^q+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(2*a*c)) + + 1/(a*(n-q)*(p+1)*(2*a*c))* + Int[x^(-q)*((a*A*c*(p*q+2*(n-q)*(p+1)+1)+a*B*c*(p*q+(n-q)*(2*p+3)+1)*x^(n-q))*(a*x^q+c*x^(2*n-q))^(p+1)),x] /; + EqQ[j-(2*n-q)]] /; +FreeQ[{a,c,A,B,q},x] && Not[IntegerQ[p]] && RationalQ[p] && p<-1 + + +(* Int[(A_+B_.*x_^q_)*(a_.*x_^j_.+b_.*x_^k_.+c_.*x_^n_.)^p_,x_Symbol] := + (a*x^j+b*x^k+c*x^n)^p/(x^(j*p)*(a+b*x^(k-j)+c*x^(2*(k-j)))^p)* + Int[x^(j*p)*(A+B*x^(k-j))*(a+b*x^(k-j)+c*x^(2*(k-j)))^p,x] /; +FreeQ[{a,b,c,A,B,j,k,p},x] && EqQ[q-(k-j)] && EqQ[n-(2*k-j)] && PosQ[k-j] && Not[IntegerQ[p]] *) + + +Int[(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_.,x_Symbol] := + Defer[Int][(A+B*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p,x] /; +FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] + + +Int[(A_+B_.*u_^j_.)*(a_.*u_^q_.+b_.*u_^n_.+c_.*u_^r_.)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(A+B*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p,x],x,u] /; +FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && LinearQ[u,x] && NeQ[u-x] + + + + + +(* ::Subsection::Closed:: *) +(*1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p*) + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + Int[x^(m+p*q)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,m,n,q},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && IntegerQ[p] && PosQ[n-q] + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + x^(m+1)*(A*(m+p*q+(n-q)*(2*p+1)+1)+B*(m+p*q+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p/((m+p*q+1)*(m+p*q+(n-q)*(2*p+1)+1)) + + (n-q)*p/((m+p*q+1)*(m+p*q+(n-q)*(2*p+1)+1))* + Int[x^(n+m)* + Simp[2*a*B*(m+p*q+1)-A*b*(m+p*q+(n-q)*(2*p+1)+1)+(b*B*(m+p*q+1)-2*A*c*(m+p*q+(n-q)*(2*p+1)+1))*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c,A,B},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q<=-(n-q) && m+p*q+1!=0 && m+p*q+(n-q)*(2*p+1)+1!=0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_.,x_Symbol] := + With[{n=q+r}, + x^(m+1)*(A*(m+p*q+(n-q)*(2*p+1)+1)+B*(m+p*q+1)*x^(n-q))*(a*x^q+c*x^(2*n-q))^p/((m+p*q+1)*(m+p*q+(n-q)*(2*p+1)+1)) + + 2*(n-q)*p/((m+p*q+1)*(m+p*q+(n-q)*(2*p+1)+1))* + Int[x^(n+m)*Simp[a*B*(m+p*q+1)-A*c*(m+p*q+(n-q)*(2*p+1)+1)*x^(n-q),x]*(a*x^q+c*x^(2*n-q))^(p-1),x] /; + EqQ[j-(2*n-q)] && PositiveIntegerQ[n] && m+p*q<=-(n-q) && m+p*q+1!=0 && m+p*q+(n-q)*(2*p+1)+1!=0] /; +FreeQ[{a,c,A,B},x] && Not[IntegerQ[p]] && RationalQ[m,p,q] && p>0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + x^(m-n+1)*(A*b-2*a*B-(b*B-2*A*c)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/((n-q)*(p+1)*(b^2-4*a*c)) + + 1/((n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(m-n)* + Simp[(m+p*q-n+q+1)*(2*a*B-A*b)+(m+p*q+2*(n-q)*(p+1)+1)*(b*B-2*A*c)*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c,A,B},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p<-1 && m+p*q>n-q-1 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_.,x_Symbol] := + With[{n=q+r}, + x^(m-n+1)*(a*B-A*c*x^(n-q))*(a*x^q+c*x^(2*n-q))^(p+1)/(2*a*c*(n-q)*(p+1)) - + 1/(2*a*c*(n-q)*(p+1))* + Int[x^(m-n)*Simp[a*B*(m+p*q-n+q+1)-A*c*(m+p*q+(n-q)*2*(p+1)+1)*x^(n-q),x]*(a*x^q+c*x^(2*n-q))^(p+1),x] /; + EqQ[j-(2*n-q)] && PositiveIntegerQ[n] && m+p*q>n-q-1] /; +FreeQ[{a,c,A,B},x] && Not[IntegerQ[p]] && RationalQ[m,p,q] && p<-1 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + x^(m+1)*(b*B*(n-q)*p+A*c*(m+p*q+(n-q)*(2*p+1)+1)+B*c*(m+p*q+2*(n-q)*p+1)*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p/ + (c*(m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p+1)+1)) + + (n-q)*p/(c*(m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p+1)+1))* + Int[x^(m+q)* + Simp[2*a*A*c*(m+p*q+(n-q)*(2*p+1)+1)-a*b*B*(m+p*q+1)+ + (2*a*B*c*(m+p*q+2*(n-q)*p+1)+A*b*c*(m+p*q+(n-q)*(2*p+1)+1)-b^2*B*(m+p*q+(n-q)*p+1))*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^(p-1),x] /; +FreeQ[{a,b,c,A,B},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p>0 && m+p*q>-(n-q)-1 && m+p*(2*n-q)+1!=0 && m+p*q+(n-q)*(2*p+1)+1!=0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_.,x_Symbol] := + With[{n=q+r}, + x^(m+1)*(A*(m+p*q+(n-q)*(2*p+1)+1)+B*(m+p*q+2*(n-q)*p+1)*x^(n-q))*(a*x^q+c*x^(2*n-q))^p/ + ((m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p+1)+1)) + + (n-q)*p/((m+p*(2*n-q)+1)*(m+p*q+(n-q)*(2*p+1)+1))* + Int[x^(m+q)*Simp[2*a*A*(m+p*q+(n-q)*(2*p+1)+1)+2*a*B*(m+p*q+2*(n-q)*p+1)*x^(n-q),x]*(a*x^q+c*x^(2*n-q))^(p-1),x] /; + EqQ[j-(2*n-q)] && PositiveIntegerQ[n] && m+p*q>-(n-q) && m+p*q+2*(n-q)*p+1!=0 && m+p*q+(n-q)*(2*p+1)+1!=0 && m+1!=n] /; +FreeQ[{a,c,A,B},x] && Not[IntegerQ[p]] && RationalQ[m,p,q] && p>0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + -x^(m-q+1)*(A*b^2-a*b*B-2*a*A*c+(A*b-2*a*B)*c*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(n-q)*(p+1)*(b^2-4*a*c)) + + 1/(a*(n-q)*(p+1)*(b^2-4*a*c))* + Int[x^(m-q)* + Simp[A*b^2*(m+p*q+(n-q)*(p+1)+1)-a*b*B*(m+p*q+1)-2*a*A*c*(m+p*q+2*(n-q)*(p+1)+1)+ + (m+p*q+(n-q)*(2*p+3)+1)*(A*b-2*a*B)*c*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^(p+1),x] /; +FreeQ[{a,b,c,A,B},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && p<-1 && m+p*q=n-q-1 && m+p*q+(n-q)*(2*p+1)+1!=0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_.,x_Symbol] := + With[{n=q+r}, + B*x^(m-n+1)*(a*x^q+c*x^(2*n-q))^(p+1)/(c*(m+p*q+(n-q)*(2*p+1)+1)) - + 1/(c*(m+p*q+(n-q)*(2*p+1)+1))* + Int[x^(m-n+q)*Simp[a*B*(m+p*q-n+q+1)-A*c*(m+p*q+(n-q)*(2*p+1)+1)*x^(n-q),x]*(a*x^q+c*x^(2*n-q))^p,x] /; + EqQ[j-(2*n-q)] && PositiveIntegerQ[n] && m+p*q>=n-q-1 && m+p*q+(n-q)*(2*p+1)+1!=0] /; +FreeQ[{a,c,A,B},x] && Not[IntegerQ[p]] && RationalQ[m,p,q] && -1<=p<0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + A*x^(m-q+1)*(a*x^q+b*x^n+c*x^(2*n-q))^(p+1)/(a*(m+p*q+1)) + + 1/(a*(m+p*q+1))* + Int[x^(m+n-q)* + Simp[a*B*(m+p*q+1)-A*b*(m+p*q+(n-q)*(p+1)+1)-A*c*(m+p*q+2*(n-q)*(p+1)+1)*x^(n-q),x]* + (a*x^q+b*x^n+c*x^(2*n-q))^p,x] /; +FreeQ[{a,b,c,A,B},x] && EqQ[r-(n-q)] && EqQ[j-(2*n-q)] && Not[IntegerQ[p]] && NeQ[b^2-4*a*c] && PositiveIntegerQ[n] && + RationalQ[m,p,q] && (-1<=p<0 || m+p*q+(n-q)*(2*p+1)+1==0) && m+p*q<=-(n-q) && m+p*q+1!=0 + + +Int[x_^m_.*(A_+B_.*x_^r_.)*(a_.*x_^q_.+c_.*x_^j_.)^p_.,x_Symbol] := + With[{n=q+r}, + A*x^(m-q+1)*(a*x^q+c*x^(2*n-q))^(p+1)/(a*(m+p*q+1)) + + 1/(a*(m+p*q+1))* + Int[x^(m+n-q)*Simp[a*B*(m+p*q+1)-A*c*(m+p*q+2*(n-q)*(p+1)+1)*x^(n-q),x]*(a*x^q+c*x^(2*n-q))^p,x] /; + EqQ[j-(2*n-q)] && PositiveIntegerQ[n] && (-1<=p<0 || m+p*q+(n-q)*(2*p+1)+1==0) && m+p*q<=-(n-q) && m+p*q+1!=0] /; +FreeQ[{a,c,A,B},x] && Not[IntegerQ[p]] && RationalQ[m,p,q] + + +Int[x_^m_.*(A_+B_.*x_^j_.)/Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[x^(m-q/2)*(A+B*x^(n-q))/Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))],x] /; +FreeQ[{a,b,c,A,B,m,n,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PosQ[n-q] && + (EqQ[m-1/2] || EqQ[m+1/2]) && EqQ[n-3] && EqQ[q-1] + + +(* Int[x_^m_.*(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))])* + Int[x^(m+q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,m,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && PositiveIntegerQ[p+1/2] && PosQ[n-q] *) + + +(* Int[x_^m_.*(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] := + x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]* + Int[x^(m+q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x] /; +FreeQ[{a,b,c,A,B,m,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && NegativeIntegerQ[p-1/2] && PosQ[n-q] *) + + +Int[x_^m_.*(A_+B_.*x_^q_)*(a_.*x_^j_.+b_.*x_^k_.+c_.*x_^n_.)^p_,x_Symbol] := + (a*x^j+b*x^k+c*x^n)^p/(x^(j*p)*(a+b*x^(k-j)+c*x^(2*(k-j)))^p)* + Int[x^(m+j*p)*(A+B*x^(k-j))*(a+b*x^(k-j)+c*x^(2*(k-j)))^p,x] /; +FreeQ[{a,b,c,A,B,j,k,m,p},x] && EqQ[q-(k-j)] && EqQ[n-(2*k-j)] && Not[IntegerQ[p]] && PosQ[k-j] + + +Int[u_^m_.*(A_+B_.*u_^j_.)*(a_.*u_^q_.+b_.*u_^n_.+c_.*u_^r_.)^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[x^m*(A+B*x^(n-q))*(a*x^q+b*x^n+c*x^(2*n-q))^p,x],x,u] /; +FreeQ[{a,b,c,A,B,m,n,p,q},x] && EqQ[j-(n-q)] && EqQ[r-(2*n-q)] && LinearQ[u,x] && NeQ[u-x] + + + +`) + +func resourcesRubi124ImproperTrinomialProductsMBytes() ([]byte, error) { + return _resourcesRubi124ImproperTrinomialProductsM, nil +} + +func resourcesRubi124ImproperTrinomialProductsM() (*asset, error) { + bytes, err := resourcesRubi124ImproperTrinomialProductsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/1.2.4 Improper trinomial products.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi13MiscellaneousAlgebraicFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*1.3 Miscellaneous Algebraic Function Rules*) + + +(* ::Subsection::Closed:: *) +(*1.3.1 u (a+b x+c x^2+d x^3)^p*) + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*a^(2*p))*Int[(3*a-b*x)^p*(3*a+2*b*x)^(2*p),x] /; +FreeQ[{a,b,d},x] && IntegerQ[p] && EqQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandToSum[(a+b*x+d*x^3)^p,x],x] /; +FreeQ[{a,b,d},x] && PositiveIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+b*x+d*x^3]}, + FreeFactors[u,x]^p*Int[DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,b,d},x] && NegativeIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*b^3*d+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p,x]] /; +FreeQ[{a,b,d},x] && NegativeIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + (a+b*x+d*x^3)^p/((3*a-b*x)^p*(3*a+2*b*x)^(2*p))*Int[(3*a-b*x)^p*(3*a+2*b*x)^(2*p),x] /; +FreeQ[{a,b,d,p},x] && Not[IntegerQ[p]] && EqQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+b*x+d*x^3],x]}, + (a+b*x+d*x^3)^p/DistributeDegree[u,p]*Int[DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,b,d,p},x] && Not[IntegerQ[p]] && NeQ[4*b^3+27*a^2*d] + + +Int[(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*b^3*d+27*a^2*d^2],3]}, + (a+b*x+d*x^3)^p/(((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p)* + Int[((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p,x]] /; +FreeQ[{a,b,d,p},x] && Not[IntegerQ[p]] && NeQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*a^(2*p))*Int[(e+f*x)^m*(3*a-b*x)^p*(3*a+2*b*x)^(2*p),x] /; +FreeQ[{a,b,d,e,f,m},x] && IntegerQ[p] && EqQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandIntegrand[(e+f*x)^m*(a+b*x+d*x^3)^p,x],x] /; +FreeQ[{a,b,d,e,f,m},x] && PositiveIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+b*x+d*x^3]}, + FreeFactors[u,x]^p*Int[(e+f*x)^m*DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,b,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*b^3*d+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[(e+f*x)^m*((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p,x]] /; +FreeQ[{a,b,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + (a+b*x+d*x^3)^p/((3*a-b*x)^p*(3*a+2*b*x)^(2*p))*Int[(e+f*x)^m*(3*a-b*x)^p*(3*a+2*b*x)^(2*p),x] /; +FreeQ[{a,b,d,e,f,m,p},x] && Not[IntegerQ[p]] && EqQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+b*x+d*x^3],x]}, + (a+b*x+d*x^3)^p/DistributeDegree[u,p]*Int[(e+f*x)^m*DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,b,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[4*b^3+27*a^2*d] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*b^3*d+27*a^2*d^2],3]}, + (a+b*x+d*x^3)^p/(((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p)* + Int[(e+f*x)^m*((6*b*d-2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + ((6*(1+I*Sqrt[3])*b*d-2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p* + ((6*(1-I*Sqrt[3])*b*d-2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)-3*d*x)^p,x]] /; +FreeQ[{a,b,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[4*b^3+27*a^2*d] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + -1/(3^(3*p)*d^(2*p))*Int[(c-3*d*x)^p*(2*c+3*d*x)^(2*p),x] /; +FreeQ[{a,c,d},x] && IntegerQ[p] && EqQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandToSum[(a+c*x^2+d*x^3)^p,x],x] /; +FreeQ[{a,c,d},x] && PositiveIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+c*x^2+d*x^3]}, + FreeFactors[u,x]^p*Int[DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,c,d},x] && NegativeIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*a*c^3+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[(c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,c,d},x] && NegativeIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + (a+c*x^2+d*x^3)^p/((c-3*d*x)^p*(2*c+3*d*x)^(2*p))*Int[(c-3*d*x)^p*(2*c+3*d*x)^(2*p),x] /; +FreeQ[{a,c,d,p},x] && Not[IntegerQ[p]] && EqQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+c*x^2+d*x^3],x]}, + (a+c*x^2+d*x^3)^p/DistributeDegree[u,p]*Int[DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,c,d,p},x] && Not[IntegerQ[p]] && NeQ[4*c^3+27*a*d^2] + + +Int[(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*a*c^3+27*a^2*d^2],3]}, + (a+c*x^2+d*x^3)^p/((c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p)* + Int[(c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,c,d,p},x] && Not[IntegerQ[p]] && NeQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + -1/(3^(3*p)*d^(2*p))*Int[(e+f*x)^m*(c-3*d*x)^p*(2*c+3*d*x)^(2*p),x] /; +FreeQ[{a,c,d,e,f,m},x] && IntegerQ[p] && EqQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandIntegrand[(e+f*x)^m*(a+c*x^2+d*x^3)^p,x],x] /; +FreeQ[{a,c,d,e,f,m},x] && PositiveIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+c*x^2+d*x^3]}, + FreeFactors[u,x]^p*Int[(e+f*x)^m*DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,c,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*a*c^3+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[(e+f*x)^m*(c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,c,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + (a+c*x^2+d*x^3)^p/((c-3*d*x)^p*(2*c+3*d*x)^(2*p))*Int[(e+f*x)^m*(c-3*d*x)^p*(2*c+3*d*x)^(2*p),x] /; +FreeQ[{a,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && EqQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+c*x^2+d*x^3],x]}, + (a+c*x^2+d*x^3)^p/DistributeDegree[u,p]*Int[(e+f*x)^m*DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[4*c^3+27*a*d^2] + + +Int[(e_.+f_.*x_)^m_.*(a_.+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3-27*a*d^2+3*Sqrt[3]*d*Sqrt[4*a*c^3+27*a^2*d^2],3]}, + (a+c*x^2+d*x^3)^p/((c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p)* + Int[(e+f*x)^m*(c-(2*c^2+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2+2^(1/3)*(1-I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2+2^(1/3)*(1+I*Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[4*c^3+27*a*d^2] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^p*b^p*c^p)*Int[(b+c*x)^(3*p),x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && EqQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^p*b^p*c^p)*Subst[Int[(3*a*b*c-b^3+c^3*x^3)^p,x],x,x+c/(3*d)] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && EqQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +(* Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[b^3-3*a*b*c,3]}, + 1/(3^p*b^p*c^p)*Int[(b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p,x]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && EqQ[c^2-3*b*d] && NeQ[b^2-3*a*c] *) + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[c^3-3*b*c*d,3]}, + 1/(3^p*b^p*c^p)*Int[(b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p,x]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && NeQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandToSum[(a+b*x+c*x^2+d*x^3)^p,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+b*x+c*x^2+d*x^3]}, + FreeFactors[u,x]^p*Int[DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*d^(2*p))*Subst[Int[(2*c^3-9*b*c*d+27*a*d^2-9*d*(c^2-3*b*d)*x+27*d^3*x^3)^p,x],x,x+c/(3*d)] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +(* Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3+9*b*c*d-27*a*d^2+3*Sqrt[3]*d*Sqrt[-b^2*c^2+4*a*c^3+4*b^3*d-18*a*b*c*d+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[(c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] *) + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + (a+b*x+c*x^2+d*x^3)^p/(b+c*x)^(3*p)*Int[(b+c*x)^(3*p),x] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] && EqQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[b^3-3*a*b*c,3]}, + (a+b*x+c*x^2+d*x^3)^p/((b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p)* + Int[(b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p,x]] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] && EqQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[c^3-3*b*c*d,3]}, + (a+b*x+c*x^2+d*x^3)^p/((b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p)* + Int[(b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p,x]] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+b*x+c*x^2+d*x^3],x]}, + (a+b*x+c*x^2+d*x^3)^p/DistributeDegree[u,p]*Int[DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +(* Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*d^(2*p))*Subst[Int[(2*c^3-9*b*c*d+27*a*d^2-9*d*(c^2-3*b*d)*x+27*d^3*x^3)^p,x],x,x+c/(3*d)] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] *) + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3+9*b*c*d-27*a*d^2+3*Sqrt[3]*d*Sqrt[-b^2*c^2+4*a*c^3+4*b^3*d-18*a*b*c*d+27*a^2*d^2],3]}, + (a+b*x+c*x^2+d*x^3)^p/ + ((c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p)* + Int[(c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,b,c,d,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[u_^p_,x_Symbol] := + Int[ExpandToSum[u,x]^p,x] /; +FreeQ[p,x] && PolyQ[u,x,3] && Not[CubicMatchQ[u,x]] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^p*b^p*c^p)*Int[(e+f*x)^m*(b+c*x)^(3*p),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && IntegerQ[p] && EqQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[b^3-3*a*b*c,3]}, + 1/(3^p*b^p*c^p)*Int[(e+f*x)^m*(b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && IntegerQ[p] && EqQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[c^3-3*b*c*d,3]}, + 1/(3^p*b^p*c^p)*Int[(e+f*x)^m*(b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && IntegerQ[p] && NeQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + Int[ExpandIntegrand[(e+f*x)^m*(a+b*x+c*x^2+d*x^3)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=Factor[a+b*x+c*x^2+d*x^3]}, + FreeFactors[u,x]^p*Int[(e+f*x)^m*DistributeDegree[NonfreeFactors[u,x],p],x] /; + ProductQ[NonfreeFactors[u,x]]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*d^(2*p))*Subst[Int[(2*c^3-9*b*c*d+27*a*d^2-9*d*(c^2-3*b*d)*x+27*d^3*x^3)^p,x],x,x+c/(3*d)] /; +FreeQ[{a,b,c,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +(* Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3+9*b*c*d-27*a*d^2+3*Sqrt[3]*d*Sqrt[-b^2*c^2+4*a*c^3+4*b^3*d-18*a*b*c*d+27*a^2*d^2],3]}, + 1/(3^(3*p)*d^(2*p))* + Int[(e+f*x)^m*(c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NegativeIntegerQ[p] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] *) + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + (a+b*x+c*x^2+d*x^3)^p/(b+c*x)^(3*p)*Int[(e+f*x)^m*(b+c*x)^(3*p),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && EqQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[b^3-3*a*b*c,3]}, + (a+b*x+c*x^2+d*x^3)^p/((b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p)* + Int[(e+f*x)^m*(b-r+c*x)^p*(b+(1-I*Sqrt[3])*r/2+c*x)^p*(b+(1+I*Sqrt[3])*r/2+c*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && EqQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[c^3-3*b*c*d,3]}, + (a+b*x+c*x^2+d*x^3)^p/((b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p)* + Int[(e+f*x)^m*(b+(c-r)*x)^p*(b+(c+(1-I*Sqrt[3])*r/2)*x)^p*(b+(c+(1+I*Sqrt[3])*r/2)*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && EqQ[b^2-3*a*c] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{u=NonfreeFactors[Factor[a+b*x+c*x^2+d*x^3],x]}, + (a+b*x+c*x^2+d*x^3)^p/DistributeDegree[u,p]*Int[(e+f*x)^m*DistributeDegree[u,p],x] /; + ProductQ[u]] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +(* Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + 1/(3^(3*p)*d^(2*p))*Subst[Int[(2*c^3-9*b*c*d+27*a*d^2-9*d*(c^2-3*b*d)*x+27*d^3*x^3)^p,x],x,x+c/(3*d)] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] *) + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3)^p_,x_Symbol] := + With[{r=Rt[-2*c^3+9*b*c*d-27*a*d^2+3*Sqrt[3]*d*Sqrt[-b^2*c^2+4*a*c^3+4*b^3*d-18*a*b*c*d+27*a^2*d^2],3]}, + (a+b*x+c*x^2+d*x^3)^p/ + ((c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p)* + Int[(e+f*x)^m*(c-(2*c^2-6*b*d+2^(1/3)*r^2)/(2^(2/3)*r)+3*d*x)^p* + (c+(2*(1+I*Sqrt[3])*c^2-6*(1+I*Sqrt[3])*b*d-I*2^(1/3)*(I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p* + (c+(2*(1-I*Sqrt[3])*c^2-6*(1-I*Sqrt[3])*b*d+I*2^(1/3)*(-I+Sqrt[3])*r^2)/(2*2^(2/3)*r)+3*d*x)^p,x]] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && Not[IntegerQ[p]] && NeQ[c^2-3*b*d] && NeQ[b^2-3*a*c] + + +Int[u_^m_.*v_^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*ExpandToSum[v,x]^p,x] /; +FreeQ[{m,p},x] && LinearQ[u,x] && PolyQ[v,x,3] && Not[LinearMatchQ[u,x] && CubicMatchQ[v,x]] + + + + + +(* ::Subsection::Closed:: *) +(*1.3.2 u (a+b x+c x^2+d x^3+e x^4)^p*) + + +Int[(f_+g_.*x_^2)/((d_+e_.*x_+d_.*x_^2)*Sqrt[a_+b_.*x_+c_.*x_^2+b_.*x_^3+a_.*x_^4]),x_Symbol] := + a*f/(d*Rt[a^2*(2*a-c),2])*ArcTan[(a*b+(4*a^2+b^2-2*a*c)*x+a*b*x^2)/(2*Rt[a^2*(2*a-c),2]*Sqrt[a+b*x+c*x^2+b*x^3+a*x^4])] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*d-a*e] && EqQ[f+g] && PosQ[a^2*(2*a-c)] + + +Int[(f_+g_.*x_^2)/((d_+e_.*x_+d_.*x_^2)*Sqrt[a_+b_.*x_+c_.*x_^2+b_.*x_^3+a_.*x_^4]),x_Symbol] := + -a*f/(d*Rt[-a^2*(2*a-c),2])*ArcTanh[(a*b+(4*a^2+b^2-2*a*c)*x+a*b*x^2)/(2*Rt[-a^2*(2*a-c),2]*Sqrt[a+b*x+c*x^2+b*x^3+a*x^4])] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*d-a*e] && EqQ[f+g] && NegQ[a^2*(2*a-c)] + + +Int[(a_.+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4)^p_,x_Symbol] := + Subst[Int[SimplifyIntegrand[(a+d^4/(256*e^3)-b*d/(8*e)+(c-3*d^2/(8*e))*x^2+e*x^4)^p,x],x],x,d/(4*e)+x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[d^3-4*c*d*e+8*b*e^2] && p=!=2 && p=!=3 + + +Int[v_^p_,x_Symbol] := + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=Coefficient[v,x,2],d=Coefficient[v,x,3],e=Coefficient[v,x,4]}, + Subst[Int[SimplifyIntegrand[(a+d^4/(256*e^3)-b*d/(8*e)+(c-3*d^2/(8*e))*x^2+e*x^4)^p,x],x],x,d/(4*e)+x] /; + EqQ[d^3-4*c*d*e+8*b*e^2] && NeQ[d]] /; +FreeQ[p,x] && PolynomialQ[v,x] && Exponent[v,x]==4 && p=!=2 && p=!=3 + + +Int[u_*(a_.+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4)^p_,x_Symbol] := + Subst[Int[SimplifyIntegrand[ReplaceAll[u,x->-d/(4*e)+x]*(a+d^4/(256*e^3)-b*d/(8*e)+(c-3*d^2/(8*e))*x^2+e*x^4)^p,x],x],x,d/(4*e)+x] /; +FreeQ[{a,b,c,d,e,p},x] && PolynomialQ[u,x] && EqQ[d^3-4*c*d*e+8*b*e^2] && Not[PositiveIntegerQ[p]] + + +Int[u_*v_^p_,x_Symbol] := + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=Coefficient[v,x,2],d=Coefficient[v,x,3],e=Coefficient[v,x,4]}, + Subst[Int[SimplifyIntegrand[ReplaceAll[u,x->-d/(4*e)+x]*(a+d^4/(256*e^3)-b*d/(8*e)+(c-3*d^2/(8*e))*x^2+e*x^4)^p,x],x],x,d/(4*e)+x] /; + EqQ[d^3-4*c*d*e+8*b*e^2] && NeQ[d]] /; +FreeQ[p,x] && PolynomialQ[u,x] && PolynomialQ[v,x] && Exponent[v,x]==4 && Not[PositiveIntegerQ[p]] + + +Int[(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4)^p_,x_Symbol] := + -16*a^2*Subst[ + Int[1/(b-4*a*x)^2*(a*(-3*b^4+16*a*b^2*c-64*a^2*b*d+256*a^3*e-32*a^2*(3*b^2-8*a*c)*x^2+256*a^4*x^4)/(b-4*a*x)^4)^p,x],x,b/(4*a)+1/x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[b^3-4*a*b*c+8*a^2*d] && IntegerQ[2*p] + + +Int[v_^p_,x_Symbol] := + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=Coefficient[v,x,2],d=Coefficient[v,x,3],e=Coefficient[v,x,4]}, + -16*a^2*Subst[ + Int[1/(b-4*a*x)^2*(a*(-3*b^4+16*a*b^2*c-64*a^2*b*d+256*a^3*e-32*a^2*(3*b^2-8*a*c)*x^2+256*a^4*x^4)/(b-4*a*x)^4)^p,x],x,b/(4*a)+1/x] /; + NeQ[a] && NeQ[b] && EqQ[b^3-4*a*b*c+8*a^2*d]] /; +FreeQ[p,x] && PolynomialQ[v,x] && Exponent[v,x]==4 && IntegerQ[2*p] + + +Int[(A_.+B_.*x_+C_.*x_^2+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Sqrt[8*a^2+b^2-4*a*c]}, + 1/q*Int[(b*A-2*a*B+2*a*D+A*q+(2*a*A-2*a*C+b*D+D*q)*x)/(2*a+(b+q)*x+2*a*x^2),x] - + 1/q*Int[(b*A-2*a*B+2*a*D-A*q+(2*a*A-2*a*C+b*D-D*q)*x)/(2*a+(b-q)*x+2*a*x^2),x]] /; +FreeQ[{a,b,c,A,B,C,D},x] && EqQ[d-b] && EqQ[e-a] && SumQ[Factor[a+b*x+c*x^2+b*x^3+a*x^4]] + + +Int[(A_.+B_.*x_+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Sqrt[8*a^2+b^2-4*a*c]}, + 1/q*Int[(b*A-2*a*B+2*a*D+A*q+(2*a*A+b*D+D*q)*x)/(2*a+(b+q)*x+2*a*x^2),x] - + 1/q*Int[(b*A-2*a*B+2*a*D-A*q+(2*a*A+b*D-D*q)*x)/(2*a+(b-q)*x+2*a*x^2),x]] /; +FreeQ[{a,b,c,A,B,D},x] && EqQ[d-b] && EqQ[e-a] && SumQ[Factor[a+b*x+c*x^2+b*x^3+a*x^4]] + + +Int[x_^m_.*(A_.+B_.*x_+C_.*x_^2+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Sqrt[8*a^2+b^2-4*a*c]}, + 1/q*Int[x^m*(b*A-2*a*B+2*a*D+A*q+(2*a*A-2*a*C+b*D+D*q)*x)/(2*a+(b+q)*x+2*a*x^2),x] - + 1/q*Int[x^m*(b*A-2*a*B+2*a*D-A*q+(2*a*A-2*a*C+b*D-D*q)*x)/(2*a+(b-q)*x+2*a*x^2),x]] /; +FreeQ[{a,b,c,A,B,C,D,m},x] && EqQ[d-b] && EqQ[e-a] && SumQ[Factor[a+b*x+c*x^2+b*x^3+a*x^4]] + + +Int[x_^m_.*(A_.+B_.*x_+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Sqrt[8*a^2+b^2-4*a*c]}, + 1/q*Int[x^m*(b*A-2*a*B+2*a*D+A*q+(2*a*A+b*D+D*q)*x)/(2*a+(b+q)*x+2*a*x^2),x] - + 1/q*Int[x^m*(b*A-2*a*B+2*a*D-A*q+(2*a*A+b*D-D*q)*x)/(2*a+(b-q)*x+2*a*x^2),x]] /; +FreeQ[{a,b,c,A,B,D,m},x] && EqQ[d-b] && EqQ[e-a] && SumQ[Factor[a+b*x+c*x^2+b*x^3+a*x^4]] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Rt[C*(2*e*(B*d-4*A*e)+C*(d^2-4*c*e)),2]}, + -2*C^2/q*ArcTanh[(C*d-B*e+2*C*e*x)/q] + + 2*C^2/q*ArcTanh[C*(4*B*c*C-3*B^2*d-4*A*C*d+12*A*B*e+4*C*(2*c*C-B*d+2*A*e)*x+4*C*(2*C*d-B*e)*x^2+8*C^2*e*x^3)/(q*(B^2-4*A*C))]] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && EqQ[B^2*d+2*C*(b*C+A*d)-2*B*(c*C+2*A*e)] && + EqQ[2*B^2*c*C-8*a*C^3-B^3*d-4*A*B*C*d+4*A*(B^2+2*A*C)*e] && PosQ[C*(2*e*(B*d-4*A*e)+C*(d^2-4*c*e))] + + +Int[(A_.+C_.*x_^2)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Rt[C*(-8*A*e^2+C*(d^2-4*c*e)),2]}, + -2*C^2/q*ArcTanh[C*(d+2*e*x)/q] + 2*C^2/q*ArcTanh[C*(A*d-2*(c*C+A*e)*x-2*C*d*x^2-2*C*e*x^3)/(A*q)]] /; +FreeQ[{a,b,c,d,e,A,C},x] && EqQ[b*C+A*d] && EqQ[a*C^2-A^2*e] && PosQ[C*(-8*A*e^2+C*(d^2-4*c*e))] + + +Int[(A_.+B_.*x_+C_.*x_^2)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Rt[-C*(2*e*(B*d-4*A*e)+C*(d^2-4*c*e)),2]}, + 2*C^2/q*ArcTan[(C*d-B*e+2*C*e*x)/q] - + 2*C^2/q*ArcTan[C*(4*B*c*C-3*B^2*d-4*A*C*d+12*A*B*e+4*C*(2*c*C-B*d+2*A*e)*x+4*C*(2*C*d-B*e)*x^2+8*C^2*e*x^3)/(q*(B^2-4*A*C))]] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && EqQ[B^2*d+2*C*(b*C+A*d)-2*B*(c*C+2*A*e)] && + EqQ[2*B^2*c*C-8*a*C^3-B^3*d-4*A*B*C*d+4*A*(B^2+2*A*C)*e] && NegQ[C*(2*e*(B*d-4*A*e)+C*(d^2-4*c*e))] + + +Int[(A_.+C_.*x_^2)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + With[{q=Rt[-C*(-8*A*e^2+C*(d^2-4*c*e)),2]}, + 2*C^2/q*ArcTan[(C*d+2*C*e*x)/q] - 2*C^2/q*ArcTan[-C*(-A*d+2*(c*C+A*e)*x+2*C*d*x^2+2*C*e*x^3)/(A*q)]] /; +FreeQ[{a,b,c,d,e,A,C},x] && EqQ[b*C+A*d] && EqQ[a*C^2-A^2*e] && NegQ[C*(-8*A*e^2+C*(d^2-4*c*e))] + + +Int[(A_.+B_.*x_+C_.*x_^2+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + D/(4*e)*Log[a+b*x+c*x^2+d*x^3+e*x^4] - + 1/(4*e)*Int[(b*D-4*A*e+2*(c*D-2*B*e)*x+(3*d*D-4*C*e)*x^2)/(a+b*x+c*x^2+d*x^3+e*x^4),x] /; +FreeQ[{a,b,c,d,e,A,B,C,D},x] && + EqQ[4*d*(c*D-2*B*e)^2+8*(3*d*D-4*C*e)*(b*d*D-b*C*e-A*d*e)-4*(c*D-2*B*e)*(3*c*d*D-4*c*C*e+2*b*D*e-8*A*e^2)] && + EqQ[8*d*(c*D-2*B*e)^3+8*d*(b*D-4*A*e)*(c*D-2*B*e)*(3*d*D-4*C*e)+8*a*(3*d*D-4*C*e)^3-8*c*(c*D-2*B*e)^2*(3*d*D-4*C*e)- + 4*e*(b*D-4*A*e)*(4*(c*D-2*B*e)^2+2*(b*D-4*A*e)*(3*d*D-4*C*e))] + + +Int[(A_.+B_.*x_+D_.*x_^3)/(a_+b_.*x_+c_.*x_^2+d_.*x_^3+e_.*x_^4),x_Symbol] := + D/(4*e)*Log[a+b*x+c*x^2+d*x^3+e*x^4] - + 1/(4*e)*Int[(b*D-4*A*e+2*(c*D-2*B*e)*x+(3*d*D)*x^2)/(a+b*x+c*x^2+d*x^3+e*x^4),x] /; +FreeQ[{a,b,c,d,e,A,B,D},x] && + EqQ[c^2*d*D^2+2*(3*d*D-4*C*e)*(b*d*D-b*C*e-A*d*e)-c*D*(3*c*d*D-4*c*C*e+2*b*D*e-8*A*e^2)] && + EqQ[54*a*d^3*D^3+6*d^2*D*(b*D-4*A*e)*(c*D-2*B*e)-6*c*d*D*(c*D-2*B*e)^2+2*d*(c*D-2*B*e)^3- + e*(b*D-4*A*e)*(6*d*D*(b*D-4*A*e)+4*(c*D-2*B*e)^2)] + + + + + +(* ::Subsection::Closed:: *) +(*1.3.3 Miscellaneous algebraic functions*) + + +Int[u_/(e_.*Sqrt[a_.+b_.*x_]+f_.*Sqrt[c_.+d_.*x_]),x_Symbol] := + c/(e*(b*c-a*d))*Int[(u*Sqrt[a+b*x])/x,x] - a/(f*(b*c-a*d))*Int[(u*Sqrt[c+d*x])/x,x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a*e^2-c*f^2] + + +Int[u_/(e_.*Sqrt[a_.+b_.*x_]+f_.*Sqrt[c_.+d_.*x_]),x_Symbol] := + -d/(e*(b*c-a*d))*Int[u*Sqrt[a+b*x],x] + b/(f*(b*c-a*d))*Int[u*Sqrt[c+d*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[b*e^2-d*f^2] + + +Int[u_/(e_.*Sqrt[a_.+b_.*x_]+f_.*Sqrt[c_.+d_.*x_]),x_Symbol] := + e*Int[(u*Sqrt[a+b*x])/(a*e^2-c*f^2+(b*e^2-d*f^2)*x),x] - + f*Int[(u*Sqrt[c+d*x])/(a*e^2-c*f^2+(b*e^2-d*f^2)*x),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a*e^2-c*f^2] && NeQ[b*e^2-d*f^2] + + +Int[u_./(d_.*x_^n_.+c_.*Sqrt[a_.+b_.*x_^p_.]),x_Symbol] := + -b/(a*d)*Int[u*x^n,x] + 1/(a*c)*Int[u*Sqrt[a+b*x^(2*n)],x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[p-2*n] && EqQ[b*c^2-d^2] + + +Int[x_^m_./(d_.*x_^n_.+c_.*Sqrt[a_.+b_.*x_^p_.]),x_Symbol] := + -d*Int[x^(m+n)/(a*c^2+(b*c^2-d^2)*x^(2*n)),x] + + c*Int[(x^m*Sqrt[a+b*x^(2*n)])/(a*c^2+(b*c^2-d^2)*x^(2*n)),x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[p-2*n] && NeQ[b*c^2-d^2] + + +Int[1/((a_+b_.*x_^3)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{r=Numerator[Rt[a/b,3]], s=Denominator[Rt[a/b,3]]}, + r/(3*a)*Int[1/((r+s*x)*Sqrt[d+e*x+f*x^2]),x] + + r/(3*a)*Int[(2*r-s*x)/((r^2-r*s*x+s^2*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,d,e,f},x] && PosQ[a/b] + + +Int[1/((a_+b_.*x_^3)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + With[{r=Numerator[Rt[a/b,3]], s=Denominator[Rt[a/b,3]]}, + r/(3*a)*Int[1/((r+s*x)*Sqrt[d+f*x^2]),x] + + r/(3*a)*Int[(2*r-s*x)/((r^2-r*s*x+s^2*x^2)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,d,f},x] && PosQ[a/b] + + +Int[1/((a_+b_.*x_^3)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol] := + With[{r=Numerator[Rt[-a/b,3]], s=Denominator[Rt[-a/b,3]]}, + r/(3*a)*Int[1/((r-s*x)*Sqrt[d+e*x+f*x^2]),x] + + r/(3*a)*Int[(2*r+s*x)/((r^2+r*s*x+s^2*x^2)*Sqrt[d+e*x+f*x^2]),x]] /; +FreeQ[{a,b,d,e,f},x] && NegQ[a/b] + + +Int[1/((a_+b_.*x_^3)*Sqrt[d_.+f_.*x_^2]),x_Symbol] := + With[{r=Numerator[Rt[-a/b,3]], s=Denominator[Rt[-a/b,3]]}, + r/(3*a)*Int[1/((r-s*x)*Sqrt[d+f*x^2]),x] + + r/(3*a)*Int[(2*r+s*x)/((r^2+r*s*x+s^2*x^2)*Sqrt[d+f*x^2]),x]] /; +FreeQ[{a,b,d,f},x] && NegQ[a/b] + + +Int[1/((d_+e_.*x_)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + d*Int[1/((d^2-e^2*x^2)*Sqrt[a+b*x^2+c*x^4]),x] - e*Int[x/((d^2-e^2*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[1/((d_+e_.*x_)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + d*Int[1/((d^2-e^2*x^2)*Sqrt[a+c*x^4]),x] - e*Int[x/((d^2-e^2*x^2)*Sqrt[a+c*x^4]),x] /; +FreeQ[{a,c,d,e},x] + + +Int[1/((d_+e_.*x_)^2*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + -e^3*Sqrt[a+b*x^2+c*x^4]/((c*d^4+b*d^2*e^2+a*e^4)*(d+e*x)) - + c/(c*d^4+b*d^2*e^2+a*e^4)*Int[(d^2-e^2*x^2)/Sqrt[a+b*x^2+c*x^4],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[c*d^4+b*d^2*e^2+a*e^4] && EqQ[2*c*d^3+b*d*e^2] + + +Int[1/((d_+e_.*x_)^2*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + -e^3*Sqrt[a+b*x^2+c*x^4]/((c*d^4+b*d^2*e^2+a*e^4)*(d+e*x)) - + c/(c*d^4+b*d^2*e^2+a*e^4)*Int[(d^2-e^2*x^2)/Sqrt[a+b*x^2+c*x^4],x] + + (2*c*d^3+b*d*e^2)/(c*d^4+b*d^2*e^2+a*e^4)*Int[1/((d+e*x)*Sqrt[a+b*x^2+c*x^4]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[c*d^4+b*d^2*e^2+a*e^4] && NeQ[2*c*d^3+b*d*e^2] + + +Int[1/((d_+e_.*x_)^2*Sqrt[a_+c_.*x_^4]),x_Symbol] := + -e^3*Sqrt[a+c*x^4]/((c*d^4+a*e^4)*(d+e*x)) - + c/(c*d^4+a*e^4)*Int[(d^2-e^2*x^2)/Sqrt[a+c*x^4],x] + + 2*c*d^3/(c*d^4+a*e^4)*Int[1/((d+e*x)*Sqrt[a+c*x^4]),x] /; +FreeQ[{a,c,d,e},x] && NeQ[c*d^4+a*e^4] + + +Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + A*Subst[Int[1/(d-(b*d-2*a*e)*x^2),x],x,x/Sqrt[a+b*x^2+c*x^4]] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[B*d+A*e] && EqQ[c*d^2-a*e^2] + + +Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + A*Subst[Int[1/(d+2*a*e*x^2),x],x,x/Sqrt[a+c*x^4]] /; +FreeQ[{a,c,d,e,A,B},x] && EqQ[B*d+A*e] && EqQ[c*d^2-a*e^2] + + +Int[(A_+B_.*x_^4)/((d_+e_.*x_^2+f_.*x_^4)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + A*Subst[Int[1/(d-(b*d-a*e)*x^2),x],x,x/Sqrt[a+b*x^2+c*x^4]] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && EqQ[c*d-a*f] && EqQ[a*B+A*c] + + +Int[(A_+B_.*x_^4)/((d_+e_.*x_^2+f_.*x_^4)*Sqrt[a_+c_.*x_^4]),x_Symbol] := + A*Subst[Int[1/(d+a*e*x^2),x],x,x/Sqrt[a+c*x^4]] /; +FreeQ[{a,c,d,e,f,A,B},x] && EqQ[c*d-a*f] && EqQ[a*B+A*c] + + +Int[(A_+B_.*x_^4)/((d_+f_.*x_^4)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] := + A*Subst[Int[1/(d-b*d*x^2),x],x,x/Sqrt[a+b*x^2+c*x^4]] /; +FreeQ[{a,b,c,d,f,A,B},x] && EqQ[c*d-a*f] && EqQ[a*B+A*c] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/(d_+e_.*x_^4),x_Symbol] := + a/d*Subst[Int[1/(1-2*b*x^2+(b^2-4*a*c)*x^4),x],x,x/Sqrt[a+b*x^2+c*x^4]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d+a*e] && PosQ[a*c] + + +Int[Sqrt[a_+b_.*x_^2+c_.*x_^4]/(d_+e_.*x_^4),x_Symbol] := + With[{q=Sqrt[b^2-4*a*c]}, + -a*Sqrt[b+q]/(2*Sqrt[2]*Rt[-a*c,2]*d)*ArcTan[Sqrt[b+q]*x*(b-q+2*c*x^2)/(2*Sqrt[2]*Rt[-a*c,2]*Sqrt[a+b*x^2+c*x^4])] + + a*Sqrt[-b+q]/(2*Sqrt[2]*Rt[-a*c,2]*d)*ArcTanh[Sqrt[-b+q]*x*(b+q+2*c*x^2)/(2*Sqrt[2]*Rt[-a*c,2]*Sqrt[a+b*x^2+c*x^4])]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c*d+a*e] && NegQ[a*c] + + +Int[1/((a_+b_.*x_)*Sqrt[c_+d_.*x_^2]*Sqrt[e_+f_.*x_^2]),x_Symbol] := + a*Int[1/((a^2-b^2*x^2)*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] - b*Int[x/((a^2-b^2*x^2)*Sqrt[c+d*x^2]*Sqrt[e+f*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] + + +Int[(g_.+h_.*x_)*Sqrt[d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2]],x_Symbol] := + 2*(f*(5*b*c*g^2-2*b^2*g*h-3*a*c*g*h+2*a*b*h^2)+c*f*(10*c*g^2-b*g*h+a*h^2)*x+9*c^2*f*g*h*x^2+3*c^2*f*h^2*x^3- + (e*g-d*h)*(5*c*g-2*b*h+c*h*x)*Sqrt[a+b*x+c*x^2])/ + (15*c^2*f*(g+h*x))*Sqrt[d+e*x+f*Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[(e*g-d*h)^2-f^2*(c*g^2-b*g*h+a*h^2)] && EqQ[2*e^2*g-2*d*e*h-f^2*(2*c*g-b*h)] + + +Int[(g_.+h_.*x_)^m_.*(u_+f_.*(j_.+k_.*Sqrt[v_]))^n_.,x_Symbol] := + Int[(g+h*x)^m*(ExpandToSum[u+f*j,x]+f*k*Sqrt[ExpandToSum[v,x]])^n,x] /; +FreeQ[{f,g,h,j,k,m,n},x] && LinearQ[u,x] && QuadraticQ[v,x] && + Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x] && (EqQ[j] || EqQ[f-1])] && + EqQ[(Coefficient[u,x,1]*g-h*(Coefficient[u,x,0]+f*j))^2-f^2*k^2*(Coefficient[v,x,2]*g^2-Coefficient[v,x,1]*g*h+Coefficient[v,x,0]*h^2)] + + +(* Int[1/(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol] := + Int[(d+e*x)/(d^2-a*f^2+(2*d*e-b*f^2)*x),x] - + f*Int[Sqrt[a+b*x+c*x^2]/(d^2-a*f^2+(2*d*e-b*f^2)*x),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e^2-c*f^2] *) + + +(* Int[1/(d_.+e_.*x_+f_.*Sqrt[a_.+c_.*x_^2]),x_Symbol] := + Int[(d+e*x)/(d^2-a*f^2+2*d*e*x),x] - + f*Int[Sqrt[a+c*x^2]/(d^2-a*f^2+2*d*e*x),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[e^2-c*f^2] *) + + +Int[(g_.+h_.*(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2])^n_)^p_.,x_Symbol] := + 2*Subst[Int[(g+h*x^n)^p*(d^2*e-(b*d-a*e)*f^2-(2*d*e-b*f^2)*x+e*x^2)/(-2*d*e+b*f^2+2*e*x)^2,x],x,d+e*x+f*Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h,n},x] && EqQ[e^2-c*f^2] && IntegerQ[p] + + +Int[(g_.+h_.*(d_.+e_.*x_+f_.*Sqrt[a_+c_.*x_^2])^n_)^p_.,x_Symbol] := + 1/(2*e)*Subst[Int[(g+h*x^n)^p*(d^2+a*f^2-2*d*x+x^2)/(d-x)^2,x],x,d+e*x+f*Sqrt[a+c*x^2]] /; +FreeQ[{a,c,d,e,f,g,h,n},x] && EqQ[e^2-c*f^2] && IntegerQ[p] + + +Int[(g_.+h_.*(u_+f_. Sqrt[v_])^n_)^p_.,x_Symbol] := + Int[(g+h*(ExpandToSum[u,x]+f*Sqrt[ExpandToSum[v,x]])^n)^p,x] /; +FreeQ[{f,g,h,n},x] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] && + EqQ[Coefficient[u,x,1]^2-Coefficient[v,x,2]*f^2] && IntegerQ[p] + + +Int[(g_.+h_.*x_)^m_.*(e_.*x_+f_.*Sqrt[a_.+c_.*x_^2])^n_.,x_Symbol] := + 1/(2^(m+1)*e^(m+1))*Subst[Int[x^(n-m-2)*(a*f^2+x^2)*(-a*f^2*h+2*e*g*x+h*x^2)^m,x],x,e*x+f*Sqrt[a+c*x^2]] /; +FreeQ[{a,c,e,f,g,h,n},x] && EqQ[e^2-c*f^2] && IntegerQ[m] + + +Int[x_^p_.*(g_+i_.*x_^2)^m_.*(e_.*x_+f_.*Sqrt[a_+c_.*x_^2])^n_.,x_Symbol] := + 1/(2^(2*m+p+1)*e^(p+1)*f^(2*m))*(i/c)^m*Subst[Int[x^(n-2*m-p-2)*(-a*f^2+x^2)^p*(a*f^2+x^2)^(2*m+1),x],x,e*x+f*Sqrt[a+c*x^2]] /; +FreeQ[{a,c,e,f,g,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && IntegersQ[p,2*m] && (IntegerQ[m] || PositiveQ[i/c]) + + +Int[(g_.+h_.*x_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2])^n_.,x_Symbol] := + 2/f^(2*m)*(i/c)^m* + Subst[Int[x^n*(d^2*e-(b*d-a*e)*f^2-(2*d*e-b*f^2)*x+e*x^2)^(2*m+1)/(-2*d*e+b*f^2+2*e*x)^(2*(m+1)),x],x,d+e*x+f*Sqrt[a+b*x+c*x^2]] /; +FreeQ[{a,b,c,d,e,f,g,h,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && EqQ[c*h-b*i] && IntegerQ[2*m] && (IntegerQ[m] || PositiveQ[i/c]) + + +Int[(g_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_+c_.*x_^2])^n_.,x_Symbol] := + 1/(2^(2*m+1)*e*f^(2*m))*(i/c)^m* + Subst[Int[x^n*(d^2+a*f^2-2*d*x+x^2)^(2*m+1)/(-d+x)^(2*(m+1)),x],x,d+e*x+f*Sqrt[a+c*x^2]] /; +FreeQ[{a,c,d,e,f,g,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && IntegerQ[2*m] && (IntegerQ[m] || PositiveQ[i/c]) + + +Int[(g_.+h_.*x_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2])^n_.,x_Symbol] := + (i/c)^(m-1/2)*Sqrt[g+h*x+i*x^2]/Sqrt[a+b*x+c*x^2]*Int[(a+b*x+c*x^2)^m*(d+e*x+f*Sqrt[a+b*x+c*x^2])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && EqQ[c*h-b*i] && PositiveIntegerQ[m+1/2] && Not[PositiveQ[i/c]] + + +Int[(g_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_+c_.*x_^2])^n_.,x_Symbol] := + (i/c)^(m-1/2)*Sqrt[g+i*x^2]/Sqrt[a+c*x^2]*Int[(a+c*x^2)^m*(d+e*x+f*Sqrt[a+c*x^2])^n,x] /; +FreeQ[{a,c,d,e,f,g,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && PositiveIntegerQ[m+1/2] && Not[PositiveQ[i/c]] + + +Int[(g_.+h_.*x_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2])^n_.,x_Symbol] := + (i/c)^(m+1/2)*Sqrt[a+b*x+c*x^2]/Sqrt[g+h*x+i*x^2]*Int[(a+b*x+c*x^2)^m*(d+e*x+f*Sqrt[a+b*x+c*x^2])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && EqQ[c*h-b*i] && NegativeIntegerQ[m-1/2] && Not[PositiveQ[i/c]] + + +Int[(g_+i_.*x_^2)^m_.*(d_.+e_.*x_+f_.*Sqrt[a_+c_.*x_^2])^n_.,x_Symbol] := + (i/c)^(m+1/2)*Sqrt[a+c*x^2]/Sqrt[g+i*x^2]*Int[(a+c*x^2)^m*(d+e*x+f*Sqrt[a+c*x^2])^n,x] /; +FreeQ[{a,c,d,e,f,g,i,n},x] && EqQ[e^2-c*f^2] && EqQ[c*g-a*i] && NegativeIntegerQ[m-1/2] && Not[PositiveQ[i/c]] + + +Int[w_^m_.*(u_+f_.*(j_.+k_.*Sqrt[v_]))^n_.,x_Symbol] := + Int[ExpandToSum[w,x]^m*(ExpandToSum[u+f*j,x]+f*k*Sqrt[ExpandToSum[v,x]])^n,x] /; +FreeQ[{f,j,k,m,n},x] && LinearQ[u,x] && QuadraticQ[{v,w},x] && + Not[LinearMatchQ[u,x] && QuadraticMatchQ[{v,w},x] && (EqQ[j] || EqQ[f-1])] && + EqQ[Coefficient[u,x,1]^2-Coefficient[v,x,2]*f^2*k^2] + + +Int[1/((a_+b_.*x_^n_.)*Sqrt[c_.*x_^2+d_.*(a_+b_.*x_^n_.)^p_.]),x_Symbol] := + 1/a*Subst[Int[1/(1-c*x^2),x],x,x/Sqrt[c*x^2+d*(a+b*x^n)^(2/n)]] /; +FreeQ[{a,b,c,d,n},x] && EqQ[p-2/n] + + +Int[Sqrt[a_+b_.*Sqrt[c_+d_.*x_^2]],x_Symbol] := + 2*b^2*d*x^3/(3*(a+b*Sqrt[c+d*x^2])^(3/2)) + 2*a*x/Sqrt[a+b*Sqrt[c+d*x^2]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2*c] + + +Int[Sqrt[a_.*x_^2+b_.*x_*Sqrt[c_+d_.*x_^2]]/(x_*Sqrt[c_+d_.*x_^2]),x_Symbol] := + Sqrt[2]*b/a*Subst[Int[1/Sqrt[1+x^2/a],x],x,a*x+b*Sqrt[c+d*x^2]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2*d] && EqQ[b^2*c+a] + + +Int[Sqrt[e_.*x_*(a_.*x_+b_.*Sqrt[c_+d_.*x_^2])]/(x_*Sqrt[c_+d_.*x_^2]),x_Symbol] := + Int[Sqrt[a*e*x^2+b*e*x*Sqrt[c+d*x^2]]/(x*Sqrt[c+d*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2*d] && EqQ[b^2*c*e+a] + + +Int[Sqrt[c_.*x_^2+d_.*Sqrt[a_+b_.*x_^4]]/Sqrt[a_+b_.*x_^4],x_Symbol] := + d*Subst[Int[1/(1-2*c*x^2),x],x,x/Sqrt[c*x^2+d*Sqrt[a+b*x^4]]] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-b*d^2] + + +Int[(c_.+d_.*x_)^m_.*Sqrt[b_.*x_^2+Sqrt[a_+e_.*x_^4]]/Sqrt[a_+e_.*x_^4],x_Symbol] := + (1-I)/2*Int[(c+d*x)^m/Sqrt[Sqrt[a]-I*b*x^2],x] + + (1+I)/2*Int[(c+d*x)^m/Sqrt[Sqrt[a]+I*b*x^2],x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[e-b^2] && PositiveQ[a] + + +Int[1/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + -q/((1+Sqrt[3])*d-c*q)*Int[1/Sqrt[a+b*x^3],x] + + d/((1+Sqrt[3])*d-c*q)*Int[(1+Sqrt[3]+q*x)/((c+d*x)*Sqrt[a+b*x^3]),x]] /; +FreeQ[{a,b,c,d},x] && PosQ[a] && PosQ[b] + + +Int[1/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + q/((1+Sqrt[3])*d+c*q)*Int[1/Sqrt[a+b*x^3],x] + + d/((1+Sqrt[3])*d+c*q)*Int[(1+Sqrt[3]-q*x)/((c+d*x)*Sqrt[a+b*x^3]),x]] /; +FreeQ[{a,b,c,d},x] && PosQ[a] && NegQ[b] + + +Int[1/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + q/((1-Sqrt[3])*d+c*q)*Int[1/Sqrt[a+b*x^3],x] + + d/((1-Sqrt[3])*d+c*q)*Int[(1-Sqrt[3]-q*x)/((c+d*x)*Sqrt[a+b*x^3]),x]] /; +FreeQ[{a,b,c,d},x] && NegQ[a] && PosQ[b] + + +Int[1/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + -q/((1-Sqrt[3])*d-c*q)*Int[1/Sqrt[a+b*x^3],x] + + d/((1-Sqrt[3])*d-c*q)*Int[(1-Sqrt[3]+q*x)/((c+d*x)*Sqrt[a+b*x^3]),x]] /; +FreeQ[{a,b,c,d},x] && NegQ[a] && NegQ[b] + + +Int[(e_+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + 4*3^(1/4)*Sqrt[2-Sqrt[3]]*f*(1+q*x)*Sqrt[(1-q*x+q^2*x^2)/(1+Sqrt[3]+q*x)^2]/ + (q*Sqrt[a+b*x^3]*Sqrt[(1+q*x)/(1+Sqrt[3]+q*x)^2])* + Subst[Int[1/(((1-Sqrt[3])*d-c*q+((1+Sqrt[3])*d-c*q)*x)*Sqrt[1-x^2]*Sqrt[7-4*Sqrt[3]+x^2]),x],x,(-1+Sqrt[3]-q*x)/(1+Sqrt[3]+q*x)] /; + EqQ[(1+Sqrt[3])*f-e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[a] && PosQ[b] + + +Int[(e_.+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + ((1+Sqrt[3])*f-e*q)/((1+Sqrt[3])*d-c*q)*Int[1/Sqrt[a+b*x^3],x] + + (d*e-c*f)/((1+Sqrt[3])*d-c*q)*Int[(1+Sqrt[3]+q*x)/((c+d*x)*Sqrt[a+b*x^3]),x] /; + NeQ[(1+Sqrt[3])*f-e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[a] && PosQ[b] + + +Int[(e_+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + -4*3^(1/4)*Sqrt[2-Sqrt[3]]*f*(1-q*x)*Sqrt[(1+q*x+q^2*x^2)/(1+Sqrt[3]-q*x)^2]/ + (q*Sqrt[a+b*x^3]*Sqrt[(1-q*x)/(1+Sqrt[3]-q*x)^2])* + Subst[Int[1/(((1-Sqrt[3])*d+c*q+((1+Sqrt[3])*d+c*q)*x)*Sqrt[1-x^2]*Sqrt[7-4*Sqrt[3]+x^2]),x],x,(-1+Sqrt[3]+q*x)/(1+Sqrt[3]-q*x)] /; + EqQ[(1+Sqrt[3])*f+e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[a] && NegQ[b] + + +Int[(e_.+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + ((1+Sqrt[3])*f+e*q)/((1+Sqrt[3])*d+c*q)*Int[1/Sqrt[a+b*x^3],x] + + (d*e-c*f)/((1+Sqrt[3])*d+c*q)*Int[(1+Sqrt[3]-q*x)/((c+d*x)*Sqrt[a+b*x^3]),x] /; + NeQ[(1+Sqrt[3])*f+e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && PosQ[a] && NegQ[b] + + +Int[(e_+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + 4*3^(1/4)*Sqrt[2+Sqrt[3]]*f*(1-q*x)*Sqrt[(1+q*x+q^2*x^2)/(1-Sqrt[3]-q*x)^2]/ + (q*Sqrt[a+b*x^3]*Sqrt[-(1-q*x)/(1-Sqrt[3]-q*x)^2])* + Subst[Int[1/(((1+Sqrt[3])*d+c*q+((1-Sqrt[3])*d+c*q)*x)*Sqrt[1-x^2]*Sqrt[7+4*Sqrt[3]+x^2]),x],x,(1+Sqrt[3]-q*x)/(-1+Sqrt[3]+q*x)] /; + EqQ[(1-Sqrt[3])*f+e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[a] && PosQ[b] + + +Int[(e_.+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[-b/a,3]}, + ((1-Sqrt[3])*f+e*q)/((1-Sqrt[3])*d+c*q)*Int[1/Sqrt[a+b*x^3],x] + + (d*e-c*f)/((1-Sqrt[3])*d+c*q)*Int[(1-Sqrt[3]-q*x)/((c+d*x)*Sqrt[a+b*x^3]),x] /; + NeQ[(1-Sqrt[3])*f+e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[a] && PosQ[b] + + +Int[(e_+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + -4*3^(1/4)*Sqrt[2+Sqrt[3]]*f*(1+q*x)*Sqrt[(1-q*x+q^2*x^2)/(1-Sqrt[3]+q*x)^2]/ + (q*Sqrt[a +b*x^3]*Sqrt[-(1+q*x)/(1-Sqrt[3]+q*x)^2])* + Subst[Int[1/(((1+Sqrt[3])*d-c*q+((1-Sqrt[3])*d-c*q)*x)*Sqrt[1-x^2]*Sqrt[7+4*Sqrt[3]+x^2]),x],x,(1+Sqrt[3]+q*x)/(-1+Sqrt[3]-q*x)] /; + EqQ[(1-Sqrt[3])*f-e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[a] && NegQ[b] + + +Int[(e_.+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol] := + With[{q=Rt[b/a,3]}, + ((1-Sqrt[3])*f-e*q)/((1-Sqrt[3])*d-c*q)*Int[1/Sqrt[a+b*x^3],x] + + (d*e-c*f)/((1-Sqrt[3])*d-c*q)*Int[(1-Sqrt[3]+q*x)/((c+d*x)*Sqrt[a+b*x^3]),x] /; + NeQ[(1-Sqrt[3])*f-e*q]] /; +FreeQ[{a,b,c,d,e,f},x] && NegQ[a] && NegQ[b] + + +Int[x_^m_./(c_+d_.*x_^n_+e_.*Sqrt[a_+b_.*x_^n_]),x_Symbol] := + 1/n*Subst[Int[x^((m+1)/n-1)/(c+d*x+e*Sqrt[a+b*x]),x],x,x^n] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[b*c-a*d,0] && IntegerQ[(m+1)/n] + + +Int[u_./(c_+d_.*x_^n_+e_.*Sqrt[a_+b_.*x_^n_]),x_Symbol] := + c*Int[u/(c^2-a*e^2+c*d*x^n),x] - a*e*Int[u/((c^2-a*e^2+c*d*x^n)*Sqrt[a+b*x^n]),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[b*c-a*d,0] + + +Int[(A_+B_.*x_^n_)/(a_+b_.*x_^2+c_.*x_^n_+d_.*x_^n2_), x_Symbol] := + A^2*(n-1)*Subst[Int[1/(a+A^2*b*(n-1)^2*x^2),x],x,x/(A*(n-1)-B*x^n)] /; +FreeQ[{a,b,c,d,A,B,n},x] && EqQ[n2-2*n] && NeQ[n-2] && EqQ[a*B^2-A^2*d*(n-1)^2] && EqQ[B*c+2*A*d*(n-1)] + + +Int[x_^m_.*(A_+B_.*x_^n_.)/(a_+b_.*x_^k_.+c_.*x_^n_.+d_.*x_^n2_), x_Symbol] := + A^2*(m-n+1)/(m+1)*Subst[Int[1/(a+A^2*b*(m-n+1)^2*x^2),x],x,x^(m+1)/(A*(m-n+1)+B*(m+1)*x^n)] /; +FreeQ[{a,b,c,d,A,B,m,n},x] && EqQ[n2-2*n] && EqQ[k-2*(m+1)] && EqQ[a*B^2*(m+1)^2-A^2*d*(m-n+1)^2] && EqQ[B*c*(m+1)-2*A*d*(m-n+1)] + + +Int[(d_+e_.*x_^n_+f_.*x_^n2_+g_*x_^n3_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*c*d-2*a*c*(c*d-a*f)-a*b*(c*e+a*g)+(b*c*(c*d+a*f)-a*b^2*g-2*a*c*(c*e-a*g))*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(b^2-4*a*c)) - + 1/(a*c*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[a*b*(c*e+a*g)-b^2*c*d*(n+n*p+1)-2*a*c*(a*f-c*d*(2*n*(p+1)+1))+ + (a*b^2*g*(n*(p+2)+1)-b*c*(c*d+a*f)*(n*(2*p+3)+1)-2*a*c*(a*g*(n+1)-c*e*(n*(2*p+3)+1)))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+e_.*x_^n_+f_.*x_^n2_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*d-2*a*(c*d-a*f)-a*b*e+(b*(c*d+a*f)-2*a*c*e)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c)) - + 1/(a*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[a*b*e-b^2*d*(n+n*p+1)-2*a*(a*f-c*d*(2*n*(p+1)+1))- + (b*(c*d+a*f)*(n*(2*p+3)+1)-2*a*c*e*(n*(2*p+3)+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+e_.*x_^n_+g_*x_^n3_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*c*d-2*a*c^2*d-a*b*(c*e+a*g)+(b*c^2*d-a*b^2*g-2*a*c*(c*e-a*g))*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(b^2-4*a*c)) - + 1/(a*c*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[a*b*(c*e+a*g)-b^2*c*d*(n+n*p+1)+2*a*c^2*d*(2*n*(p+1)+1)+ + (a*b^2*g*(n*(p+2)+1)-b*c^2*d*(n*(2*p+3)+1)-2*a*c*(a*g*(n+1)-c*e*(n*(2*p+3)+1)))*x^n,x],x] /; +FreeQ[{a,b,c,d,e,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+f_.*x_^n2_+g_*x_^n3_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*c*d-2*a*c*(c*d-a*f)-a^2*b*g+(b*c*(c*d+a*f)-a*b^2*g+2*a^2*c*g)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(b^2-4*a*c)) - + 1/(a*c*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[a^2*b*g-b^2*c*d*(n+n*p+1)-2*a*c*(a*f-c*d*(2*n*(p+1)+1))+ + (a*b^2*g*(n*(p+2)+1)-b*c*(c*d+a*f)*(n*(2*p+3)+1)-2*a^2*c*g*(n+1))*x^n,x],x] /; +FreeQ[{a,b,c,d,f,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+f_.*x_^n2_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*d-2*a*(c*d-a*f)+b*(c*d+a*f)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/(a*n*(p+1)*(b^2-4*a*c)) + + 1/(a*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[b^2*d*(n+n*p+1)+2*a*(a*f-c*d*(2*n*(p+1)+1))+b*(c*d+a*f)*(n*(2*p+3)+1)*x^n,x],x] /; +FreeQ[{a,b,c,d,f,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+g_*x_^n3_)*(a_+b_.*x_^n_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(b^2*c*d-2*a*c^2*d-a^2*b*g+(b*c^2*d-a*b^2*g+2*a^2*c*g)*x^n)*(a+b*x^n+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(b^2-4*a*c)) - + 1/(a*c*n*(p+1)*(b^2-4*a*c))*Int[(a+b*x^n+c*x^(2*n))^(p+1)* + Simp[a^2*b*g-b^2*c*d*(n+n*p+1)+2*a*c^2*d*(2*n*(p+1)+1)+ + (a*b^2*g*(n*(p+2)+1)-b*c^2*d*(n*(2*p+3)+1)-2*a*c*(a*g*(n+1)))*x^n,x],x] /; +FreeQ[{a,b,c,d,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NeQ[b^2-4*a*c] && NegativeIntegerQ[p+1] + + +Int[(d_+e_.*x_^n_+f_.*x_^n2_+g_*x_^n3_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(-2*a*c*(c*d-a*f)+(-2*a*c*(c*e-a*g))*x^n)*(a+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(-4*a*c)) - + 1/(a*c*n*(p+1)*(-4*a*c))*Int[(a+c*x^(2*n))^(p+1)* + Simp[-2*a*c*(a*f-c*d*(2*n*(p+1)+1))+ + (-2*a*c*(a*g*(n+1)-c*e*(n*(2*p+3)+1)))*x^n,x],x] /; +FreeQ[{a,c,d,e,f,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NegativeIntegerQ[p+1] + + +Int[(d_+e_.*x_^n_+f_.*x_^n2_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(-2*a*(c*d-a*f)+(-2*a*c*e)*x^n)*(a+c*x^(2*n))^(p+1)/(a*n*(p+1)*(-4*a*c)) - + 1/(a*n*(p+1)*(-4*a*c))*Int[(a+c*x^(2*n))^(p+1)* + Simp[-2*a*(a*f-c*d*(2*n*(p+1)+1))- + (-2*a*c*e*(n*(2*p+3)+1))*x^n,x],x] /; +FreeQ[{a,c,d,e,f,n},x] && EqQ[n2-2*n] && NegativeIntegerQ[p+1] + + +Int[(d_+e_.*x_^n_+g_*x_^n3_)*(a_+c_.*x_^n2_)^p_,x_Symbol] := + -x*(-2*a*c^2*d+(-2*a*c*(c*e-a*g))*x^n)*(a+c*x^(2*n))^(p+1)/ + (a*c*n*(p+1)*(-4*a*c)) - + 1/(a*c*n*(p+1)*(-4*a*c))*Int[(a+c*x^(2*n))^(p+1)* + Simp[2*a*c^2*d*(2*n*(p+1)+1)+ + (-2*a*c*(a*g*(n+1)-c*e*(n*(2*p+3)+1)))*x^n,x],x] /; +FreeQ[{a,c,d,e,g,n},x] && EqQ[n2-2*n] && EqQ[n3-3*n] && NegativeIntegerQ[p+1] + + +Int[(a_.+b_.*x_^2+c_.*x_^4)/(d_+e_.*x_^2+f_.*x_^4+g_.*x_^6),x_Symbol] := + With[{q=Rt[(-a*c*f^2+12*a^2*g^2+f*(3*c^2*d-2*a*b*g))/(c*g*(3*c*d-a*f)),2], + r=Rt[(a*c*f^2+4*g*(b*c*d+a^2*g)-f*(3*c^2*d+2*a*b*g))/(c*g*(3*c*d-a*f)),2]}, + c/(g*q)*ArcTan[(r+2*x)/q] - + c/(g*q)*ArcTan[(r-2*x)/q] - + c/(g*q)*ArcTan[(3*c*d-a*f)*x/(g*q*(b*c*d-2*a^2*g)*(b*c*d-a*b*f+4*a^2*g))* + (b*c^2*d*f-a*b^2*f*g-2*a^2*c*f*g+6*a^2*b*g^2+c*(3*c^2*d*f-a*c*f^2-b*c*d*g+2*a^2*g^2)*x^2+c^2*g*(3*c*d-a*f)*x^4)]] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[9*c^3*d^2-c*(b^2+6*a*c)*d*f+a^2*c*f^2+2*a*b*(3*c*d+a*f)*g-12*a^3*g^2] && + EqQ[3*c^4*d^2*e-3*a^2*c^2*d*f*g+a^3*c*f^2*g+2*a^3*g^2*(b*f-6*a*g)-c^3*d*(2*b*d*f+a*e*f-12*a*d*g)] && + NeQ[3*c*d-a*f] && NeQ[b*c*d-2*a^2*g] && NeQ[b*c*d-a*b*f+4*a^2*g] && + PosQ[(-a*c*f^2+12*a^2*g^2+f*(3*c^2*d-2*a*b*g))/(c*g*(3*c*d-a*f))] + + +Int[(a_.+c_.*x_^4)/(d_+e_.*x_^2+f_.*x_^4+g_.*x_^6),x_Symbol] := + With[{q=Rt[(-a*c*f^2+12*a^2*g^2+3*f*c^2*d)/(c*g*(3*c*d-a*f)),2], + r=Rt[(a*c*f^2+4*a^2*g^2-3*c^2*d*f)/(c*g*(3*c*d-a*f)),2]}, + c/(g*q)*ArcTan[(r+2*x)/q] - + c/(g*q)*ArcTan[(r-2*x)/q] - + c/(g*q)*ArcTan[(c*(3*c*d-a*f)*x*(2*a^2*f*g-(3*c^2*d*f-a*c*f^2+2*a^2*g^2)*x^2-c*(3*c*d-a*f)*g*x^4))/(8*a^4*g^3*q)]] /; +FreeQ[{a,c,d,e,f,g},x] && EqQ[9*c^3*d^2-6*a*c^2*d*f+a^2*c*f^2-12*a^3*g^2] && + EqQ[3*c^4*d^2*e-3*a^2*c^2*d*f*g+a^3*c*f^2*g-12*a^4*g^3-a*c^3*d*(e*f-12*d*g)] && + NeQ[3*c*d-a*f] && PosQ[(-a*c*f^2+12*a^2*g^2+3*c^2*d*f)/(c*g*(3*c*d-a*f))] + + +If[ShowSteps, + +Int[u_*v_^p_,x_Symbol] := + With[{m=Exponent[u,x],n=Exponent[v,x]}, + Module[{c=Coefficient[u,x,m]/(Coefficient[v,x,n]*(m+1+n*p)),w}, + w=Apart[u-c*x^(m-n)*((m-n+1)*v+(p+1)*x*D[v,x]),x]; + If[EqQ[w], + ShowStep[" +If p>1, 11, 10 && IntegerQ[2*m] && Not[$UseGamma===True] + + +Int[(c_.+d_.*x_)^m_*(b_.*F_^(g_.*(e_.+f_.*x_)))^n_.,x_Symbol] := + (c+d*x)^(m+1)*(b*F^(g*(e+f*x)))^n/(d*(m+1)) - + f*g*n*Log[F]/(d*(m+1))*Int[(c+d*x)^(m+1)*(b*F^(g*(e+f*x)))^n,x] /; +FreeQ[{F,b,c,d,e,f,g,n},x] && RationalQ[m] && m<-1 && IntegerQ[2*m] && Not[$UseGamma===True] + + +Int[F_^(g_.*(e_.+f_.*x_))/(c_.+d_.*x_),x_Symbol] := + F^(g*(e-c*f/d))/d*ExpIntegralEi[f*g*(c+d*x)*Log[F]/d] /; +FreeQ[{F,c,d,e,f,g},x] && Not[$UseGamma===True] + + +Int[(c_.+d_.*x_)^m_.*F_^(g_.*(e_.+f_.*x_)),x_Symbol] := + (-d)^m*F^(g*(e-c*f/d))/(f^(m+1)*g^(m+1)*Log[F]^(m+1))*Gamma[m+1,-f*g*Log[F]/d*(c+d*x)] /; +FreeQ[{F,c,d,e,f,g},x] && IntegerQ[m] + + +Int[F_^(g_.*(e_.+f_.*x_))/Sqrt[c_.+d_.*x_],x_Symbol] := + 2/d*Subst[Int[F^(g*(e-c*f/d)+f*g*x^2/d),x],x,Sqrt[c+d*x]] /; +FreeQ[{F,c,d,e,f,g},x] && Not[$UseGamma===True] + + +Int[(c_.+d_.*x_)^m_*F_^(g_.*(e_.+f_.*x_)),x_Symbol] := + -F^(g*(e-c*f/d))*(c+d*x)^FracPart[m]/(d*(-f*g*Log[F]/d)^(IntPart[m]+1)*(-f*g*Log[F]*(c+d*x)/d)^FracPart[m])* + Gamma[m+1,(-f*g*Log[F]/d)*(c+d*x)] /; +FreeQ[{F,c,d,e,f,g,m},x] && Not[IntegerQ[m]] + + +Int[(c_.+d_.*x_)^m_.*(b_.*F_^(g_.*(e_.+f_.*x_)))^n_,x_Symbol] := + (b*F^(g*(e+f*x)))^n/F^(g*n*(e+f*x))*Int[(c+d*x)^m*F^(g*n*(e+f*x)),x] /; +FreeQ[{F,b,c,d,e,f,g,m,n},x] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_.,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(a+b*(F^(g*(e+f*x)))^n)^p,x],x] /; +FreeQ[{F,a,b,c,d,e,f,g,m,n},x] && PositiveIntegerQ[p] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.),x_Symbol] := + -(c+d*x)^m/(a*f*g*n*Log[F])*Log[1+a/(b*(F^(g*(e+f*x)))^n)] + + d*m/(a*f*g*n*Log[F])*Int[(c+d*x)^(m-1)*Log[1+a/(b*(F^(g*(e+f*x)))^n)],x] /; +FreeQ[{F,a,b,c,d,e,f,g,n},x] && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_,x_Symbol] := + 1/a*Int[(c+d*x)^m*(a+b*(F^(g*(e+f*x)))^n)^(p+1),x] - + b/a*Int[(c+d*x)^m*(F^(g*(e+f*x)))^n*(a+b*(F^(g*(e+f*x)))^n)^p,x] /; +FreeQ[{F,a,b,c,d,e,f,g,n},x] && NegativeIntegerQ[p] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_,x_Symbol] := + With[{u=IntHide[(a+b*(F^(g*(e+f*x)))^n)^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e,f,g,n},x] && RationalQ[m,p] && m>0 && p<-1 + + +Int[u_^m_.*(a_.+b_.*(F_^(g_.*v_))^n_.)^p_.,x_Symbol] := + Int[NormalizePowerOfLinear[u,x]^m*(a+b*(F^(g*ExpandToSum[v,x]))^n)^p,x] /; +FreeQ[{F,a,b,g,n,p},x] && LinearQ[v,x] && PowerOfLinearQ[u,x] && Not[LinearMatchQ[v,x] && PowerOfLinearMatchQ[u,x]] && IntegerQ[m] + + +Int[u_^m_.*(a_.+b_.*(F_^(g_.*v_))^n_.)^p_.,x_Symbol] := + Module[{uu=NormalizePowerOfLinear[u,x],z}, + z=If[PowerQ[uu] && FreeQ[uu[[2]],x], uu[[1]]^(m*uu[[2]]), uu^m]; + uu^m/z*Int[z*(a+b*(F^(g*ExpandToSum[v,x]))^n)^p,x]] /; +FreeQ[{F,a,b,g,m,n,p},x] && LinearQ[v,x] && PowerOfLinearQ[u,x] && Not[LinearMatchQ[v,x] && PowerOfLinearMatchQ[u,x]] && + Not[IntegerQ[m]] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_.,x_Symbol] := + Defer[Int][(c+d*x)^m*(a+b*(F^(g*(e+f*x)))^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + + + + +(* ::Subsection::Closed:: *) +(*2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p*) + + +Int[(c_.+d_.*x_)^m_.*(F_^(g_.*(e_.+f_.*x_)))^n_./(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.),x_Symbol] := + (c+d*x)^m/(b*f*g*n*Log[F])*Log[1+b*(F^(g*(e+f*x)))^n/a] - + d*m/(b*f*g*n*Log[F])*Int[(c+d*x)^(m-1)*Log[1+b*(F^(g*(e+f*x)))^n/a],x] /; +FreeQ[{F,a,b,c,d,e,f,g,n},x] && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*(F_^(g_.*(e_.+f_.*x_)))^n_.*(a_.+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_.,x_Symbol] := + (c+d*x)^m*(a+b*(F^(g*(e+f*x)))^n)^(p+1)/(b*f*g*n*(p+1)*Log[F]) - + d*m/(b*f*g*n*(p+1)*Log[F])*Int[(c+d*x)^(m-1)*(a+b*(F^(g*(e+f*x)))^n)^(p+1),x] /; +FreeQ[{F,a,b,c,d,e,f,g,m,n,p},x] && NeQ[p+1] + + +Int[(c_.+d_.*x_)^m_.*(F_^(g_.*(e_.+f_.*x_)))^n_.*(a_.+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_.,x_Symbol] := + Defer[Int][(c+d*x)^m*(F^(g*(e+f*x)))^n*(a+b*(F^(g*(e+f*x)))^n)^p,x] /; +FreeQ[{F,a,b,c,d,e,f,g,m,n,p},x] + + +Int[(c_.+d_.*x_)^m_.*(k_.*G_^(j_.*(h_.+i_.*x_)))^q_.*(a_.+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.)^p_.,x_Symbol] := + (k*G^(j*(h+i*x)))^q/(F^(g*(e+f*x)))^n*Int[(c+d*x)^m*(F^(g*(e+f*x)))^n*(a+b*(F^(g*(e+f*x)))^n)^p,x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,i,j,k,m,n,p,q},x] && EqQ[f*g*n*Log[F]-i*j*q*Log[G]] && NeQ[(k*G^(j*(h+i*x)))^q-(F^(g*(e+f*x)))^n] + + + + + +(* ::Subsection::Closed:: *) +(*2.3 Miscellaneous exponentials*) + + +Int[(F_^(c_.*(a_.+b_.*x_)))^n_.,x_Symbol] := + (F^(c*(a+b*x)))^n/(b*c*n*Log[F]) /; +FreeQ[{F,a,b,c,n},x] + + +Int[u_*F_^(c_.*v_),x_Symbol] := + Int[ExpandIntegrand[u*F^(c*ExpandToSum[v,x]),x],x] /; +FreeQ[{F,c},x] && PolynomialQ[u,x] && LinearQ[v,x] && $UseGamma===True + + +Int[u_*F_^(c_.*v_),x_Symbol] := + Int[ExpandIntegrand[F^(c*ExpandToSum[v,x]),u,x],x] /; +FreeQ[{F,c},x] && PolynomialQ[u,x] && LinearQ[v,x] && Not[$UseGamma===True] + + +Int[u_^m_.*F_^(c_.*v_)*w_,x_Symbol] := + Coefficient[w,x,1]*u^(m+1)*F^(c*v)/(Coefficient[v,x,1]*c*Coefficient[u,x,1]*Log[F]) /; +FreeQ[{F,c,m},x] && LinearQ[{u,v,w},x] && + EqQ[Coefficient[u,x,1]*Coefficient[w,x,1]*(m+1)- + Coefficient[v,x,1]*c*(Coefficient[u,x,1]*Coefficient[w,x,0]-Coefficient[u,x,0]*Coefficient[w,x,1])*Log[F]] + + +Int[w_*u_^m_.*F_^(c_.*v_),x_Symbol] := + Int[ExpandIntegrand[w*NormalizePowerOfLinear[u,x]^m*F^(c*ExpandToSum[v,x]),x],x] /; +FreeQ[{F,c},x] && PolynomialQ[w,x] && LinearQ[v,x] && PowerOfLinearQ[u,x] && IntegerQ[m] && $UseGamma===True + + +Int[w_*u_^m_.*F_^(c_.*v_),x_Symbol] := + Int[ExpandIntegrand[F^(c*ExpandToSum[v,x]),w*NormalizePowerOfLinear[u,x]^m,x],x] /; +FreeQ[{F,c},x] && PolynomialQ[w,x] && LinearQ[v,x] && PowerOfLinearQ[u,x] && IntegerQ[m] && Not[$UseGamma===True] + + +Int[w_*u_^m_.*F_^(c_.*v_),x_Symbol] := + Module[{uu=NormalizePowerOfLinear[u,x],z}, + z=If[PowerQ[uu] && FreeQ[uu[[2]],x], uu[[1]]^(m*uu[[2]]), uu^m]; + uu^m/z*Int[ExpandIntegrand[w*z*F^(c*ExpandToSum[v,x]),x],x]] /; +FreeQ[{F,c,m},x] && PolynomialQ[w,x] && LinearQ[v,x] && PowerOfLinearQ[u,x] && Not[IntegerQ[m]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Log[d_.*x_]^n_.*(e_+h_.*(f_.+g_.*x_)*Log[d_.*x_]),x_Symbol] := + e*x*F^(c*(a+b*x))*Log[d*x]^(n+1)/(n+1) /; +FreeQ[{F,a,b,c,d,e,f,g,h,n},x] && EqQ[e-f*h*(n+1)] && EqQ[g*h*(n+1)-b*c*e*Log[F]] && NeQ[n+1] + + +Int[x_^m_.*F_^(c_.*(a_.+b_.*x_))*Log[d_.*x_]^n_.*(e_+h_.*(f_.+g_.*x_)*Log[d_.*x_]),x_Symbol] := + e*x^(m+1)*F^(c*(a+b*x))*Log[d*x]^(n+1)/(n+1) /; +FreeQ[{F,a,b,c,d,e,f,g,h,m,n},x] && EqQ[e*(m+1)-f*h*(n+1)] && EqQ[g*h*(n+1)-b*c*e*Log[F]] && NeQ[n+1] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)),x_Symbol] := + F^(a+b*(c+d*x))/(b*d*Log[F]) /; +FreeQ[{F,a,b,c,d},x] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^2),x_Symbol] := + F^a*Sqrt[Pi]*Erfi[(c+d*x)*Rt[b*Log[F],2]]/(2*d*Rt[b*Log[F],2]) /; +FreeQ[{F,a,b,c,d},x] && PosQ[b] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^2),x_Symbol] := + F^a*Sqrt[Pi]*Erf[(c+d*x)*Rt[-b*Log[F],2]]/(2*d*Rt[-b*Log[F],2]) /; +FreeQ[{F,a,b,c,d},x] && NegQ[b] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + (c+d*x)*F^(a+b*(c+d*x)^n)/d - + b*n*Log[F]*Int[(c+d*x)^n*F^(a+b*(c+d*x)^n),x] /; +FreeQ[{F,a,b,c,d},x] && IntegerQ[2/n] && NegativeIntegerQ[n] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + With[{k=Denominator[n]}, + k/d*Subst[Int[x^(k-1)*F^(a+b*x^(k*n)),x],x,(c+d*x)^(1/k)]] /; +FreeQ[{F,a,b,c,d},x] && IntegerQ[2/n] && Not[IntegerQ[n]] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + -F^a*(c+d*x)*Gamma[1/n,-b*(c+d*x)^n*Log[F]]/(d*n*(-b*(c+d*x)^n*Log[F])^(1/n)) /; +FreeQ[{F,a,b,c,d,n},x] && Not[IntegerQ[2/n]] + + +Int[(e_.+f_.*x_)^m_.*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + (e+f*x)^n*F^(a+b*(c+d*x)^n)/(b*f*n*(c+d*x)^n*Log[F]) /; +FreeQ[{F,a,b,c,d,e,f,n},x] && EqQ[m-(n-1)] && EqQ[d*e-c*f] + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^n_)/(e_.+f_.*x_),x_Symbol] := + F^a*ExpIntegralEi[b*(c+d*x)^n*Log[F]]/(f*n) /; +FreeQ[{F,a,b,c,d,e,f,n},x] && EqQ[d*e-c*f] + + +Int[(c_.+d_.*x_)^m_.*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + 1/(d*(m+1))*Subst[Int[F^(a+b*x^2),x],x,(c+d*x)^(m+1)] /; +FreeQ[{F,a,b,c,d,m,n},x] && EqQ[n-2*(m+1)] + + +Int[(c_.+d_.*x_)^m_.*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + (c+d*x)^(m-n+1)*F^(a+b*(c+d*x)^n)/(b*d*n*Log[F]) - + (m-n+1)/(b*n*Log[F])*Int[(c+d*x)^(m-n)*F^(a+b*(c+d*x)^n),x] /; +FreeQ[{F,a,b,c,d},x] && RationalQ[m] && IntegerQ[2*(m+1)/n] && 0<(m+1)/n<5 && IntegerQ[n] && (00 && m<-1 || 0<-n<=m+1) + + +Int[(c_.+d_.*x_)^m_.*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + (c+d*x)^(m+1)*F^(a+b*(c+d*x)^n)/(d*(m+1)) - + b*n*Log[F]/(m+1)*Int[(c+d*x)^Simplify[m+n]*F^(a+b*(c+d*x)^n),x] /; +FreeQ[{F,a,b,c,d,m,n},x] && IntegerQ[2*Simplify[(m+1)/n]] && -41 + + +Int[(e_.+f_.*x_)^m_*F_^(a_.+b_.*(c_.+d_.*x_)^2),x_Symbol] := + f*(e+f*x)^(m+1)*F^(a+b*(c+d*x)^2)/((m+1)*f^2) + + 2*b*d*(d*e-c*f)*Log[F]/(f^2*(m+1))*Int[(e+f*x)^(m+1)*F^(a+b*(c+d*x)^2),x] - + 2*b*d^2*Log[F]/(f^2*(m+1))*Int[(e+f*x)^(m+2)*F^(a+b*(c+d*x)^2),x] /; +FreeQ[{F,a,b,c,d,e,f},x] && NeQ[d*e-c*f] && RationalQ[m] && m<-1 + + +Int[(e_.+f_.*x_)^m_*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + (e+f*x)^(m+1)*F^(a+b*(c+d*x)^n)/(f*(m+1)) - + b*d*n*Log[F]/(f*(m+1))*Int[(e+f*x)^(m+1)*(c+d*x)^(n-1)*F^(a+b*(c+d*x)^n),x] /; +FreeQ[{F,a,b,c,d,e,f},x] && NeQ[d*e-c*f] && IntegerQ[n] && n>2 && RationalQ[m] && m<-1 + + +Int[F_^(a_.+b_./(c_.+d_.*x_))/(e_.+f_.*x_),x_Symbol] := + d/f*Int[F^(a+b/(c+d*x))/(c+d*x),x] - + (d*e-c*f)/f*Int[F^(a+b/(c+d*x))/((c+d*x)*(e+f*x)),x] /; +FreeQ[{F,a,b,c,d,e,f},x] && NeQ[d*e-c*f] + + +Int[(e_.+f_.*x_)^m_*F_^(a_.+b_./(c_.+d_.*x_)),x_Symbol] := + (e+f*x)^(m+1)*F^(a+b/(c+d*x))/(f*(m+1)) + + b*d*Log[F]/(f*(m+1))*Int[(e+f*x)^(m+1)*F^(a+b/(c+d*x))/(c+d*x)^2,x] /; +FreeQ[{F,a,b,c,d,e,f},x] && NeQ[d*e-c*f] && IntegerQ[m] && m<-1 + + +Int[F_^(a_.+b_.*(c_.+d_.*x_)^n_)/(e_.+f_.*x_),x_Symbol] := + Defer[Int][F^(a+b*(c+d*x)^n)/(e+f*x),x] /; +FreeQ[{F,a,b,c,d,e,f,n},x] && NeQ[d*e-c*f] + + +Int[u_^m_.*F_^v_,x_Symbol] := + Int[ExpandToSum[u,x]^m*F^ExpandToSum[v,x],x] /; +FreeQ[{F,m},x] && LinearQ[u,x] && BinomialQ[v,x] && Not[LinearMatchQ[u,x] && BinomialMatchQ[v,x]] + + +Int[u_*F_^(a_.+b_.*(c_.+d_.*x_)^n_),x_Symbol] := + Int[ExpandLinearProduct[F^(a+b*(c+d*x)^n),u,c,d,x],x] /; +FreeQ[{F,a,b,c,d,n},x] && PolynomialQ[u,x] + + +Int[u_.*F_^(a_.+b_.*v_),x_Symbol] := + Int[u*F^(a+b*NormalizePowerOfLinear[v,x]),x] /; +FreeQ[{F,a,b},x] && PolynomialQ[u,x] && PowerOfLinearQ[v,x] && Not[PowerOfLinearMatchQ[v,x]] + + +(* Int[u_.*F_^(a_.+b_.*v_^n_),x_Symbol] := + Int[u*F^(a+b*ExpandToSum[v,x]^n),x] /; +FreeQ[{F,a,b,n},x] && PolynomialQ[u,x] && LinearQ[v,x] && Not[LinearMatchQ[v,x]] *) + + +(* Int[u_.*F_^u_,x_Symbol] := + Int[u*F^ExpandToSum[u,x],x] /; +FreeQ[F,x] && PolynomialQ[u,x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] *) + + +Int[F_^(a_.+b_./(c_.+d_.*x_))/((e_.+f_.*x_)*(g_.+h_.*x_)),x_Symbol] := + -d/(f*(d*g-c*h))*Subst[Int[F^(a-b*h/(d*g-c*h)+d*b*x/(d*g-c*h))/x,x],x,(g+h*x)/(c+d*x)] /; +FreeQ[{F,a,b,c,d,e,f},x] && EqQ[d*e-c*f] + + +Int[(g_.+h_.*x_)^m_.*F_^(e_.+f_.*(a_.+b_.*x_)/(c_.+d_.*x_)),x_Symbol] := + F^(e+f*b/d)*Int[(g+h*x)^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && EqQ[b*c-a*d] + + +Int[(g_.+h_.*x_)^m_.*F_^(e_.+f_.*(a_.+b_.*x_)/(c_.+d_.*x_)),x_Symbol] := + Int[(g+h*x)^m*F^((d*e+b*f)/d-f*(b*c-a*d)/(d*(c+d*x))),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && NeQ[b*c-a*d] && EqQ[d*g-c*h] + + +Int[F_^(e_.+f_.*(a_.+b_.*x_)/(c_.+d_.*x_))/(g_.+h_.*x_),x_Symbol] := + d/h*Int[F^(e+f*(a+b*x)/(c+d*x))/(c+d*x),x] - + (d*g-c*h)/h*Int[F^(e+f*(a+b*x)/(c+d*x))/((c+d*x)*(g+h*x)),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h},x] && NeQ[b*c-a*d] && NeQ[d*g-c*h] + + +Int[(g_.+h_.*x_)^m_*F_^(e_.+f_.*(a_.+b_.*x_)/(c_.+d_.*x_)),x_Symbol] := + (g+h*x)^(m+1)*F^(e+f*(a+b*x)/(c+d*x))/(h*(m+1)) - + f*(b*c-a*d)*Log[F]/(h*(m+1))*Int[(g+h*x)^(m+1)*F^(e+f*(a+b*x)/(c+d*x))/(c+d*x)^2,x] /; +FreeQ[{F,a,b,c,d,e,f,g,h},x] && NeQ[b*c-a*d] && NeQ[d*g-c*h] && IntegerQ[m] && m<-1 + + +Int[F_^(e_.+f_.*(a_.+b_.*x_)/(c_.+d_.*x_))/((g_.+h_.*x_)*(i_.+j_.*x_)),x_Symbol] := + -d/(h*(d*i-c*j))*Subst[Int[F^(e+f*(b*i-a*j)/(d*i-c*j)-(b*c-a*d)*f*x/(d*i-c*j))/x,x],x,(i+j*x)/(c+d*x)] /; +FreeQ[{F,a,b,c,d,e,f,g,h},x] && EqQ[d*g-c*h] + + +Int[F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + F^(a-b^2/(4*c))*Int[F^((b+2*c*x)^2/(4*c)),x] /; +FreeQ[{F,a,b,c},x] + + +Int[F_^v_,x_Symbol] := + Int[F^ExpandToSum[v,x],x] /; +FreeQ[F,x] && QuadraticQ[v,x] && Not[QuadraticMatchQ[v,x]] + + +Int[(d_.+e_.*x_)*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*F^(a+b*x+c*x^2)/(2*c*Log[F]) /; +FreeQ[{F,a,b,c,d,e},x] && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m-1)*F^(a+b*x+c*x^2)/(2*c*Log[F]) - + (m-1)*e^2/(2*c*Log[F])*Int[(d+e*x)^(m-2)*F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e},x] && EqQ[b*e-2*c*d] && RationalQ[m] && m>1 + + +Int[F_^(a_.+b_.*x_+c_.*x_^2)/(d_.+e_.*x_),x_Symbol] := + 1/(2*e)*F^(a-b^2/(4*c))*ExpIntegralEi[(b+2*c*x)^2*Log[F]/(4*c)] /; +FreeQ[{F,a,b,c,d,e},x] && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + (d+e*x)^(m+1)*F^(a+b*x+c*x^2)/(e*(m+1)) - + 2*c*Log[F]/(e^2*(m+1))*Int[(d+e*x)^(m+2)*F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e},x] && EqQ[b*e-2*c*d] && RationalQ[m] && m<-1 + + +Int[(d_.+e_.*x_)*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*F^(a+b*x+c*x^2)/(2*c*Log[F]) - + (b*e-2*c*d)/(2*c)*Int[F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + e*(d+e*x)^(m-1)*F^(a+b*x+c*x^2)/(2*c*Log[F]) - + (b*e-2*c*d)/(2*c)*Int[(d+e*x)^(m-1)*F^(a+b*x+c*x^2),x] - + (m-1)*e^2/(2*c*Log[F])*Int[(d+e*x)^(m-2)*F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[b*e-2*c*d] && RationalQ[m] && m>1 + + +Int[(d_.+e_.*x_)^m_*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + (d+e*x)^(m+1)*F^(a+b*x+c*x^2)/(e*(m+1)) - + (b*e-2*c*d)*Log[F]/(e^2*(m+1))*Int[(d+e*x)^(m+1)*F^(a+b*x+c*x^2),x] - + 2*c*Log[F]/(e^2*(m+1))*Int[(d+e*x)^(m+2)*F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[b*e-2*c*d] && RationalQ[m] && m<-1 + + +Int[(d_.+e_.*x_)^m_.*F_^(a_.+b_.*x_+c_.*x_^2),x_Symbol] := + Defer[Int][(d+e*x)^m*F^(a+b*x+c*x^2),x] /; +FreeQ[{F,a,b,c,d,e,m},x] + + +Int[u_^m_.*F_^v_,x_Symbol] := + Int[ExpandToSum[u,x]^m*F^ExpandToSum[v,x],x] /; +FreeQ[{F,m},x] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + +Int[x_^m_.*F_^(e_.*(c_.+d_.*x_))*(a_.+b_.*F_^v_)^p_,x_Symbol] := + With[{u=IntHide[F^(e*(c+d*x))*(a+b*F^v)^p,x]}, + Dist[x^m,u,x] - m*Int[x^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e},x] && EqQ[2*e*(c+d*x)-v] && RationalQ[m] && m>0 && NegativeIntegerQ[p] + + +Int[(F_^(e_.*(c_.+d_.*x_)))^n_.*(a_+b_.*(F_^(e_.*(c_.+d_.*x_)))^n_.)^p_.,x_Symbol] := + 1/(d*e*n*Log[F])*Subst[Int[(a+b*x)^p,x],x,(F^(e*(c+d*x)))^n] /; +FreeQ[{F,a,b,c,d,e,n,p},x] + + +Int[(G_^(h_.(f_.+g_.*x_)))^m_.*(a_+b_.*(F_^(e_.*(c_.+d_.*x_)))^n_.)^p_.,x_Symbol] := + (G^(h*(f+g*x)))^m/(F^(e*(c+d*x)))^n*Int[(F^(e*(c+d*x)))^n*(a+b*(F^(e*(c+d*x)))^n)^p,x] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[d*e*n*Log[F]-g*h*m*Log[G]] + + +Int[G_^(h_.(f_.+g_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + With[{m=FullSimplify[g*h*Log[G]/(d*e*Log[F])]}, + Denominator[m]*G^(f*h-c*g*h/d)/(d*e*Log[F])*Subst[Int[x^(Numerator[m]-1)*(a+b*x^Denominator[m])^p,x],x,F^(e*(c+d*x)/Denominator[m])] /; + RationalQ[m] && Abs[m]>=1] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h,p},x] + + +Int[G_^(h_.(f_.+g_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + With[{m=FullSimplify[d*e*Log[F]/(g*h*Log[G])]}, + Denominator[m]/(g*h*Log[G])*Subst[Int[x^(Denominator[m]-1)*(a+b*F^(c*e-d*e*f/g)*x^Numerator[m])^p,x],x,G^(h*(f+g*x)/Denominator[m])] /; + RationalQ[m] && Abs[m]>1] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h,p},x] + + +Int[G_^(h_.(f_.+g_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + Int[Expand[G^(h*(f+g*x))*(a+b*F^(e*(c+d*x)))^p,x],x] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[p] + + +Int[G_^(h_.(f_.+g_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_,x_Symbol] := + a^p*G^(h*(f+g*x))/(g*h*Log[G])*Hypergeometric2F1[-p,g*h*Log[G]/(d*e*Log[F]),g*h*Log[G]/(d*e*Log[F])+1,Simplify[-b/a*F^(e*(c+d*x))]] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h,p},x] && (NegativeIntegerQ[p] || RationalQ[a] && a>0) + + +Int[G_^(h_.(f_.+g_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_,x_Symbol] := + (a+b*F^(e*(c+d*x)))^p/(1+(b/a)*F^(e*(c+d*x)))^p*Int[G^(h*(f+g*x))*(1+b/a*F^(e*(c+d*x)))^p,x] /; +FreeQ[{F,G,a,b,c,d,e,f,g,h,p},x] && Not[NegativeIntegerQ[p] || RationalQ[a] && a>0] + + +Int[G_^(h_. u_)*(a_+b_.*F_^(e_.*v_))^p_,x_Symbol] := + Int[G^(h*ExpandToSum[u,x])*(a+b*F^(e*ExpandToSum[v,x]))^p,x] /; +FreeQ[{F,G,a,b,e,h,p},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[G_^(h_.(f_.+g_.*x_))*H_^(t_.(r_.+s_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + With[{m=FullSimplify[(g*h*Log[G]+s*t*Log[H])/(d*e*Log[F])]}, + Denominator[m]*G^(f*h-c*g*h/d)*H^(r*t-c*s*t/d)/(d*e*Log[F])* + Subst[Int[x^(Numerator[m]-1)*(a+b*x^Denominator[m])^p,x],x,F^(e*(c+d*x)/Denominator[m])] /; + RationalQ[m]] /; +FreeQ[{F,G,H,a,b,c,d,e,f,g,h,r,s,t,p},x] + + +Int[G_^(h_.(f_.+g_.*x_))*H_^(t_.(r_.+s_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + G^((f-c*g/d)*h)*Int[H^(t*(r+s*x))*(b+a*F^(-e*(c+d*x)))^p,x] /; +FreeQ[{F,G,H,a,b,c,d,e,f,g,h,r,s,t},x] && EqQ[d*e*p*Log[F]+g*h*Log[G]] && IntegerQ[p] + + +Int[G_^(h_.(f_.+g_.*x_))*H_^(t_.(r_.+s_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + Int[Expand[G^(h*(f+g*x))*H^(t*(r+s*x))*(a+b*F^(e*(c+d*x)))^p,x],x] /; +FreeQ[{F,G,H,a,b,c,d,e,f,g,h,r,s,t},x] && PositiveIntegerQ[p] + + +Int[G_^(h_.(f_.+g_.*x_))*H_^(t_.(r_.+s_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_,x_Symbol] := + a^p*G^(h*(f+g*x))*H^(t*(r+s*x))/(g*h*Log[G]+s*t*Log[H])* + Hypergeometric2F1[-p,(g*h*Log[G]+s*t*Log[H])/(d*e*Log[F]),(g*h*Log[G]+s*t*Log[H])/(d*e*Log[F])+1,Simplify[-b/a*F^(e*(c+d*x))]] /; +FreeQ[{F,G,H,a,b,c,d,e,f,g,h,r,s,t},x] && NegativeIntegerQ[p] + + +Int[G_^(h_.(f_.+g_.*x_))*H_^(t_.(r_.+s_.*x_))*(a_+b_.*F_^(e_.*(c_.+d_.*x_)))^p_,x_Symbol] := + G^(h*(f+g*x))*H^(t*(r+s*x))*(a+b*F^(e*(c+d*x)))^p/((g*h*Log[G]+s*t*Log[H])*((a+b*F^(e*(c+d*x)))/a)^p)* + Hypergeometric2F1[-p,(g*h*Log[G]+s*t*Log[H])/(d*e*Log[F]),(g*h*Log[G]+s*t*Log[H])/(d*e*Log[F])+1,Simplify[-b/a*F^(e*(c+d*x))]] /; +FreeQ[{F,G,H,a,b,c,d,e,f,g,h,r,s,t,p},x] && Not[IntegerQ[p]] + + +Int[G_^(h_. u_)*H_^(t_. w_)*(a_+b_.*F_^(e_.*v_))^p_,x_Symbol] := + Int[G^(h*ExpandToSum[u,x])*H^(t*ExpandToSum[w,x])*(a+b*F^(e*ExpandToSum[v,x]))^p,x] /; +FreeQ[{F,G,H,a,b,e,h,t,p},x] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]] + + +Int[F_^(e_.*(c_.+d_.*x_))*(a_.*x_^n_.+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + (a*x^n+b*F^(e*(c+d*x)))^(p+1)/(b*d*e*(p+1)*Log[F]) - + a*n/(b*d*e*Log[F])*Int[x^(n-1)*(a*x^n+b*F^(e*(c+d*x)))^p,x] /; +FreeQ[{F,a,b,c,d,e,n,p},x] && NeQ[p+1] + + +Int[x_^m_.*F_^(e_.*(c_.+d_.*x_))*(a_.*x_^n_.+b_.*F_^(e_.*(c_.+d_.*x_)))^p_.,x_Symbol] := + x^m*(a*x^n+b*F^(e*(c+d*x)))^(p+1)/(b*d*e*(p+1)*Log[F]) - + a*n/(b*d*e*Log[F])*Int[x^(m+n-1)*(a*x^n+b*F^(e*(c+d*x)))^p,x] - + m/(b*d*e*(p+1)*Log[F])*Int[x^(m-1)*(a*x^n+b*F^(e*(c+d*x)))^(p+1),x] /; +FreeQ[{F,a,b,c,d,e,m,n,p},x] && NeQ[p+1] + + +Int[(f_.+g_.*x_)^m_./(a_.+b_.*F_^u_+c_.*F_^v_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[(f+g*x)^m/(b-q+2*c*F^u),x] - 2*c/q*Int[(f+g*x)^m/(b+q+2*c*F^u),x]] /; +FreeQ[{F,a,b,c,f,g},x] && EqQ[v-2*u] && LinearQ[u,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*F_^u_/(a_.+b_.*F_^u_+c_.*F_^v_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[(f+g*x)^m*F^u/(b-q+2*c*F^u),x] - 2*c/q*Int[(f+g*x)^m*F^u/(b+q+2*c*F^u),x]] /; +FreeQ[{F,a,b,c,f,g},x] && EqQ[v-2*u] && LinearQ[u,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*(h_+i_.*F_^u_)/(a_.+b_.*F_^u_+c_.*F_^v_),x_Symbol] := + With[{q=Rt[b^2-4*a*c,2]}, + (Simplify[(2*c*h-b*i)/q]+i)*Int[(f+g*x)^m/(b-q+2*c*F^u),x] - + (Simplify[(2*c*h-b*i)/q]-i)*Int[(f+g*x)^m/(b+q+2*c*F^u),x]] /; +FreeQ[{F,a,b,c,f,g,h,i},x] && EqQ[v-2*u] && LinearQ[u,x] && NeQ[b^2-4*a*c] && PositiveIntegerQ[m] + + +Int[x_^m_./(a_.*F_^(c_.+d_.*x_)+b_.*F_^v_),x_Symbol] := + With[{u=IntHide[1/(a*F^(c+d*x)+b*F^v),x]}, + x^m*u - m*Int[x^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d},x] && EqQ[(c+d*x)+v] && RationalQ[m] && m>0 + + +Int[u_/(a_+b_.*F_^v_+c_.*F_^w_),x_Symbol] := + Int[u*F^v/(c+a*F^v+b*F^(2*v)),x] /; +FreeQ[{F,a,b,c},x] && LinearQ[v,x] && LinearQ[w,x] && EqQ[v+w] && + If[RationalQ[Coefficient[v,x,1]], Coefficient[v,x,1]>0, LeafCount[v]0 && n<0 && n!=-1 + + +Int[Log[a_+b_.*(F_^(e_.*(c_.+d_.*x_)))^n_.],x_Symbol] := + 1/(d*e*n*Log[F])*Subst[Int[Log[a+b*x]/x,x],x,(F^(e*(c+d*x)))^n] /; +FreeQ[{F,a,b,c,d,e,n},x] && PositiveQ[a] + + +Int[Log[a_+b_.*(F_^(e_.*(c_.+d_.*x_)))^n_.],x_Symbol] := + x*Log[a+b*(F^(e*(c+d*x)))^n] - b*d*e*n*Log[F]*Int[x*(F^(e*(c+d*x)))^n/(a+b*(F^(e*(c+d*x)))^n),x] /; +FreeQ[{F,a,b,c,d,e,n},x] && Not[PositiveQ[a]] + + +(* Int[u_.*(a_.*F_^v_)^n_,x_Symbol] := + a^n*Int[u*F^(n*v),x] /; +FreeQ[{F,a},x] && IntegerQ[n] *) + + +Int[u_.*(a_.*F_^v_)^n_,x_Symbol] := + (a*F^v)^n/F^(n*v)*Int[u*F^(n*v),x] /; +FreeQ[{F,a,n},x] && Not[IntegerQ[n]] + + +Int[u_,x_Symbol] := + With[{v=FunctionOfExponential[u,x]}, + v/D[v,x]*Subst[Int[FunctionOfExponentialFunction[u,x]/x,x],x,v]] /; +FunctionOfExponentialQ[u,x] + + +Int[u_.*(a_.*F_^v_+b_.*F_^w_)^n_,x_Symbol] := + Int[u*F^(n*v)*(a+b*F^ExpandToSum[w-v,x])^n,x] /; +FreeQ[{F,a,b,n},x] && NegativeIntegerQ[n] && LinearQ[{v,w},x] + + +Int[u_.*(a_.*F_^v_+b_.*G_^w_)^n_,x_Symbol] := + Int[u*F^(n*v)*(a+b*E^ExpandToSum[Log[G]*w-Log[F]*v,x])^n,x] /; +FreeQ[{F,G,a,b,n},x] && NegativeIntegerQ[n] && LinearQ[{v,w},x] + + +Int[u_.*(a_.*F_^v_+b_.*F_^w_)^n_,x_Symbol] := + (a*F^v+b*F^w)^n/(F^(n*v)*(a+b*F^ExpandToSum[w-v,x])^n)*Int[u*F^(n*v)*(a+b*F^ExpandToSum[w-v,x])^n,x] /; +FreeQ[{F,a,b,n},x] && Not[IntegerQ[n]] && LinearQ[{v,w},x] + + +Int[u_.*(a_.*F_^v_+b_.*G_^w_)^n_,x_Symbol] := + (a*F^v+b*G^w)^n/(F^(n*v)*(a+b*E^ExpandToSum[Log[G]*w-Log[F]*v,x])^n)*Int[u*F^(n*v)*(a+b*E^ExpandToSum[Log[G]*w-Log[F]*v,x])^n,x] /; +FreeQ[{F,G,a,b,n},x] && Not[IntegerQ[n]] && LinearQ[{v,w},x] + + +Int[u_.*F_^v_*G_^w_,x_Symbol] := + Int[u*NormalizeIntegrand[E^(v*Log[F]+w*Log[G]),x],x] /; +FreeQ[{F,G},x] && (BinomialQ[v+w,x] || PolynomialQ[v+w,x] && Exponent[v+w,x]<=2) + + +Int[F_^u_*(v_+w_)*y_.,x_Symbol] := + With[{z=v*y/(Log[F]*D[u,x])}, + F^u*z /; + EqQ[D[z,x]-w*y]] /; +FreeQ[F,x] + + +Int[F_^u_*v_^n_.*w_,x_Symbol] := + With[{z=Log[F]*v*D[u,x]+(n+1)*D[v,x]}, + Coefficient[w,x,Exponent[w,x]]/Coefficient[z,x,Exponent[z,x]]*F^u*v^(n+1) /; + Exponent[w,x]==Exponent[z,x] && EqQ[w*Coefficient[z,x,Exponent[z,x]]-z*Coefficient[w,x,Exponent[w,x]]]] /; +FreeQ[{F,n},x] && PolynomialQ[u,x] && PolynomialQ[v,x] && PolynomialQ[w,x] + + + +`) + +func resourcesRubi2ExponentialsMBytes() ([]byte, error) { + return _resourcesRubi2ExponentialsM, nil +} + +func resourcesRubi2ExponentialsM() (*asset, error) { + bytes, err := resourcesRubi2ExponentialsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/2 Exponentials.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi3LogarithmsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Logarithm Function Rules*) + + +(* ::Subsection::Closed:: *) +(*3.1 u (a+b log(c (d (e+f x)^p)^q))^n*) + + +(* Int[(e_.+f_.*x_)^m_*Log[a_+b_.*x_]*(c_+d_.*x_)^n_.,x_Symbol] := + Int[ExpandIntegrand[Log[a+b*x],(e+f*x)^m*(c+d*x)^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && FractionQ[m] && IGtQ[n,0] *) + + +Int[(e_.+f_.*x_)^m_*Log[a_+b_.*x_]*(c_+d_.*x_)^n_,x_Symbol] := + With[{k=Denominator[m]}, + k/f*Subst[Int[x^(k*(m+1)-1)*Log[-(b*e-a*f)/f+b*x^k/f]*(-(d*e-c*f)/f+d*x^k/f)^n,x],x,(e+f*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e,f,n},x] && FractionQ[m] && ILtQ[n,0] + + +Int[Log[g_.*(h_.*(b_.*x_)^p_.)^q_.]*Log[c_+d_.*x_]/x_,x_Symbol] := + -Log[g*(h*(b*x)^p)^q]*PolyLog[2,-d*x] + p*q*PolyLog[3,-d*x] /; +FreeQ[{b,c,d,g,h,p,q},x] && EqQ[c,1] + + +Int[Log[g_.*(h_.*(b_.*x_)^p_.)^q_.]*Log[cd_*(c_+d_.*x_)]/x_,x_Symbol] := + -Log[g*(h*(b*x)^p)^q]*PolyLog[2,-cd*d*x] + p*q*PolyLog[3,-cd*d*x] /; +FreeQ[{b,c,cd,d,g,h,p,q},x] && EqQ[cd*c,1] + + +Int[Log[g_.*(h_.*(b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_+d_.*x_)^r_.)^s_.]/x_,x_Symbol] := + r*s*Int[Log[g*(h*(b*x)^p)^q]*Log[1+d*x]/x,x] - + (r*s*Log[1+d*x]-Log[i*(j*(1+d*x)^r)^s])*Int[Log[g*(h*(b*x)^p)^q]/x,x]/; +FreeQ[{b,c,d,g,h,i,j,p,q,r,s},x] && EqQ[c,1] && NeQ[1+d*x,i*(j*(1+d*x)^r)^s] + + +Int[Log[g_.*(h_.*(b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.]/x_,x_Symbol] := + Log[g*(h*(b*x)^p)^q]^2*Log[i*(j*(c+d*x)^r)^s]/(2*p*q) - d*r*s/(2*p*q)*Int[Log[g*(h*(b*x)^p)^q]^2/(c+d*x),x]/; +FreeQ[{b,c,d,g,h,i,j,p,q,r,s},x] && NeQ[c,1] + + +Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.]/x_,x_Symbol] := + Log[x]*Log[g*(h*(a+b*x)^p)^q]*Log[i*(j*(c+d*x)^r)^s] - + d*r*s*Int[Log[x]*Log[g*(h*(a+b*x)^p)^q]/(c+d*x),x] - + d*p*q*Int[Log[x]*Log[i*(j*(c+d*x)^r)^s]/(c+d*x),x]/; +FreeQ[{a,b,c,d,g,h,i,j,p,q,r,s},x] && EqQ[b*c-a*d,0] + + +(*Int[Log[a_+b_.*x_]*Log[c_+d_.*x_]/x_,x_Symbol] :=*) + (*Log[-b*x/a]*Log[a+b*x]*Log[c+d*x] + *) + (*1/2*Log[-b*x/a]*Log[a*(c+d*x)/(c*(a+b*x))]^2 + *) + (*1/2*Log[-(b*c-a*d)/(d*(a+b*x))]*Log[a*(c+d*x)/(c*(a+b*x))]^2 - *) + (*1/2*Log[a*(c+d*x)/(c*(a+b*x))]^2*Log[(b*c-a*d)*x/(c*(a+b*x))] - *) + (*Log[-b*x/a]*Log[a+b*x]*Log[1+d*x/c] + *) + (*Log[-(d*x/c)]*Log[a+b*x]*Log[1+d*x/c] - *) + (*Log[-b*x/a]*Log[a*(c+d*x)/(c*(a+b*x))]*Log[1+d*x/c] + *) + (*Log[-d*x/c]*Log[a*(c+d*x)/(c*(a+b*x))]*Log[1+d*x/c] + *) + (*1/2*Log[-b*x/a]*Log[1+d*x/c]^2 - *) + (*1/2*Log[-d*x/c]*Log[1+d*x/c]^2 + *) + (*Log[c+d*x]*PolyLog[2,1+b*x/a] - *) + (*Log[a*(c+d*x)/(c*(a+b*x))]*PolyLog[2,1+b*x/a] - *) + (*Log[a*(c+d*x)/(c*(a+b*x))]*PolyLog[2,a*(c+d*x)/(c*(a+b*x))] + *) + (*Log[a*(c+d*x)/(c*(a+b*x))]*PolyLog[2,b*(c+d*x)/(d*(a+b*x))] + *) + (*Log[a+b*x]*PolyLog[2,1+d*x/c] + *) + (*Log[a*(c+d*x)/(c*(a+b*x))]*PolyLog[2,1+d*x/c] - *) + (*PolyLog[3,1+b*x/a] +*) + (*PolyLog[3,a*(c+d*x)/(c*(a+b*x))] - *) + (*PolyLog[3,b*(c+d*x)/(d*(a+b*x))] - *) + (*PolyLog[3,1+d*x/c]/;*) +(*FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d,0]*) + + +Int[Log[v_]*Log[w_]/x_,x_Symbol] := + Int[Log[ExpandToSum[v,x]]*Log[ExpandToSum[w,x]]/x,x] /; +LinearQ[{v,w},x] && Not[LinearMatchQ[{v,w},x]] + + +Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.]/x_,x_Symbol] := + p*q*Int[Log[a+b*x]*Log[i*(j*(c+d*x)^r)^s]/x,x] - + (p*q*Log[a+b*x]-Log[g*(h*(a+b*x)^p)^q])*Int[Log[i*(j*(c+d*x)^r)^s]/x,x]/; +FreeQ[{a,b,c,d,g,h,i,j,p,q,r,s},x] && NeQ[b*c-a*d,0] && NeQ[a+b*x,g*(h*(a+b*x)^p)^q] + + +Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.]/(e_+f_.*x_),x_Symbol] := + 1/f*Subst[Int[Log[g*(h*Simp[-(b*e-a*f)/f+b*x/f,x]^p)^q]*Log[i*(j*Simp[-(d*e-c*f)/f+d*x/f,x]^r)^s]/x,x],x,e+f*x]/; +FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q,r,s},x] + + +Int[(e_.+f_.*x_)^m_.*Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.],x_Symbol] := + (e+f*x)^(m+1)*Log[g*(h*(a+b*x)^p)^q]*Log[i*(j*(c+d*x)^r)^s]/(f*(m+1)) - + d*r*s/(f*(m+1))*Int[(e+f*x)^(m+1)*Log[g*(h*(a+b*x)^p)^q]/(c+d*x),x] - + b*p*q/(f*(m+1))*Int[(e+f*x)^(m+1)*Log[i*(j*(c+d*x)^r)^s]/(a+b*x),x]/; +FreeQ[{a,b,c,d,e,f,g,h,i,j,m,p,q,r,s},x] && m!=-1 && RationalQ[m] + + +Int[(e_.+f_.*x_)^m_.*Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.],x_Symbol] := + Defer[Int][(e+f*x)^m*Log[g*(h*(a+b*x)^p)^q]*Log[i*(j*(c+d*x)^r)^s],x]/; +FreeQ[{a,b,c,d,e,f,g,h,i,j,m,p,q,r,s},x] + + +Int[Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.],x_Symbol] := + (e+f*x)*Log[c*(d*(e+f*x)^p)^q]/f - p*q*x /; +FreeQ[{c,d,e,f,p,q},x] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/f - b*n*p*q*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && GtQ[n,0] + + +Int[1/Log[d_.*(e_.+f_.*x_)],x_Symbol] := + LogIntegral[d*(e+f*x)]/(d*f) /; +FreeQ[{d,e,f},x] + + +Int[1/(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]),x_Symbol] := + (e+f*x)/(b*f*p*q*E^(a/(b*p*q))*(c*(d*(e+f*x)^p)^q)^(1/(p*q)))*ExpIntegralEi[(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q)] /; +FreeQ[{a,b,c,d,e,f,p,q},x] + + +Int[1/Sqrt[a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]],x_Symbol] := + Sqrt[Pi]*Rt[b*p*q,2]*(e+f*x)/(b*f*p*q*E^(a/(b*p*q))*(c*(d*(e+f*x)^p)^q)^(1/(p*q)))* + Erfi[Sqrt[a+b*Log[c*(d*(e+f*x)^p)^q]]/Rt[b*p*q,2]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && PosQ[b*p*q] + + +Int[1/Sqrt[a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]],x_Symbol] := + Sqrt[Pi]*Rt[-b*p*q,2]*(e+f*x)/(b*f*p*q*E^(a/(b*p*q))*(c*(d*(e+f*x)^p)^q)^(1/(p*q)))* + Erf[Sqrt[a+b*Log[c*(d*(e+f*x)^p)^q]]/Rt[-b*p*q,2]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && NegQ[b*p*q] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(b*f*p*q*(n+1)) - + 1/(b*p*q*(n+1))*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && LtQ[n,-1] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/ + (f*E^(a/(b*p*q))*(c*(d*(e+f*x)^p)^q)^(1/(p*q))*(-(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q))^n)* + Gamma[n+1,-(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q)] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && Not[IntegerQ[2*n]] + + +Int[1/((g_.+h_.*x_)*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])),x_Symbol] := + Log[RemoveContent[a+b*Log[c*(d*(e+f*x)^p)^q],x]]/(b*h*p*q) /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && EqQ[f*g-e*h,0] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_./(g_.+h_.*x_),x_Symbol] := + (a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(b*h*p*q*(n+1)) /; +FreeQ[{a,b,c,d,e,f,g,h,n,p,q},x] && EqQ[f*g-e*h,0] && NeQ[n,-1] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + (g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/(h*(m+1)) - + b*n*p*q/(m+1)*Int[(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] && GtQ[n,0] + + +Int[(g_.+h_.*x_)^m_./Log[d_.*(e_.+f_.*x_)^p_.],x_Symbol] := + (h/f)^(p-1)*LogIntegral[d*(e+f*x)^p]/(d*f*p) /; +FreeQ[{d,e,f,g,h,m,p},x] && EqQ[m-(p-1),0] && EqQ[f*g-e*h,0] && (IntegerQ[p] || PositiveQ[h/f]) + + +Int[(g_.+h_.*x_)^m_/Log[d_.*(e_.+f_.*x_)^p_.],x_Symbol] := + (g+h*x)^(p-1)/(e+f*x)^(p-1)*Int[(e+f*x)^(p-1)/Log[d*(e+f*x)^p],x] /; +FreeQ[{d,e,f,g,h,m,p},x] && EqQ[m-(p-1),0] && EqQ[f*g-e*h,0] && Not[IntegerQ[p] || PositiveQ[h/f]] + + +Int[(g_.+h_.*x_)^m_./(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]),x_Symbol] := + (g+h*x)^(m+1)/(b*h*p*q*E^(a*(m+1)/(b*p*q))*(c*(d*(e+f*x)^p)^q)^((m+1)/(p*q)))* + ExpIntegralEi[(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q)] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] + + +Int[(g_.+h_.*x_)^m_./Sqrt[a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]],x_Symbol] := + Sqrt[Pi]*(g+h*x)^(m+1)/(b*h*p*q*Rt[(m+1)/(b*p*q),2]*E^(a*(m+1)/(b*p*q))*(c*(d*(e+f*x)^p)^q)^((m+1)/(p*q)))* + Erfi[Rt[(m+1)/(b*p*q),2]*Sqrt[a+b*Log[c*(d*(e+f*x)^p)^q]]] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] && PosQ[(m+1)/(b*p*q)] + + +Int[(g_.+h_.*x_)^m_./Sqrt[a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]],x_Symbol] := + Sqrt[Pi]*(g+h*x)^(m+1)/(b*h*p*q*Rt[-(m+1)/(b*p*q),2]*E^(a*(m+1)/(b*p*q))*(c*(d*(e+f*x)^p)^q)^((m+1)/(p*q)))* + Erf[Rt[-(m+1)/(b*p*q),2]*Sqrt[a+b*Log[c*(d*(e+f*x)^p)^q]]] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] && NegQ[(m+1)/(b*p*q)] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(b*h*p*q*(n+1)) - + (m+1)/(b*p*q*(n+1))*Int[(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] && LtQ[n,-1] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + (g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/ + (h*(m+1)*E^(a*(m+1)/(b*p*q))*(c*(d*(e+f*x)^p)^q)^((m+1)/(p*q))*(-(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q))^n)* + Gamma[n+1,-(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])/(b*p*q)] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[f*g-e*h,0] && NeQ[m,-1] + + +Int[Log[c_.*(e_.+f_.*x_)]/(g_.+h_.*x_),x_Symbol] := + -1/h*PolyLog[2,-Together[c*f/h]*(g+h*x)] /; +FreeQ[{c,e,f,g,h},x] && EqQ[h+c*(f*g-e*h),0] + + +Int[(a_.+b_.*Log[c_.*(e_.+f_.*x_)])/(g_.+h_.*x_),x_Symbol] := + (a+b*Log[c*(e-f*g/h)])*Log[g+h*x]/h + b*Int[Log[-h*(e+f*x)/(f*g-e*h)]/(g+h*x),x] /; +FreeQ[{a,b,c,e,f,g,h},x] && NeQ[h+c*(f*g-e*h),0] && PositiveQ[c*(e-f*g/h)] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_./(g_.+h_.*x_),x_Symbol] := + (a+b*Log[c*(d*(e+f*x)^p)^q])^n/h*Log[f*(g+h*x)/(f*g-e*h)] - + b*f*n*p*q/h*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1)*Log[f*(g+h*x)/(f*g-e*h)]/(e+f*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && NeQ[f*g-e*h,0] && PositiveIntegerQ[n] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]),x_Symbol] := + (g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])/(h*(m+1)) - + b*f*p*q/(h*(m+1))*Int[(g+h*x)^(m+1)/(e+f*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && NeQ[f*g-e*h,0] && NeQ[m,-1] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_/(g_.+h_.*x_)^2,x_Symbol] := + (e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/((f*g-e*h)*(g+h*x)) - + b*f*n*p*q/(f*g-e*h)*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1)/(g+h*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && NeQ[f*g-e*h,0] && GtQ[n,0] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/(h*(m+1)) - + b*f*n*p*q/(h*(m+1))*Int[(g+h*x)^(m+1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1)/(e+f*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q},x] && NeQ[f*g-e*h,0] && GtQ[n,0] && NeQ[m,-1] && IntegersQ[2*m,2*n] && + (n==1 || Not[PositiveIntegerQ[m]] || n==2 && NeQ[m,1]) + + +Int[(g_.+h_.*x_)^m_./(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]),x_Symbol] := + Int[ExpandIntegrand[(g+h*x)^m/(a+b*Log[c*(d*(e+f*x)^p)^q]),x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && NeQ[f*g-e*h,0] && PositiveIntegerQ[m] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + (e+f*x)*(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(b*f*p*q*(n+1)) + + m*(f*g-e*h)/(b*f*p*q*(n+1))*Int[(g+h*x)^(m-1)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1),x] - + (m+1)/(b*p*q*(n+1))*Int[(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && NeQ[f*g-e*h,0] && LtQ[n,-1] && GtQ[m,0] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + Int[ExpandIntegrand[(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,h,n,p,q},x] && NeQ[f*g-e*h,0] && PositiveIntegerQ[m] + + +Int[u_^m_.*(a_.+b_.*Log[c_.*(d_.*v_^p_)^q_.])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Log[c*(d*ExpandToSum[v,x]^p)^q])^n,x] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + Defer[Int][(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p,q},x] + + +Int[Log[c_./(e_.+f_.*x_)]/((g_.+h_.*x_)*(i_.+j_.*x_)),x_Symbol] := + f/(h*(f*i-e*j))*PolyLog[2,Simplify[f*(i+j*x)/(j*(e+f*x))]] /; +FreeQ[{c,e,f,g,h,i,j},x] && EqQ[f*g-e*h,0] && EqQ[f*i+j*(c-e),0] + + +Int[(a_+b_.*Log[c_./(e_.+f_.*x_)])/((g_.+h_.*x_)*(i_.+j_.*x_)),x_Symbol] := + a*Int[1/((g+h*x)*(i+j*x)),x] + b*Int[Log[c/(e+f*x)]/((g+h*x)*(i+j*x)),x] /; +FreeQ[{a,b,c,e,f,g,h,i,j},x] && EqQ[f*g-e*h,0] && EqQ[f*i+j*(c-e),0] + + +Int[(i_.+j_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/(g_.+h_.*x_),x_Symbol] := + With[{u=IntHide[(i+j*x)^m/(g+h*x),x]}, + Dist[a+b*Log[c*(d*(e+f*x)^p)^q],u] - b*h*p*q*Int[SimplifyIntegrand[u/(g+h*x),x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q},x] && EqQ[f*g-e*h,0] && IntegerQ[m+1/2] + + +Int[(i_.+j_.*x_)^m_.*(a_.+b_.*Log[c_.*(e_.+f_.*x_)])^n_./(g_.+h_.*x_),x_Symbol] := + 1/(c^m*f^m*h)*Subst[Int[(a+b*x)^n*(c*f*i-c*e*j+j*E^x)^m,x],x,Log[c*(e+f*x)]] /; +FreeQ[{a,b,c,e,f,g,h,i,j,n},x] && EqQ[f*g-e*h,0] && PositiveIntegerQ[m] && (IntegerQ[n] || m>0) + + +Int[(i_.+j_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_./(g_.+h_.*x_),x_Symbol] := + With[{u=ExpandIntegrand[(a+b*Log[c*(d*(e+f*x)^p)^q])^n,(i+j*x)^m/(g+h*x),x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q},x] && IntegerQ[m] && PositiveIntegerQ[n] + + +Int[(i_.+j_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_./(g_.+h_.*x_),x_Symbol] := + Defer[Int][(i+j*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^n/(g+h*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,m,n,p,q},x] + + +Int[Log[j_.*(k_.+m_.*x_)]/(g_.+h_.*x_)*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + -PolyLog[2,Together[1-j*(k+m*x)]]/h*(a+b*Log[c*(d*(e+f*x)^p)^q])^n + + b*f*n*p*q/h*Int[PolyLog[2,Together[1-j*(k+m*x)]]/(e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,j,k,m,p,q},x] && EqQ[h-h*j*k+g*j*m,0] && GtQ[n,0] + + +Int[Log[i_.*(j_.*(k_.+m_.*x_)^r_.)^s_.]*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_./(g_.+h_.*x_),x_Symbol] := + Log[i*(j*(k+m*x)^r)^s]*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(b*h*p*q*(n+1)) - + m*r*s/(b*h*p*q*(n+1))*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^(n+1)/(k+m*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,k,m,n,p,q,r,s},x] && EqQ[f*g-e*h,0] && NeQ[n,-1] + + +Int[Log[c_./(e_.+f_.*x_)]/(g_+h_.*x_^2),x_Symbol] := + -f/(2*e*h)*PolyLog[2,Simplify[(-e+f*x)/(e+f*x)]] /; +FreeQ[{c,e,f,g,h},x] && EqQ[f^2*g+e^2*h,0] && EqQ[c-2*e,0] + + +Int[(a_.+b_.*Log[c_./(e_.+f_.*x_)])/(g_+h_.*x_^2),x_Symbol] := + (a+b*Log[c/(2*e)])*Int[1/(g+h*x^2),x] + b*Int[Log[2*e/(e+f*x)]/(g+h*x^2),x] /; +FreeQ[{c,e,f,g,h},x] && EqQ[f^2*g+e^2*h,0] && PositiveQ[c/(2*e)] && (NeQ[c-2*e,0] || NeQ[a,0]) + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/(g_.+h_.*x_+i_.*x_^2),x_Symbol] := + e*f*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])/((e+f*x)*(f*g+e*i*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,p,q},x] && EqQ[f^2*g-e*f*h+e^2*i,0] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_+f_.*x_)^p_.)^q_.])/(g_+i_.*x_^2),x_Symbol] := + e*f*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])/((e+f*x)*(f*g+e*i*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,i,p,q},x] && EqQ[f^2*g+e^2*i,0] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/Sqrt[g_+h_.*x_^2],x_Symbol] := + With[{u=IntHide[1/Sqrt[g+h*x^2],x]}, + u*(a+b*Log[c*(d*(e+f*x)^p)^q]) - b*f*p*q*Int[SimplifyIntegrand[u/(e+f*x),x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && PositiveQ[g] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/(Sqrt[g1_+h1_.*x_]*Sqrt[g2_+h2_.*x_]),x_Symbol] := + With[{u=IntHide[1/Sqrt[g1*g2+h1*h2*x^2],x]}, + u*(a+b*Log[c*(d*(e+f*x)^p)^q]) - b*f*p*q*Int[SimplifyIntegrand[u/(e+f*x),x],x]] /; +FreeQ[{a,b,c,d,e,f,g1,h1,g2,h2,p,q},x] && EqQ[g2*h1+g1*h2,0] && PositiveQ[g1] && PositiveQ[g2] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/Sqrt[g_+h_.*x_^2],x_Symbol] := + Sqrt[1+h/g*x^2]/Sqrt[g+h*x^2]*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])/Sqrt[1+h/g*x^2],x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && Not[PositiveQ[g]] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])/(Sqrt[g1_+h1_.*x_]*Sqrt[g2_+h2_.*x_]),x_Symbol] := + Sqrt[1+h1*h2/(g1*g2)*x^2]/(Sqrt[g1+h1*x]*Sqrt[g2+h2*x])*Int[(a+b*Log[c*(d*(e+f*x)^p)^q])/Sqrt[1+h1*h2/(g1*g2)*x^2],x] /; +FreeQ[{a,b,c,d,e,f,g1,h1,g2,h2,p,q},x] && EqQ[g2*h1+g1*h2,0] + + +Int[Log[1+i_.*(j_.+k_.*x_)^m_.]/(g_.+h_.*x_)*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + -PolyLog[2,-i*(j+k*x)^m]/(h*m)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n + + b*f*n*p*q/(h*m)*Int[PolyLog[2,-i*(j+k*x)^m]/(e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,k,m,p,q},x] && GtQ[n,0] && EqQ[h*j-g*k,0] + + +Int[PolyLog[r_,i_.*(j_.+k_.*x_)^m_.]/(g_.+h_.*x_)*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + PolyLog[r+1,i*(j+k*x)^m]/(h*m)*(a+b*Log[c*(d*(e+f*x)^p)^q])^n - + b*f*n*p*q/(h*m)*Int[PolyLog[r+1,i*(j+k*x)^m]/(e+f*x)*(a+b*Log[c*(d*(e+f*x)^p)^q])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,k,m,p,q,r},x] && GtQ[n,0] && EqQ[h*j-g*k,0] + + +Int[Px_.*F_[g_.+h_.*x_]^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.]),x_Symbol] := + With[{u=IntHide[Px*F[g+h*x]^m,x]}, + Dist[(a+b*Log[c*(d*(e+f*x)^p)^q]),u,x] - b*f*p*q*Int[SimplifyIntegrand[u/(e+f*x),x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q},x] && PolynomialQ[Px,x] && PositiveIntegerQ[m] && + MemberQ[{Log,ArcSin,ArcCos,ArcTan,ArcCot,ArcSinh, ArcCosh,ArcTanh, ArcCoth},F] + + +Int[(a_.+b_.*Log[c_.*(d_.*(e_+f_.*x_^m_)^p_.)^q_.])^n_./x_,x_Symbol] := + 1/m*Subst[Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^n/x,x],x,x^m] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*Log[c_.*(d_.*(x_^m_*(f_+e_.*x_^r_.))^p_.)^q_.])^n_./x_,x_Symbol] := + 1/m*Subst[Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^n/x,x],x,x^m] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[m+r,0] && PositiveIntegerQ[n] + + +Int[x_^r1_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_^r_)^p_.)^q_.])^n_.,x_Symbol] := + 1/r*Subst[Int[(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x],x,x^r] /; +FreeQ[{a,b,c,d,e,f,n,p,q,r},x] && EqQ[r1-(r-1),0] + + +Int[x_^r1_.*(g_.+h_.*x_^r_)^m_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_^r_)^p_.)^q_.])^n_.,x_Symbol] := + 1/r*Subst[Int[(g+h*x)^m*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x],x,x^r] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p,q,r},x] && EqQ[r1-(r-1),0] + + +Int[(a_.+b_.*Log[c_.*x_^n_.])/(d_+e_.*x_^2),x_Symbol] := + With[{u=IntHide[1/(d+e*x^2),x]}, + Dist[(a+b*Log[c*x^n]),u] - b*n*Int[u/x,x]] /; +FreeQ[{a,b,c,d,e,n},x] + + +Int[Log[c_.*(a_.+b_.*x_^mn_)]/(x_*(d_+e_.*x_^n_.)),x_Symbol] := + 1/(d*n)*PolyLog[2,-Together[b*c*(d+e*x^n)/(d*x^n)]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n+mn,0] && EqQ[d-a*c*d+b*c*e,0] + + +Int[Log[c_.*x_^mn_*(b_.+a_.*x_^n_.)]/(x_*(d_+e_.*x_^n_.)),x_Symbol] := + 1/(d*n)*PolyLog[2,-Together[b*c*(d+e*x^n)/(d*x^n)]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n+mn,0] && EqQ[d-a*c*d+b*c*e,0] + + +Int[Px_*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + Int[ExpandIntegrand[Px*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,n,p,q},x] && PolynomialQ[Px,x] + + +Int[RFx_*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(a+b*Log[c*(d*(e+f*x)^p)^q])^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[RFx*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,p,q},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[u_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_+g_.*x_^2)^p_.)^q_.])^n_.,x_Symbol] := + Int[u*(a+b*Log[c*(d*(f+2*g*x)^(2*p)/(4^p*g^p))^q])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,q,n},x] && EqQ[f^2-4*e*g,0] && IntegerQ[p] + + +Int[u_.*(a_.+b_.*Log[c_.*(d_.*v_^p_.)^q_.])^n_.,x_Symbol] := + Int[u*(a+b*Log[c*(d*ExpandToSum[v,x]^p)^q])^n,x] /; +FreeQ[{a,b,c,d,n,p,q},x] && LinearQ[v,x] && + Not[MatchQ[c*(d*v^p)^q, cc_.*(dd_.*(e_.+f_.*x)^pp_.)^qq_. /; FreeQ[{cc,dd,e,f,pp,qq},x]]] + + +Int[Log[a_.*(b_.*(c_.*x_^n_.)^p_)^q_]^r_.,x_Symbol] := + Subst[Int[Log[x^(n*p*q)]^r,x],x^(n*p*q),a*(b*(c*x^n)^p)^q] /; +FreeQ[{a,b,c,n,p,q,r},x] + + +Int[x_^m_.*Log[a_.*(b_.*(c_.*x_^n_.)^p_)^q_]^r_.,x_Symbol] := + Subst[Int[x^m*Log[x^(n*p*q)]^r,x],x^(n*p*q),a*(b*(c*x^n)^p)^q] /; +FreeQ[{a,b,c,m,n,p,q,r},x] && NeQ[m,-1] && Not[x^(n*p*q)===a*(b*(c*x^n)^p)^q] + + + + + +(* ::Subsection::Closed:: *) +(*3.2 u log(e (f (a+b x)^p (c+d x)^q)^r)^s*) + + +Int[u_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + Int[u*Log[e*(b^p*f/d^p*(c+d*x)^(p+q))^r]^s,x] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && EqQ[b*c-a*d,0] && IntegerQ[p] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + (a+b*x)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/b - + r*s/b*Int[(b*c*p+a*d*q+b*d*(p+q)*x)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] + + +Int[Log[e_.*(a_.+b_.*x_)/(c_.+d_.*x_)]/(g_.+h_.*x_),x_Symbol] := + 1/h*PolyLog[2,Together[c-a*e]/(c+d*x)] /; +FreeQ[{a,b,c,d,e,g,h},x] && NeQ[b*c-a*d,0] && EqQ[d*g-c*h,0] && EqQ[d-b*e,0] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_./(g_.+h_.*x_),x_Symbol] := + -Log[-(b*c-a*d)/(d*(a+b*x))]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/h + + p*r*s (b*c-a*d)/h*Int[Log[-(b*c-a*d)/(d*(a+b*x))]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((a+b*x)*(c+d*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[b*g-a*h,0] && EqQ[p+q,0] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]/(g_.+h_.*x_),x_Symbol] := + Log[g+h*x]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]/h - + b*p*r/h*Int[Log[g+h*x]/(a+b*x),x] - + d*q*r/h*Int[Log[g+h*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r},x] && NeQ[b*c-a*d,0] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_/(g_.+h_.*x_),x_Symbol] := + d/h*Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(c+d*x),x] - + (d*g-c*h)/h*Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/((c+d*x)*(g+h*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,1] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_./(g_.+h_.*x_)^2,x_Symbol] := + (a+b*x)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/((b*g-a*h)*(g+h*x)) - + p*r*s*(b*c-a*d)/(b*g-a*h)*Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((c+d*x)*(g+h*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[p+q,0] && NeQ[b*g-a*h,0] + + +Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_/(g_.+h_.*x_)^3,x_Symbol] := + d/(d*g-c*h)*Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(g+h*x)^2,x] - + h/(d*g-c*h)*Int[(c+d*x)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(g+h*x)^3,x] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[p+q,0] && EqQ[b*g-a*h,0] && NeQ[d*g-c*h,0] + + +Int[(g_.+h_.*x_)^m_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + (g+h*x)^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(h*(m+1)) - + p*r*s*(b*c-a*d)/(h*(m+1))*Int[(g+h*x)^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((a+b*x)*(c+d*x)),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && NeQ[m,-1] && EqQ[p+q,0] + + +Int[(g_.+h_.*x_)^m_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + (g+h*x)^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(h*(m+1)) - + b*p*r*s/(h*(m+1))*Int[(g+h*x)^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/(a+b*x),x] - + d*q*r*s/(h*(m+1))*Int[(g+h*x)^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && NeQ[m,-1] && NeQ[p+q,0] + + +Int[1/((g_.+h_.*x_)^2*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]),x_Symbol] := + b*(c+d*x)*(e*(f*(a+b*x)^p*(c+d*x)^q)^r)^(1/(p*r))/(h*p*r*(b*c-a*d)*(g+h*x))* + ExpIntegralEi[-1/(p*r)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]] /; +FreeQ[{a,b,c,d,e,f,g,h,p,q,r},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] && EqQ[b*g-a*h,0] + + +Int[Log[i_.*(j_.*(h_.*x_)^t_.)^u_.]^m_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]/x_,x_Symbol] := + Log[i*(j*(h*x)^t)^u]^(m+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]/(t*u*(m+1)) - + b*p*r/(t*u*(m+1))*Int[Log[i*(j*(h*x)^t)^u]^(m+1)/(a+b*x),x] - + d*q*r/(t*u*(m+1))*Int[Log[i*(j*(h*x)^t)^u]^(m+1)/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f,h,i,j,m,p,q,r,t,u},x] && NeQ[b*c-a*d,0] && IGtQ[m,0] + + +Int[Log[i_.*(j_.*(h_.*x_)^t_.)^u_.]^m_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_./x_,x_Symbol] := + Defer[Int][Log[i*(j*(h*x)^t)^u]^m*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/x,x] /; +FreeQ[{a,b,c,d,e,f,h,i,j,m,p,q,r,s,t,u},x] && NeQ[b*c-a*d,0] + + +Int[u_*Log[e_.*(c_.+d_.*x_)/(a_.+b_.*x_)],x_Symbol] := + With[{g=Coeff[Simplify[1/(u*(a+b*x))],x,0],h=Coeff[Simplify[1/(u*(a+b*x))],x,1]}, + -(b-d*e)*PolyLog[2,(b-d*e)*(g+h*x)/(h*(a+b*x))]/(e*h*(b*c-a*d)) /; + EqQ[g*(b-d*e)-h*(a-c*e),0]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*c-a*d,0] && LinearQ[Simplify[1/(u*(a+b*x))],x] + + +Int[u_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{g=Coeff[Simplify[1/(u*(a+b*x))],x,0],h=Coeff[Simplify[1/(u*(a+b*x))],x,1]}, + -Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(b*g-a*h)*Log[-(b*c-a*d)*(g+h*x)/((d*g-c*h)*(a+b*x))] + + p*r*s*(b*c-a*d)/(b*g-a*h)* + Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((a+b*x)*(c+d*x))*Log[-(b*c-a*d)*(g+h*x)/((d*g-c*h)*(a+b*x))],x] /; + NeQ[b*g-a*h,0] && NeQ[d*g-c*h,0]] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[p+q,0] && LinearQ[Simplify[1/(u*(a+b*x))],x] + + +Int[u_/Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_Symbol] := + With[{h=Simplify[u*(a+b*x)*(c+d*x)]}, + h*Log[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]]/(p*r*(b*c-a*d)) /; + FreeQ[h,x]] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] + + +Int[u_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{h=Simplify[u*(a+b*x)*(c+d*x)]}, + h*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s+1)/(p*r*(s+1)*(b*c-a*d)) /; + FreeQ[h,x]] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] && NeQ[s,-1] + + +Int[u_*Log[v_]*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{g=Simplify[(v-1)*(c+d*x)/(a+b*x)],h=Simplify[u*(a+b*x)*(c+d*x)]}, + -h*PolyLog[2,-g*(a+b*x)/(c+d*x)]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(b*c-a*d) + + h*p*r*s*Int[PolyLog[2,-g*(a+b*x)/(c+d*x)]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((a+b*x)*(c+d*x)),x] /; + FreeQ[{g,h},x]] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[p+q,0] + + +Int[v_*Log[i_.*(j_.*(g_.+h_.*x_)^t_.)^u_.]*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{k=Simplify[v*(a+b*x)*(c+d*x)]}, + k*Log[i*(j*(g+h*x)^t)^u]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s+1)/(p*r*(s+1)*(b*c-a*d)) - + k*h*t*u/(p*r*(s+1)*(b*c-a*d))*Int[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s+1)/(g+h*x),x] /; + FreeQ[k,x]] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q,r,s,t,u},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] && NeQ[s,-1] + + +Int[u_*PolyLog[n_,v_]*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{g=Simplify[v*(c+d*x)/(a+b*x)],h=Simplify[u*(a+b*x)*(c+d*x)]}, + h*PolyLog[n+1,g*(a+b*x)/(c+d*x)]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/(b*c-a*d) - + h*p*r*s*Int[PolyLog[n+1,g*(a+b*x)/(c+d*x)]*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1)/((a+b*x)*(c+d*x)),x] /; + FreeQ[{g,h},x]] /; +FreeQ[{a,b,c,d,e,f,n,p,q,r,s},x] && NeQ[b*c-a*d,0] && IGtQ[s,0] && EqQ[p+q,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^(n+1)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s/((m+1)*(b*c-a*d)) - + p*r*s*(b*c-a*d)/((m+1)*(b*c-a*d))*Int[(a+b*x)^m*(c+d*x)^n*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^(s-1),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r,s},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] && EqQ[m+n+2,0] && NeQ[m,-1] && IGtQ[s,0] + + +Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_./Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_Symbol] := + (a+b*x)^(m+1)*(c+d*x)^(n+1)/(p*r*(b*c-a*d)*(e*(f*(a+b*x)^p*(c+d*x)^q)^r)^((m+1)/(p*r)))* + ExpIntegralEi[(m+1)/(p*r)*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q,r},x] && NeQ[b*c-a*d,0] && EqQ[p+q,0] && EqQ[m+n+2,0] && NeQ[m,-1] + + +Int[RFx_.*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_Symbol] := + p*r*Int[RFx*Log[a+b*x],x] + + q*r*Int[RFx*Log[c+d*x],x] - + (p*r*Log[a+b*x]+q*r*Log[c+d*x] - Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r])*Int[RFx,x] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && RationalFunctionQ[RFx,x] && NeQ[b*c-a*d,0] && + Not[MatchQ[RFx,u_.*(a+b*x)^m_.*(c+d*x)^n_. /; IntegersQ[m,n]]] + + +(* Int[RFx_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_Symbol] := + With[{u=IntHide[RFx,x]}, + u*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r] - b*p*r*Int[u/(a+b*x),x] - d*q*r*Int[u/(c+d*x),x] /; + NonsumQ[u]] /; +FreeQ[{a,b,c,d,e,f,p,q,r},x] && RationalFunctionQ[RFx,x] && NeQ[b*c-a*d,0] *) + + +Int[RFx_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + With[{u=ExpandIntegrand[Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && RationalFunctionQ[RFx,x] && IGtQ[s,0] + + +Int[RFx_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^s_.,x_Symbol] := + Defer[Int][RFx*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]^s,x] /; +FreeQ[{a,b,c,d,e,f,p,q,r,s},x] && RationalFunctionQ[RFx,x] + + +Int[u_.*Log[e_.*(f_.*v_^p_.*w_^q_.)^r_.]^s_.,x_Symbol] := + Int[u*Log[e*(f*ExpandToSum[v,x]^p*ExpandToSum[w,x]^q)^r]^s,x] /; +FreeQ[{e,f,p,q,r,s},x] && LinearQ[{v,w},x] && Not[LinearMatchQ[{v,w},x]] + + +Int[u_.*Log[e_.*(f_.*(g_+v_/w_))^r_.]^s_.,x_Symbol] := + Int[u*Log[e*(f*ExpandToSum[g*w+v,x]/ExpandToSum[w,x])^r]^s,x] /; +FreeQ[{e,f,g,r,s},x] && LinearQ[{v,w},x] + + +(* Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^r_.)^s_.]/(e_+f_.*x_),x_Symbol] := + 1/f*Subst[Int[Log[g*(h*Simp[-(b*e-a*f)/f+b*x/f,x]^p)^q]*Log[i*(j*Simp[-(d*e-c*f)/f+d*x/f,x]^r)^s]/x,x],x,e+f*x] /; +FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q,r,s},x] *) + + + + + +(* ::Subsection::Closed:: *) +(*3.3 Miscellaneous logarithms*) + + +Int[Log[c_.*(a_.+b_.*x_^n_)^p_.],x_Symbol] := + x*Log[c*(a+b*x^n)^p] - + b*n*p*Int[x^n/(a+b*x^n),x] /; +FreeQ[{a,b,c,n,p},x] + + +Int[Log[c_.*v_^p_.],x_Symbol] := + Int[Log[c*ExpandToSum[v,x]^p],x] /; +FreeQ[{c,p},x] && BinomialQ[v,x] && Not[BinomialMatchQ[v,x]] + + +Int[(a_.+b_.*Log[c_.*(d_.+e_.*x_^n_)^p_.])/(f_.+g_.*x_),x_Symbol] := + Log[f+g*x]*(a+b*Log[c*(d+e*x^n)^p])/g - + b*e*n*p/g*Int[x^(n-1)*Log[f+g*x]/(d+e*x^n),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] + + +Int[(f_.+g_.*x_)^m_.*(a_.+b_.*Log[c_.*(d_.+e_.*x_^n_)^p_.]),x_Symbol] := + (f+g*x)^(m+1)*(a+b*Log[c*(d+e*x^n)^p])/(g*(m+1)) - + b*e*n*p/(g*(m+1))*Int[x^(n-1)*(f+g*x)^(m+1)/(d+e*x^n),x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[m+1] + + +Int[u_^m_.*(a_.+b_.*Log[c_.*v_^p_.]),x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Log[c*ExpandToSum[v,x]^p]),x] /; +FreeQ[{a,b,c,m,p},x] && LinearQ[u,x] && BinomialQ[v,x] && Not[LinearMatchQ[u,x] && BinomialMatchQ[v,x]] + + +Int[ArcSin[f_.+g_.*x_]^m_.*(a_.+b_.*Log[c_.*(d_.+e_.*x_^n_)^p_.]),x_Symbol] := + With[{w=IntHide[ArcSin[f+g*x]^m,x]}, + Dist[a+b*Log[c*(d+e*x^n)^p],w,x] - + b*e*n*p*Int[SimplifyIntegrand[x^(n-1)*w/(d+e*x^n),x],x]] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && PositiveIntegerQ[m] + + +Int[(a_.+b_.*Log[c_.*(d_.+e_.*x_^2)^p_.])/(f_.+g_.*x_^2),x_Symbol] := + With[{u=IntHide[1/(f+g*x^2),x]}, + u*(a+b*Log[c*(d+e*x^2)^p]) - 2*b*e*p*Int[(x*u)/(d+e*x^2),x]] /; +FreeQ[{a,b,c,d,e,f,g,p},x] + + +Int[(a_.+b_.*Log[c_.*(d_+e_.*x_^2)^p_.])^n_,x_Symbol] := + x*(a+b*Log[c*(d+e*x^2)^p])^n - + 2*b*e*n*p*Int[x^2*(a+b*Log[c*(d+e*x^2)^p])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && PositiveIntegerQ[n] + + +Int[x_^m_.*(a_.+b_.*Log[c_.*(d_+e_.*x_^2)^p_.])^n_,x_Symbol] := + 1/2*Subst[Int[x^((m-1)/2)*(a+b*Log[c*(d+e*x)^p])^n,x],x,x^2] /; +FreeQ[{a,b,c,d,e,p},x] && PositiveIntegerQ[n] && IntegerQ[(m-1)/2] + + +Int[x_^m_.*(a_.+b_.*Log[c_.*(d_+e_.*x_^2)^p_.])^n_,x_Symbol] := + x^(m+1)*(a+b*Log[c*(d+e*x^2)^p])^n/(m+1) - + 2*b*e*n*p/(m+1)*Int[x^(m+2)*(a+b*Log[c*(d+e*x^2)^p])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && PositiveIntegerQ[n] && Not[IntegerQ[(m-1)/2]] + + +Int[u_*Log[v_],x_Symbol] := + With[{w=DerivativeDivides[v,u*(1-v),x]}, + w*PolyLog[2,Together[1-v]] /; + Not[FalseQ[w]]] + + +Int[(a_.+b_.*Log[u_])*Log[v_]*w_,x_Symbol] := + With[{z=DerivativeDivides[v,w*(1-v),x]}, + z*(a+b*Log[u])*PolyLog[2,Together[1-v]] - + b*Int[SimplifyIntegrand[z*PolyLog[2,Together[1-v]]*D[u,x]/u,x],x] /; + Not[FalseQ[z]]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] + + +Int[Log[c_.*(a_+b_.*(d_.+e_.*x_)^n_)^p_.],x_Symbol] := + (d+e*x)*Log[c*(a+b*(d+e*x)^n)^p]/e - + b*n*p*Int[1/(b+a*(d+e*x)^(-n)),x] /; +FreeQ[{a,b,c,d,e,p},x] && RationalQ[n] && n<0 + + +Int[Log[c_.*(a_+b_.*(d_.+e_.*x_)^n_.)^p_.],x_Symbol] := + (d+e*x)*Log[c*(a+b*(d+e*x)^n)^p]/e - n*p*x + + a*n*p*Int[1/(a+b*(d+e*x)^n),x] /; +FreeQ[{a,b,c,d,e,n,p},x] && Not[RationalQ[n] && n<0] + + +Int[(a_.+b_.*Log[c_.*(d_+e_./(f_.+g_.*x_))^p_.])^n_.,x_Symbol] := + (e+d*(f+g*x))*(a+b*Log[c*(d+e/(f+g*x))^p])^n/(d*g) - + b*e*n*p/(d*g)*Subst[Int[(a+b*Log[c*(d+e*x)^p])^(n-1)/x,x],x,1/(f+g*x)] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*Log[c_.*RFx_^p_.])^n_.,x_Symbol] := + x*(a+b*Log[c*RFx^p])^n - + b*n*p*Int[SimplifyIntegrand[x*(a+b*Log[c*RFx^p])^(n-1)*D[RFx,x]/RFx,x],x] /; +FreeQ[{a,b,c,p},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*Log[c_.*RFx_^p_.])^n_./(d_.+e_.*x_),x_Symbol] := + Log[d+e*x]*(a+b*Log[c*RFx^p])^n/e - + b*n*p/e*Int[Log[d+e*x]*(a+b*Log[c*RFx^p])^(n-1)*D[RFx,x]/RFx,x] /; +FreeQ[{a,b,c,d,e,p},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*Log[c_.*RFx_^p_.])^n_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*Log[c*RFx^p])^n/(e*(m+1)) - + b*n*p/(e*(m+1))*Int[SimplifyIntegrand[(d+e*x)^(m+1)*(a+b*Log[c*RFx^p])^(n-1)*D[RFx,x]/RFx,x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && (n==1 || IntegerQ[m]) && NeQ[m+1] + + +Int[Log[c_.*RFx_^n_.]/(d_+e_.*x_^2),x_Symbol] := + With[{u=IntHide[1/(d+e*x^2),x]}, + u*Log[c*RFx^n] - n*Int[SimplifyIntegrand[u*D[RFx,x]/RFx,x],x]] /; +FreeQ[{c,d,e,n},x] && RationalFunctionQ[RFx,x] && Not[PolynomialQ[RFx,x]] + + +Int[Log[c_.*Px_^n_.]/Qx_,x_Symbol] := + With[{u=IntHide[1/Qx,x]}, + u*Log[c*Px^n] - n*Int[SimplifyIntegrand[u*D[Px,x]/Px,x],x]] /; +FreeQ[{c,n},x] && QuadraticQ[{Qx,Px},x] && EqQ[D[Px/Qx,x]] + + +Int[RGx_*(a_.+b_.*Log[c_.*RFx_^p_.])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(a+b*Log[c*RFx^p])^n,RGx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,p},x] && RationalFunctionQ[RFx,x] && RationalFunctionQ[RGx,x] && PositiveIntegerQ[n] + + +Int[RGx_*(a_.+b_.*Log[c_.*RFx_^p_.])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[RGx*(a+b*Log[c*RFx^p])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,p},x] && RationalFunctionQ[RFx,x] && RationalFunctionQ[RGx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_.+b_.*Log[u_]),x_Symbol] := + With[{lst=SubstForFractionalPowerOfLinear[RFx*(a+b*Log[u]),x]}, + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])] /; + Not[FalseQ[lst]]] /; +FreeQ[{a,b},x] && RationalFunctionQ[RFx,x] + + +Int[(f_.+g_.*x_)^m_.*Log[1+e_.*(F_^(c_.*(a_.+b_.*x_)))^n_.],x_Symbol] := + -(f+g*x)^m*PolyLog[2,-e*(F^(c*(a+b*x)))^n]/(b*c*n*Log[F]) + + g*m/(b*c*n*Log[F])*Int[(f+g*x)^(m-1)*PolyLog[2,-e*(F^(c*(a+b*x)))^n],x] /; +FreeQ[{F,a,b,c,e,f,g,n},x] && RationalQ[m] && m>0 + + +Int[(f_.+g_.*x_)^m_.*Log[d_+e_.*(F_^(c_.*(a_.+b_.*x_)))^n_.],x_Symbol] := + (f+g*x)^(m+1)*Log[d+e*(F^(c*(a+b*x)))^n]/(g*(m+1)) - + (f+g*x)^(m+1)*Log[1+e/d*(F^(c*(a+b*x)))^n]/(g*(m+1)) + + Int[(f+g*x)^m*Log[1+e/d*(F^(c*(a+b*x)))^n],x] /; +FreeQ[{F,a,b,c,d,e,f,g,n},x] && RationalQ[m] && m>0 && NeQ[d-1] + + +Int[Log[d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2]],x_Symbol] := + x*Log[d+e*x+f*Sqrt[a+b*x+c*x^2]] + + f^2*(b^2-4*a*c)/2*Int[x/((2*d*e-b*f^2)*(a+b*x+c*x^2)-f*(b*d-2*a*e+(2*c*d-b*e)*x)*Sqrt[a+b*x+c*x^2]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e^2-c*f^2] + + +Int[Log[d_.+e_.*x_+f_.*Sqrt[a_.+c_.*x_^2]],x_Symbol] := + x*Log[d+e*x+f*Sqrt[a+c*x^2]] - + a*c*f^2*Int[x/(d*e*(a+c*x^2)+f*(a*e-c*d*x)*Sqrt[a+c*x^2]),x] /; +FreeQ[{a,c,d,e,f},x] && EqQ[e^2-c*f^2] + + +Int[(g_.*x_)^m_.*Log[d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2]],x_Symbol] := + (g*x)^(m+1)*Log[d+e*x+f*Sqrt[a+b*x+c*x^2]]/(g*(m+1)) + + f^2*(b^2-4*a*c)/(2*g*(m+1))*Int[(g*x)^(m+1)/((2*d*e-b*f^2)*(a+b*x+c*x^2)-f*(b*d-2*a*e+(2*c*d-b*e)*x)*Sqrt[a+b*x+c*x^2]),x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && EqQ[e^2-c*f^2] && NeQ[m+1] && IntegerQ[2*m] + + +Int[(g_.*x_)^m_.*Log[d_.+e_.*x_+f_.*Sqrt[a_.+c_.*x_^2]],x_Symbol] := + (g*x)^(m+1)*Log[d+e*x+f*Sqrt[a+c*x^2]]/(g*(m+1)) - + a*c*f^2/(g*(m+1))*Int[(g*x)^(m+1)/(d*e*(a+c*x^2)+f*(a*e-c*d*x)*Sqrt[a+c*x^2]),x] /; +FreeQ[{a,c,d,e,f,g,m},x] && EqQ[e^2-c*f^2] && NeQ[m+1] && IntegerQ[2*m] + + +Int[v_.*Log[d_.+e_.*x_+f_.*Sqrt[u_]],x_Symbol] := + Int[v*Log[d+e*x+f*Sqrt[ExpandToSum[u,x]]],x] /; +FreeQ[{d,e,f},x] && QuadraticQ[u,x] && Not[QuadraticMatchQ[u,x]] && (EqQ[v-1] || MatchQ[v,(g_.*x)^m_. /; FreeQ[{g,m},x]]) + + +Int[Log[u_],x_Symbol] := + x*Log[u] - Int[SimplifyIntegrand[x*D[u,x]/u,x],x] /; +InverseFunctionFreeQ[u,x] + + +Int[Log[u_]/(a_.+b_.*x_),x_Symbol] := + Log[a+b*x]*Log[u]/b - + 1/b*Int[SimplifyIntegrand[Log[a+b*x]*D[u,x]/u,x],x] /; +FreeQ[{a,b},x] && RationalFunctionQ[D[u,x]/u,x] && (NeQ[a] || Not[BinomialQ[u,x] && EqQ[BinomialDegree[u,x]^2-1]]) + + +Int[(a_.+b_.*x_)^m_.*Log[u_],x_Symbol] := + (a+b*x)^(m+1)*Log[u]/(b*(m+1)) - + 1/(b*(m+1))*Int[SimplifyIntegrand[(a+b*x)^(m+1)*D[u,x]/u,x],x] /; +FreeQ[{a,b,m},x] && InverseFunctionFreeQ[u,x] && NeQ[m+1] (* && Not[FunctionOfQ[x^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1,x]] *) + + +Int[Log[u_]/Qx_,x_Symbol] := + With[{v=IntHide[1/Qx,x]}, + v*Log[u] - Int[SimplifyIntegrand[v*D[u,x]/u,x],x]] /; +QuadraticQ[Qx,x] && InverseFunctionFreeQ[u,x] + + +(* Int[x_^m_.*Px_.*Log[u_],x_Symbol] := + With[{v=IntHide[x^m*Px,x]}, + Dist[Log[u],v] - Int[SimplifyIntegrand[v*D[u,x]/u,x],x]] /; +FreeQ[m,x] && PolynomialQ[Px,x] && InverseFunctionFreeQ[u,x] *) + + +Int[u_^(a_.*x_)*Log[u_],x_Symbol] := + u^(a*x)/a - Int[SimplifyIntegrand[x*u^(a*x-1)*D[u,x],x],x] /; +FreeQ[a,x] && InverseFunctionFreeQ[u,x] + + +Int[v_*Log[u_],x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[Log[u],w,x] - + Int[SimplifyIntegrand[w*D[u,x]/u,x],x] /; + InverseFunctionFreeQ[w,x]] /; +InverseFunctionFreeQ[u,x] + + +Int[Log[v_]*Log[w_],x_Symbol] := + x*Log[v]*Log[w] - + Int[SimplifyIntegrand[x*Log[w]*D[v,x]/v,x],x] - + Int[SimplifyIntegrand[x*Log[v]*D[w,x]/w,x],x] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + +Int[u_*Log[v_]*Log[w_],x_Symbol] := + With[{z=IntHide[u,x]}, + Dist[Log[v]*Log[w],z,x] - + Int[SimplifyIntegrand[z*Log[w]*D[v,x]/v,x],x] - + Int[SimplifyIntegrand[z*Log[v]*D[w,x]/w,x],x] /; + InverseFunctionFreeQ[z,x]] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + +Int[Log[a_.*Log[b_.*x_^n_.]^p_.],x_Symbol] := + x*Log[a*Log[b*x^n]^p] - + n*p*Int[1/Log[b*x^n],x] /; +FreeQ[{a,b,n,p},x] + + +Int[Log[a_.*Log[b_.*x_^n_.]^p_.]/x_,x_Symbol] := + Log[b*x^n]*(-p+Log[a*Log[b*x^n]^p])/n /; +FreeQ[{a,b,n,p},x] + + +Int[x_^m_.*Log[a_.*Log[b_.*x_^n_.]^p_.],x_Symbol] := + x^(m+1)*Log[a*Log[b*x^n]^p]/(m+1) - + n*p/(m+1)*Int[x^m/Log[b*x^n],x] /; +FreeQ[{a,b,m,n,p},x] && NeQ[m+1] + + +Int[(A_.+B_.*Log[c_.+d_.*x_])/Sqrt[a_+b_.*Log[c_.+d_.*x_]],x_Symbol] := + (b*A-a*B)/b*Int[1/Sqrt[a+b*Log[c+d*x]],x] + + B/b*Int[Sqrt[a+b*Log[c+d*x]],x] /; +FreeQ[{a,b,c,d,A,B},x] && NeQ[b*A-a*B] + + +Int[f_^(a_.*Log[u_]),x_Symbol] := + Int[u^(a*Log[f]),x] /; +FreeQ[{a,f},x] + + +(* If[ShowSteps, + +Int[u_/x_,x_Symbol] := + With[{lst=FunctionOfLog[u,x]}, + ShowStep["","Int[F[Log[a*x^n]]/x,x]","Subst[Int[F[x],x],x,Log[a*x^n]]/n",Hold[ + 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]]]] /; + Not[FalseQ[lst]]] /; +SimplifyFlag && NonsumQ[u], + +Int[u_/x_,x_Symbol] := + With[{lst=FunctionOfLog[u,x]}, + 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]] /; + Not[FalseQ[lst]]] /; +NonsumQ[u]] *) + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfLog[Cancel[x*u],x]}, + ShowStep["","Int[F[Log[a*x^n]]/x,x]","Subst[Int[F[x],x],x,Log[a*x^n]]/n",Hold[ + 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]]]] /; + Not[FalseQ[lst]]] /; +SimplifyFlag && NonsumQ[u], + +Int[u_,x_Symbol] := + With[{lst=FunctionOfLog[Cancel[x*u],x]}, + 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]] /; + Not[FalseQ[lst]]] /; +NonsumQ[u]] + + +Int[u_.*Log[Gamma[v_]],x_Symbol] := + (Log[Gamma[v]]-LogGamma[v])*Int[u,x] + Int[u*LogGamma[v],x] + + +Int[u_.*(a_. w_+b_. w_*Log[v_]^n_.)^p_.,x_Symbol] := + Int[u*w^p*(a+b*Log[v]^n)^p,x] /; +FreeQ[{a,b,n},x] && IntegerQ[p] + + +Int[u_.*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + Defer[Int][u*(a+b*Log[c*(d*(e+f*x)^p)^q])^n,x] /; +FreeQ[{a,b,c,d,e,f,n,p,q},x] && AlgebraicFunctionQ[u,x] + + + +`) + +func resourcesRubi3LogarithmsMBytes() ([]byte, error) { + return _resourcesRubi3LogarithmsM, nil +} + +func resourcesRubi3LogarithmsM() (*asset, error) { + bytes, err := resourcesRubi3LogarithmsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/3 Logarithms.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi41SineM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Trig Function Product Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*4.1.0 (a sin)^m (b trg)^n*) + + +If[ShowSteps, + +Int[u_,x_Symbol] := + Int[DeactivateTrig[u,x],x] /; +SimplifyFlag && FunctionOfTrigOfLinearQ[u,x], + +Int[u_,x_Symbol] := + Int[DeactivateTrig[u,x],x] /; +FunctionOfTrigOfLinearQ[u,x]] + + +Int[(a_.*sin[e_.+f_.*x_])^m_.*(b_.*cos[e_.+f_.*x_])^n_.,x_Symbol] := + (a*Sin[e+f*x])^(m+1)*(b*Cos[e+f*x])^(n+1)/(a*b*f*(m+1)) /; +FreeQ[{a,b,e,f,m,n},x] && EqQ[m+n+2,0] && NeQ[m,-1] + + +Int[(a_.*sin[e_.+f_.*x_])^m_.*cos[e_.+f_.*x_]^n_.,x_Symbol] := + 1/(a*f)*Subst[Int[x^m*(1-x^2/a^2)^((n-1)/2),x],x,a*Sin[e+f*x]] /; +FreeQ[{a,e,f,m},x] && IntegerQ[(n-1)/2] && Not[IntegerQ[(m-1)/2] && 01 || m==1 && n==-3/2) && IntegersQ[2*m,2*n] + + +Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+1)/(b*f*(n+1)) - + (m+n+1)/(b^2*(n+1))*Int[(a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+2),x] /; +FreeQ[{a,b,e,f,m},x] && RationalQ[n] && n<-1 && IntegersQ[2*m,2*n] + + +Int[(a_.*sec[e_.+f_.*x_])^m_*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n-1)/(f*m) - + b^2*(n-1)/(a^2*m)*Int[(a*Sec[e+f*x])^(m+2)*(b*Tan[e+f*x])^(n-2),x] /; +FreeQ[{a,b,e,f},x] && RationalQ[m,n] && n>1 && (m<-1 || m==-1 && n==3/2) && IntegersQ[2*m,2*n] + + +Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n-1)/(f*(m+n-1)) - + b^2*(n-1)/(m+n-1)*Int[(a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n-2),x] /; +FreeQ[{a,b,e,f,m},x] && RationalQ[n] && n>1 && NeQ[m+n-1] && IntegersQ[2*m,2*n] + + +Int[(a_.*sec[e_.+f_.*x_])^m_*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + -(a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+1)/(b*f*m) + + (m+n+1)/(a^2*m)*Int[(a*Sec[e+f*x])^(m+2)*(b*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,n},x] && RationalQ[m] && (m<-1 || m==-1 && RationalQ[n] && n==-1/2) && IntegersQ[2*m,2*n] + + +Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a^2*(a*Sec[e+f*x])^(m-2)*(b*Tan[e+f*x])^(n+1)/(b*f*(m+n-1)) + + a^2*(m-2)/(m+n-1)*Int[(a*Sec[e+f*x])^(m-2)*(b*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,n},x] && RationalQ[m] && (m>1 || m==1 && RationalQ[n] && n==1/2) && NeQ[m+n-1] && IntegersQ[2*m,2*n] + + +Int[sec[e_.+f_.*x_]/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol]:= + Sqrt[Sin[e+f*x]]/(Sqrt[Cos[e+f*x]]*Sqrt[b*Tan[e+f*x]])*Int[1/(Sqrt[Cos[e+f*x]]*Sqrt[Sin[e+f*x]]),x] /; +FreeQ[{b,e,f},x] + + +Int[Sqrt[b_.*tan[e_.+f_.*x_]]/sec[e_.+f_.*x_],x_Symbol]:= + Sqrt[Cos[e+f*x]]*Sqrt[b*Tan[e+f*x]]/Sqrt[Sin[e+f*x]]*Int[Sqrt[Cos[e+f*x]]*Sqrt[Sin[e+f*x]],x] /; +FreeQ[{b,e,f},x] + + +Int[(a_.*sec[e_.+f_.*x_])^m_*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol]:= + a^(m+n)*(b*Tan[e+f*x])^n/((a*Sec[e+f*x])^n*(b*Sin[e+f*x])^n)*Int[(b*Sin[e+f*x])^n/Cos[e+f*x]^(m+n),x] /; +FreeQ[{a,b,e,f,m,n},x] && IntegerQ[n+1/2] && IntegerQ[m+1/2] + + +(* Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol]:= + (a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+1)*(Cos[e+f*x]^2)^((m+n+1)/2)/(b*f*(b*Sin[e+f*x])^(n+1))* + Subst[Int[x^n/(1-x^2/b^2)^((m+n+1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m,n},x] && Not[IntegerQ[(n-1)/2]] && Not[IntegerQ[m/2]] *) + + +Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol]:= + (a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+1)*(Cos[e+f*x]^2)^((m+n+1)/2)/(b*f*(n+1))* + Hypergeometric2F1[(n+1)/2,(m+n+1)/2,(n+3)/2,Sin[e+f*x]^2] /; +FreeQ[{a,b,e,f,m,n},x] && Not[IntegerQ[(n-1)/2]] && Not[IntegerQ[m/2]] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (a*Csc[e+f*x])^FracPart[m]*(Sin[e+f*x]/a)^FracPart[m]*Int[(b*Tan[e+f*x])^n/(Sin[e+f*x]/a)^m,x] /; +FreeQ[{a,b,e,f,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.0 (a csc)^m (b sec)^n*) + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + a*b*(a*Csc[e+f*x])^(m-1)*(b*Sec[e+f*x])^(n-1)/(f*(n-1)) /; +FreeQ[{a,b,e,f,m,n},x] && EqQ[m+n-2] && NeQ[n-1] + + +Int[csc[e_.+f_.*x_]^m_.*sec[e_.+f_.*x_]^n_.,x_Symbol] := + 1/f*Subst[Int[(1+x^2)^((m+n)/2-1)/x^m,x],x,Tan[e+f*x]] /; +FreeQ[{e,f},x] && IntegersQ[m,n,(m+n)/2] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*sec[e_.+f_.*x_]^n_.,x_Symbol] := + -1/(f*a^n)*Subst[Int[x^(m+n-1)/(-1+x^2/a^2)^((n+1)/2),x],x,a*Csc[e+f*x]] /; +FreeQ[{a,e,f,m},x] && IntegerQ[(n+1)/2] && Not[IntegerQ[(m+1)/2] && 01 && n<-1 && IntegersQ[2*m,2*n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + b*(a*Csc[e+f*x])^(m+1)*(b*Sec[e+f*x])^(n-1)/(f*a*(n-1)) + + b^2*(m+1)/(a^2*(n-1))*Int[(a*Csc[e+f*x])^(m+2)*(b*Sec[e+f*x])^(n-2),x] /; +FreeQ[{a,b,e,f},x] && RationalQ[m,n] && m<-1 && n>1 && IntegersQ[2*m,2*n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_.,x_Symbol] := + -a*b*(a*Csc[e+f*x])^(m-1)*(b*Sec[e+f*x])^(n-1)/(f*(m-1)) + + a^2*(m+n-2)/(m-1)*Int[(a*Csc[e+f*x])^(m-2)*(b*Sec[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,n},x] && GtQ[m,1] && IntegersQ[2*m,2*n] && Not[n>m] + + +Int[(a_.*csc[e_.+f_.*x_])^m_.*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + a*b*(a*Csc[e+f*x])^(m-1)*(b*Sec[e+f*x])^(n-1)/(f*(n-1)) + + b^2*(m+n-2)/(n-1)*Int[(a*Csc[e+f*x])^m*(b*Sec[e+f*x])^(n-2),x] /; +FreeQ[{a,b,e,f,m},x] && GtQ[n,1] && IntegersQ[2*m,2*n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_.,x_Symbol] := + b*(a*Csc[e+f*x])^(m+1)*(b*Sec[e+f*x])^(n-1)/(a*f*(m+n)) + + (m+1)/(a^2*(m+n))*Int[(a*Csc[e+f*x])^(m+2)*(b*Sec[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,n},x] && RationalQ[m] && m<-1 && NeQ[m+n] && IntegersQ[2*m,2*n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_.*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + -a*(a*Csc[e+f*x])^(m-1)*(b*Sec[e+f*x])^(n+1)/(b*f*(m+n)) + + (n+1)/(b^2*(m+n))*Int[(a*Csc[e+f*x])^m*(b*Sec[e+f*x])^(n+2),x] /; +FreeQ[{a,b,e,f,m},x] && RationalQ[n] && n<-1 && NeQ[m+n] && IntegersQ[2*m,2*n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + (a*Csc[e+f*x])^m*(b*Sec[e+f*x])^n/Tan[e+f*x]^n*Int[Tan[e+f*x]^n,x] /; +FreeQ[{a,b,e,f,m,n},x] && Not[IntegerQ[n]] && EqQ[m+n] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + (a*Csc[e+f*x])^m*(b*Sec[e+f*x])^n*(a*Sin[e+f*x])^m*(b*Cos[e+f*x])^n*Int[(a*Sin[e+f*x])^(-m)*(b*Cos[e+f*x])^(-n),x] /; +FreeQ[{a,b,e,f,m,n},x] && IntegerQ[m-1/2] && IntegerQ[n-1/2] + + +Int[(a_.*csc[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_])^n_,x_Symbol] := + a^2/b^2*(a*Csc[e+f*x])^(m-1)*(b*Sec[e+f*x])^(n+1)*(a*Sin[e+f*x])^(m-1)*(b*Cos[e+f*x])^(n+1)* + Int[(a*Sin[e+f*x])^(-m)*(b*Cos[e+f*x])^(-n),x] /; +FreeQ[{a,b,e,f,m,n},x] && Not[SimplerQ[-m,-n]] + + +Int[(a_.*sec[e_.+f_.*x_])^m_*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a^2/b^2*(a*Sec[e+f*x])^(m-1)*(b*Csc[e+f*x])^(n+1)*(a*Cos[e+f*x])^(m-1)*(b*Sin[e+f*x])^(n+1)* + Int[(a*Cos[e+f*x])^(-m)*(b*Sin[e+f*x])^(-n),x] /; +FreeQ[{a,b,e,f,m,n},x] + + + + + +(* ::Section:: *) +(*Sine Function Integration Rules*) + + +(* ::Subsection::Closed:: *) +(*4.1.1.1 (a+b sin)^n*) + + +Int[sin[c_.+d_.*x_]^n_,x_Symbol] := + -1/d*Subst[Int[Expand[(1-x^2)^((n-1)/2),x],x],x,Cos[c+d*x]] /; +FreeQ[{c,d},x] && PositiveIntegerQ[(n-1)/2] + + +Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := +(* -Cot[c+d*x]*(c*Sin[c+d*x])^n/(d*n) + b^2*(n-1)/n*Int[(b*Sin[c+d*x])^(n-2),x] *) + -b*Cos[c+d*x]*(b*Sin[c+d*x])^(n-1)/(d*n) + b^2*(n-1)/n*Int[(b*Sin[c+d*x])^(n-2),x] /; +FreeQ[{b,c,d},x] && IntegerQ[2*n] && n>1 + + +Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + Cos[c+d*x]*(b*Sin[c+d*x])^(n+1)/(b*d*(n+1)) + + (n+2)/(b^2*(n+1))*Int[(b*Sin[c+d*x])^(n+2),x] /; +FreeQ[{b,c,d},x] && IntegerQ[2*n] && n<-1 + + +Int[sin[c_.+Pi/2+d_.*x_],x_Symbol] := + Sin[c+d*x]/d /; +FreeQ[{c,d},x] + + +Int[sin[c_.+d_.*x_],x_Symbol] := + -Cos[c+d*x]/d /; +FreeQ[{c,d},x] + + +(* Int[1/sin[c_.+d_.*x_],x_Symbol] := + Int[Csc[c+d*x],x] /; +FreeQ[{c,d},x] *) + + +Int[Sqrt[sin[c_.+d_.*x_]],x_Symbol] := + 2/d*EllipticE[1/2*(c-Pi/2+d*x),2] /; +FreeQ[{c,d},x] + + +Int[Sqrt[b_*sin[c_.+d_.*x_]],x_Symbol] := + Sqrt[b*Sin[c+d*x]]/Sqrt[Sin[c+d*x]]*Int[Sqrt[Sin[c+d*x]],x] /; +FreeQ[{b,c,d},x] + + +Int[1/Sqrt[sin[c_.+d_.*x_]],x_Symbol] := + 2/d*EllipticF[1/2*(c-Pi/2+d*x),2] /; +FreeQ[{c,d},x] + + +Int[1/Sqrt[b_*sin[c_.+d_.*x_]],x_Symbol] := + Sqrt[Sin[c+d*x]]/Sqrt[b*Sin[c+d*x]]*Int[1/Sqrt[Sin[c+d*x]],x] /; +FreeQ[{b,c,d},x] + + +(* Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + Cos[c+d*x]/(b*d*Sqrt[Cos[c+d*x]^2])*Subst[Int[x^n/Sqrt[1-x^2/b^2],x],x,b*Sin[c+d*x]] /; +FreeQ[{b,c,d,n},x] && Not[IntegerQ[2*n]] *) + + +Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + Cos[c+d*x]*(b*Sin[c+d*x])^(n+1)/(b*d*(n+1)*Sqrt[Cos[c+d*x]^2])*Hypergeometric2F1[1/2,(n+1)/2,(n+3)/2,Sin[c+d*x]^2] /; +FreeQ[{b,c,d,n},x] && Not[IntegerQ[2*n]] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^2,x_Symbol] := + (2*a^2+b^2)*x/2 - 2*a*b*Cos[c+d*x]/d - b^2*Cos[c+d*x]*Sin[c+d*x]/(2*d) /; +FreeQ[{a,b,c,d},x] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(a+b*sin[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && PositiveIntegerQ[n] + + +Int[Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + -2*b*Cos[c+d*x]/(d*Sqrt[a+b*Sin[c+d*x]]) /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -b*Cos[c+d*x]*(a+b*Sin[c+d*x])^(n-1)/(d*n) + + a*(2*n-1)/n*Int[(a+b*Sin[c+d*x])^(n-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] && PositiveIntegerQ[n-1/2] + + +Int[1/(a_+b_.*sin[c_.+d_.*x_]),x_Symbol] := + -Cos[c+d*x]/(d*(b+a*Sin[c+d*x])) /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] + + +Int[1/Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + -2/d*Subst[Int[1/(2*a-x^2),x],x,b*Cos[c+d*x]/Sqrt[a+b*Sin[c+d*x]]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + b*Cos[c+d*x]*(a+b*Sin[c+d*x])^n/(a*d*(2*n+1)) + + (n+1)/(a*(2*n+1))*Int[(a+b*Sin[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] && RationalQ[n] && n<-1 && IntegerQ[2*n] + + +(* Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a^2*Cos[c+d*x]/(d*Sqrt[a+b*Sin[c+d*x]]*Sqrt[a-b*Sin[c+d*x]])*Subst[Int[(a+b*x)^(n-1/2)/Sqrt[a-b*x],x],x,Sin[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[2*n]] *) + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -2^(n+1/2)*a^(n-1/2)*b*Cos[c+d*x]/(d*Sqrt[a+b*Sin[c+d*x]])*Hypergeometric2F1[1/2,1/2-n,3/2,1/2*(1-b*Sin[c+d*x]/a)] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[2*n]] && PositiveQ[a] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a^IntPart[n]*(a+b*Sin[c+d*x])^FracPart[n]/(1+b/a*Sin[c+d*x])^FracPart[n]*Int[(1+b/a*Sin[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[2*n]] && Not[PositiveQ[a]] + + +Int[Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + 2*Sqrt[a+b]/d*EllipticE[1/2*(c-Pi/2+d*x),2*b/(a+b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && PositiveQ[a+b] + + +Int[Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + 2*Sqrt[a-b]/d*EllipticE[1/2*(c+Pi/2+d*x),-2*b/(a-b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && PositiveQ[a-b] + + +Int[Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + Sqrt[a+b*Sin[c+d*x]]/Sqrt[(a+b*Sin[c+d*x])/(a+b)]*Int[Sqrt[a/(a+b)+b/(a+b)*Sin[c+d*x]],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && Not[PositiveQ[a+b]] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -b*Cos[c+d*x]*(a+b*Sin[c+d*x])^(n-1)/(d*n) + + 1/n*Int[(a+b*Sin[c+d*x])^(n-2)*Simp[a^2*n+b^2*(n-1)+a*b*(2*n-1)*Sin[c+d*x],x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && RationalQ[n] && n>1 && IntegerQ[2*n] + + +Int[1/(a_+b_.*sin[c_.+d_.*x_]),x_Symbol] := + With[{q=Rt[a^2-b^2,2]}, + x/q + 2/(d*q)*ArcTan[b*Cos[c+d*x]/(a+q+b*Sin[c+d*x])]] /; +FreeQ[{a,b,c,d},x] && PositiveQ[a^2-b^2] && PosQ[a] + + +Int[1/(a_+b_.*sin[c_.+d_.*x_]),x_Symbol] := + With[{q=Rt[a^2-b^2,2]}, + -x/q - 2/(d*q)*ArcTan[b*Cos[c+d*x]/(a-q+b*Sin[c+d*x])]] /; +FreeQ[{a,b,c,d},x] && PositiveQ[a^2-b^2] && NegQ[a] + + +Int[1/(a_+b_.*sin[c_.+Pi/2+d_.*x_]),x_Symbol] := + With[{e=FreeFactors[Tan[(c+d*x)/2],x]}, + 2*e/d*Subst[Int[1/(a+b+(a-b)*e^2*x^2),x],x,Tan[(c+d*x)/2]/e]] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[1/(a_+b_.*sin[c_.+d_.*x_]),x_Symbol] := + With[{e=FreeFactors[Tan[(c+d*x)/2],x]}, + 2*e/d*Subst[Int[1/(a+2*b*e*x+a*e^2*x^2),x],x,Tan[(c+d*x)/2]/e]] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[1/Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + 2/(d*Sqrt[a+b])*EllipticF[1/2*(c-Pi/2+d*x),2*b/(a+b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && PositiveQ[a+b] + + +Int[1/Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + 2/(d*Sqrt[a-b])*EllipticF[1/2*(c+Pi/2+d*x),-2*b/(a-b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && PositiveQ[a-b] + + +Int[1/Sqrt[a_+b_.*sin[c_.+d_.*x_]],x_Symbol] := + Sqrt[(a+b*Sin[c+d*x])/(a+b)]/Sqrt[a+b*Sin[c+d*x]]*Int[1/Sqrt[a/(a+b)+b/(a+b)*Sin[c+d*x]],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && Not[PositiveQ[a+b]] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -b*Cos[c+d*x]*(a+b*Sin[c+d*x])^(n+1)/(d*(n+1)*(a^2-b^2)) + + 1/((n+1)*(a^2-b^2))*Int[(a+b*Sin[c+d*x])^(n+1)*Simp[a*(n+1)-b*(n+2)*Sin[c+d*x],x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && RationalQ[n] && n<-1 && IntegerQ[2*n] + + +Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + Cos[c+d*x]/(d*Sqrt[1+Sin[c+d*x]]*Sqrt[1-Sin[c+d*x]])*Subst[Int[(a+b*x)^n/(Sqrt[1+x]*Sqrt[1-x]),x],x,Sin[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && NeQ[a^2-b^2] && Not[IntegerQ[2*n]] + + +Int[(a_+b_.*sin[c_.+d_.*x_]*cos[c_.+d_.*x_])^n_,x_Symbol] := + Int[(a+b*Sin[2*c+2*d*x]/2)^n,x] /; +FreeQ[{a,b,c,d,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.1.2 (g cos)^p (a+b sin)^m*) + + +Int[cos[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + 1/(b^p*f)*Subst[Int[(a+x)^(m+(p-1)/2)*(a-x)^((p-1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && IntegerQ[(p-1)/2] && EqQ[a^2-b^2] && (p>=-1 || Not[IntegerQ[m+1/2] && EqQ[a^2-b^2]]) + + +Int[cos[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + 1/(b^p*f)*Subst[Int[(a+x)^m*(b^2-x^2)^((p-1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && IntegerQ[(p-1)/2] && NeQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)/(f*g*(p+1)) + a*Int[(g*Cos[e+f*x])^p,x] /; +FreeQ[{a,b,e,f,g,p},x] && (IntegerQ[2*p] || NeQ[a^2-b^2]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a/g)^(2*m)*Int[(g*Cos[e+f*x])^(2*m+p)/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && IntegerQ[m] && RationalQ[p] && p<-1 && 2*m+p>=0 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*m) /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && EqQ[Simplify[m+p+1]] && Not[NegativeIntegerQ[p]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*Simplify[2*m+p+1]) + + Simplify[m+p+1]/(a*Simplify[2*m+p+1])*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && NegativeIntegerQ[Simplify[m+p+1]] && NeQ[2*m+p+1] && Not[PositiveIntegerQ[m]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)/(f*g*(m-1)) /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && EqQ[2*m+p-1] && NeQ[m-1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)/(f*g*(m+p)) + + a*(2*m+p-1)/(m+p)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[Simplify[(2*m+p-1)/2]] && NeQ[m+p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*(p+1)) + + a*(m+p+1)/(g^2*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-1),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[m,p] && m>0 && p<=-2*m && IntegersQ[m+1/2,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -2*b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)/(f*g*(p+1)) + + b^2*(2*m+p-1)/(g^2*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-2),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[m,p] && m>1 && p<-1 && IntegersQ[2*m,2*p] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[g_.*cos[e_.+f_.*x_]],x_Symbol] := + a*Sqrt[1+Cos[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(a+a*Cos[e+f*x]+b*Sin[e+f*x])*Int[Sqrt[1+Cos[e+f*x]]/Sqrt[g*Cos[e+f*x]],x] + + b*Sqrt[1+Cos[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(a+a*Cos[e+f*x]+b*Sin[e+f*x])*Int[Sin[e+f*x]/(Sqrt[g*Cos[e+f*x]]*Sqrt[1+Cos[e+f*x]]),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)/(f*g*(m+p)) + + a*(2*m+p-1)/(m+p)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && RationalQ[m] && m>0 && NeQ[m+p] && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+p)) + + g^2*(p-1)/(a*(m+p))*Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[m,p] && m<-1 && p>1 && (m>-2 || EqQ[2*m+p+1] || m==-2 && IntegerQ[p]) && + NeQ[m+p] && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + 2*g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)/(b*f*(2*m+p+1)) + + g^2*(p-1)/(b^2*(2*m+p+1))*Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^(m+2),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[m,p] && m<=-2 && p>1 && NeQ[2*m+p+1] && Not[NegativeIntegerQ[m+p+1]] && + IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*(2*m+p+1)) + + (m+p+1)/(a*(2*m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1 && NeQ[2*m+p+1] && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)/(b*f*(p-1)) + g^2/a*Int[(g*Cos[e+f*x])^(p-2),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[p] && p>1 && IntegerQ[2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)/(a*f*g*(p-1)*(a+b*Sin[e+f*x])) + + p/(a*(p-1))*Int[(g*Cos[e+f*x])^p,x] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && Not[RationalQ[p] && p>=1] && IntegerQ[2*p] + + +Int[Sqrt[g_.*cos[e_.+f_.*x_]]/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + g*Sqrt[1+Cos[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(a+a*Cos[e+f*x]+b*Sin[e+f*x])*Int[Sqrt[1+Cos[e+f*x]]/Sqrt[g*Cos[e+f*x]],x] - + g*Sqrt[1+Cos[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(b+b*Cos[e+f*x]+a*Sin[e+f*x])*Int[Sin[e+f*x]/(Sqrt[g*Cos[e+f*x]]*Sqrt[1+Cos[e+f*x]]),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^(3/2)/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + g*Sqrt[g*Cos[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(b*f) + + g^2/(2*a)*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[g*Cos[e+f*x]],x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + -2*b*(g*Cos[e+f*x])^(p+1)/(f*g*(2*p-1)*(a+b*Sin[e+f*x])^(3/2)) + + 2*a*(p-2)/(2*p-1)*Int[(g*Cos[e+f*x])^p/(a+b*Sin[e+f*x])^(3/2),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[p] && p>2 && IntegerQ[2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)/(a*f*g*(p+1)*Sqrt[a+b*Sin[e+f*x]]) + + a*(2*p+1)/(2*g^2*(p+1))*Int[(g*Cos[e+f*x])^(p+2)/(a+b*Sin[e+f*x])^(3/2),x] /; +FreeQ[{a,b,e,f,g},x] && EqQ[a^2-b^2] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + a^m*(g*Cos[e+f*x])^(p+1)/(f*g*(1+Sin[e+f*x])^((p+1)/2)*(1-Sin[e+f*x])^((p+1)/2))* + Subst[Int[(1+b/a*x)^(m+(p-1)/2)*(1-b/a*x)^((p-1)/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && IntegerQ[m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + a^2*(g*Cos[e+f*x])^(p+1)/(f*g*(a+b*Sin[e+f*x])^((p+1)/2)*(a-b*Sin[e+f*x])^((p+1)/2))* + Subst[Int[(a+b*x)^(m+(p-1)/2)*(a-b*x)^((p-1)/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m*Sin[e+f*x]/(f*g*(p+1)) + + 1/(g^2*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-1)*(a*(p+2)+b*(m+p+2)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m,p] && 01 && p<-1 && (IntegersQ[2*m,2*p] || IntegerQ[m]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)/(f*g*(m+p)) + + 1/(m+p)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-2)*(b^2*(m-1)+a^2*(m+p)+a*b*(2*m+p-1)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,p},x] && NeQ[a^2-b^2] && RationalQ[m] && m>1 && NeQ[m+p] && (IntegersQ[2*m,2*p] || IntegerQ[m]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+1)) + + g^2*(p-1)/(b*(m+1))*Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^(m+1)*Sin[e+f*x],x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m,p] && m<-1 && p>1 && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(f*g*(a^2-b^2)*(m+1)) + + 1/((a^2-b^2)*(m+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1)*(a*(m+1)-b*(m+p+2)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,p},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+p)) + + g^2*(p-1)/(b*(m+p))*Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^m*(b+a*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,m},x] && NeQ[a^2-b^2] && RationalQ[p] && p>1 && NeQ[m+p] && IntegersQ[2*m,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)*(b-a*Sin[e+f*x])/(f*g*(a^2-b^2)*(p+1)) + + 1/(g^2*(a^2-b^2)*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m*(a^2*(p+2)-b^2*(m+p+2)+a*b*(m+p+3)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,m},x] && NeQ[a^2-b^2] && RationalQ[p] && p<-1 && IntegersQ[2*m,2*p] + + +Int[1/(Sqrt[g_.*cos[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]),x_Symbol] := + 2*Sqrt[2]*Sqrt[g*Cos[e+f*x]]*Sqrt[(a+b*Sin[e+f*x])/((a-b)*(1-Sin[e+f*x]))]/ + (f*g*Sqrt[a+b*Sin[e+f*x]]*Sqrt[(1+Cos[e+f*x]+Sin[e+f*x])/(1+Cos[e+f*x]-Sin[e+f*x])])* + Subst[Int[1/Sqrt[1+(a+b)*x^4/(a-b)],x],x,Sqrt[(1+Cos[e+f*x]+Sin[e+f*x])/(1+Cos[e+f*x]-Sin[e+f*x])]] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(1-Sin[e+f*x])*(a+b*Sin[e+f*x])^(m+1)*(-(a-b)*(1-Sin[e+f*x])/((a+b)*(1+Sin[e+f*x])))^(m/2)/ + (f*(a+b)*(m+1))* + Hypergeometric2F1[m+1,m/2+1,m+2,2*(a+b*Sin[e+f*x])/((a+b)*(1+Sin[e+f*x]))] /; +FreeQ[{a,b,e,f,g,m,p},x] && NeQ[a^2-b^2] && EqQ[m+p+1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(f*g*(a-b)*(p+1)) + + a/(g^2*(a-b))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m/(1-Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && NeQ[a^2-b^2] && EqQ[m+p+2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(f*g*(a-b)*(p+1)) - + b*(m+p+2)/(g^2*(a-b)*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m,x] + + a/(g^2*(a-b))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m/(1-Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && NeQ[a^2-b^2] && NegativeIntegerQ[m+p+2] + + +Int[Sqrt[g_.*cos[e_.+f_.*x_]]/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + With[{q=Rt[-a^2+b^2,2]}, + a*g/(2*b)*Int[1/(Sqrt[g*Cos[e+f*x]]*(q+b*Cos[e+f*x])),x] - + a*g/(2*b)*Int[1/(Sqrt[g*Cos[e+f*x]]*(q-b*Cos[e+f*x])),x] + + b*g/f*Subst[Int[Sqrt[x]/(g^2*(a^2-b^2)+b^2*x^2),x],x,g*Cos[e+f*x]]] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[1/(Sqrt[g_.*cos[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + With[{q=Rt[-a^2+b^2,2]}, + -a/(2*q)*Int[1/(Sqrt[g*Cos[e+f*x]]*(q+b*Cos[e+f*x])),x] - + a/(2*q)*Int[1/(Sqrt[g*Cos[e+f*x]]*(q-b*Cos[e+f*x])),x] + + b*g/f*Subst[Int[1/(Sqrt[x]*(g^2*(a^2-b^2)+b^2*x^2)),x],x,g*Cos[e+f*x]]] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)/ + (b*f*(m+p)*(-b*(1-Sin[e+f*x])/(a+b*Sin[e+f*x]))^((p-1)/2)*(b*(1+Sin[e+f*x])/(a+b*Sin[e+f*x]))^((p-1)/2))* + AppellF1[-p-m,(1-p)/2,(1-p)/2,1-p-m,(a+b)/(a+b*Sin[e+f*x]),(a-b)/(a+b*Sin[e+f*x])] /; +FreeQ[{a,b,e,f,g,p},x] && NeQ[a^2-b^2] && NegativeIntegerQ[m] && Not[PositiveIntegerQ[m+p+1]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)/(f*(1-(a+b*Sin[e+f*x])/(a-b))^((p-1)/2)*(1-(a+b*Sin[e+f*x])/(a+b))^((p-1)/2))* + Subst[Int[(-b/(a-b)-b*x/(a-b))^((p-1)/2)*(b/(a+b)-b*x/(a+b))^((p-1)/2)*(a+b*x)^m,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,e,f,g,m,p},x] && NeQ[a^2-b^2] && Not[PositiveIntegerQ[m]] + + +Int[(g_.*sec[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + g^(2*IntPart[p])*(g*Cos[e+f*x])^FracPart[p]*(g*Sec[e+f*x])^FracPart[p]*Int[(a+b*Sin[e+f*x])^m/(g*Cos[e+f*x])^p,x] /; +FreeQ[{a,b,e,f,g,m,p},x] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.1.3 (g tan)^p (a+b sin)^m*) + + +Int[(g_.*tan[e_.+f_.*x_])^p_./(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + 1/a*Int[Sec[e+f*x]^2*(g*Tan[e+f*x])^p,x] - 1/(b*g)*Int[Sec[e+f*x]*(g*Tan[e+f*x])^(p+1),x] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && NeQ[p+1] + + +Int[tan[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + 1/f*Subst[Int[x^p*(a+x)^(m-(p+1)/2)/(a-x)^((p+1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[(p+1)/2] + + +Int[tan[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + a^p*Int[Sin[e+f*x]^p/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && IntegersQ[m,p] && p==2*m + + +Int[tan[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a^p*Int[ExpandIntegrand[Sin[e+f*x]^p*(a+b*Sin[e+f*x])^(m-p/2)/(a-b*Sin[e+f*x])^(p/2),x],x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && IntegersQ[m,p/2] && (p<0 || m-p/2>0) + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + Int[ExpandIntegrand[(g*Tan[e+f*x])^p,(a+b*Sin[e+f*x])^m,x],x] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a^(2*m)*Int[ExpandIntegrand[(g*Tan[e+f*x])^p*Sec[e+f*x]^(-m),(a*Sec[e+f*x]-b*Tan[e+f*x])^(-m),x],x] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && NegativeIntegerQ[m] + + +Int[tan[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(a+b*Sin[e+f*x])^m/(a*f*(2*m-1)*Cos[e+f*x]) - + 1/(a^2*(2*m-1))*Int[(a+b*Sin[e+f*x])^(m+1)*(a*m-b*(2*m-1)*Sin[e+f*x])/Cos[e+f*x]^2,x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && RationalQ[m] && m<0 + + +Int[tan[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -(a+b*Sin[e+f*x])^(m+1)/(b*f*m*Cos[e+f*x]) + + 1/(b*m)*Int[(a+b*Sin[e+f*x])^m*(b*(m+1)+a*Sin[e+f*x])/Cos[e+f*x]^2,x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && Not[RationalQ[m] && m<0] + + +Int[tan[e_.+f_.*x_]^4*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[(a+b*Sin[e+f*x])^m,x] - Int[(a+b*Sin[e+f*x])^m*(1-2*Sin[e+f*x]^2)/Cos[e+f*x]^4,x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[m-1/2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^2,x_Symbol] := + -(a+b*Sin[e+f*x])^(m+1)/(a*f*Tan[e+f*x]) + + 1/b^2*Int[(a+b*Sin[e+f*x])^(m+1)*(b*m-a*(m+1)*Sin[e+f*x])/Sin[e+f*x],x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && IntegerQ[m-1/2] && m<-1 + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_./tan[e_.+f_.*x_]^2,x_Symbol] := + -(a+b*Sin[e+f*x])^m/(f*Tan[e+f*x]) + + 1/a*Int[(a+b*Sin[e+f*x])^m*(b*m-a*(m+1)*Sin[e+f*x])/Sin[e+f*x],x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[m-1/2] && Not[m<-1] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := + -2/(a*b)*Int[(a+b*Sin[e+f*x])^(m+2)/Sin[e+f*x]^3,x] + + 1/a^2*Int[(a+b*Sin[e+f*x])^(m+2)*(1+Sin[e+f*x]^2)/Sin[e+f*x]^4,x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && IntegerQ[m-1/2] && m<-1 + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := + Int[(a+b*Sin[e+f*x])^m,x] + Int[(a+b*Sin[e+f*x])^m*(1-2*Sin[e+f*x]^2)/Sin[e+f*x]^4,x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[m-1/2] && Not[m<-1] + + +Int[tan[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]]/(b*f*Cos[e+f*x])* + Subst[Int[x^p*(a+x)^(m-(p+1)/2)/(a-x)^((p+1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && IntegerQ[p/2] + + +Int[(g_.*tan[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (g*Tan[e+f*x])^(p+1)*(a-b*Sin[e+f*x])^((p+1)/2)*(a+b*Sin[e+f*x])^((p+1)/2)/(f*g*(b*Sin[e+f*x])^(p+1))* + Subst[Int[x^p*(a+x)^(m-(p+1)/2)/(a-x)^((p+1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && Not[IntegerQ[p]] + + +Int[tan[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + 1/f*Subst[Int[(x^p*(a+x)^m)/(b^2-x^2)^((p+1)/2),x],x,b*Sin[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && IntegerQ[(p+1)/2] + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + Int[ExpandIntegrand[(g*Tan[e+f*x])^p,(a+b*Sin[e+f*x])^m,x],x] /; +FreeQ[{a,b,e,f,g,p},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^2,x_Symbol] := + Int[(a+b*Sin[e+f*x])^m*(1-Sin[e+f*x]^2)/Sin[e+f*x]^2,x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := + -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a*f*Sin[e+f*x]^3) - + (3*a^2+b^2*(m-2))*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a^2*b*f*(m+1)*Sin[e+f*x]^2) - + 1/(3*a^2*b*(m+1))*Int[(a+b*Sin[e+f*x])^(m+1)/Sin[e+f*x]^3* + Simp[6*a^2-b^2*(m-1)*(m-2)+a*b*(m+1)*Sin[e+f*x]-(3*a^2-b^2*m*(m-2))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegerQ[2*m] + + +(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := + -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a*f*Sin[e+f*x]^3) - + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*m*Sin[e+f*x]^2) - + 1/(3*a*b*m)*Int[(a+b*Sin[e+f*x])^m/Sin[e+f*x]^3* + Simp[6*a^2-b^2*m*(m-2)+a*b*(m+3)*Sin[e+f*x]-(3*a^2-b^2*m*(m-1))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && Not[RationalQ[m] && m<-1] && IntegerQ[2*m] *) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := + -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a*f*Sin[e+f*x]^3) - + b*(m-2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(6*a^2*f*Sin[e+f*x]^2) - + 1/(6*a^2)*Int[(a+b*Sin[e+f*x])^m/Sin[e+f*x]^2* + Simp[8*a^2-b^2*(m-1)*(m-2)+a*b*m*Sin[e+f*x]-(6*a^2-b^2*m*(m-2))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && Not[RationalQ[m] && m<-1] && IntegerQ[2*m] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^6,x_Symbol] := + -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(5*a*f*Sin[e+f*x]^5) - + b*(m-4)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(20*a^2*f*Sin[e+f*x]^4) + + a*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b^2*f*m*(m-1)*Sin[e+f*x]^3) + + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*m*Sin[e+f*x]^2) + + 1/(20*a^2*b^2*m*(m-1))*Int[(a+b*Sin[e+f*x])^m/Sin[e+f*x]^4* + Simp[60*a^4-44*a^2*b^2*(m-1)*m+b^4*m*(m-1)*(m-3)*(m-4)+a*b*m*(20*a^2-b^2*m*(m-1))*Sin[e+f*x]- + (40*a^4+b^4*m*(m-1)*(m-2)*(m-4)-20*a^2*b^2*(m-1)*(2*m+1))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && NeQ[m-1] && IntegerQ[2*m] + + +Int[(g_.*tan[e_.+f_.*x_])^p_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + a/(a^2-b^2)*Int[(g*Tan[e+f*x])^p/Sin[e+f*x]^2,x] - + b*g/(a^2-b^2)*Int[(g*Tan[e+f*x])^(p-1)/Cos[e+f*x],x] - + a^2*g^2/(a^2-b^2)*Int[(g*Tan[e+f*x])^(p-2)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*p] && p>1 + + +Int[(g_.*tan[e_.+f_.*x_])^p_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + 1/a*Int[(g*Tan[e+f*x])^p/Cos[e+f*x]^2,x] - + b/(a^2*g)*Int[(g*Tan[e+f*x])^(p+1)/Cos[e+f*x],x] - + (a^2-b^2)/(a^2*g^2)*Int[(g*Tan[e+f*x])^(p+2)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*p] && p<-1 + + +Int[Sqrt[g_.*tan[e_.+f_.*x_]]/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + Sqrt[Cos[e+f*x]]*Sqrt[g*Tan[e+f*x]]/Sqrt[Sin[e+f*x]]*Int[Sqrt[Sin[e+f*x]]/(Sqrt[Cos[e+f*x]]*(a+b*Sin[e+f*x])),x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[1/(Sqrt[g_*tan[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + Sqrt[Sin[e+f*x]]/(Sqrt[Cos[e+f*x]]*Sqrt[g*Tan[e+f*x]])*Int[Sqrt[Cos[e+f*x]]/(Sqrt[Sin[e+f*x]]*(a+b*Sin[e+f*x])),x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[tan[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[ExpandIntegrand[Sin[e+f*x]^p*(a+b*Sin[e+f*x])^m/(1-Sin[e+f*x]^2)^(p/2),x],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && IntegersQ[m,p/2] + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + Defer[Int][(g*Tan[e+f*x])^p*(a+b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,e,f,g,m,p},x] + + +Int[(g_.*cot[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.,x_Symbol] := + g^(2*IntPart[p])*(g*Cot[e+f*x])^FracPart[p]*(g*Tan[e+f*x])^FracPart[p]*Int[(a+b*Sin[e+f*x])^m/(g*Tan[e+f*x])^p,x] /; +FreeQ[{a,b,e,f,g,m,p},x] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.2.1 (a+b sin)^m (c+d sin)^n*) + + +Int[(a_+b_.*sin[e_.+f_.*x_])*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + (2*a*c+b*d)*x/2 - (b*c+a*d)*Cos[e+f*x]/f - b*d*Cos[e+f*x]*Sin[e+f*x]/(2*f) /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])/(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + b*x/d - (b*c-a*d)/d*Int[1/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^m*c^m*Int[Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m] && Not[IntegerQ[n] && (m<0 && n>0 || 00] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n/(f*(m+n)) + + a*(2*m-1)/(m+n)*Int[(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && PositiveIntegerQ[m-1/2] && Not[RationalQ[n] && n<-1] && + Not[PositiveIntegerQ[n-1/2] && n0] + + +Int[1/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + Cos[e+f*x]/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]])*Int[1/Cos[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*(2*m+1)) /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && EqQ[m+n+1] && NeQ[m+1/2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*(2*m+1)) + + (m+n+1)/(a*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && NegativeIntegerQ[Simplify[m+n+1]] && NeQ[m+1/2] && + (SumSimplerQ[m,1] || Not[SumSimplerQ[n,1]]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*(2*m+1)) + + (m+n+1)/(a*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[m] && m<-1 && + Not[RationalQ[n] && m0 && IntegerQ[2*m] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -(b*c-a*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*Simp[(a*c-b*d)*(m+1)-(b*c-a*d)*(m+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegerQ[2*m] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_]),x_Symbol] := + c*Cos[e+f*x]/(f*Sqrt[1+Sin[e+f*x]]*Sqrt[1-Sin[e+f*x]])*Subst[Int[(a+b*x)^m*Sqrt[1+d/c*x]/Sqrt[1-d/c*x],x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && Not[IntegerQ[2*m]] && EqQ[c^2-d^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + (b*c-a*d)/b*Int[(a+b*Sin[e+f*x])^m,x] + d/b*Int[(a+b*Sin[e+f*x])^(m+1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandTrig[(a+b*sin[e+f*x])^m*(d*sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] && RationalQ[n] + + +Int[sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*Cos[e+f*x]*(a+b*Sin[e+f*x])^m/(a*f*(2*m+1)) - + 1/(a^2*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*(a*m-b*(2*m+1)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 + + +Int[sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[(a+b*Sin[e+f*x])^m*(b*(m+1)-a*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && Not[RationalQ[m] && m<-1/2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^2,x_Symbol] := + (b*c-a*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])/(a*f*(2*m+1)) + + 1/(a*b*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*Simp[a*c*d*(m-1)+b*(d^2+c^2*(m+1))+d*(a*d*(m-1)+b*c*(m+2))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^2,x_Symbol] := + -d^2*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[(a+b*Sin[e+f*x])^m*Simp[b*(d^2*(m+1)+c^2*(m+2))-d*(a*d-2*b*c*(m+2))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && Not[RationalQ[m] && m<-1] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*(b*c-a*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(n+1)*(b*c+a*d)) + + b^2/(d*(n+1)*(b*c+a*d))*Int[(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)* + Simp[a*c*(m-2)-b*d*(m-2*n-4)-(b*c*(m-1)-a*d*(m+2*n+1))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m>1 && n<-1 && + (IntegersQ[2*m,2*n] || IntegerQ[m+1/2] || IntegerQ[m] && EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n)) + + 1/(d*(m+n))*Int[(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^n* + Simp[a*b*c*(m-2)+b^2*d*(n+1)+a^2*d*(m+n)-b*(b*c*(m-1)-a*d*(3*m+2*n-2))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>1 && + Not[RationalQ[n] && n<-1] && (IntegersQ[2*m,2*n] || IntegerQ[m+1/2] || IntegerQ[m] && EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*(2*m+1)) - + 1/(a*b*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n-1)*Simp[a*d*n-b*c*(m+1)-b*d*(m+n+1)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m<-1 && 01 && + (IntegersQ[2*m,2*n] || IntegerQ[m] && EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b^2*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(a*f*(2*m+1)*(b*c-a*d)) + + 1/(a*(2*m+1)*(b*c-a*d))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n* + Simp[b*c*(m+1)-a*d*(2*m+n+2)+b*d*(m+n+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m<-1 && + Not[RationalQ[n] && n>0] && (IntegersQ[2*m,2*n] || IntegerQ[m] && EqQ[c]) + + +Int[(c_.+d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -(b*c-a*d)*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n-1)/(a*f*(a+b*Sin[e+f*x])) - + d/(a*b)*Int[(c+d*Sin[e+f*x])^(n-2)*Simp[b*d*(n-1)-a*c*n+(b*c*(n-1)-a*d*n)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n>1 && + (IntegerQ[2*n] || EqQ[c]) + + +Int[(c_.+d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -b^2*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(a*f*(b*c-a*d)*(a+b*Sin[e+f*x])) + + d/(a*(b*c-a*d))*Int[(c+d*Sin[e+f*x])^n*(a*n-b*(n+1)*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n<0 && (IntegerQ[2*n] || EqQ[c]) + + +Int[(c_.+d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -b*Cos[e+f*x]*(c+d*Sin[e+f*x])^n/(a*f*(a+b*Sin[e+f*x])) + + d*n/(a*b)*Int[(c+d*Sin[e+f*x])^(n-1)*(a-b*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && (IntegerQ[2*n] || EqQ[c]) + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -2*b*Cos[e+f*x]*(c+d*Sin[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Sin[e+f*x]]) + + 2*n*(b*c+a*d)/(b*(2*n+1))*Int[Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n>0 && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(c_.+d_.*sin[e_.+f_.*x_])^(3/2),x_Symbol] := + -2*b^2*Cos[e+f*x]/(f*(b*c+a*d)*Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]) /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (b*c-a*d)*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(f*(n+1)*(c^2-d^2)*Sqrt[a+b*Sin[e+f*x]]) + + (2*n+3)*(b*c-a*d)/(2*b*(n+1)*(c^2-d^2))*Int[Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n<-1 && NeQ[2*n+3] && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -2*b/f*Subst[Int[1/(b*c+a*d-d*x^2),x],x,b*Cos[e+f*x]/Sqrt[a+b*Sin[e+f*x]]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[d_.*sin[e_.+f_.*x_]],x_Symbol] := + -2/f*Subst[Int[1/Sqrt[1-x^2/a],x],x,b*Cos[e+f*x]/Sqrt[a+b*Sin[e+f*x]]] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && EqQ[d-a/b] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[c_.+d_.*sin[e_.+f_.*x_]],x_Symbol] := + -2*b/f*Subst[Int[1/(b+d*x^2),x],x,b*Cos[e+f*x]/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + a^2*Cos[e+f*x]/(f*Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]])*Subst[Int[(c+d*x)^n/Sqrt[a-b*x],x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && Not[IntegerQ[2*n]] + + +Int[Sqrt[c_.+d_.*sin[e_.+f_.*x_]]/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + d/b*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + (b*c-a*d)/b*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(c_.+d_.*sin[e_.+f_.*x_])^n_/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + -2*d*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n-1)/(f*(2*n-1)*Sqrt[a+b*Sin[e+f*x]]) - + 1/(b*(2*n-1))*Int[(c+d*Sin[e+f*x])^(n-2)/Sqrt[a+b*Sin[e+f*x]]* + Simp[a*c*d-b*(2*d^2*(n-1)+c^2*(2*n-1))+d*(a*d-b*c*(4*n-3))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n>1 && IntegerQ[2*n] + + +Int[(c_.+d_.*sin[e_.+f_.*x_])^n_/Sqrt[a_+b_.*sin[e_.+f_.*x_]],x_Symbol] := + -d*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(f*(n+1)*(c^2-d^2)*Sqrt[a+b*Sin[e+f*x]]) - + 1/(2*b*(n+1)*(c^2-d^2))*Int[(c+d*Sin[e+f*x])^(n+1)*Simp[a*d-2*b*c*(n+1)+b*d*(2*n+3)*Sin[e+f*x],x]/Sqrt[a+b*Sin[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n<-1 && IntegerQ[2*n] + + +Int[1/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + b/(b*c-a*d)*Int[1/Sqrt[a+b*Sin[e+f*x]],x] - d/(b*c-a*d)*Int[Sqrt[a+b*Sin[e+f*x]]/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -Sqrt[2]/(Sqrt[a]*f)*Subst[Int[1/Sqrt[1-x^2],x],x,b*Cos[e+f*x]/(a+b*Sin[e+f*x])] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && EqQ[d-a/b] && PositiveQ[a] + + +Int[1/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_.+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -2*a/f*Subst[Int[1/(2*b^2-(a*c-b*d)*x^2),x],x,b*Cos[e+f*x]/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -d*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n-1)/(f*(m+n)) + + 1/(b*(m+n))*Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n-2)* + Simp[d*(a*c*m+b*d*(n-1))+b*c^2*(m+n)+d*(a*d*m+b*c*(m+2*n-1))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n>1 && IntegerQ[n] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^m*Cos[e+f*x]/(f*Sqrt[1+Sin[e+f*x]]*Sqrt[1-Sin[e+f*x]])*Subst[Int[(1+b/a*x)^(m-1/2)*(c+d*x)^n/Sqrt[1-b/a*x],x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && IntegerQ[m] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*(d/b)^n*Cos[e+f*x]/(f*Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]])* + Subst[Int[(a-x)^n*(2*a-x)^(m-1/2)/Sqrt[x],x],x,a-b*Sin[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && PositiveQ[a] && PositiveQ[d/b] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + (d/b)^IntPart[n]*(d*Sin[e+f*x])^FracPart[n]/(b*Sin[e+f*x])^FracPart[n]*Int[(a+b*Sin[e+f*x])^m*(b*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && PositiveQ[a] && Not[PositiveQ[d/b]] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^IntPart[m]*(a+b*Sin[e+f*x])^FracPart[m]/(1+b/a*Sin[e+f*x])^FracPart[m]* + Int[(1+b/a*Sin[e+f*x])^m*(d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && Not[PositiveQ[a]] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^2*Cos[e+f*x]/(f*Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]])*Subst[Int[(a+b*x)^(m-1/2)*(c+d*x)^n/Sqrt[a-b*x],x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && Not[IntegerQ[m]] + + +Int[(b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^2,x_Symbol] := + 2*c*d/b*Int[(b*Sin[e+f*x])^(m+1),x] + Int[(b*Sin[e+f*x])^m*(c^2+d^2*Sin[e+f*x]^2),x] /; +FreeQ[{b,c,d,e,f,m},x] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^2,x_Symbol] := + -(b^2*c^2-2*a*b*c*d+a^2*d^2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+1)*(a^2-b^2)) - + 1/(b*(m+1)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)* + Simp[b*(m+1)*(2*b*c*d-a*(c^2+d^2))+(a^2*d^2-2*a*b*c*d*(m+2)+b^2*(d^2*(m+1)+c^2*(m+2)))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^2,x_Symbol] := + -d^2*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[(a+b*Sin[e+f*x])^m*Simp[b*(d^2*(m+1)+c^2*(m+2))-d*(a*d-2*b*c*(m+2))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && Not[RationalQ[m] && m<-1] + + +(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandTrig[(a+b*sin[e+f*x])^m*(d*sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] *) + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -(b^2*c^2-2*a*b*c*d+a^2*d^2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(n+1)*(c^2-d^2)) + + 1/(d*(n+1)*(c^2-d^2))*Int[(a+b*Sin[e+f*x])^(m-3)*(c+d*Sin[e+f*x])^(n+1)* + Simp[b*(m-2)*(b*c-a*d)^2+a*d*(n+1)*(c*(a^2+b^2)-2*a*b*d)+ + (b*(n+1)*(a*b*c^2+c*d*(a^2+b^2)-3*a*b*d^2)-a*(n+2)*(b*c-a*d)^2)*Sin[e+f*x]+ + b*(b^2*(c^2-d^2)-m*(b*c-a*d)^2+d*n*(2*a*b*c-d*(a^2+b^2)))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m>2 && n<-1 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n)) + + 1/(d*(m+n))*Int[(a+b*Sin[e+f*x])^(m-3)*(c+d*Sin[e+f*x])^n* + Simp[a^3*d*(m+n)+b^2*(b*c*(m-2)+a*d*(n+1))- + b*(a*b*c-b^2*d*(m+n-1)-3*a^2*d*(m+n))*Sin[e+f*x]- + b^2*(b*c*(m-1)-a*d*(3*m+2*n-2))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>2 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) && Not[IntegerQ[n] && n>2 && (Not[IntegerQ[m]] || EqQ[a] && NeQ[c])] + + +Int[Sqrt[d_.*sin[e_.+f_.*x_]]/(a_+b_.*sin[e_.+f_.*x_])^(3/2),x_Symbol] := + -2*a*d*Cos[e+f*x]/(f*(a^2-b^2)*Sqrt[a+b*Sin[e+f*x]]*Sqrt[d*Sin[e+f*x]]) - + d^2/(a^2-b^2)*Int[Sqrt[a+b*Sin[e+f*x]]/(d*Sin[e+f*x])^(3/2),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[Sqrt[c_+d_.*sin[e_.+f_.*x_]]/(a_.+b_.*sin[e_.+f_.*x_])^(3/2),x_Symbol] := + (c-d)/(a-b)*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] - + (b*c-a*d)/(a-b)*Int[(1+Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n-1)* + Simp[a*c*(m+1)+b*d*n+(a*d*(m+1)-b*c*(m+2))*Sin[e+f*x]-b*d*(m+n+2)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m<-1 && 00] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n/(f*g*(m+n+p)) + + a*(2*m+p-1)/(m+n+p)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && PositiveIntegerQ[Simplify[m+p/2-1/2]] && Not[RationalQ[n] && n<-1] && + Not[PositiveIntegerQ[Simplify[n+p/2-1/2]] && m-n>0] && Not[NegativeIntegerQ[Simplify[m+n+p]] && Simplify[2*m+n+3*p/2+1]>0] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a^IntPart[m]*c^IntPart[m]*(a+b*Sin[e+f*x])^FracPart[m]*(c+d*Sin[e+f*x])^FracPart[m]/ + (g^(2*IntPart[m])*(g*Cos[e+f*x])^(2*FracPart[m]))*Int[(g*Cos[e+f*x])^(2*m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && EqQ[2*m+p+1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*g*(m-n)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && EqQ[m+n+p+1] && NeQ[m-n] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*g*(2*m+p+1)) + + (m+n+p+1)/(a*(2*m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && NegativeIntegerQ[Simplify[m+n+p+1]] && NeQ[2*m+p+1] && + (SumSimplerQ[m,1] || Not[SumSimplerQ[n,1]]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -2*b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n/(f*g*(2*n+p+1)) - + b*(2*m+p-1)/(d*(2*n+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[m,n] && m>0 && n<-1 && NeQ[2*n+p+1] && + IntegersQ[2*m,2*n,2*p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n/(f*g*(m+n+p)) + + a*(2*m+p-1)/(m+n+p)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[m] && m>0 && NeQ[m+n+p] && + Not[RationalQ[n] && 0-1 && p<-1 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -d*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(f*g*(m+p+1)) + + (a*d*m+b*c*(m+p+1))/(b*(m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[Simplify[(2*m+p+1)/2]] && NeQ[m+p+1] + + +Int[cos[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + 2*(b*c-a*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b^2*f*(2*m+3)) + + 1/(b^3*(2*m+3))*Int[(a+b*Sin[e+f*x])^(m+2)*(b*c+2*a*d*(m+1)-b*d*(2*m+3)*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-3/2 + + +Int[cos[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + d*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+2)/(b^2*f*(m+3)) - + 1/(b^2*(m+3))*Int[(a+b*Sin[e+f*x])^(m+1)*(b*d*(m+2)-a*c*(m+3)+(b*c*(m+3)-a*d*(m+4))*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && -3/2<=m<0 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + (b*c-a*d)*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*(2*m+p+1)) + + (a*d*m+b*c*(m+p+1))/(a*b*(2*m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1),x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[a^2-b^2] && (RationalQ[m] && m<-1 || NegativeIntegerQ[Simplify[m+p]]) && NeQ[2*m+p+1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -d*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(f*g*(m+p+1)) + + (a*d*m+b*c*(m+p+1))/(b*(m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && EqQ[a^2-b^2] && NeQ[m+p+1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m*(d+c*Sin[e+f*x])/(f*g*(p+1)) + + 1/(g^2*(p+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-1)*Simp[a*c*(p+2)+b*d*m+b*c*(m+p+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m,p] && m>0 && p<-1 && IntegerQ[2*m] && + Not[m==1 && NeQ[c^2-d^2] && SimplerQ[c+d*x,a+b*x]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -d*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(f*g*(m+p+1)) + + 1/(m+p+1)*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m-1)*Simp[a*c*(m+p+1)+b*d*m+(a*d*m+b*c*(m+p+1))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[a^2-b^2] && RationalQ[m] && m>0 && Not[RationalQ[p] && p<-1] && IntegerQ[2*m] && + Not[m==1 && NeQ[c^2-d^2] && SimplerQ[c+d*x,a+b*x]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)*(b*c*(m+p+1)-a*d*p+b*d*(m+1)*Sin[e+f*x])/(b^2*f*(m+1)*(m+p+1)) + + g^2*(p-1)/(b^2*(m+1)*(m+p+1))* + Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^(m+1)*Simp[b*d*(m+1)+(b*c*(m+p+1)-a*d*p)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m,p] && m<-1 && p>1 && NeQ[m+p+1] && IntegerQ[2*m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + -(b*c-a*d)*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(f*g*(a^2-b^2)*(m+1)) + + 1/((a^2-b^2)*(m+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1)*Simp[(a*c-b*d)*(m+1)-(b*c-a*d)*(m+p+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegerQ[2*m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)*(a+b*Sin[e+f*x])^(m+1)*(b*c*(m+p+1)-a*d*p+b*d*(m+p)*Sin[e+f*x])/(b^2*f*(m+p)*(m+p+1)) + + g^2*(p-1)/(b^2*(m+p)*(m+p+1))* + Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^m*Simp[b*(a*d*m+b*c*(m+p+1))+(a*b*c*(m+p+1)-d*(a^2*p-b^2*(m+p)))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[a^2-b^2] && RationalQ[p] && p>1 && NeQ[m+p] && NeQ[m+p+1] && IntegerQ[2*m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + (g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)*(b*c-a*d-(a*c-b*d)*Sin[e+f*x])/(f*g*(a^2-b^2)*(p+1)) + + 1/(g^2*(a^2-b^2)*(p+1))* + Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m*Simp[c*(a^2*(p+2)-b^2*(m+p+2))+a*b*d*m+b*(a*c-b*d)*(m+p+3)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,g,m},x] && NeQ[a^2-b^2] && RationalQ[p] && p<-1 && IntegerQ[2*m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(c_.+d_.*sin[e_.+f_.*x_])/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + d/b*Int[(g*Cos[e+f*x])^p,x] + (b*c-a*d)/b*Int[(g*Cos[e+f*x])^p/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_]),x_Symbol] := + c*g*(g*Cos[e+f*x])^(p-1)/(f*(1+Sin[e+f*x])^((p-1)/2)*(1-Sin[e+f*x])^((p-1)/2))* + Subst[Int[(1+d/c*x)^((p+1)/2)*(1-d/c*x)^((p-1)/2)*(a+b*x)^m,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[cos[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + a^(2*m)*Int[(d*Sin[e+f*x])^n/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && IntegersQ[m,p] && 2*m+p==0 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(2*b*f*g*(m+1)) + + a/(2*g^2)*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-1),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && EqQ[m-p] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*m) - + 1/g^2*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && EqQ[m+p+1] + + +Int[cos[e_.+f_.*x_]^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + 1/a^p*Int[ExpandTrig[(d*sin[e+f*x])^n*(a-b*sin[e+f*x])^(p/2)*(a+b*sin[e+f*x])^(m+p/2),x],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && IntegersQ[m,n,p/2] && (m>0 && p>0 && -m-p2 && p<0 && m+p/2>0) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[ExpandTrig[(g*cos[e+f*x])^p,(d*sin[e+f*x])^n*(a+b*sin[e+f*x])^m,x],x] /; +FreeQ[{a,b,d,e,f,g,n,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[cos[e_.+f_.*x_]^2*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + 1/b^2*Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^(m+1)*(a-b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && (NegativeIntegerQ[m] || Not[PositiveIntegerQ[n]]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a/g)^(2*m)*Int[(g*Cos[e+f*x])^(2*m+p)*(d*Sin[e+f*x])^n/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,g,n,p},x] && EqQ[a^2-b^2] && NegativeIntegerQ[m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a/g)^(2*m)*Int[(g*Cos[e+f*x])^(2*m+p)*(d*Sin[e+f*x])^n/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,g,n},x] && EqQ[a^2-b^2] && IntegerQ[m] && RationalQ[p] && (2*m+p==0 || 2*m+p>0 && p<-1) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + b*(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^m/(a*f*g*(2*m+p+1)) - + 1/(a^2*(2*m+p+1))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^(m+1)*(a*m-b*(2*m+p+1)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,p},x] && EqQ[a^2-b^2] && RationalQ[m] && m<=-1/2 && NeQ[2*m+p+1] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*sin[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -(g*Cos[e+f*x])^(p+1)*(a+b*Sin[e+f*x])^(m+1)/(b*f*g*(m+p+2)) + + 1/(b*(m+p+2))*Int[(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^m*(b*(m+1)-a*(p+1)*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f,g,m,p},x] && EqQ[a^2-b^2] && NeQ[m+p+2] + + +Int[cos[e_.+f_.*x_]^2*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + 1/b^2*Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^(m+1)*(a-b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && IntegersQ[2*m,2*n] + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + -2/(a*b*d)*Int[(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+2),x] + + 1/a^2*Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^(m+2)*(1+Sin[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + 1/d^4*Int[(d*Sin[e+f*x])^(n+4)*(a+b*Sin[e+f*x])^m,x] + + Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^m*(1-2*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[PositiveIntegerQ[m]] + + +Int[cos[e_.+f_.*x_]^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a^m*Cos[e+f*x]/(f*Sqrt[1+Sin[e+f*x]]*Sqrt[1-Sin[e+f*x]])* + Subst[Int[(d*x)^n*(1+b/a*x)^(m+(p-1)/2)*(1-b/a*x)^((p-1)/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && IntegerQ[p/2] && IntegerQ[m] + + +Int[cos[e_.+f_.*x_]^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Cos[e+f*x]/(a^(p-2)*f*Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]])* + Subst[Int[(d*x)^n(a+b*x)^(m+p/2-1/2)*(a-b*x)^(p/2-1/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && IntegerQ[p/2] && Not[IntegerQ[m]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[ExpandTrig[(g*cos[e+f*x])^p,(d*sin[e+f*x])^n*(a+b*sin[e+f*x])^m,x],x] /; +FreeQ[{a,b,d,e,f,g,n,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] && (IntegerQ[p] || PositiveIntegerQ[n]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a^m*g*(g*Cos[e+f*x])^(p-1)/(f*(1+Sin[e+f*x])^((p-1)/2)*(1-Sin[e+f*x])^((p-1)/2))* + Subst[Int[(d*x)^n*(1+b/a*x)^(m+(p-1)/2)*(1-b/a*x)^((p-1)/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,d,e,f,n,p},x] && EqQ[a^2-b^2] && IntegerQ[m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)/(f*(a+b*Sin[e+f*x])^((p-1)/2)*(a-b*Sin[e+f*x])^((p-1)/2))* + Subst[Int[(d*x)^n*(a+b*x)^(m+(p-1)/2)*(a-b*x)^((p-1)/2),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n,p},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_/Sqrt[d_.*sin[e_.+f_.*x_]],x_Symbol] := + -g*(g*Cos[e+f*x])^(p-1)*Sqrt[d*Sin[e+f*x]]*(a+b*Sin[e+f*x])^(m+1)/(a*d*f*(m+1)) + + g^2*(2*m+3)/(2*a*(m+1))*Int[(g*Cos[e+f*x])^(p-2)*(a+b*Sin[e+f*x])^(m+1)/Sqrt[d*Sin[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && EqQ[m+p+1/2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_/Sqrt[d_.*sin[e_.+f_.*x_]],x_Symbol] := + 2*(g*Cos[e+f*x])^(p+1)*Sqrt[d*Sin[e+f*x]]*(a+b*Sin[e+f*x])^m/(d*f*g*(2*m+1)) + + 2*a*m/(g^2*(2*m+1))*Int[(g*Cos[e+f*x])^(p+2)*(a+b*Sin[e+f*x])^(m-1)/Sqrt[d*Sin[e+f*x]],x] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] && RationalQ[m] && m>0 && EqQ[m+p+3/2] + + +Int[cos[e_.+f_.*x_]^2*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^m*(1-Sin[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && NeQ[a^2-b^2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) + + +(* Int[cos[e_.+f_.*x_]^4*sin[e_.+f_.*x_]^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a^2-b^2)*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(a*b^2*d*(m+1)) - + (a^2*(n+1)-b^2*(m+n+2))*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+2)/(a^2*b^2*d*(n+1)*(m+1)) + + 1/(a^2*b*(n+1)*(m+1))*Int[Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)* + Simp[a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+3)+a*b*(m+1)*Sin[e+f*x]-(a^2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] && m<-1 && n<-1 *) + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Cos[e+f*x]*(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(a*d*f*(n+1)) - + (a^2*(n+1)-b^2*(m+n+2))*Cos[e+f*x]*(d*Sin[e+f*x])^(n+2)*(a+b*Sin[e+f*x])^(m+1)/(a^2*b*d^2*f*(n+1)*(m+1)) + + 1/(a^2*b*d*(n+1)*(m+1))*Int[(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+1)* + Simp[a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+3)+a*b*(m+1)*Sin[e+f*x]-(a^2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] && m<-1 && n<-1 + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a^2-b^2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+1)/(a*b^2*d*f*(m+1)) + + (a^2*(n-m+1)-b^2*(m+n+2))*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+2)*(d*Sin[e+f*x])^(n+1)/(a^2*b^2*d*f*(m+1)*(m+2)) - + 1/(a^2*b^2*(m+1)*(m+2))*Int[(a+b*Sin[e+f*x])^(m+2)*(d*Sin[e+f*x])^n* + Simp[a^2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+3)+a*b*(m+2)*Sin[e+f*x]-(a^2*(n+2)*(n+3)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] && m<-1 && Not[n<-1] && (m<-2 || EqQ[m+n+4]) + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + (a^2-b^2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+1)/(a*b^2*d*f*(m+1)) - + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+2)*(d*Sin[e+f*x])^(n+1)/(b^2*d*f*(m+n+4)) - + 1/(a*b^2*(m+1)*(m+n+4))*Int[(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^n* + Simp[a^2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+4)+a*b*(m+1)*Sin[e+f*x]-(a^2*(n+2)*(n+3)-b^2*(m+n+3)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] && m<-1 && Not[n<-1] && NeQ[m+n+4] + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+1)/(a*d*f*(n+1)) - + b*(m+n+2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+2)/(a^2*d^2*f*(n+1)*(n+2)) - + 1/(a^2*d^2*(n+1)*(n+2))*Int[(a+b*Sin[e+f*x])^m*(d*Sin[e+f*x])^(n+2)* + Simp[a^2*n*(n+2)-b^2*(m+n+2)*(m+n+3)+a*b*m*Sin[e+f*x]-(a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,m},x] && NeQ[a^2-b^2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) && Not[m<-1] && RationalQ[n] && n<-1 && + (n<-2 || EqQ[m+n+4]) + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+1)/(a*d*f*(n+1)) - + Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(d*Sin[e+f*x])^(n+2)/(b*d^2*f*(m+n+4)) + + 1/(a*b*d*(n+1)*(m+n+4))*Int[(a+b*Sin[e+f*x])^m*(d*Sin[e+f*x])^(n+1)* + Simp[a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+4)+a*b*(m+3)*Sin[e+f*x]-(a^2*(n+1)*(n+3)-b^2*(m+n+3)*(m+n+4))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,m},x] && NeQ[a^2-b^2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) && Not[m<-1] && RationalQ[n] && n<-1 && + NeQ[m+n+4] + + +Int[cos[e_.+f_.*x_]^4*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + a*(n+3)*Cos[e+f*x]*(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(b^2*d*f*(m+n+3)*(m+n+4)) - + Cos[e+f*x]*(d*Sin[e+f*x])^(n+2)*(a+b*Sin[e+f*x])^(m+1)/(b*d^2*f*(m+n+4)) - + 1/(b^2*(m+n+3)*(m+n+4))*Int[(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^m* + Simp[a^2*(n+1)*(n+3)-b^2*(m+n+3)*(m+n+4)+a*b*m*Sin[e+f*x]-(a^2*(n+2)*(n+3)-b^2*(m+n+3)*(m+n+5))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,m,n},x] && NeQ[a^2-b^2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) && Not[m<-1] && Not[RationalQ[n] && n<-1] && + NeQ[m+n+3] && NeQ[m+n+4] + + +Int[cos[e_.+f_.*x_]^6*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Cos[e+f*x]*(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(a*d*f*(n+1)) - + b*(m+n+2)*Cos[e+f*x]*(d*Sin[e+f*x])^(n+2)*(a+b*Sin[e+f*x])^(m+1)/(a^2*d^2*f*(n+1)*(n+2)) - + a*(n+5)*Cos[e+f*x]*(d*Sin[e+f*x])^(n+3)*(a+b*Sin[e+f*x])^(m+1)/(b^2*d^3*f*(m+n+5)*(m+n+6)) + + Cos[e+f*x]*(d*Sin[e+f*x])^(n+4)*(a+b*Sin[e+f*x])^(m+1)/(b*d^4*f*(m+n+6)) + + 1/(a^2*b^2*d^2*(n+1)*(n+2)*(m+n+5)*(m+n+6))* + Int[(d*Sin[e+f*x])^(n+2)*(a+b*Sin[e+f*x])^m* + Simp[a^4*(n+1)*(n+2)*(n+3)*(n+5)-a^2*b^2*(n+2)*(2*n+1)*(m+n+5)*(m+n+6)+b^4*(m+n+2)*(m+n+3)*(m+n+5)*(m+n+6) + + a*b*m*(a^2*(n+1)*(n+2)-b^2*(m+n+5)*(m+n+6))*Sin[e+f*x] - + (a^4*(n+1)*(n+2)*(4+n)*(n+5)+b^4*(m+n+2)*(m+n+4)*(m+n+5)*(m+n+6)-a^2*b^2*(n+1)*(n+2)*(m+n+5)*(2*n+2*m+13))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,m,n},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] && NeQ[n+1] && NeQ[n+2] && NeQ[m+n+5] && + NeQ[m+n+6] && Not[PositiveIntegerQ[m]] + + +Int[cos[e_.+f_.*x_]^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + Int[ExpandTrig[(d*sin[e+f*x])^n*(a+b*sin[e+f*x])^m*(1-sin[e+f*x]^2)^(p/2),x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && IntegersQ[m,2*n,p/2] && (m<-1 || m==-1 && p>0) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*sin[e_.+f_.*x_]^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + Int[ExpandTrig[(g*cos[e+f*x])^p,sin[e+f*x]^n/(a+b*sin[e+f*x]),x],x] /; +FreeQ[{a,b,e,f,g,p},x] && NeQ[a^2-b^2] && IntegerQ[n] && (n<0 || PositiveIntegerQ[p+1/2]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + g^2/a*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n,x] - + b*g^2/(a^2*d)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+1),x] - + g^2*(a^2-b^2)/(a^2*d^2)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+2)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p>1 && (n<=-2 || n==-3/2 && p==3/2) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + g^2/(a*b)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n*(b-a*Sin[e+f*x]),x] + + g^2*(a^2-b^2)/(a*b*d)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+1)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p>1 && (n<-1 || p==3/2 && n==-1/2) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + g^2/b^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n*(a-b*Sin[e+f*x]),x] - + g^2*(a^2-b^2)/b^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p>1 + + +(* Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + g^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n/(a+b*Sin[e+f*x]),x] - + g^2/d^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+2)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p>1 *) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + a*d^2/(a^2-b^2)*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n-2),x] - + b*d/(a^2-b^2)*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n-1),x] - + a^2*d^2/(g^2*(a^2-b^2))*Int[(g*Cos[e+f*x])^(p+2)*(d*Sin[e+f*x])^(n-2)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p<-1 && n>1 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -d/(a^2-b^2)*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n-1)*(b-a*Sin[e+f*x]),x] + + a*b*d/(g^2*(a^2-b^2))*Int[(g*Cos[e+f*x])^(p+2)*(d*Sin[e+f*x])^(n-1)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p<-1 && n>0 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + 1/(a^2-b^2)*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^n*(a-b*Sin[e+f*x]),x] - + b^2/(g^2*(a^2-b^2))*Int[(g*Cos[e+f*x])^(p+2)*(d*Sin[e+f*x])^n/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && p<-1 + + +Int[Sqrt[g_.*cos[e_.+f_.*x_]]/(Sqrt[sin[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + -4*Sqrt[2]*g/f*Subst[Int[x^2/(((a+b)*g^2+(a-b)*x^4)*Sqrt[1-x^4/g^2]),x],x,Sqrt[g*Cos[e+f*x]]/Sqrt[1+Sin[e+f*x]]] /; +FreeQ[{a,b,e,f,g},x] && NeQ[a^2-b^2] + + +Int[Sqrt[g_.*cos[e_.+f_.*x_]]/(Sqrt[d_*sin[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + Sqrt[Sin[e+f*x]]/Sqrt[d*Sin[e+f*x]]*Int[Sqrt[g*Cos[e+f*x]]/(Sqrt[Sin[e+f*x]]*(a+b*Sin[e+f*x])),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] + + +Int[Sqrt[d_.*sin[e_.+f_.*x_]]/(Sqrt[cos[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + With[{q=Rt[-a^2+b^2,2]}, + 2*Sqrt[2]*d*(b+q)/(f*q)*Subst[Int[1/((d*(b+q)+a*x^2)*Sqrt[1-x^4/d^2]),x],x,Sqrt[d*Sin[e+f*x]]/Sqrt[1+Cos[e+f*x]]] - + 2*Sqrt[2]*d*(b-q)/(f*q)*Subst[Int[1/((d*(b-q)+a*x^2)*Sqrt[1-x^4/d^2]),x],x,Sqrt[d*Sin[e+f*x]]/Sqrt[1+Cos[e+f*x]]]] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[Sqrt[d_.*sin[e_.+f_.*x_]]/(Sqrt[g_.*cos[e_.+f_.*x_]]*(a_+b_.*sin[e_.+f_.*x_])),x_Symbol] := + Sqrt[Cos[e+f*x]]/Sqrt[g*Cos[e+f*x]]*Int[Sqrt[d*Sin[e+f*x]]/(Sqrt[Cos[e+f*x]]*(a+b*Sin[e+f*x])),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + d/b*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n-1),x] - + a*d/b*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n-1)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && -10 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + 1/a*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^n,x] - + b/(a*d)*Int[(g*Cos[e+f*x])^p*(d*Sin[e+f*x])^(n+1)/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[2*n,2*p] && -10 || IntegerQ[n]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_*(a_+b_.*sin[e_.+f_.*x_])^m_,x_Symbol] := + g^2/a*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n*(a+b*Sin[e+f*x])^(m+1),x] - + b*g^2/(a^2*d)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+1)*(a+b*Sin[e+f*x])^(m+1),x] - + g^2*(a^2-b^2)/(a^2*d^2)*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+2)*(a+b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2] && IntegersQ[m,2*n,2*p] && m<0 && p>1 && (n<=-2 || m==-1 && n==-3/2 && p==3/2) + + +Int[cos[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + a^(2*m)*Int[(c+d*Sin[e+f*x])^n/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[a^2-b^2] && IntegersQ[m,p] && 2*m+p==0 + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (a/g)^(2*m)*Int[(g*Cos[e+f*x])^(2*m+p)*(c+d*Sin[e+f*x])^n/(a-b*Sin[e+f*x])^m,x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[a^2-b^2] && IntegerQ[m] && RationalQ[p] && (2*m+p==0 || 2*m+p>0 && p<-1) + + +Int[cos[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + 1/b^2*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n*(a-b*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[a^2-b^2] && IntegersQ[2*m,2*n] + + +Int[cos[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + a^m*Cos[e+f*x]/(f*Sqrt[1+Sin[e+f*x]]*Sqrt[1-Sin[e+f*x]])* + Subst[Int[(1+b/a*x)^(m+(p-1)/2)*(1-b/a*x)^((p-1)/2)*(c+d*x)^n,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[a^2-b^2] && IntegerQ[p/2] && IntegerQ[m] + + +Int[cos[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Cos[e+f*x]/(a^(p-2)*f*Sqrt[a+b*Sin[e+f*x]]*Sqrt[a-b*Sin[e+f*x]])* + Subst[Int[(a+b*x)^(m+p/2-1/2)*(a-b*x)^(p/2-1/2)*(c+d*x)^n,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[a^2-b^2] && IntegerQ[p/2] && Not[IntegerQ[m]] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(g*cos[e+f*x])^p,(a+b*sin[e+f*x])^m*(c+d*sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] && (IntegerQ[p] || PositiveIntegerQ[n]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + a^m*g*(g*Cos[e+f*x])^(p-1)/(f*(1+Sin[e+f*x])^((p-1)/2)*(1-Sin[e+f*x])^((p-1)/2))* + Subst[Int[(1+b/a*x)^(m+(p-1)/2)*(1-b/a*x)^((p-1)/2)*(c+d*x)^n,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[a^2-b^2] && IntegerQ[m] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + g*(g*Cos[e+f*x])^(p-1)/(f*(a+b*Sin[e+f*x])^((p-1)/2)*(a-b*Sin[e+f*x])^((p-1)/2))* + Subst[Int[(a+b*x)^(m+(p-1)/2)*(a-b*x)^((p-1)/2)*(c+d*x)^n,x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] + + +Int[cos[e_.+f_.*x_]^2*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n*(1-Sin[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[a^2-b^2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[cos[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(a+b*sin[e+f*x])^m*(c+d*sin[e+f*x])^n*(1-sin[e+f*x]^2)^(p/2),x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[a^2-b^2] && PositiveIntegerQ[p/2] && (PositiveIntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(g*cos[e+f*x])^p*(a+b*sin[e+f*x])^m*(c+d*sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[a^2-b^2] && IntegersQ[2*m,2*n] + + +Int[(g_.*cos[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + Defer[Int][(g*Cos[e+f*x])^p*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[a^2-b^2] + + +Int[(g_.*sec[e_.+f_.*x_])^p_*(a_.+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + g^(2*IntPart[p])*(g*Cos[e+f*x])^FracPart[p]*(g*Sec[e+f*x])^FracPart[p]* + Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(g*Cos[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && Not[IntegerQ[p]] + + +Int[(g_.*csc[e_.+f_.*x_])^p_*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_Symbol] := + g^(2*IntPart[p])*(g*Sin[e+f*x])^FracPart[p]*(g*Csc[e+f*x])^FracPart[p]* + Int[(a+b*Cos[e+f*x])^m*(c+d*Cos[e+f*x])^n/(g*Sin[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n*) + + +Int[Sqrt[g_.*sin[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(c_+d_.*sin[e_.+f_.*x_]),x_Symbol] := + g/d*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[g*Sin[e+f*x]],x] - + c*g/d*Int[Sqrt[a+b*Sin[e+f*x]]/(Sqrt[g*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[Sqrt[g_.*sin[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(c_+d_.*sin[e_.+f_.*x_]),x_Symbol] := + b/d*Int[Sqrt[g*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] - + (b*c-a*d)/d*Int[Sqrt[g*Sin[e+f*x]]/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(Sqrt[g_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + -2*b/f*Subst[Int[1/(b*c+a*d+c*g*x^2),x],x,b*Cos[e+f*x]/(Sqrt[g*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(Sqrt[sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + -Sqrt[a+b]/(c*f)*EllipticE[ArcSin[Cos[e+f*x]/(1+Sin[e+f*x])],-(a-b)/(a+b)] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[d-c] && PositiveQ[b^2-a^2] && PositiveQ[b] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(Sqrt[g_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + -Sqrt[a+b*Sin[e+f*x]]*Sqrt[d*Sin[e+f*x]/(c+d*Sin[e+f*x])]/ + (d*f*Sqrt[g*Sin[e+f*x]]*Sqrt[c^2*(a+b*Sin[e+f*x])/((a*c+b*d)*(c+d*Sin[e+f*x]))])* + EllipticE[ArcSin[c*Cos[e+f*x]/(c+d*Sin[e+f*x])],(b*c-a*d)/(b*c+a*d)] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(Sqrt[g_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + a/c*Int[1/(Sqrt[g*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]),x] + + (b*c-a*d)/(c*g)*Int[Sqrt[g*Sin[e+f*x]]/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + 1/c*Int[Sqrt[a+b*Sin[e+f*x]]/Sin[e+f*x],x] - + d/c*Int[Sqrt[a+b*Sin[e+f*x]]/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + a/c*Int[1/(Sin[e+f*x]*Sqrt[a+b*Sin[e+f*x]]),x] + + (b*c-a*d)/c*Int[1/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[Sqrt[g_.*sin[e_.+f_.*x_]]/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + -a*g/(b*c-a*d)*Int[1/(Sqrt[g*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]),x] + + c*g/(b*c-a*d)*Int[Sqrt[a+b*Sin[e+f*x]]/(Sqrt[g*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[Sqrt[g_.*sin[e_.+f_.*x_]]/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + 2*Sqrt[-Cot[e+f*x]^2]*Sqrt[g*Sin[e+f*x]]/(f*(c+d)*Cot[e+f*x]*Sqrt[a+b*Sin[e+f*x]])*Sqrt[(b+a*Csc[e+f*x])/(a+b)]* + EllipticPi[2*c/(c+d),ArcSin[Sqrt[1-Csc[e+f*x]]/Sqrt[2]],2*a/(a+b)] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(Sqrt[g_.*sin[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + b/(b*c-a*d)*Int[1/(Sqrt[g*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]),x] - + d/(b*c-a*d)*Int[Sqrt[a+b*Sin[e+f*x]]/(Sqrt[g*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[1/(Sqrt[g_.*sin[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + 1/c*Int[1/(Sqrt[g*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]),x] - + d/(c*g)*Int[Sqrt[g*Sin[e+f*x]]/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(sin[e_.+f_.*x_]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + d^2/(c*(b*c-a*d))*Int[Sqrt[a+b*Sin[e+f*x]]/(c+d*Sin[e+f*x]),x] + + 1/(c*(b*c-a*d))*Int[(b*c-a*d-b*d*Sin[e+f*x])/(Sin[e+f*x]*Sqrt[a+b*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[1/(sin[e_.+f_.*x_]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_+d_.*sin[e_.+f_.*x_])),x_Symbol] := + 1/c*Int[1/(Sin[e+f*x]*Sqrt[a+b*Sin[e+f*x]]),x] - d/c*Int[1/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -d/c*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + 1/c*Int[Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]/Sin[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[b*c+a*d] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -2*a/f*Subst[Int[1/(1-a*c*x^2),x],x,Cos[e+f*x]/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[b*c+a*d] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + (b*c-a*d)/c*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] + + a/c*Int[Sqrt[c+d*Sin[e+f*x]]/(Sin[e+f*x]*Sqrt[a+b*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/(sin[e_.+f_.*x_]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -2*(a+b*Sin[e+f*x])/(c*f*Rt[(a+b)/(c+d),2]*Cos[e+f*x])* + Sqrt[-(b*c-a*d)*(1-Sin[e+f*x])/((c+d)*(a+b*Sin[e+f*x]))]*Sqrt[(b*c-a*d)*(1+Sin[e+f*x])/((c-d)*(a+b*Sin[e+f*x]))]* + EllipticPi[a*(c+d)/(c*(a+b)),ArcSin[Rt[(a+b)/(c+d),2]*Sqrt[c+d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]]],(a-b)*(c+d)/((a+b)*(c-d))] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(sin[e_.+f_.*x_]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + Cos[e+f*x]/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]])*Int[1/(Cos[e+f*x]*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[1/(sin[e_.+f_.*x_]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -b/a*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] + + 1/a*Int[Sqrt[a+b*Sin[e+f*x]]/(Sin[e+f*x]*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (NeQ[a^2-b^2] || NeQ[c^2-d^2]) + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]/sin[e_.+f_.*x_],x_Symbol] := + Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]/Cos[e+f*x]*Int[Cot[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]/sin[e_.+f_.*x_],x_Symbol] := + d*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + c*Int[Sqrt[a+b*Sin[e+f*x]]/(Sin[e+f*x]*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (NeQ[a^2-b^2] || NeQ[c^2-d^2]) + + +Int[sin[e_.+f_.*x_]^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^n*c^n*Int[Tan[e+f*x]^p*(a+b*Sin[e+f*x])^(m-n),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && EqQ[p+2*n] && IntegerQ[n] + + +Int[(g_.*sin[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Sqrt[a-b*Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]/(f*Cos[e+f*x])* + Subst[Int[(g*x)^p*(a+b*x)^(m-1/2)*(c+d*x)^n/Sqrt[a-b*x],x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && IntegerQ[m-1/2] + + +Int[(g_.*sin[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(g*sin[e+f*x])^p*(a+b*sin[e+f*x])^m*(c+d*sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && NeQ[b*c-a*d] && (IntegersQ[m,n] || IntegersQ[m,p] || IntegersQ[n,p]) && NeQ[p-2] + + +Int[(g_.*sin[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + Defer[Int][(g*Sin[e+f*x])^p*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[p-2] + + +Int[(g_.*sin[e_.+f_.*x_])^p_.*(a_.+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + g^(m+n)*Int[(g*Sin[e+f*x])^(p-m-n)*(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b*c-a*d] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n] + + +Int[(g_.*sin[e_.+f_.*x_])^p_.*(a_.+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + (g*Csc[e+f*x])^p*(g*Sin[e+f*x])^p*Int[(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/(g*Csc[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]] + + +Int[(g_.*sin[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + g^n*Int[(g*Sin[e+f*x])^(p-n)*(a+b*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && IntegerQ[n] + + +Int[sin[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Int[(b+a*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/Csc[e+f*x]^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p] + + +Int[(g_.*sin[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Csc[e+f*x]^p*(g*Sin[e+f*x])^p*Int[(b+a*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/Csc[e+f*x]^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]] + + +Int[(g_.*sin[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + (g*Sin[e+f*x])^n*(c+d*Csc[e+f*x])^n/(d+c*Sin[e+f*x])^n*Int[(g*Sin[e+f*x])^(p-n)*(a+b*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && Not[IntegerQ[n]] && Not[IntegerQ[m]] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_.+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + g^(m+n)*Int[(g*Csc[e+f*x])^(p-m-n)*(b+a*Csc[e+f*x])^m*(d+c*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b*c-a*d] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_.+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + (g*Csc[e+f*x])^p*(g*Sin[e+f*x])^p*Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(g*Sin[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + g^m*Int[(g*Csc[e+f*x])^(p-m)*(b+a*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && IntegerQ[m] + + +Int[csc[e_.+f_.*x_]^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Int[(a+b*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^(n+p),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && Not[IntegerQ[m]] && IntegerQ[n] && IntegerQ[p] + + +Int[(g_.*csc[e_.+f_.*x_])^p_*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Sin[e+f*x]^p*(g*Csc[e+f*x])^p*Int[(a+b*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^(n+p),x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && Not[IntegerQ[m]] && IntegerQ[n] && Not[IntegerQ[p]] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + (a+b*Sin[e+f*x])^m*(g*Csc[e+f*x])^m/(b+a*Csc[e+f*x])^m* + Int[(g*Csc[e+f*x])^(p-m)*(b+a*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin)*) + + +Int[sin[e_.+f_.*x_]^n_.*(a_+b_.*sin[e_.+f_.*x_])^m_.*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + Int[ExpandTrig[sin[e+f*x]^n*(a+b*sin[e+f*x])^m*(A+B*sin[e+f*x]),x],x] /; +FreeQ[{a,b,e,f,A,B},x] && EqQ[A*b+a*B] && EqQ[a^2-b^2] && IntegerQ[m] && IntegerQ[n] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_+d_.*sin[e_.+f_.*x_])^n_.*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + a^m*c^m*Int[Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m)*(A+B*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m] && Not[IntegerQ[n] && (m<0 && n>0 || 01/2 && n<-1 && + IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -b*B*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n+1)) + + 1/(d*(m+n+1))*Int[(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n* + Simp[a*A*d*(m+n+1)+B*(a*c*(m-1)+b*d*(n+1))+(A*b*d*(m+n+1)-B*(b*c*m-a*d*(2*m+n)))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>1/2 && + Not[RationalQ[n] && n<-1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(a*f*(2*m+1)) - + 1/(a*b*(2*m+1))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n-1)* + Simp[A*(a*d*n-b*c*(m+1))-B*(a*c*m+b*d*n)-d*(a*B*(m-n)+A*b*(m+n+1))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m<-1/2 && n>0 && + IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + b*(A*b-a*B)*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(a*f*(2*m+1)*(b*c-a*d)) + + 1/(a*(2*m+1)*(b*c-a*d))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n* + Simp[B*(a*c*m+b*d*(n+1))+A*(b*c*(m+1)-a*d*(2*m+n+2))+d*(A*b-a*B)*(m+n+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m<-1/2 && + Not[RationalQ[n] && n>0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c]) + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -2*b*B*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(d*f*(2*n+3)*Sqrt[a+b*Sin[e+f*x]]) /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && EqQ[A*b*d*(2*n+3)-B*(b*c-2*a*d*(n+1))] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -b^2*(B*c-A*d)*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(d*f*(n+1)*(b*c+a*d)*Sqrt[a+b*Sin[e+f*x]]) + + (A*b*d*(2*n+3)-B*(b*c-2*a*d*(n+1)))/(2*d*(n+1)*(b*c+a*d))*Int[Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n<-1 + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -2*b*B*Cos[e+f*x]*(c+d*Sin[e+f*x])^(n+1)/(d*f*(2*n+3)*Sqrt[a+b*Sin[e+f*x]]) + + (A*b*d*(2*n+3)-B*(b*c-2*a*d*(n+1)))/(b*d*(2*n+3))*Int[Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && Not[RationalQ[n] && n<-1] + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_.+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + (A*b-a*B)/b*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] + + B/b*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -B*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n/(f*(m+n+1)) + + 1/(b*(m+n+1))*Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n-1)* + Simp[A*b*c*(m+n+1)+B*(a*c*m+b*d*n)+(A*b*d*(m+n+1)+B*(a*d*m+b*c*n))*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n>0 && + (IntegerQ[n] || EqQ[m+1/2]) + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + (B*c-A*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(f*(n+1)*(c^2-d^2)) + + 1/(b*(n+1)*(c^2-d^2))*Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)* + Simp[A*(a*d*m+b*c*(n+1))-B*(a*c*m+b*d*(n+1))+b*(B*c-A*d)*(m+n+2)*Sin[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n<-1 && + (IntegerQ[n] || EqQ[m+1/2]) + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + (A*b-a*B)/(b*c-a*d)*Int[1/Sqrt[a+b*Sin[e+f*x]],x] + + (B*c-A*d)/(b*c-a*d)*Int[Sqrt[a+b*Sin[e+f*x]]/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(A_.+B_.*sin[e_.+f_.*x_])/(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + B/d*Int[(a+b*Sin[e+f*x])^m,x] - (B*c-A*d)/d*Int[(a+b*Sin[e+f*x])^m/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && NeQ[m+1/2] + + +Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)/b*Int[(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n,x] + + B/b*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && NeQ[A*b+a*B] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -(b*c-a*d)*(B*c-A*d)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(n+1)*(c^2-d^2)) + + 1/(d*(n+1)*(c^2-d^2))*Int[(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^(n+1)* + Simp[b*(b*c-a*d)*(B*c-A*d)*(m-1)+a*d*(a*A*c+b*B*c-(A*b+a*B)*d)*(n+1)+ + (b*(b*d*(B*c-A*d)+a*(A*c*d+B*(c^2-2*d^2)))*(n+1)-a*(b*c-a*d)*(B*c-A*d)*(n+2))*Sin[e+f*x]+ + b*(d*(A*b*c+a*B*c-a*A*d)*(m+n+1)-b*B*(c^2*m+d^2*(n+1)))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m>1 && n<-1 + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -b*B*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n+1)) + + 1/(d*(m+n+1))*Int[(a+b*Sin[e+f*x])^(m-2)*(c+d*Sin[e+f*x])^n* + Simp[a^2*A*d*(m+n+1)+b*B*(b*c*(m-1)+a*d*(n+1))+ + (a*d*(2*A*b+a*B)*(m+n+1)-b*B*(a*c-b*d*(m+n)))*Sin[e+f*x]+ + b*(A*b*d*(m+n+1)-B*(b*c*m-a*d*(2*m+n)))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>1 && + Not[IntegerQ[n] && n>1 && (Not[IntegerQ[m]] || EqQ[a] && NeQ[c])] + + +Int[Sqrt[c_+d_.*sin[e_.+f_.*x_]]*(A_.+B_.*sin[e_.+f_.*x_])/(b_.*sin[e_.+f_.*x_])^(3/2),x_Symbol] := + B*d/b^2*Int[Sqrt[b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + Int[(A*c+(B*c+A*d)*Sin[e+f*x])/((b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{b,c,d,e,f,A,B},x] && NeQ[c^2-d^2] + + +Int[Sqrt[c_.+d_.*sin[e_.+f_.*x_]]*(A_.+B_.*sin[e_.+f_.*x_])/(a_+b_.*sin[e_.+f_.*x_])^(3/2),x_Symbol] := + B/b*Int[Sqrt[c+d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] + + (A*b-a*B)/b*Int[Sqrt[c+d*Sin[e+f*x]]/(a+b*Sin[e+f*x])^(3/2),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/((a_+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[d_.*sin[e_.+f_.*x_]]),x_Symbol] := + 2*(A*b-a*B)*Cos[e+f*x]/(f*(a^2-b^2)*Sqrt[a+b*Sin[e+f*x]]*Sqrt[d*Sin[e+f*x]]) + + d/(a^2-b^2)*Int[(A*b-a*B+(a*A-b*B)*Sin[e+f*x])/(Sqrt[a+b*Sin[e+f*x]]*(d*Sin[e+f*x])^(3/2)),x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[a^2-b^2] + + +Int[(A_+B_.*sin[e_.+f_.*x_])/((b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -2*A*(c-d)*Tan[e+f*x]/(f*b*c^2)*Rt[(c+d)/b,2]*Sqrt[c*(1+Csc[e+f*x])/(c-d)]*Sqrt[c*(1-Csc[e+f*x])/(c+d)]* + EllipticE[ArcSin[Sqrt[c+d*Sin[e+f*x]]/Sqrt[b*Sin[e+f*x]]/Rt[(c+d)/b,2]],-(c+d)/(c-d)] /; +FreeQ[{b,c,d,e,f,A,B},x] && NeQ[c^2-d^2] && EqQ[A-B] && PosQ[(c+d)/b] + + +Int[(A_+B_.*sin[e_.+f_.*x_])/((b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -Sqrt[-b*Sin[e+f*x]]/Sqrt[b*Sin[e+f*x]]*Int[(A+B*Sin[e+f*x])/((-b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{b,c,d,e,f,A,B},x] && NeQ[c^2-d^2] && EqQ[A-B] && NegQ[(c+d)/b] + + +Int[(A_+B_.*sin[e_.+f_.*x_])/((a_+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -2*A*(c-d)*(a+b*Sin[e+f*x])/(f*(b*c-a*d)^2*Rt[(a+b)/(c+d),2]*Cos[e+f*x])* + Sqrt[(b*c-a*d)*(1+Sin[e+f*x])/((c-d)*(a+b*Sin[e+f*x]))]* + Sqrt[-(b*c-a*d)*(1-Sin[e+f*x])/((c+d)*(a+b*Sin[e+f*x]))]* + EllipticE[ArcSin[Rt[(a+b)/(c+d),2]*Sqrt[c+d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]]],(a-b)*(c+d)/((a+b)*(c-d))] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && EqQ[A-B] && PosQ[(a+b)/(c+d)] + + +Int[(A_+B_.*sin[e_.+f_.*x_])/((a_+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + Sqrt[-c-d*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]]*Int[(A+B*Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[-c-d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && EqQ[A-B] && NegQ[(a+b)/(c+d)] + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/((a_.+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + (A-B)/(a-b)*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] - + (A*b-a*B)/(a-b)*Int[(1+Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && NeQ[A-B] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + (B*a-A*b)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n-1)* + Simp[c*(a*A-b*B)*(m+1)+d*n*(A*b-a*B)+(d*(a*A-b*B)*(m+1)-c*(A*b-a*B)*(m+2))*Sin[e+f*x]-d*(A*b-a*B)*(m+n+2)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m<-1 && n>0 + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -(A*b^2-a*b*B)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(1+n)/(f*(m+1)*(b*c-a*d)*(a^2-b^2)) + + 1/((m+1)*(b*c-a*d)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n* + Simp[(a*A-b*B)*(b*c-a*d)*(m+1)+b*d*(A*b-a*B)*(m+n+2)+ + (A*b-a*B)*(a*d*(m+1)-b*c*(m+2))*Sin[e+f*x]- + b*d*(A*b-a*B)*(m+n+3)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m<-1 && + (EqQ[a] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && n<-1 && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a])]) + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/((a_.+b_.*sin[e_.+f_.*x_])*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + (A*b-a*B)/(b*c-a*d)*Int[1/(a+b*Sin[e+f*x]),x] + (B*c-A*d)/(b*c-a*d)*Int[1/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(A_.+B_.*sin[e_.+f_.*x_])/(c_.+d_.*sin[e_.+f_.*x_]),x_Symbol] := + B/d*Int[(a+b*Sin[e+f*x])^m,x] - (B*c-A*d)/d*Int[(a+b*Sin[e+f*x])^m/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_.+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + -2*B*Cos[e+f*x]*Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])^n/(f*(2*n+3)) + + 1/(2*n+3)*Int[(c+d*Sin[e+f*x])^(n-1)/Sqrt[a+b*Sin[e+f*x]]* + Simp[a*A*c*(2*n+3)+B*(b*c+2*a*d*n)+ + (B*(a*c+b*d)*(2*n+1)+A*(b*c+a*d)*(2*n+3))*Sin[e+f*x]+ + (A*b*d*(2*n+3)+B*(a*d+2*b*c*n))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[n] && n^2==1/4 + + +Int[(A_+B_.*sin[e_.+f_.*x_])/(Sqrt[sin[e_.+f_.*x_]]*Sqrt[a_+b_.*sin[e_.+f_.*x_]]),x_Symbol] := + 4*A/(f*Sqrt[a+b])*EllipticPi[-1,-ArcSin[Cos[e+f*x]/(1+Sin[e+f*x])],-(a-b)/(a+b)] /; +FreeQ[{a,b,e,f,A,B},x] && PositiveQ[b] && PositiveQ[b^2-a^2] && EqQ[A-B] + + +Int[(A_+B_.*sin[e_.+f_.*x_])/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[d_*sin[e_.+f_.*x_]]),x_Symbol] := + Sqrt[Sin[e+f*x]]/Sqrt[d*Sin[e+f*x]]*Int[(A+B*Sin[e+f*x])/(Sqrt[Sin[e+f*x]]*Sqrt[a+b*Sin[e+f*x]]),x] /; +FreeQ[{a,b,e,f,d,A,B},x] && PositiveQ[b] && PositiveQ[b^2-a^2] && EqQ[A-B] + + +Int[(A_.+B_.*sin[e_.+f_.*x_])/(Sqrt[a_+b_.*sin[e_.+f_.*x_]]*Sqrt[c_.+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + B/d*Int[Sqrt[c+d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] - + (B*c-A*d)/d*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_.*(A_.+B_.*sin[e_.+f_.*x_]),x_Symbol] := + Defer[Int][(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n*(A+B*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_])^p_,x_Symbol] := + a^m*c^m*Int[Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m)*(A+B*Sin[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m] && + Not[IntegerQ[n] && (m<0 && n>0 || 00 || 00 && n<-1 + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -(c^2*C+A*d^2)*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(d*f*(n+1)*(c^2-d^2)) + + 1/(d*(n+1)*(c^2-d^2))*Int[(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^(n+1)* + Simp[A*d*(b*d*m+a*c*(n+1))+c*C*(b*c*m+a*d*(n+1)) - + (A*d*(a*d*(n+2)-b*c*(n+1))-C*(b*c*d*(n+1)-a*(c^2+d^2*(n+1))))*Sin[e+f*x] - + b*(A*d^2*(m+n+2)+C*(c^2*(m+1)+d^2*(n+1)))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m,n] && m>0 && n<-1 + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_])^n_.*(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -C*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n+2)) + + 1/(d*(m+n+2))*Int[(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n* + Simp[a*A*d*(m+n+2)+C*(b*c*m+a*d*(n+1))+ + (d*(A*b+a*B)*(m+n+2)-C*(a*c-b*d*(m+n+1)))*Sin[e+f*x]+ + (C*(a*d*m-b*c*(m+1))+b*B*d*(m+n+2))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>0 && + Not[IntegerQ[n] && n>0 && (Not[IntegerQ[m]] || EqQ[a] && NeQ[c])] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_.*(c_.+d_.*sin[e_.+f_.*x_])^n_.*(A_.+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -C*Cos[e+f*x]*(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^(n+1)/(d*f*(m+n+2)) + + 1/(d*(m+n+2))*Int[(a+b*Sin[e+f*x])^(m-1)*(c+d*Sin[e+f*x])^n* + Simp[a*A*d*(m+n+2)+C*(b*c*m+a*d*(n+1))+(A*b*d*(m+n+2)-C*(a*c-b*d*(m+n+1)))*Sin[e+f*x]+C*(a*d*m-b*c*(m+1))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m>0 && + Not[IntegerQ[n] && n>0 && (Not[IntegerQ[m]] || EqQ[a] && NeQ[c])] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2)/((a_+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[d_.*sin[e_.+f_.*x_]]),x_Symbol] := + C/(b*d)*Int[Sqrt[d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] + + 1/b*Int[(A*b+(b*B-a*C)*Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,d,e,f,A,B,C},x] && NeQ[a^2-b^2] + + +Int[(A_.+C_.*sin[e_.+f_.*x_]^2)/((a_+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[d_.*sin[e_.+f_.*x_]]),x_Symbol] := + C/(b*d)*Int[Sqrt[d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] + + 1/b*Int[(A*b-a*C*Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,d,e,f,A,C},x] && NeQ[a^2-b^2] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2)/((a_.+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_.+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + C/b^2*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + 1/b^2*Int[(A*b^2-a^2*C+b*(b*B-2*a*C)*Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+C_.*sin[e_.+f_.*x_]^2)/((a_.+b_.*sin[e_.+f_.*x_])^(3/2)*Sqrt[c_.+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + C/b^2*Int[Sqrt[a+b*Sin[e+f*x]]/Sqrt[c+d*Sin[e+f*x]],x] + + 1/b^2*Int[(A*b^2-a^2*C-2*a*b*C*Sin[e+f*x])/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -(A*b^2-a*b*B+a^2*C)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n+1)/(f*(m+1)*(b*c-a*d)*(a^2-b^2)) + + 1/((m+1)*(b*c-a*d)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n* + Simp[(m+1)*(b*c-a*d)*(a*A-b*B+a*C)+d*(A*b^2-a*b*B+a^2*C)*(m+n+2) - + (c*(A*b^2-a*b*B+a^2*C)+(m+1)*(b*c-a*d)*(A*b-a*B+b*C))*Sin[e+f*x] - + d*(A*b^2-a*b*B+a^2*C)*(m+n+3)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m<-1 && + (EqQ[a] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && n<-1 && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a])]) + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -(A*b^2+a^2*C)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^(n+1)/(f*(m+1)*(b*c-a*d)*(a^2-b^2)) + + 1/((m+1)*(b*c-a*d)*(a^2-b^2))*Int[(a+b*Sin[e+f*x])^(m+1)*(c+d*Sin[e+f*x])^n* + Simp[a*(m+1)*(b*c-a*d)*(A+C)+d*(A*b^2+a^2*C)*(m+n+2) - + (c*(A*b^2+a^2*C)+b*(m+1)*(b*c-a*d)*(A+C))*Sin[e+f*x] - + d*(A*b^2+a^2*C)*(m+n+3)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && RationalQ[m] && m<-1 && + (EqQ[a] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && n<-1 && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a])]) + + +Int[(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2)/((a_+b_.*sin[e_.+f_.*x_])*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + C*x/(b*d) + + (A*b^2-a*b*B+a^2*C)/(b*(b*c-a*d))*Int[1/(a+b*Sin[e+f*x]),x] - + (c^2*C-B*c*d+A*d^2)/(d*(b*c-a*d))*Int[1/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+C_.*sin[e_.+f_.*x_]^2)/((a_+b_.*sin[e_.+f_.*x_])*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + C*x/(b*d) + + (A*b^2+a^2*C)/(b*(b*c-a*d))*Int[1/(a+b*Sin[e+f*x]),x] - + (c^2*C+A*d^2)/(d*(b*c-a*d))*Int[1/(c+d*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2)/(Sqrt[a_.+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + C/(b*d)*Int[Sqrt[a+b*Sin[e+f*x]],x] - + 1/(b*d)*Int[Simp[a*c*C-A*b*d+(b*c*C-b*B*d+a*C*d)*Sin[e+f*x],x]/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+C_.*sin[e_.+f_.*x_]^2)/(Sqrt[a_.+b_.*sin[e_.+f_.*x_]]*(c_.+d_.*sin[e_.+f_.*x_])),x_Symbol] := + C/(b*d)*Int[Sqrt[a+b*Sin[e+f*x]],x] - + 1/(b*d)*Int[Simp[a*c*C-A*b*d+(b*c*C+a*C*d)*Sin[e+f*x],x]/(Sqrt[a+b*Sin[e+f*x]]*(c+d*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2)/(Sqrt[a_.+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -C*Cos[e+f*x]*Sqrt[c+d*Sin[e+f*x]]/(d*f*Sqrt[a+b*Sin[e+f*x]]) + + 1/(2*d)*Int[1/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]])* + Simp[2*a*A*d-C*(b*c-a*d)-2*(a*c*C-d*(A*b+a*B))*Sin[e+f*x]+(2*b*B*d-C*(b*c+a*d))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(A_.+C_.*sin[e_.+f_.*x_]^2)/(Sqrt[a_.+b_.*sin[e_.+f_.*x_]]*Sqrt[c_+d_.*sin[e_.+f_.*x_]]),x_Symbol] := + -C*Cos[e+f*x]*Sqrt[c+d*Sin[e+f*x]]/(d*f*Sqrt[a+b*Sin[e+f*x]]) + + 1/(2*d)*Int[1/((a+b*Sin[e+f*x])^(3/2)*Sqrt[c+d*Sin[e+f*x]])* + Simp[2*a*A*d-C*(b*c-a*d)-2*(a*c*C-A*b*d)*Sin[e+f*x]-C*(b*c+a*d)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + Defer[Int][(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n*(A+B*Sin[e+f*x]+C*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_.+b_.*sin[e_.+f_.*x_])^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_*(A_.+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + Defer[Int][(a+b*Sin[e+f*x])^m*(c+d*Sin[e+f*x])^n*(A+C*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(b_.*sin[e_.+f_.*x_]^p_)^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_.*(A_.+B_.*sin[e_.+f_.*x_]+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + (b*Sin[e+f*x]^p)^m/(b*Sin[e+f*x])^(m*p)*Int[(b*Sin[e+f*x])^(m*p)*(c+d*Sin[e+f*x])^n*(A+B*Sin[e+f*x]+C*Sin[e+f*x]^2),x] /; +FreeQ[{b,c,d,e,f,A,B,C,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(b_.*cos[e_.+f_.*x_]^p_)^m_*(c_.+d_.*cos[e_.+f_.*x_])^n_.*(A_.+B_.*cos[e_.+f_.*x_]+C_.*cos[e_.+f_.*x_]^2),x_Symbol] := + (b*Cos[e+f*x]^p)^m/(b*Cos[e+f*x])^(m*p)*Int[(b*Cos[e+f*x])^(m*p)*(c+d*Cos[e+f*x])^n*(A+B*Cos[e+f*x]+C*Cos[e+f*x]^2),x] /; +FreeQ[{b,c,d,e,f,A,B,C,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(b_.*sin[e_.+f_.*x_]^p_)^m_*(c_.+d_.*sin[e_.+f_.*x_])^n_.*(A_.+C_.*sin[e_.+f_.*x_]^2),x_Symbol] := + (b*Sin[e+f*x]^p)^m/(b*Sin[e+f*x])^(m*p)*Int[(b*Sin[e+f*x])^(m*p)*(c+d*Sin[e+f*x])^n*(A+C*Sin[e+f*x]^2),x] /; +FreeQ[{b,c,d,e,f,A,C,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(b_.*cos[e_.+f_.*x_]^p_)^m_*(c_.+d_.*cos[e_.+f_.*x_])^n_.*(A_.+C_.*cos[e_.+f_.*x_]^2),x_Symbol] := + (b*Cos[e+f*x]^p)^m/(b*Cos[e+f*x])^(m*p)*Int[(b*Cos[e+f*x])^(m*p)*(c+d*Cos[e+f*x])^n*(A+C*Cos[e+f*x]^2),x] /; +FreeQ[{b,c,d,e,f,A,C,m,n,p},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.5 trig^m (a cos+b sin)^n*) + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(b*d*n) /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2+b^2,0] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -1/d*Subst[Int[(a^2+b^2-x^2)^((n-1)/2),x],x,b*Cos[c+d*x]-a*Sin[c+d*x]] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && PositiveIntegerQ[(n-1)/2] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -(b*Cos[c+d*x]-a*Sin[c+d*x])*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*n) + + (n-1)*(a^2+b^2)/n*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && Not[OddQ[n]] && GtQ[n,1] + + +Int[1/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + -1/d*Subst[Int[1/(a^2+b^2-x^2),x],x,b*Cos[c+d*x]-a*Sin[c+d*x]] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[1/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^2,x_Symbol] := + Sin[c+d*x]/(a*d*(a*Cos[c+d*x]+b*Sin[c+d*x])) /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + (b*Cos[c+d*x]-a*Sin[c+d*x])*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/(d*(n+1)*(a^2+b^2)) + + (n+2)/((n+1)*(a^2+b^2))*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] && NeQ[n,-2] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + (a^2+b^2)^(n/2)*Int[(Cos[c+d*x-ArcTan[a,b]])^n,x] /; +FreeQ[{a,b,c,d,n},x] && Not[GeQ[n,1] || LeQ[n,-1]] && PositiveQ[a^2+b^2] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + (a*Cos[c+d*x]+b*Sin[c+d*x])^n/((a*Cos[c+d*x]+b*Sin[c+d*x])/Sqrt[a^2+b^2])^n*Int[Cos[c+d*x-ArcTan[a,b]]^n,x] /; +FreeQ[{a,b,c,d,n},x] && Not[GeQ[n,1] || LeQ[n,-1]] && Not[PositiveQ[a^2+b^2] || EqQ[a^2+b^2,0]] + + +Int[sin[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -a*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*(n-1)*Sin[c+d*x]^(n-1)) + + 2*b*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/Sin[c+d*x]^(n-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && GtQ[n,1] + + +Int[cos[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + b*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*(n-1)*Cos[c+d*x]^(n-1)) + + 2*a*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/Cos[c+d*x]^(n-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && GtQ[n,1] + + +Int[sin[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(2*b*d*n*Sin[c+d*x]^n) + + 1/(2*b)*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/Sin[c+d*x]^(n+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && LtQ[n,0] + + +Int[cos[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -b*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(2*a*d*n*Cos[c+d*x]^n) + + 1/(2*a)*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/Cos[c+d*x]^(n+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && LtQ[n,0] + + +Int[sin[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(2*b*d*n*Sin[c+d*x]^n)*Hypergeometric2F1[1,n,n+1,(b+a*Cot[c+d*x])/(2*b)] /; +FreeQ[{a,b,c,d,n},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && Not[IntegerQ[n]] + + +Int[cos[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -b*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(2*a*d*n*Cos[c+d*x]^n)*Hypergeometric2F1[1,n,n+1,(a+b*Tan[c+d*x])/(2*a)] /; +FreeQ[{a,b,c,d,n},x] && EqQ[m+n,0] && EqQ[a^2+b^2,0] && Not[IntegerQ[n]] + + +Int[sin[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_.,x_Symbol] := + Int[(b+a*Cot[c+d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && IntegerQ[n] && NeQ[a^2+b^2,0] + + +Int[cos[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_.,x_Symbol] := + Int[(a+b*Tan[c+d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && EqQ[m+n,0] && IntegerQ[n] && NeQ[a^2+b^2,0] + + +Int[sin[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + 1/d*Subst[Int[x^m*(a+b*x)^n/(1+x^2)^((m+n+2)/2),x],x,Tan[c+d*x]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[n] && IntegerQ[(m+n)/2] && n!=-1 && Not[n>0 && m>1] + + +Int[cos[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -1/d*Subst[Int[x^m*(b+a*x)^n/(1+x^2)^((m+n+2)/2),x],x,Cot[c+d*x]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[n] && IntegerQ[(m+n)/2] && n!=-1 && Not[n>0 && m>1] + + +Int[sin[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_.,x_Symbol] := + Int[ExpandTrig[sin[c+d*x]^m*(a*cos[c+d*x]+b*sin[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && PositiveIntegerQ[n] + + +Int[cos[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_.,x_Symbol] := + Int[ExpandTrig[cos[c+d*x]^m*(a*cos[c+d*x]+b*sin[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && PositiveIntegerQ[n] + + +Int[sin[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a^n*b^n*Int[Sin[c+d*x]^m*(b*Cos[c+d*x]+a*Sin[c+d*x])^(-n),x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[a^2+b^2,0] && NegativeIntegerQ[n] + + +Int[cos[c_.+d_.*x_]^m_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + a^n*b^n*Int[Cos[c+d*x]^m*(b*Cos[c+d*x]+a*Sin[c+d*x])^(-n),x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[a^2+b^2,0] && NegativeIntegerQ[n] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_/sin[c_.+d_.*x_],x_Symbol] := + a*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*(n-1)) + + b*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1),x] + + a^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2)/Sin[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_/cos[c_.+d_.*x_],x_Symbol] := + -b*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*(n-1)) + + a*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1),x] + + b^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2)/Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] + + +Int[sin[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -(a^2+b^2)*Int[Sin[c+d*x]^(m+2)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x] + + 2*b*Int[Sin[c+d*x]^(m+1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1),x] + + a^2*Int[Sin[c+d*x]^m*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && GtQ[n,1] && LtQ[m,-1] + + +Int[cos[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + -(a^2+b^2)*Int[Cos[c+d*x]^(m+2)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x] + + 2*a*Int[Cos[c+d*x]^(m+1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1),x] + + b^2*Int[Cos[c+d*x]^m*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && GtQ[n,1] && LtQ[m,-1] + + +Int[sin[c_.+d_.*x_]/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + b*x/(a^2+b^2) - + a/(a^2+b^2)*Int[(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[cos[c_.+d_.*x_]/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + a*x/(a^2+b^2) + + b/(a^2+b^2)*Int[(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[sin[c_.+d_.*x_]^m_/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + -a*Sin[c+d*x]^(m-1)/(d*(a^2+b^2)*(m-1)) + + b/(a^2+b^2)*Int[Sin[c+d*x]^(m-1),x] + + a^2/(a^2+b^2)*Int[Sin[c+d*x]^(m-2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && GtQ[m,1] + + +Int[cos[c_.+d_.*x_]^m_/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + b*Cos[c+d*x]^(m-1)/(d*(a^2+b^2)*(m-1)) + + a/(a^2+b^2)*Int[Cos[c+d*x]^(m-1),x] + + b^2/(a^2+b^2)*Int[Cos[c+d*x]^(m-2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && GtQ[m,1] + + +Int[1/(sin[c_.+d_.*x_]*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])),x_Symbol] := + 1/a*Int[Cot[c+d*x],x] - + 1/a*Int[(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[1/(cos[c_.+d_.*x_]*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])),x_Symbol] := + 1/b*Int[Tan[c+d*x],x] + + 1/b*Int[(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] + + +Int[sin[c_.+d_.*x_]^m_/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + Sin[c+d*x]^(m+1)/(a*d*(m+1)) - + b/a^2*Int[Sin[c+d*x]^(m+1),x] + + (a^2+b^2)/a^2*Int[Sin[c+d*x]^(m+2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[m,-1] + + +Int[cos[c_.+d_.*x_]^m_/(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + -Cos[c+d*x]^(m+1)/(b*d*(m+1)) - + a/b^2*Int[Cos[c+d*x]^(m+1),x] + + (a^2+b^2)/b^2*Int[Cos[c+d*x]^(m+2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[m,-1] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_/sin[c_.+d_.*x_],x_Symbol] := + -(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/(a*d*(n+1)) - + b/a^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1),x] + + 1/a^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2)/Sin[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] + + +Int[(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_/cos[c_.+d_.*x_],x_Symbol] := + (a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/(b*d*(n+1)) - + a/b^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1),x] + + 1/b^2*Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2)/Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] + + +Int[sin[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + (a^2+b^2)/a^2*Int[Sin[c+d*x]^(m+2)*(a*Cos[c+d*x]+b*Sin[c+d*x])^n,x] - + 2*b/a^2*Int[Sin[c+d*x]^(m+1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1),x] + + 1/a^2*Int[Sin[c+d*x]^m*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] && LtQ[m,-1] + + +Int[cos[c_.+d_.*x_]^m_*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := + (a^2+b^2)/b^2*Int[Cos[c+d*x]^(m+2)*(a*Cos[c+d*x]+b*Sin[c+d*x])^n,x] - + 2*a/b^2*Int[Cos[c+d*x]^(m+1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1),x] + + 1/b^2*Int[Cos[c+d*x]^m*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && LtQ[n,-1] && LtQ[m,-1] + + +Int[cos[c_.+d_.*x_]^m_.*sin[c_.+d_.*x_]^n_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^p_.,x_Symbol] := + Int[ExpandTrig[cos[c+d*x]^m*sin[c+d*x]^n*(a*cos[c+d*x]+b*sin[c+d*x])^p,x],x] /; +FreeQ[{a,b,c,d,m,n},x] && PositiveIntegerQ[p] + + +Int[cos[c_.+d_.*x_]^m_.*sin[c_.+d_.*x_]^n_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + a^p*b^p*Int[Cos[c+d*x]^m*Sin[c+d*x]^n*(b*Cos[c+d*x]+a*Sin[c+d*x])^(-p),x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[a^2+b^2,0] && NegativeIntegerQ[p] + + +Int[cos[c_.+d_.*x_]^m_.*sin[c_.+d_.*x_]^n_./(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + b/(a^2+b^2)*Int[Cos[c+d*x]^m*Sin[c+d*x]^(n-1),x] + + a/(a^2+b^2)*Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^n,x] - + a*b/(a^2+b^2)*Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^(n-1)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && IntegersQ[m,n] && m>0 && n>0 + + +Int[cos[c_.+d_.*x_]^m_.*sin[c_.+d_.*x_]^n_./(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_]),x_Symbol] := + Int[ExpandTrig[cos[c+d*x]^m*sin[c+d*x]^n/(a*cos[c+d*x]+b*sin[c+d*x]),x],x] /; +FreeQ[{a,b,c,d,m,n},x] && IntegersQ[m,n] + + +Int[cos[c_.+d_.*x_]^m_.*sin[c_.+d_.*x_]^n_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + b/(a^2+b^2)*Int[Cos[c+d*x]^m*Sin[c+d*x]^(n-1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(p+1),x] + + a/(a^2+b^2)*Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^n*(a*Cos[c+d*x]+b*Sin[c+d*x])^(p+1),x] - + a*b/(a^2+b^2)*Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^(n-1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2,0] && IntegersQ[m,n,p] && m>0 && n>0 && p<0 + + + + + +(* ::Subsection::Closed:: *) +(*4.1.6 (a+b cos+c sin)^n*) + + +Int[Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + -2*(c*Cos[d+e*x]-b*Sin[d+e*x])/(e*Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2-c^2] + + +Int[(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + -(c*Cos[d+e*x]-b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)/(e*n) + + a*(2*n-1)/n*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2-c^2] && RationalQ[n] && n>0 + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + -(c-a*Sin[d+e*x])/(c*e*(c*Cos[d+e*x]-b*Sin[d+e*x])) /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2-c^2] + + +Int[1/Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + Int[1/Sqrt[a+Sqrt[b^2+c^2]*Cos[d+e*x-ArcTan[b,c]]],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2-c^2] + + +Int[(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + (c*Cos[d+e*x]-b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(2*n+1)) + + (n+1)/(a*(2*n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 + + +Int[Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + b/(c*e)*Subst[Int[Sqrt[a+x]/x,x],x,b*Cos[d+e*x]+c*Sin[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[b^2+c^2] + + +Int[Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + Int[Sqrt[a+Sqrt[b^2+c^2]*Cos[d+e*x-ArcTan[b,c]]],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2+c^2] && PositiveQ[a+Sqrt[b^2+c^2]] + + +Int[Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/Sqrt[(a+b*Cos[d+e*x]+c*Sin[d+e*x])/(a+Sqrt[b^2+c^2])]* + Int[Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2])*Cos[d+e*x-ArcTan[b,c]]],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] && NeQ[b^2+c^2] && Not[PositiveQ[a+Sqrt[b^2+c^2]]] + + +Int[(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + -(c*Cos[d+e*x]-b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)/(e*n) + + 1/n*Int[Simp[n*a^2+(n-1)*(b^2+c^2)+a*b*(2*n-1)*Cos[d+e*x]+a*c*(2*n-1)*Sin[d+e*x],x]* + (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] && RationalQ[n] && n>1 + + +(* Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + x/Sqrt[a^2-b^2-c^2] + + 2/(e*Sqrt[a^2-b^2-c^2])*ArcTan[(c*Cos[d+e*x]-b*Sin[d+e*x])/(a+Sqrt[a^2-b^2-c^2]+b*Cos[d+e*x]+c*Sin[d+e*x])] /; +FreeQ[{a,b,c,d,e},x] && PositiveQ[a^2-b^2-c^2] *) + + +(* Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + Log[RemoveContent[b^2+c^2+(a*b-c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a*c+b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ + (2*e*Rt[-a^2+b^2+c^2,2]) - + Log[RemoveContent[b^2+c^2+(a*b+c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a*c-b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ + (2*e*Rt[-a^2+b^2+c^2,2]) /; +FreeQ[{a,b,c,d,e},x] && NegativeQ[a^2-b^2-c^2] *) + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + Module[{f=FreeFactors[Cot[(d+e*x)/2],x]}, + -f/e*Subst[Int[1/(a+c*f*x),x],x,Cot[(d+e*x)/2]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a+b] + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + Module[{f=FreeFactors[Tan[(d+e*x)/2+Pi/4],x]}, + f/e*Subst[Int[1/(a+b*f*x),x],x,Tan[(d+e*x)/2+Pi/4]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a+c] + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + Module[{f=FreeFactors[Cot[(d+e*x)/2+Pi/4],x]}, + -f/e*Subst[Int[1/(a+b*f*x),x],x,Cot[(d+e*x)/2+Pi/4]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[a-c] && NeQ[a-b] + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + Module[{f=FreeFactors[Tan[(d+e*x)/2],x]}, + 2*f/e*Subst[Int[1/(a+b+2*c*f*x+(a-b)*f^2*x^2),x],x,Tan[(d+e*x)/2]/f]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] + + +Int[1/Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + b/(c*e)*Subst[Int[1/(x*Sqrt[a+x]),x],x,b*Cos[d+e*x]+c*Sin[d+e*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[b^2+c^2] + + +Int[1/Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + Int[1/Sqrt[a+Sqrt[b^2+c^2]*Cos[d+e*x-ArcTan[b,c]]],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b^2+c^2] && PositiveQ[a+Sqrt[b^2+c^2]] + + +Int[1/Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + Sqrt[(a+b*Cos[d+e*x]+c*Sin[d+e*x])/(a+Sqrt[b^2+c^2])]/Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]* + Int[1/Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2])*Cos[d+e*x-ArcTan[b,c]]],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] && NeQ[b^2+c^2] && Not[PositiveQ[a+Sqrt[b^2+c^2]]] + + +Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^(3/2),x_Symbol] := + 2*(c*Cos[d+e*x]-b*Sin[d+e*x])/(e*(a^2-b^2-c^2)*Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]) + + 1/(a^2-b^2-c^2)*Int[Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] + + +Int[(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + (-c*Cos[d+e*x]+b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/(e*(n+1)*(a^2-b^2-c^2)) + + 1/((n+1)*(a^2-b^2-c^2))* + Int[(a*(n+1)-b*(n+2)*Cos[d+e*x]-c*(n+2)*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 && n!=-3/2 + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + (2*a*A-b*B-c*C)*x/(2*a^2) - (b*B+c*C)*(b*Cos[d+e*x]-c*Sin[d+e*x])/(2*a*b*c*e) + + (a^2*(b*B-c*C)-2*a*A*b^2+b^2*(b*B+c*C))*Log[RemoveContent[a+b*Cos[d+e*x]+c*Sin[d+e*x],x]]/(2*a^2*b*c*e) /; +FreeQ[{a,b,c,d,e,A,B,C},x] && EqQ[b^2+c^2] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + (2*a*A-c*C)*x/(2*a^2) - C*Cos[d+e*x]/(2*a*e) + c*C*Sin[d+e*x]/(2*a*b*e) + + (-a^2*C+2*a*c*A+b^2*C)*Log[RemoveContent[a+b*Cos[d+e*x]+c*Sin[d+e*x],x]]/(2*a^2*b*e) /; +FreeQ[{a,b,c,d,e,A,C},x] && EqQ[b^2+c^2] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + (2*a*A-b*B)*x/(2*a^2) - b*B*Cos[d+e*x]/(2*a*c*e) + B*Sin[d+e*x]/(2*a*e) + + (a^2*B-2*a*b*A+b^2*B)*Log[RemoveContent[a+b*Cos[d+e*x]+c*Sin[d+e*x],x]]/(2*a^2*c*e) /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2+c^2] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + (b*B+c*C)*x/(b^2+c^2) + (c*B-b*C)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; +FreeQ[{a,b,c,d,e,A,B,C},x] && NeQ[b^2+c^2] && EqQ[A*(b^2+c^2)-a*(b*B+c*C)] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + c*C*x/(b^2+c^2) - b*C*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; +FreeQ[{a,b,c,d,e,A,C},x] && NeQ[b^2+c^2] && EqQ[A*(b^2+c^2)-a*c*C] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + b*B*x/(b^2+c^2) + c*B*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2+c^2] && EqQ[A*(b^2+c^2)-a*b*B] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + (b*B+c*C)*x/(b^2+c^2) + (c*B-b*C)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + + (A*(b^2+c^2)-a*(b*B+c*C))/(b^2+c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && NeQ[b^2+c^2] && NeQ[A*(b^2+c^2)-a*(b*B+c*C)] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + c*C*(d+e*x)/(e*(b^2+c^2)) - b*C*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + + (A*(b^2+c^2)-a*c*C)/(b^2+c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,C},x] && NeQ[b^2+c^2] && NeQ[A*(b^2+c^2)-a*c*C] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := + b*B*(d+e*x)/(e*(b^2+c^2)) + + c*B*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + + (A*(b^2+c^2)-a*b*B)/(b^2+c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2+c^2] && NeQ[A*(b^2+c^2)-a*b*B] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c-b*C-a*C*Cos[d+e*x]+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) /; +FreeQ[{a,b,c,d,e,A,B,C,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && EqQ[(b*B+c*C)*n+a*A*(n+1)] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + -(b*C+a*C*Cos[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) /; +FreeQ[{a,b,c,d,e,A,C,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && EqQ[c*C*n+a*A*(n+1)] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) /; +FreeQ[{a,b,c,d,e,A,B,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && EqQ[b*B*n+a*A*(n+1)] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c-b*C-a*C*Cos[d+e*x]+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + ((b*B+c*C)*n+a*A*(n+1))/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e,A,B,C,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && NeQ[(b*B+c*C)*n+a*A*(n+1)] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + -(b*C+a*C*Cos[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + (c*C*n+a*A*(n+1))/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e,A,C,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && NeQ[c*C*n+a*A*(n+1)] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + (b*B*n+a*A*(n+1))/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e,A,B,n},x] && NeQ[n+1] && EqQ[a^2-b^2-c^2] && NeQ[b*B*n+a*A*(n+1)] + + +Int[(B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])*(b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (c*B-b*C)*(b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/(e*(n+1)*(b^2+c^2)) /; +FreeQ[{b,c,d,e,B,C},x] && NeQ[n+1] && NeQ[b^2+c^2] && EqQ[b*B+c*C] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c-b*C-a*C*Cos[d+e*x]+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + 1/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)* + Simp[a*(b*B+c*C)*n+a^2*A*(n+1)+ + (n*(a^2*B-B*c^2+b*c*C)+a*b*A*(n+1))*Cos[d+e*x]+ + (n*(b*B*c+a^2*C-b^2*C)+a*c*A*(n+1))*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && RationalQ[n] && n>0 && NeQ[a^2-b^2-c^2] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + -(b*C+a*C*Cos[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + 1/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)* + Simp[a*c*C*n+a^2*A*(n+1)+(c*b*C*n+a*b*A*(n+1))*Cos[d+e*x]+(a^2*C*n-b^2*C*n+a*c*A*(n+1))*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,C},x] && RationalQ[n] && n>0 && NeQ[a^2-b^2-c^2] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])*(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_.,x_Symbol] := + (B*c+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + + 1/(a*(n+1))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)* + Simp[a*b*B*n+a^2*A*(n+1)+(a^2*B*n-c^2*B*n+a*b*A*(n+1))*Cos[d+e*x]+(b*c*B*n+a*c*A*(n+1))*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && RationalQ[n] && n>0 && NeQ[a^2-b^2-c^2] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/Sqrt[a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]],x_Symbol] := + B/b*Int[Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]],x] + + (A*b-a*B)/b*Int[1/Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]],x] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && EqQ[B*c-b*C] && NeQ[A*b-a*B] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + (c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/ + (e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; +FreeQ[{a,b,c,d,e,A,B,C},x] && NeQ[a^2-b^2-c^2] && EqQ[a*A-b*B-c*C] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + -(b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; +FreeQ[{a,b,c,d,e,A,C},x] && NeQ[a^2-b^2-c^2] && EqQ[a*A-c*C] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + (c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[a^2-b^2-c^2] && EqQ[a*A-b*B] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + (c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/ + (e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + + (a*A-b*B-c*C)/(a^2-b^2-c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && NeQ[a^2-b^2-c^2] && NeQ[a*A-b*B-c*C] + + +Int[(A_.+C_.*sin[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + -(b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + + (a*A-c*C)/(a^2-b^2-c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,C},x] && NeQ[a^2-b^2-c^2] && NeQ[a*A-c*C] + + +Int[(A_.+B_.*cos[d_.+e_.*x_])/(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^2,x_Symbol] := + (c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + + (a*A-b*B)/(a^2-b^2-c^2)*Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[a^2-b^2-c^2] && NeQ[a*A-b*B] + + +Int[(A_.+B_.*cos[d_.+e_.*x_]+C_.*sin[d_.+e_.*x_])*(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + -(c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ + (e*(n+1)*(a^2-b^2-c^2)) + + 1/((n+1)*(a^2-b^2-c^2))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)* + Simp[(n+1)*(a*A-b*B-c*C)+(n+2)*(a*B-b*A)*Cos[d+e*x]+(n+2)*(a*C-c*A)*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,B,C},x] && RationalQ[n] && n<-1 && NeQ[a^2-b^2-c^2] && n!=-2 + + +Int[(A_.+C_.*sin[d_.+e_.*x_])*(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + (b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ + (e*(n+1)*(a^2-b^2-c^2)) + + 1/((n+1)*(a^2-b^2-c^2))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)* + Simp[(n+1)*(a*A-c*C)-(n+2)*b*A*Cos[d+e*x]+(n+2)*(a*C-c*A)*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,C},x] && RationalQ[n] && n<-1 && NeQ[a^2-b^2-c^2] && n!=-2 + + +Int[(A_.+B_.*cos[d_.+e_.*x_])*(a_.+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_])^n_,x_Symbol] := + -(c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ + (e*(n+1)*(a^2-b^2-c^2)) + + 1/((n+1)*(a^2-b^2-c^2))*Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)* + Simp[(n+1)*(a*A-b*B)+(n+2)*(a*B-b*A)*Cos[d+e*x]-(n+2)*c*A*Sin[d+e*x],x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && RationalQ[n] && n<-1 && NeQ[a^2-b^2-c^2] && n!=-2 + + +Int[1/(a_.+b_.*sec[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_]),x_Symbol] := + Int[Cos[d+e*x]/(b+a*Cos[d+e*x]+c*Sin[d+e*x]),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[1/(a_.+b_.*csc[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_]),x_Symbol] := + Int[Sin[d+e*x]/(b+a*Sin[d+e*x]+c*Cos[d+e*x]),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[cos[d_.+e_.*x_]^n_.*(a_.+b_.*sec[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_])^n_.,x_Symbol] := + Int[(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[n] + + +Int[sin[d_.+e_.*x_]^n_.*(a_.+b_.*csc[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_])^n_.,x_Symbol] := + Int[(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[n] + + +Int[cos[d_.+e_.*x_]^n_*(a_.+b_.*sec[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_])^n_,x_Symbol] := + Cos[d+e*x]^n*(a+b*Sec[d+e*x]+c*Tan[d+e*x])^n/(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n*Int[(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[sin[d_.+e_.*x_]^n_*(a_.+b_.*csc[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_])^n_,x_Symbol] := + Sin[d+e*x]^n*(a+b*Csc[d+e*x]+c*Cot[d+e*x])^n/(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n*Int[(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[sec[d_.+e_.*x_]^n_.*(a_.+b_.*sec[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_])^m_,x_Symbol] := + Int[1/(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[m+n] && IntegerQ[n] + + +Int[csc[d_.+e_.*x_]^n_.*(a_.+b_.*csc[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_])^m_,x_Symbol] := + Int[1/(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[m+n] && IntegerQ[n] + + +Int[sec[d_.+e_.*x_]^n_.*(a_.+b_.*sec[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_])^m_,x_Symbol] := + Sec[d+e*x]^n*(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n/(a+b*Sec[d+e*x]+c*Tan[d+e*x])^n*Int[1/(b+a*Cos[d+e*x]+c*Sin[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[m+n] && Not[IntegerQ[n]] + + +Int[csc[d_.+e_.*x_]^n_.*(a_.+b_.*csc[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_])^m_,x_Symbol] := + Csc[d+e*x]^n*(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n/(a+b*Csc[d+e*x]+c*Cot[d+e*x])^n*Int[1/(b+a*Sin[d+e*x]+c*Cos[d+e*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[m+n] && Not[IntegerQ[n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.7 (d trig)^m (a+b (c sin)^n)^p*) + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)*(A_.+B_.*sin[e_.+f_.*x_]^2),x_Symbol] := + (4*A*(2*a+b)+B*(4*a+3*b))*x/8 - + (4*A*b+B*(4*a+3*b))*Cos[e+f*x]*Sin[e+f*x]/(8*f) - + b*B*Cos[e+f*x]*Sin[e+f*x]^3/(4*f) /; +FreeQ[{a,b,e,f,A,B},x] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^p_*(A_.+B_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -B*Cos[e+f*x]*Sin[e+f*x]*(a+b*Sin[e+f*x]^2)^p/(2*f*(p+1)) + + 1/(2*(p+1))*Int[(a+b*Sin[e+f*x]^2)^(p-1)* + Simp[a*B+2*a*A*(p+1)+(2*A*b*(p+1)+B*(b+2*a*p+2*b*p))*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f,A,B},x] && GtQ[p,0] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]^2)/(a_+b_.*sin[e_.+f_.*x_]^2),x_Symbol] := + B*x/b + (A*b-a*B)/b*Int[1/(a+b*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,e,f,A,B},x] + + +Int[(A_.+B_.*sin[e_.+f_.*x_]^2)/Sqrt[a_+b_.*sin[e_.+f_.*x_]^2],x_Symbol] := + B/b*Int[Sqrt[a+b*Sin[e+f*x]^2],x] + (A*b-a*B)/b*Int[1/Sqrt[a+b*Sin[e+f*x]^2],x] /; +FreeQ[{a,b,e,f,A,B},x] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^p_*(A_.+B_.*sin[e_.+f_.*x_]^2),x_Symbol] := + -(A*b-a*B)*Cos[e+f*x]*Sin[e+f*x]*(a+b*Sin[e+f*x]^2)^(p+1)/(2*a*f*(a+b)*(p+1)) - + 1/(2*a*(a+b)*(p+1))*Int[(a+b*Sin[e+f*x]^2)^(p+1)* + Simp[a*B-A*(2*a*(p+1)+b*(2*p+3))+2*(A*b-a*B)*(p+2)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f,A,B},x] && LtQ[p,-1] && NeQ[a+b,0] + + +Int[(a_.+b_.*sin[e_.+f_.*x_]^2)^p_*(A_.+B_.*sin[e_.+f_.*x_]^2),x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(a+b*Sin[e+f*x]^2)^p*(Sec[e+f*x]^2)^p/(f*(a+(a+b)*Tan[e+f*x]^2)^p)* + Subst[Int[(a+(a+b)*ff^2*x^2)^p*(A+(A+B)*ff^2*x^2)/(1+ff^2*x^2)^(p+2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,A,B},x] && Not[IntegerQ[p]] + + +Int[u_.*(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + a^p*Int[ActivateTrig[u*cos[e+f*x]^(2*p)],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a+b,0] && IntegerQ[p] + + +Int[u_.*(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + Int[ActivateTrig[u*(a*cos[e+f*x]^2)^p],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a+b,0] + + +Int[(b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + -Cot[e+f*x]*(b*Sin[e+f*x]^2)^p/(2*f*p) + + b*(2*p-1)/(2*p)*Int[(b*Sin[e+f*x]^2)^(p-1),x] /; +FreeQ[{b,e,f},x] && Not[IntegerQ[p]] && GtQ[p,1] + + +Int[(b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + Cot[e+f*x]*(b*Sin[e+f*x]^2)^(p+1)/(b*f*(2*p+1)) + + 2*(p+1)/(b*(2*p+1))*Int[(b*Sin[e+f*x]^2)^(p+1),x] /; +FreeQ[{b,e,f},x] && Not[IntegerQ[p]] && LtQ[p,-1] + + +Int[tan[e_.+f_.*x_]^m_.*(b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x]^2,x]}, + ff^((m+1)/2)/(2*f)*Subst[Int[x^((m-1)/2)*(b*ff^(n/2)*x^(n/2))^p/(1-ff*x)^((m+1)/2),x],x,Sin[e+f*x]^2/ff]] /; +FreeQ[{b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] + + +Int[tan[e_.+f_.*x_]^m_.*(b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[x^m*(b*(c*ff*x)^n)^p/(1-ff^2*x^2)^((m+1)/2),x],x,Sin[e+f*x]/ff]] /; +FreeQ[{b,c,e,f,n,p},x] && ILtQ[(m-1)/2,0] + + +Int[u_.*(b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + (b*ff^n)^IntPart[p]*(b*Sin[e+f*x]^n)^FracPart[p]/(Sin[e+f*x]/ff)^(n*FracPart[p])* + Int[ActivateTrig[u]*(Sin[e+f*x]/ff)^(n*p),x]] /; +FreeQ[{b,e,f,n,p},x] && Not[IntegerQ[p]] && IntegerQ[n] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[u_.*(b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + b^IntPart[p]*(b*(c*Sin[e+f*x])^n)^FracPart[p]/(c*Sin[e+f*x])^(n*FracPart[p])* + Int[ActivateTrig[u]*(c*Sin[e+f*x])^(n*p),x] /; +FreeQ[{b,c,e,f,n,p},x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]^2],x_Symbol] := + Sqrt[a]/f*EllipticE[e+f*x,-b/a] /; +FreeQ[{a,b,e,f},x] && GtQ[a,0] + + +Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]^2],x_Symbol] := + Sqrt[a+b*Sin[e+f*x]^2]/Sqrt[1+b*Sin[e+f*x]^2/a]*Int[Sqrt[1+(b*Sin[e+f*x]^2)/a],x] /; +FreeQ[{a,b,e,f},x] && Not[GtQ[a,0]] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^2,x_Symbol] := + (8*a^2+8*a*b+3*b^2)*x/8 - + b*(8*a+3*b)*Cos[e+f*x]*Sin[e+f*x]/(8*f) - + b^2*Cos[e+f*x]*Sin[e+f*x]^3/(4*f) /; +FreeQ[{a,b,e,f},x] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + -b*Cos[e+f*x]*Sin[e+f*x]*(a+b*Sin[e+f*x]^2)^(p-1)/(2*f*p) + + 1/(2*p)*Int[(a+b*Sin[e+f*x]^2)^(p-2)*Simp[a*(b+2*a*p)+b*(2*a+b)*(2*p-1)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a+b,0] && GtQ[p,1] + + +Int[1/(a_+b_.*sin[e_.+f_.*x_]^2),x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[1/(a+(a+b)*ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] + + +Int[1/Sqrt[a_+b_.*sin[e_.+f_.*x_]^2],x_Symbol] := + 1/(Sqrt[a]*f)*EllipticF[e+f*x,-b/a] /; +FreeQ[{a,b,e,f},x] && GtQ[a,0] + + +Int[1/Sqrt[a_+b_.*sin[e_.+f_.*x_]^2],x_Symbol] := + Sqrt[1+b*Sin[e+f*x]^2/a]/Sqrt[a+b*Sin[e+f*x]^2]*Int[1/Sqrt[1+(b*Sin[e+f*x]^2)/a],x] /; +FreeQ[{a,b,e,f},x] && Not[GtQ[a,0]] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + -b*Cos[e+f*x]*Sin[e+f*x]*(a+b*Sin[e+f*x]^2)^(p+1)/(2*a*f*(p+1)*(a+b)) + + 1/(2*a*(p+1)*(a+b))*Int[(a+b*Sin[e+f*x]^2)^(p+1)*Simp[2*a*(p+1)+b*(2*p+3)-2*b*(p+2)*Sin[e+f*x]^2,x],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a+b,0] && LtQ[p,-1] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(a+b*Sin[e+f*x]^2)^p*(Sec[e+f*x]^2)^p/(f*(a+(a+b)*Tan[e+f*x]^2)^p)* + Subst[Int[(a+(a+b)*ff^2*x^2)^p/(1+ff^2*x^2)^(p+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && Not[IntegerQ[p]] + + +(* Int[(a_+b_.*sin[e_.+f_.*x_]^4)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + -ff/f*Subst[Int[(a+b+2*a*ff^2*x^2+a*ff^4*x^4)^p/(1+ff^2*x^2)^(2*p+1),x],x,Cot[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[p] *) + + +Int[(a_+b_.*sin[e_.+f_.*x_]^4)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+2*a*ff^2*x^2+(a+b)*ff^4*x^4)^p/(1+ff^2*x^2)^(2*p+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[p] + + +Int[(a_+b_.*sin[e_.+f_.*x_]^4)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(a+b*Sin[e+f*x]^4)^p*(Sec[e+f*x]^2)^(2*p)/(f*(a+2*a*Tan[e+f*x]^2+(a+b)*Tan[e+f*x]^4)^p)* + Subst[Int[(a+2*a*ff^2*x^2+(a+b)*ff^4*x^4)^p/(1+ff^2*x^2)^(2*p+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[p-1/2] + + +Int[1/(a_+b_.*sin[e_.+f_.*x_]^n_),x_Symbol] := + Module[{k}, + Dist[2/(a*n),Sum[Int[1/(1-Sin[e+f*x]^2/((-1)^(4*k/n)*Rt[-a/b,n/2])),x],{k,1,n/2}],x]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[n/2] + + +(* Int[(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + -ff/f*Subst[Int[(b+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(n*p/2+1),x],x,Cot[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[n/2] && IGtQ[p,0] *) + + +Int[(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(b*ff^n*x^n+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[n/2] && IGtQ[p,0] + + +Int[(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + Int[ExpandTrig[(a+b*(c*sin[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,e,f,n},x] && (IGtQ[p,0] || EqQ[p,-1] && IntegerQ[n]) + + +Int[(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + Defer[Int][(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,e,f,n,p},x] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Cos[e+f*x],x]}, + -ff/f*Subst[Int[(1-ff^2*x^2)^((m-1)/2)*ExpandToSum[a+b*(1-ff^2*x^2)^(n/2),x]^p,x],x,Cos[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && (EqQ[n,2] || EqQ[n,4]) + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Cos[e+f*x],x]}, + -ff/f*Subst[Int[(1-ff^2*x^2)^((m-1)/2)*(a+b*(1-ff^2*x^2)^(n/2))^p,x],x,Cos[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[x^m*ExpandToSum[a*(1+ff^2*x^2)^(n/2)+b*ff^n*x^n,x]^p/(1+ff^2*x^2)^(m/2+n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && (EqQ[n,2] || EqQ[n,4]) && IntegerQ[p] + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[x^m*(a*(1+ff^2*x^2)^(n/2)+b*ff^n*x^n)^p/(1+ff^2*x^2)^(m/2+n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p] + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff^(m+1)*(a+b*Sin[e+f*x]^n)^p*(Sec[e+f*x]^2)^(n*p/2)/(f*Apart[a*(1+Tan[e+f*x]^2)^(n/2)+b*Tan[e+f*x]^n]^p)* + Subst[Int[x^m*ExpandToSum[a*(1+ff^2*x^2)^(n/2)+b*ff^n*x^n,x]^p/(1+ff^2*x^2)^(m/2+n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[m/2] && (EqQ[n,2] || EqQ[n,4] && IntegerQ[p-1/2]) + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + Int[ExpandTrig[sin[e+f*x]^m*(a+b*sin[e+f*x]^n)^p,x],x] /; +FreeQ[{a,b,e,f},x] && IntegersQ[m,p] && (EqQ[n,2] || EqQ[n,4] || p>0 || p==-1 && IntegerQ[n]) + + +Int[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*sin[e+f*x])^m*(a+b*(c*sin[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Sin[e+f*x])^m*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[cos[e_.+f_.*x_]^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff/f*Subst[Int[(1-ff^2*x^2)^((m-1)/2)*(a+b*(c*ff*x)^n)^p,x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && IntegerQ[(m-1)/2] && (EqQ[n,2] || EqQ[n,4] || m>0 || IGtQ[p,0] || IntegersQ[m,p]) + + +Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[ExpandToSum[b*ff^n*x^n+a*(1+ff^2*x^2)^(n/2),x]^p/(1+ff^2*x^2)^(m/2+n*(p/2)+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && (EqQ[n,2] || EqQ[n,4]) && IntegerQ[p] + + +Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(b*ff^n*x^n+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(m/2+n*(p/2)+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p] + + +Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(a+b*Sin[e+f*x]^2)^p*(Sec[e+f*x]^2)^p/(f*(a+(a+b)*Tan[e+f*x]^2)^p)* + Subst[Int[(a+(a+b)*ff^2*x^2)^p/(1+ff^2*x^2)^(m/2+p+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[m/2] && Not[IntegerQ[p]] + + +Int[cos[e_.+f_.*x_]^m_/(a_+b_.*sin[e_.+f_.*x_]^n_),x_Symbol] := + Int[Expand[(1-Sin[e+f*x]^2)^(m/2)/(a+b*Sin[e+f*x]^n),x],x] /; +FreeQ[{a,b,e,f},x] && IGtQ[m/2,0] && IntegerQ[(n-1)/2] + + +(* Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + Int[ExpandTrig[(1-sin[e+f*x]^2)^(m/2)*(a+b*sin[e+f*x]^n)^p,x],x] /; +FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && IntegerQ[(n-1)/2] && IntegerQ[p] && m<0 && p<-1 *) + + +Int[(d_.*cos[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*cos[e+f*x])^m*(a+b*(c*sin[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*cos[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Cos[e+f*x])^m*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[tan[e_.+f_.*x_]^m_.*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x]^2,x]}, + ff^((m+1)/2)/(2*f)*Subst[Int[x^((m-1)/2)*(a+b*ff^(n/2)*x^(n/2))^p/(1-ff*x)^((m+1)/2),x],x,Sin[e+f*x]^2/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] + + +Int[tan[e_.+f_.*x_]^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[x^m*(a+b*(c*ff*x)^n)^p/(1-ff^2*x^2)^((m+1)/2),x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && ILtQ[(m-1)/2,0] + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(d*ff*x)^m*ExpandToSum[a*(1+ff^2*x^2)^(n/2)+b*ff^n*x^n,x]^p/(1+ff^2*x^2)^(n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,d,e,f,m},x] && (EqQ[n,2] || EqQ[n,4]) && IntegerQ[p] + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(a+b*Sin[e+f*x]^n)^p*(Sec[e+f*x]^2)^(n*p/2)/(f*Apart[a*(1+Tan[e+f*x]^2)^(n/2)+b*Tan[e+f*x]^n]^p)* + Subst[Int[(d*ff*x)^m*ExpandToSum[a*(1+ff^2*x^2)^(n/2)+b*ff^n*x^n,x]^p/(1+ff^2*x^2)^(n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,d,e,f,m},x] && (EqQ[n,2] || EqQ[n,4] && IntegerQ[p-1/2]) && Not[IntegerQ[p]] + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[(d*x)^m*(b*ff^n*x^n+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(n*p/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,d,e,f,m},x] && IntegerQ[n/2] && IGtQ[p,0] + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*tan[e+f*x])^m*(a+b*(c*sin[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Tan[e+f*x])^m*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*cot[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Cot[e+f*x])^FracPart[m]*(Tan[e+f*x]/d)^FracPart[m]*Int[(Tan[e+f*x]/d)^(-m)*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Sec[e+f*x])^FracPart[m]*(Cos[e+f*x]/d)^FracPart[m]*Int[(Cos[e+f*x]/d)^(-m)*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(d_.*csc[e_.+f_.*x_])^m_*(a_+b_.*sin[e_.+f_.*x_]^n_.)^p_.,x_Symbol] := + d^(n*p)*Int[(d*Csc[e+f*x])^(m-n*p)*(b+a*Csc[e+f*x]^n)^p,x] /; +FreeQ[{a,b,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && IntegersQ[n,p] + + +Int[(d_.*csc[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sin[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Csc[e+f*x])^FracPart[m]*(Sin[e+f*x]/d)^FracPart[m]*Int[(Sin[e+f*x]/d)^(-m)*(a+b*(c*Sin[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.8 trig^m (a+b cos^p+c sin^q)^n*) + + +Int[sin[d_.+e_.*x_]^m_*(a_+b_.*cos[d_.+e_.*x_]^p_+c_.*sin[d_.+e_.*x_]^q_)^n_,x_Symbol] := + Module[{f=FreeFactors[Cot[d+e*x],x]}, + -f/e*Subst[Int[ExpandToSum[c+b*(1+f^2*x^2)^(q/2-p/2)+a*(1+f^2*x^2)^(q/2),x]^n/(1+f^2*x^2)^(m/2+n*q/2+1),x],x,Cot[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e},x] && EvenQ[m] && EvenQ[p] && EvenQ[q] && IntegerQ[n] && 00 + + +Int[(c_.+d_.*x_)^m_*sin[e_.+f_.*x_],x_Symbol] := + (c+d*x)^(m+1)*Sin[e+f*x]/(d*(m+1)) - + f/(d*(m+1))*Int[(c+d*x)^(m+1)*Cos[e+f*x],x] /; +FreeQ[{c,d,e,f},x] && RationalQ[m] && m<-1 + + +Int[sin[e_.+f_.*Complex[0,fz_]*x_]/(c_.+d_.*x_),x_Symbol] := + I*SinhIntegral[c*f*fz/d+f*fz*x]/d /; +FreeQ[{c,d,e,f,fz},x] && EqQ[d*e-c*f*fz*I] + + +Int[sin[e_.+f_.*x_]/(c_.+d_.*x_),x_Symbol] := + SinIntegral[e+f*x]/d /; +FreeQ[{c,d,e,f},x] && EqQ[d*e-c*f] + + +Int[sin[e_.+f_.*Complex[0,fz_]*x_]/(c_.+d_.*x_),x_Symbol] := + CoshIntegral[-c*f*fz/d-f*fz*x]/d /; +FreeQ[{c,d,e,f,fz},x] && EqQ[d*(e-Pi/2)-c*f*fz*I] && NegQ[c*f*fz/d] + + +Int[sin[e_.+f_.*Complex[0,fz_]*x_]/(c_.+d_.*x_),x_Symbol] := + CoshIntegral[c*f*fz/d+f*fz*x]/d /; +FreeQ[{c,d,e,f,fz},x] && EqQ[d*(e-Pi/2)-c*f*fz*I] + + +Int[sin[e_.+f_.*x_]/(c_.+d_.*x_),x_Symbol] := + CosIntegral[e-Pi/2+f*x]/d /; +FreeQ[{c,d,e,f},x] && EqQ[d*(e-Pi/2)-c*f] + + +Int[sin[e_.+f_.*x_]/(c_.+d_.*x_),x_Symbol] := + Cos[(d*e-c*f)/d]*Int[Sin[c*f/d+f*x]/(c+d*x),x] + + Sin[(d*e-c*f)/d]*Int[Cos[c*f/d+f*x]/(c+d*x),x] /; +FreeQ[{c,d,e,f},x] && NeQ[d*e-c*f] + + +Int[sin[e_.+Pi/2+f_.*x_]/Sqrt[c_.+d_.*x_],x_Symbol] := + 2/d*Subst[Int[Cos[f*x^2/d],x],x,Sqrt[c+d*x]] /; +FreeQ[{c,d,e,f},x] && ComplexFreeQ[f] && EqQ[d*e-c*f] + + +Int[sin[e_.+f_.*x_]/Sqrt[c_.+d_.*x_],x_Symbol] := + 2/d*Subst[Int[Sin[f*x^2/d],x],x,Sqrt[c+d*x]] /; +FreeQ[{c,d,e,f},x] && ComplexFreeQ[f] && EqQ[d*e-c*f] + + +Int[sin[e_.+f_.*x_]/Sqrt[c_.+d_.*x_],x_Symbol] := + Cos[(d*e-c*f)/d]*Int[Sin[c*f/d+f*x]/Sqrt[c+d*x],x] + + Sin[(d*e-c*f)/d]*Int[Cos[c*f/d+f*x]/Sqrt[c+d*x],x] /; +FreeQ[{c,d,e,f},x] && ComplexFreeQ[f] && NeQ[d*e-c*f] + + +Int[(c_.+d_.*x_)^m_.*sin[e_.+k_.*Pi+f_.*x_],x_Symbol] := + I/2*Int[(c+d*x)^m*E^(-I*k*Pi)*E^(-I*(e+f*x)),x] - I/2*Int[(c+d*x)^m*E^(I*k*Pi)*E^(I*(e+f*x)),x] /; +FreeQ[{c,d,e,f,m},x] && IntegerQ[2*k] + + +Int[(c_.+d_.*x_)^m_.*sin[e_.+f_.*x_],x_Symbol] := + I/2*Int[(c+d*x)^m*E^(-I*(e+f*x)),x] - I/2*Int[(c+d*x)^m*E^(I*(e+f*x)),x] /; +FreeQ[{c,d,e,f,m},x] + + +Int[(c_.+d_.*x_)*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + d*(b*Sin[e+f*x])^n/(f^2*n^2) - + b*(c+d*x)*Cos[e+f*x]*(b*Sin[e+f*x])^(n-1)/(f*n) + + b^2*(n-1)/n*Int[(c+d*x)*(b*Sin[e+f*x])^(n-2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[n] && n>1 + + +Int[(c_.+d_.*x_)^m_*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + d*m*(c+d*x)^(m-1)*(b*Sin[e+f*x])^n/(f^2*n^2) - + b*(c+d*x)^m*Cos[e+f*x]*(b*Sin[e+f*x])^(n-1)/(f*n) + + b^2*(n-1)/n*Int[(c+d*x)^m*(b*Sin[e+f*x])^(n-2),x] - + d^2*m*(m-1)/(f^2*n^2)*Int[(c+d*x)^(m-2)*(b*Sin[e+f*x])^n,x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n>1 && m>1 + + +Int[(c_.+d_.*x_)^m_*sin[e_.+f_.*x_]^n_,x_Symbol] := + Int[ExpandTrigReduce[(c+d*x)^m,Sin[e+f*x]^n,x],x] /; +FreeQ[{c,d,e,f,m},x] && IntegerQ[n] && n>1 && (Not[RationalQ[m]] || -1<=m<1) + + +Int[(c_.+d_.*x_)^m_*sin[e_.+f_.*x_]^n_,x_Symbol] := + (c+d*x)^(m+1)*Sin[e+f*x]^n/(d*(m+1)) - + f*n/(d*(m+1))*Int[ExpandTrigReduce[(c+d*x)^(m+1),Cos[e+f*x]*Sin[e+f*x]^(n-1),x],x] /; +FreeQ[{c,d,e,f,m},x] && IntegerQ[n] && n>1 && RationalQ[m] && -2<=m<-1 + + +Int[(c_.+d_.*x_)^m_*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (c+d*x)^(m+1)*(b*Sin[e+f*x])^n/(d*(m+1)) - + b*f*n*(c+d*x)^(m+2)*Cos[e+f*x]*(b*Sin[e+f*x])^(n-1)/(d^2*(m+1)*(m+2)) - + f^2*n^2/(d^2*(m+1)*(m+2))*Int[(c+d*x)^(m+2)*(b*Sin[e+f*x])^n,x] + + b^2*f^2*n*(n-1)/(d^2*(m+1)*(m+2))*Int[(c+d*x)^(m+2)*(b*Sin[e+f*x])^(n-2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n>1 && m<-2 + + +Int[(c_.+d_.*x_)*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (c+d*x)*Cos[e+f*x]*(b*Sin[e+f*x])^(n+1)/(b*f*(n+1)) - + d*(b*Sin[e+f*x])^(n+2)/(b^2*f^2*(n+1)*(n+2)) + + (n+2)/(b^2*(n+1))*Int[(c+d*x)*(b*Sin[e+f*x])^(n+2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[n] && n<-1 && n!=-2 + + +Int[(c_.+d_.*x_)^m_.*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (c+d*x)^m*Cos[e+f*x]*(b*Sin[e+f*x])^(n+1)/(b*f*(n+1)) - + d*m*(c+d*x)^(m-1)*(b*Sin[e+f*x])^(n+2)/(b^2*f^2*(n+1)*(n+2)) + + (n+2)/(b^2*(n+1))*Int[(c+d*x)^m*(b*Sin[e+f*x])^(n+2),x] + + d^2*m*(m-1)/(b^2*f^2*(n+1)*(n+2))*Int[(c+d*x)^(m-2)*(b*Sin[e+f*x])^(n+2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n<-1 && n!=-2 && m>1 + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(a+b*Sin[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[n] && (n==1 || PositiveIntegerQ[m] || NeQ[a^2-b^2]) + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + (2*a)^n*Int[(c+d*x)^m*Sin[1/2*(e+Pi*a/(2*b))+f*x/2]^(2*n),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[n] && (n>0 || PositiveIntegerQ[m]) + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (2*a)^IntPart[n]*(a+b*Sin[e+f*x])^FracPart[n]/Sin[e/2+a*Pi/(4*b)+f*x/2]^(2*FracPart[n])* + Int[(c+d*x)^m*Sin[e/2+a*Pi/(4*b)+f*x/2]^(2*n),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[n+1/2] && (n>0 || PositiveIntegerQ[m]) + + +(* Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + (2*a)^n*Int[(c+d*x)^m*Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*n),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[n] && (n>0 || PositiveIntegerQ[m]) *) + + +(* Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + (2*a)^IntPart[n]*(a+b*Sin[e+f*x])^FracPart[n]/Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*FracPart[n])* + Int[(c+d*x)^m*Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*n),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2] && IntegerQ[n+1/2] && (n>0 || PositiveIntegerQ[m]) *) + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+k_.*Pi+f_.*Complex[0,fz_]*x_]),x_Symbol] := + 2*Int[(c+d*x)^m*E^(-I*Pi*(k-1/2))*E^(-I*e+f*fz*x)/(b+2*a*E^(-I*Pi*(k-1/2))*E^(-I*e+f*fz*x)-b*E^(-2*I*k*Pi)*E^(2*(-I*e+f*fz*x))),x] /; +FreeQ[{a,b,c,d,e,f,fz},x] && IntegerQ[2*k] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+k_.*Pi+f_.*x_]),x_Symbol] := + 2*Int[(c+d*x)^m*E^(I*Pi*(k-1/2))*E^(I*(e+f*x))/(b+2*a*E^(I*Pi*(k-1/2))*E^(I*(e+f*x))-b*E^(2*I*k*Pi)*E^(2*I*(e+f*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && IntegerQ[2*k] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +(* Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*Complex[0,fz_]*x_]),x_Symbol] := + 2*I*Int[(c+d*x)^m*E^(-I*e+f*fz*x)/(b+2*I*a*E^(-I*e+f*fz*x)-b*E^(2*(-I*e+f*fz*x))),x] /; +FreeQ[{a,b,c,d,e,f,fz},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] *) + + +(* Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + -2*I*Int[(c+d*x)^m*E^(I*(e+f*x))/(b-2*I*a*E^(I*(e+f*x))-b*E^(2*I*(e+f*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] *) + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*Complex[0,fz_]*x_]),x_Symbol] := + 2*Int[(c+d*x)^m*E^(-I*e+f*fz*x)/(-I*b+2*a*E^(-I*e+f*fz*x)+I*b*E^(2*(-I*e+f*fz*x))),x] /; +FreeQ[{a,b,c,d,e,f,fz},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := + 2*Int[(c+d*x)^m*E^(I*(e+f*x))/(I*b+2*a*E^(I*(e+f*x))-I*b*E^(2*I*(e+f*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*x_])^2,x_Symbol] := + b*(c+d*x)^m*Cos[e+f*x]/(f*(a^2-b^2)*(a+b*Sin[e+f*x])) + + a/(a^2-b^2)*Int[(c+d*x)^m/(a+b*Sin[e+f*x]),x] - + b*d*m/(f*(a^2-b^2))*Int[(c+d*x)^(m-1)*Cos[e+f*x]/(a+b*Sin[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := + -b*(c+d*x)^m*Cos[e+f*x]*(a+b*Sin[e+f*x])^(n+1)/(f*(n+1)*(a^2-b^2)) + + a/(a^2-b^2)*Int[(c+d*x)^m*(a+b*Sin[e+f*x])^(n+1),x] + + b*d*m/(f*(n+1)*(a^2-b^2))*Int[(c+d*x)^(m-1)*Cos[e+f*x]*(a+b*Sin[e+f*x])^(n+1),x] - + b*(n+2)/((n+1)*(a^2-b^2))*Int[(c+d*x)^m*Sin[e+f*x]*(a+b*Sin[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2] && NegativeIntegerQ[n+2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := + Defer[Int][(c+d*x)^m*(a+b*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[u_^m_.*(a_.+b_.*Sin[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Sin[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*(a_.+b_.*Cos[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Cos[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.11 (e x)^m (a+b x^n)^p sin*) + + +Int[(a_+b_.*x_^n_)^p_.*Sin[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sin[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_.*Cos[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cos[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + x^(-n+1)*(a+b*x^n)^(p+1)*Sin[c+d*x]/(b*n*(p+1)) - + (-n+1)/(b*n*(p+1))*Int[x^(-n)*(a+b*x^n)^(p+1)*Sin[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(-n+1)*(a+b*x^n)^(p+1)*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && p<-1 && n>2 + + +Int[(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + x^(-n+1)*(a+b*x^n)^(p+1)*Cos[c+d*x]/(b*n*(p+1)) - + (-n+1)/(b*n*(p+1))*Int[x^(-n)*(a+b*x^n)^(p+1)*Cos[c+d*x],x] + + d/(b*n*(p+1))*Int[x^(-n+1)*(a+b*x^n)^(p+1)*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && p<-1 && n>2 + + +Int[(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sin[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cos[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + Int[x^(n*p)*(b+a*x^(-n))^p*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + Int[x^(n*p)*(b+a*x^(-n))^p*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + Defer[Int][(a+b*x^n)^p*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + Defer[Int][(a+b*x^n)^p*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Sin[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sin[c+d*x],(e*x)^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && PositiveIntegerQ[p] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Cos[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cos[c+d*x],(e*x)^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && PositiveIntegerQ[p] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + e^m*(a+b*x^n)^(p+1)*Sin[c+d*x]/(b*n*(p+1)) - + d*e^m/(b*n*(p+1))*Int[(a+b*x^n)^(p+1)*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && IntegerQ[p] && EqQ[m-n+1] && RationalQ[p] && p<-1 && (IntegerQ[n] || PositiveQ[e]) + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + e^m*(a+b*x^n)^(p+1)*Cos[c+d*x]/(b*n*(p+1)) + + d*e^m/(b*n*(p+1))*Int[(a+b*x^n)^(p+1)*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && IntegerQ[p] && EqQ[m-n+1] && RationalQ[p] && p<-1 && (IntegerQ[n] || PositiveQ[e]) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + x^(m-n+1)*(a+b*x^n)^(p+1)*Sin[c+d*x]/(b*n*(p+1)) - + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*(a+b*x^n)^(p+1)*Sin[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(m-n+1)*(a+b*x^n)^(p+1)*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && RationalQ[m] && p<-1 && (m-n+1>0 || n>2) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + x^(m-n+1)*(a+b*x^n)^(p+1)*Cos[c+d*x]/(b*n*(p+1)) - + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*(a+b*x^n)^(p+1)*Cos[c+d*x],x] + + d/(b*n*(p+1))*Int[x^(m-n+1)*(a+b*x^n)^(p+1)*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && RationalQ[m] && p<-1 && (m-n+1>0 || n>2) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sin[c+d*x],x^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && IntegerQ[m] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cos[c+d*x],x^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && IntegerQ[m] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sin[c_.+d_.*x_],x_Symbol] := + Int[x^(m+n*p)*(b+a*x^(-n))^p*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d,m},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cos[c_.+d_.*x_],x_Symbol] := + Int[x^(m+n*p)*(b+a*x^(-n))^p*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d,m},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Sin[c_.+d_.*x_],x_Symbol] := + Defer[Int][(e*x)^m*(a+b*x^n)^p*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Cos[c_.+d_.*x_],x_Symbol] := + Defer[Int][(e*x)^m*(a+b*x^n)^p*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + + + + +(* ::Subsection::Closed:: *) +(*4.1.12 (e x)^m (a+b sin(c+d x^n))^p*) + + +Int[Sin[d_.*x_^2],x_Symbol] := + Sqrt[Pi/2]/Rt[d,2]*FresnelS[Sqrt[2/Pi]*Rt[d,2]*x] /; +FreeQ[d,x] + + +Int[Cos[d_.*x_^2],x_Symbol] := + Sqrt[Pi/2]/Rt[d,2]*FresnelC[Sqrt[2/Pi]*Rt[d,2]*x] /; +FreeQ[d,x] + + +Int[Sin[c_+d_.*x_^2],x_Symbol] := + Sin[c]*Int[Cos[d*x^2],x] + Cos[c]*Int[Sin[d*x^2],x] /; +FreeQ[{c,d},x] + + +Int[Cos[c_+d_.*x_^2],x_Symbol] := + Cos[c]*Int[Cos[d*x^2],x] - Sin[c]*Int[Sin[d*x^2],x] /; +FreeQ[{c,d},x] + + +Int[Sin[c_.+d_.*x_^n_],x_Symbol] := + I/2*Int[E^(-c*I-d*I*x^n),x] - I/2*Int[E^(c*I+d*I*x^n),x] /; +FreeQ[{c,d},x] && IntegerQ[n] && n>2 + + +Int[Cos[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(-c*I-d*I*x^n),x] + 1/2*Int[E^(c*I+d*I*x^n),x] /; +FreeQ[{c,d},x] && IntegerQ[n] && n>2 + + +Int[(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Sin[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && n>1 && p>1 + + +Int[(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Cos[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && n>1 && p>1 + + +Int[(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(a+b*Sin[c+d*x^(-n)])^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(a+b*Cos[c+d*x^(-n)])^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Module[{k=Denominator[n]}, + k*Subst[Int[x^(k-1)*(a+b*Sin[c+d*x^(k*n)])^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && FractionQ[n] + + +Int[(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Module[{k=Denominator[n]}, + k*Subst[Int[x^(k-1)*(a+b*Cos[c+d*x^(k*n)])^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && FractionQ[n] + + +Int[Sin[c_.+d_.*x_^n_],x_Symbol] := + I/2*Int[E^(-c*I-d*I*x^n),x] - I/2*Int[E^(c*I+d*I*x^n),x] /; +FreeQ[{c,d,n},x] + + +Int[Cos[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(-c*I-d*I*x^n),x] + 1/2*Int[E^(c*I+d*I*x^n),x] /; +FreeQ[{c,d,n},x] + + +Int[(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Sin[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Cos[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_.+b_.*Sin[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Sin[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n},x] && IntegerQ[p] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Cos[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Cos[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n},x] && IntegerQ[p] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Sin[c_.+d_.*u_^n_])^p_,x_Symbol] := + Defer[Int][(a+b*Sin[c+d*u^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] + + +Int[(a_.+b_.*Cos[c_.+d_.*u_^n_])^p_,x_Symbol] := + Defer[Int][(a+b*Cos[c+d*u^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] + + +Int[(a_.+b_.*Sin[u_])^p_.,x_Symbol] := + Int[(a+b*Sin[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Cos[u_])^p_.,x_Symbol] := + Int[(a+b*Cos[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[Sin[d_.*x_^n_]/x_,x_Symbol] := + SinIntegral[d*x^n]/n /; +FreeQ[{d,n},x] + + +Int[Cos[d_.*x_^n_]/x_,x_Symbol] := + CosIntegral[d*x^n]/n /; +FreeQ[{d,n},x] + + +Int[Sin[c_+d_.*x_^n_]/x_,x_Symbol] := + Sin[c]*Int[Cos[d*x^n]/x,x] + Cos[c]*Int[Sin[d*x^n]/x,x] /; +FreeQ[{c,d,n},x] + + +Int[Cos[c_+d_.*x_^n_]/x_,x_Symbol] := + Cos[c]*Int[Cos[d*x^n]/x,x] - Sin[c]*Int[Sin[d*x^n]/x,x] /; +FreeQ[{c,d,n},x] + + +Int[x_^m_.*(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Sin[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] && (EqQ[p,1] || EqQ[m,n-1] || IntegerQ[p] && Simplify[(m+1)/n]>0) + + +Int[x_^m_.*(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Cos[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] && (EqQ[p,1] || EqQ[m,n-1] || IntegerQ[p] && Simplify[(m+1)/n]>0) + + +Int[(e_*x_)^m_*(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Sin[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_*x_)^m_*(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Cos[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[x_^m_.*Sin[a_.+b_.*x_^n_],x_Symbol] := + 2/n*Subst[Int[Sin[a+b*x^2],x],x,x^(n/2)] /; +FreeQ[{a,b,m,n},x] && EqQ[m-(n/2-1)] + + +Int[x_^m_.*Cos[a_.+b_.*x_^n_],x_Symbol] := + 2/n*Subst[Int[Cos[a+b*x^2],x],x,x^(n/2)] /; +FreeQ[{a,b,m,n},x] && EqQ[m-(n/2-1)] + + +Int[(e_.*x_)^m_.*Sin[c_.+d_.*x_^n_],x_Symbol] := + -e^(n-1)*(e*x)^(m-n+1)*Cos[c+d*x^n]/(d*n) + + e^n*(m-n+1)/(d*n)*Int[(e*x)^(m-n)*Cos[c+d*x^n],x] /; +FreeQ[{c,d,e},x] && PositiveIntegerQ[n] && RationalQ[m] && 01 && NeQ[n-1] + + +Int[x_^m_.*Cos[a_.+b_.*x_^n_]^p_,x_Symbol] := + -Cos[a+b*x^n]^p/((n-1)*x^(n-1)) - + b*n*p/(n-1)*Int[Cos[a+b*x^n]^(p-1)*Sin[a+b*x^n],x] /; +FreeQ[{a,b},x] && IntegersQ[n,p] && EqQ[m+n] && p>1 && NeQ[n-1] + + +Int[x_^m_.*Sin[a_.+b_.*x_^n_]^p_,x_Symbol] := + n*Sin[a+b*x^n]^p/(b^2*n^2*p^2) - + x^n*Cos[a+b*x^n]*Sin[a+b*x^n]^(p-1)/(b*n*p) + + (p-1)/p*Int[x^m*Sin[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p>1 + + +Int[x_^m_.*Cos[a_.+b_.*x_^n_]^p_,x_Symbol] := + n*Cos[a+b*x^n]^p/(b^2*n^2*p^2) + + x^n*Sin[a+b*x^n]*Cos[a+b*x^n]^(p-1)/(b*n*p) + + (p-1)/p*Int[x^m*Cos[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p>1 + + +Int[x_^m_.*Sin[a_.+b_.*x_^n_]^p_,x_Symbol] := + (m-n+1)*x^(m-2*n+1)*Sin[a+b*x^n]^p/(b^2*n^2*p^2) - + x^(m-n+1)*Cos[a+b*x^n]*Sin[a+b*x^n]^(p-1)/(b*n*p) + + (p-1)/p*Int[x^m*Sin[a+b*x^n]^(p-2),x] - + (m-n+1)*(m-2*n+1)/(b^2*n^2*p^2)*Int[x^(m-2*n)*Sin[a+b*x^n]^p,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p>1 && 0<2*n1 && 0<2*n1 && 0<2*n<1-m && NeQ[m+n+1] + + +Int[x_^m_.*Cos[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m+1)*Cos[a+b*x^n]^p/(m+1) + + b*n*p*x^(m+n+1)*Sin[a+b*x^n]*Cos[a+b*x^n]^(p-1)/((m+1)*(m+n+1)) - + b^2*n^2*p^2/((m+1)*(m+n+1))*Int[x^(m+2*n)*Cos[a+b*x^n]^p,x] + + b^2*n^2*p*(p-1)/((m+1)*(m+n+1))*Int[x^(m+2*n)*Cos[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p>1 && 0<2*n<1-m && NeQ[m+n+1] + + +Int[(e_.*x_)^m_*(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_.,x_Symbol] := + With[{k=Denominator[m]}, + k/e*Subst[Int[x^(k*(m+1)-1)*(a+b*Sin[c+d*x^(k*n)/e^n])^p,x],x,(e*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] && FractionQ[m] + + +Int[(e_.*x_)^m_*(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_.,x_Symbol] := + With[{k=Denominator[m]}, + k/e*Subst[Int[x^(k*(m+1)-1)*(a+b*Cos[c+d*x^(k*n)/e^n])^p,x],x,(e*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] && FractionQ[m] + + +Int[(e_.*x_)^m_.*(a_.+b_.*Sin[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(e*x)^m,(a+b*Sin[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && p>1 + + +Int[(e_.*x_)^m_.*(a_.+b_.*Cos[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(e*x)^m,(a+b*Cos[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && p>1 + + +Int[x_^m_.*Sin[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^n*Cos[a+b*x^n]*Sin[a+b*x^n]^(p+1)/(b*n*(p+1)) - + n*Sin[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (p+2)/(p+1)*Int[x^m*Sin[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p<-1 && p!=-2 + + +Int[x_^m_.*Cos[a_.+b_.*x_^n_]^p_,x_Symbol] := + -x^n*Sin[a+b*x^n]*Cos[a+b*x^n]^(p+1)/(b*n*(p+1)) - + n*Cos[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (p+2)/(p+1)*Int[x^m*Cos[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p<-1 && p!=-2 + + +Int[x_^m_.*Sin[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m-n+1)*Cos[a+b*x^n]*Sin[a+b*x^n]^(p+1)/(b*n*(p+1)) - + (m-n+1)*x^(m-2*n+1)*Sin[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (p+2)/(p+1)*Int[x^m*Sin[a+b*x^n]^(p+2),x] + + (m-n+1)*(m-2*n+1)/(b^2*n^2*(p+1)*(p+2))*Int[x^(m-2*n)*Sin[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p<-1 && p!=-2 && 0<2*n1 + + +Int[Cos[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[Cos[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[Sin[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Sin[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[Cos[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Cos[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[Sin[v_]^n_.,x_Symbol] := + Int[Sin[ExpandToSum[v,x]]^n,x] /; +PositiveIntegerQ[n] && QuadraticQ[v,x] && Not[QuadraticMatchQ[v,x]] + + +Int[Cos[v_]^n_.,x_Symbol] := + Int[Cos[ExpandToSum[v,x]]^n,x] /; +PositiveIntegerQ[n] && QuadraticQ[v,x] && Not[QuadraticMatchQ[v,x]] + + +Int[(d_+e_.*x_)*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*Cos[a+b*x+c*x^2]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] + + +Int[(d_+e_.*x_)*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Sin[a+b*x+c*x^2]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)^m_*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*(d+e*x)^(m-1)*Cos[a+b*x+c*x^2]/(2*c) + + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Cos[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] && GtQ[m,1] + + +Int[(d_.+e_.*x_)^m_*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Sin[a+b*x+c*x^2]/(2*c) - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Sin[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] && GtQ[m,1] + + +Int[(d_.+e_.*x_)^m_*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Sin[a+b*x+c*x^2]/(e*(m+1)) - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Cos[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)^m_*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Cos[a+b*x+c*x^2]/(e*(m+1)) + + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Sin[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*Cos[a+b*x+c*x^2]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Sin[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Sin[a+b*x+c*x^2]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Cos[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e,0] + + +Int[(d_.+e_.*x_)^m_*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*(d+e*x)^(m-1)*Cos[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[(d+e*x)^(m-1)*Sin[a+b*x+c*x^2],x] + + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Cos[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d,0] && GtQ[m,1] + + +Int[(d_.+e_.*x_)^m_*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Sin[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[(d+e*x)^(m-1)*Cos[a+b*x+c*x^2],x] - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Sin[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d,0] && GtQ[m,1] + + +Int[(d_.+e_.*x_)^m_*Sin[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Sin[a+b*x+c*x^2]/(e*(m+1)) - + (b*e-2*c*d)/(e^2*(m+1))*Int[(d+e*x)^(m+1)*Cos[a+b*x+c*x^2],x] - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Cos[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)^m_*Cos[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Cos[a+b*x+c*x^2]/(e*(m+1)) + + (b*e-2*c*d)/(e^2*(m+1))*Int[(d+e*x)^(m+1)*Sin[a+b*x+c*x^2],x] + + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Sin[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d,0] && LtQ[m,-1] + + +Int[(d_.+e_.*x_)^m_.*Sin[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[(d+e*x)^m,Sin[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IGtQ[n,1] + + +Int[(d_.+e_.*x_)^m_.*Cos[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[(d+e*x)^m,Cos[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IGtQ[n,1] + + +Int[(d_.+e_.*x_)^m_.*Sin[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Sin[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + +Int[(d_.+e_.*x_)^m_.*Cos[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Cos[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + +Int[u_^m_.*Sin[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Sin[ExpandToSum[v,x]]^n,x] /; +FreeQ[m,x] && PositiveIntegerQ[n] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + +Int[u_^m_.*Cos[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Cos[ExpandToSum[v,x]]^n,x] /; +FreeQ[m,x] && PositiveIntegerQ[n] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + + +`) + +func resourcesRubi41SineMBytes() ([]byte, error) { + return _resourcesRubi41SineM, nil +} + +func resourcesRubi41SineM() (*asset, error) { + bytes, err := resourcesRubi41SineMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/4.1 Sine.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi42TangentM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Tangent Function Rules*) + + +(* ::Subsection::Closed:: *) +(*4.2.1.1 (a+b tan)^n*) + + +Int[(b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b*(b*Tan[c+d*x])^(n-1)/(d*(n-1)) - + b^2*Int[(b*Tan[c+d*x])^(n-2),x] /; +FreeQ[{b,c,d},x] && RationalQ[n] && n>1 + + +Int[(b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + (b*Tan[c+d*x])^(n+1)/(b*d*(n+1)) - + 1/b^2*Int[(b*Tan[c+d*x])^(n+2),x] /; +FreeQ[{b,c,d},x] && RationalQ[n] && n<-1 + + +Int[tan[c_.+d_.*x_],x_Symbol] := + -Log[RemoveContent[Cos[c+d*x],x]]/d /; +FreeQ[{c,d},x] + + +(* Int[1/tan[c_.+d_.*x_],x_Symbol] := + Log[RemoveContent[Sin[c+d*x],x]]/d /; +FreeQ[{c,d},x] *) + + +Int[(b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b/d*Subst[Int[x^n/(b^2+x^2),x],x,b*Tan[c+d*x]] /; +FreeQ[{b,c,d,n},x] && Not[IntegerQ[n]] + + +Int[(a_+b_.*tan[c_.+d_.*x_])^2,x_Symbol] := + (a^2-b^2)*x + b^2*Tan[c+d*x]/d + 2*a*b*Int[Tan[c+d*x],x] /; +FreeQ[{a,b,c,d},x] + + +(* Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(a+b*Tan[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[n] *) + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b*(a+b*Tan[c+d*x])^(n-1)/(d*(n-1)) + + 2*a*Int[(a+b*Tan[c+d*x])^(n-1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2+b^2] && RationalQ[n] && n>1 + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + a*(a+b*Tan[c+d*x])^n/(2*b*d*n) + + 1/(2*a)*Int[(a+b*Tan[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2+b^2] && RationalQ[n] && n<0 + + +Int[Sqrt[a_+b_.*tan[c_.+d_.*x_]],x_Symbol] := + -2*b/d*Subst[Int[1/(2*a-x^2),x],x,Sqrt[a+b*Tan[c+d*x]]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2+b^2] + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + -b/d*Subst[Int[(a+x)^(n-1)/(a-x),x],x,b*Tan[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2+b^2] + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b*(a+b*Tan[c+d*x])^(n-1)/(d*(n-1)) + + Int[(a^2-b^2+2*a*b*Tan[c+d*x])*(a+b*Tan[c+d*x])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2] && RationalQ[n] && n>1 + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b*(a+b*Tan[c+d*x])^(n+1)/(d*(n+1)*(a^2+b^2)) + + 1/(a^2+b^2)*Int[(a-b*Tan[c+d*x])*(a+b*Tan[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2] && RationalQ[n] && n<-1 + + +Int[1/(a_+b_.*tan[c_.+d_.*x_]),x_Symbol] := + a*x/(a^2+b^2) + b/(a^2+b^2)*Int[(b-a*Tan[c+d*x])/(a+b*Tan[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2+b^2] + + +Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := + b/d*Subst[Int[(a+x)^n/(b^2+x^2),x],x,b*Tan[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && NeQ[a^2+b^2] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.1.2 (d sec)^m (a+b tan)^n*) + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + b*(d*Sec[e+f*x])^m/(f*m) + a*Int[(d*Sec[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,m},x] && (IntegerQ[2*m] || NeQ[a^2+b^2]) + + +Int[sec[e_.+f_.*x_]^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 1/(a^(m-2)*b*f)*Subst[Int[(a-x)^(m/2-1)*(a+x)^(n+m/2-1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,e,f,n},x] && EqQ[a^2+b^2] && IntegerQ[m/2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^n/(a*f*m) /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && EqQ[Simplify[m+n]] + + +Int[sec[e_.+f_.*x_]/Sqrt[a_+b_.*tan[e_.+f_.*x_]],x_Symbol] := + -2*a/(b*f)*Subst[Int[1/(2-a*x^2),x],x,Sec[e+f*x]/Sqrt[a+b*Tan[e+f*x]]] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2+b^2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^n/(a*f*m) + + a/(2*d^2)*Int[(d*Sec[e+f*x])^(m+2)*(a+b*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] && EqQ[m/2+n] && RationalQ[m,n] && n>0 + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 2*d^2*(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+1)/(b*f*(m-2)) + + 2*d^2/a*Int[(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] && EqQ[m/2+n] && RationalQ[m,n] && n<-1 + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (a/d)^(2*IntPart[n])*(a+b*Tan[e+f*x])^FracPart[n]*(a-b*Tan[e+f*x])^FracPart[n]/(d*Sec[e+f*x])^(2*FracPart[n])* + Int[1/(a-b*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && EqQ[Simplify[m/2+n]] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 2*b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1)/(f*m) /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && EqQ[Simplify[m/2+n-1]] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1)/(f*(m+n-1)) + + a*(m+2*n-2)/(m+n-1)*Int[(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && PositiveIntegerQ[Simplify[m/2+n-1]] && Not[IntegerQ[n]] + + +Int[Sqrt[d_.*sec[e_.+f_.*x_]]*Sqrt[a_+b_.*tan[e_.+f_.*x_]],x_Symbol] := + -4*b*d^2/f*Subst[Int[x^2/(a^2+d^2*x^4),x],x,Sqrt[a+b*Tan[e+f*x]]/Sqrt[d*Sec[e+f*x]]] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 2*b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1)/(f*m) - + b^2*(m+2*n-2)/(d^2*m)*Int[(d*Sec[e+f*x])^(m+2)*(a+b*Tan[e+f*x])^(n-2),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m,n] && n>1 && + (PositiveIntegerQ[n/2] && NegativeIntegerQ[m-1/2] || n==2 && m<0 || m<=-1 && m+n>0 || NegativeIntegerQ[m] && m/2+n-1<0 && IntegerQ[n] || n==3/2 && m==-1/2) && IntegerQ[2*m] + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^n/(a*f*m) + + a*(m+n)/(m*d^2)*Int[(d*Sec[e+f*x])^(m+2)*(a+b*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m,n] && n>0 && m<-1 && IntegersQ[2*m,2*n] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1)/(f*(m+n-1)) + + a*(m+2*n-2)/(m+n-1)*Int[(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2] && RationalQ[n] && n>0 && NeQ[m+n-1] && IntegersQ[2*m,2*n] + + +Int[(d_.*sec[e_.+f_.*x_])^(3/2)/Sqrt[a_+b_.*tan[e_.+f_.*x_]],x_Symbol] := + d*Sec[e+f*x]/(Sqrt[a-b*Tan[e+f*x]]*Sqrt[a+b*Tan[e+f*x]])*Int[Sqrt[d*Sec[e+f*x]]*Sqrt[a-b*Tan[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 2*d^2*(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+1)/(b*f*(m+2*n)) - + d^2*(m-2)/(b^2*(m+2*n))*Int[(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+2),x] /; +FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2] && RationalQ[n] && n<-1 && + (NegativeIntegerQ[n/2] && PositiveIntegerQ[m-1/2] || n==-2 || PositiveIntegerQ[m+n] || IntegersQ[n,m+1/2] && 2*m+n+1>0) && IntegerQ[2*m] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + d^2*(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+1)/(b*f*(m+n-1)) + + d^2*(m-2)/(a*(m+n-1))*Int[(d*Sec[e+f*x])^(m-2)*(a+b*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m,n] && n<0 && m>1 && Not[NegativeIntegerQ[m+n]] && NeQ[m+n-1] && IntegersQ[2*m,2*n] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^n/(b*f*(m+2*n)) + + Simplify[m+n]/(a*(m+2*n))*Int[(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2] && RationalQ[n] && n<0 && NeQ[m+2*n] && IntegersQ[2*m,2*n] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1)/(f*Simplify[m+n-1]) + + a*(m+2*n-2)/Simplify[m+n-1]*Int[(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && PositiveIntegerQ[Simplify[m+n-1]] && RationalQ[n] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^n/(b*f*(m+2*n)) + + Simplify[m+n]/(a*(m+2*n))*Int[(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] && NegativeIntegerQ[Simplify[m+n]] && NeQ[m+2*n] + + +(* Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a^n*(d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(1+x/a)^(n+m/2-1)*(1-x/a)^(m/2-1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2] && IntegerQ[n] *) + + +(* Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(a+x)^n*(1+x^2/b^2)^(m/2-1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] *) + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + (d*Sec[e+f*x])^m/((a+b*Tan[e+f*x])^(m/2)*(a-b*Tan[e+f*x])^(m/2))*Int[(a+b*Tan[e+f*x])^(m/2+n)*(a-b*Tan[e+f*x])^(m/2),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2] + + +Int[sec[e_.+f_.*x_]^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + 1/(b*f)*Subst[Int[(a+x)^n*(1+x^2/b^2)^(m/2-1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,e,f,n},x] && NeQ[a^2+b^2] && IntegerQ[m/2] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^2/sec[e_.+f_.*x_],x_Symbol] := + b^2*ArcTanh[Sin[e+f*x]]/f - 2*a*b*Cos[e+f*x]/f + (a^2-b^2)*Sin[e+f*x]/f /; +FreeQ[{a,b,e,f},x] && NeQ[a^2+b^2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^2,x_Symbol] := + b*(d*Sec[e+f*x])^m*(a+b*Tan[e+f*x])/(f*(m+1)) + + 1/(m+1)*Int[(d*Sec[e+f*x])^m*(a^2*(m+1)-b^2+a*b*(m+2)*Tan[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,m},x] && NeQ[a^2+b^2] && NeQ[m+1] + + +Int[sec[e_.+f_.*x_]/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + -1/f*Subst[Int[1/(a^2+b^2-x^2),x],x,(b-a*Tan[e+f*x])/Sec[e+f*x]] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2+b^2] + + +Int[(d_.*sec[e_.+f_.*x_])^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + -d^2/b^2*Int[(d*Sec[e+f*x])^(m-2)*(a-b*Tan[e+f*x]),x] + + d^2*(a^2+b^2)/b^2*Int[(d*Sec[e+f*x])^(m-2)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2+b^2] && PositiveIntegerQ[m-1] + + +Int[(d_.*sec[e_.+f_.*x_])^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(a^2+b^2)*Int[(d*Sec[e+f*x])^m*(a-b*Tan[e+f*x]),x] + + b^2/(d^2*(a^2+b^2))*Int[(d*Sec[e+f*x])^(m+2)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2+b^2] && NegativeIntegerQ[m] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + d^(2*IntPart[m/2])*(d*Sec[e+f*x])^(2*FracPart[m/2])/(b*f*(Sec[e+f*x]^2)^FracPart[m/2])* + Subst[Int[(a+x)^n*(1+x^2/b^2)^(m/2-1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && NeQ[a^2+b^2] && Not[IntegerQ[m/2]] + + +Int[Sqrt[a_+b_.*tan[e_.+f_.*x_]]/Sqrt[d_. cos[e_.+f_.*x_]],x_Symbol] := + -4*b/f*Subst[Int[x^2/(a^2*d^2+x^4),x],x,Sqrt[d*Cos[e+f*x]]*Sqrt[a+b*Tan[e+f*x]]] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] + + +Int[1/((d_. cos[e_.+f_.*x_])^(3/2)*Sqrt[a_+b_.*tan[e_.+f_.*x_]]),x_Symbol] := + 1/(d*Cos[e+f*x]*Sqrt[a-b*Tan[e+f*x]]*Sqrt[a+b*Tan[e+f*x]])*Int[Sqrt[a-b*Tan[e+f*x]]/Sqrt[d*Cos[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2+b^2] + + +Int[(d_.*cos[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + (d*Cos[e+f*x])^m*(d*Sec[e+f*x])^m*Int[(a+b*Tan[e+f*x])^n/(d*Sec[e+f*x])^m,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.1.3 (d sin)^m (a+b tan)^n*) + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b/f*Subst[Int[x^m*(a+x)^n/(b^2+x^2)^(m/2+1),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,e,f,n},x] && IntegerQ[m/2] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + Int[Expand[Sin[e+f*x]^m*(a+b*Tan[e+f*x])^n,x],x] /; +FreeQ[{a,b,e,f},x] && IntegerQ[(m-1)/2] && PositiveIntegerQ[n] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + Int[Sin[e+f*x]^m*(a*Cos[e+f*x]+b*Sin[e+f*x])^n/Cos[e+f*x]^n,x] /; +FreeQ[{a,b,e,f},x] && IntegerQ[(m-1)/2] && NegativeIntegerQ[n] && (m<5 && n>-4 || m==5 && n==-1) + + +Int[(d_.*csc[e_.+f_.*x_])^m_*(a_.+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + (d*Csc[e+f*x])^FracPart[m]*(Sin[e+f*x]/d)^FracPart[m]*Int[(a+b*Tan[e+f*x])^n/(Sin[e+f*x]/d)^m,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && Not[IntegerQ[m]] + + +Int[cos[e_.+f_.*x_]^m_.*sin[e_.+f_.*x_]^p_.*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + Int[Cos[e+f*x]^(m-n)*Sin[e+f*x]^p*(a*Cos[e+f*x]+b*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,m,p},x] && IntegerQ[n] + + +Int[sin[e_.+f_.*x_]^m_.*cos[e_.+f_.*x_]^p_.*(a_+b_.*cot[e_.+f_.*x_])^n_.,x_Symbol] := + Int[Sin[e+f*x]^(m-n)*Cos[e+f*x]^p*(a*Sin[e+f*x]+b*Cos[e+f*x])^n,x] /; +FreeQ[{a,b,e,f,m,p},x] && IntegerQ[n] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.2.1 (a+b tan)^m (c+d tan)^n*) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + a^m*c^m*Int[Sec[e+f*x]^(2*m)*(c+d*Tan[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2+b^2] && IntegerQ[m] && Not[IntegerQ[n] && n>0 && (m<0 || n0 + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + (b*c-a*d)*(a+b*Tan[e+f*x])^(m+1)/(f*(m+1)*(a^2+b^2)) + + 1/(a^2+b^2)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[a*c+b*d-(b*c-a*d)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && RationalQ[m] && m<-1 + + +Int[(c_+d_.*tan[e_.+f_.*x_])/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + c/(b*f)*Log[RemoveContent[a*Cos[e+f*x]+b*Sin[e+f*x],x]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && EqQ[a*c+b*d] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])/(a_.+b_.*tan[e_.+f_.*x_]),x_Symbol] := + (a*c+b*d)*x/(a^2+b^2) + (b*c-a*d)/(a^2+b^2)*Int[(b-a*Tan[e+f*x])/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[a*c+b*d] + + +Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := + -2*d^2/f*Subst[Int[1/(2*c*d+b*x^2),x],x,(c-d*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]]] /; +FreeQ[{b,c,d,e,f},x] && EqQ[c^2-d^2] + + +(* Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := + (c+d)/2*Int[(1+Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] + + (c-d)/2*Int[(1-Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] /; +FreeQ[{b,c,d,e,f},x] && NeQ[c^2+d^2] && NeQ[c^2-d^2] *) + + +Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := + 2*c^2/f*Subst[Int[1/(b*c-d*x^2),x],x,Sqrt[b*Tan[e+f*x]]] /; +FreeQ[{b,c,d,e,f},x] && EqQ[c^2+d^2] + + +(* Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := + (c+I*d)/2*Int[(1-I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] + (c-I*d)/2*Int[(1+I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] /; +FreeQ[{b,c,d,e,f},x] && NeQ[c^2-d^2] && NeQ[c^2+d^2] *) + + +Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := + 2/f*Subst[Int[(b*c+d*x^2)/(b^2+x^4),x],x,Sqrt[b*Tan[e+f*x]]] /; +FreeQ[{b,c,d,e,f},x] && NeQ[c^2-d^2] && NeQ[c^2+d^2] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])/Sqrt[a_+b_.*tan[e_.+f_.*x_]],x_Symbol] := + -2*d^2/f*Subst[Int[1/(2*b*c*d-4*a*d^2+x^2),x],x,(b*c-2*a*d-b*d*Tan[e+f*x])/Sqrt[a+b*Tan[e+f*x]]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && EqQ[2*a*c*d-b*(c^2-d^2)] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])/Sqrt[a_+b_.*tan[e_.+f_.*x_]],x_Symbol] := + With[{q=Rt[a^2+b^2,2]}, + 1/(2*q)*Int[(a*c+b*d+c*q+(b*c-a*d+d*q)*Tan[e+f*x])/Sqrt[a+b*Tan[e+f*x]],x] - + 1/(2*q)*Int[(a*c+b*d-c*q+(b*c-a*d-d*q)*Tan[e+f*x])/Sqrt[a+b*Tan[e+f*x]],x]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && NeQ[2*a*c*d-b*(c^2-d^2)] && + (PerfectSquareQ[a^2+b^2] || RationalQ[a,b,c,d]) + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_+d_.*tan[e_.+f_.*x_]),x_Symbol] := + c*d/f*Subst[Int[(a+b/d*x)^m/(d^2+c*x),x],x,d*Tan[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && EqQ[c^2+d^2] + + +Int[(b_.*tan[e_.+f_.*x_])^m_*(c_+d_.*tan[e_.+f_.*x_]),x_Symbol] := + c*Int[(b*Tan[e+f*x])^m,x] + d/b*Int[(b*Tan[e+f*x])^(m+1),x] /; +FreeQ[{b,c,d,e,f,m},x] && NeQ[c^2+d^2] && Not[IntegerQ[2*m]] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + (c+I*d)/2*Int[(a+b*Tan[e+f*x])^m*(1-I*Tan[e+f*x]),x] + + (c-I*d)/2*Int[(a+b*Tan[e+f*x])^m*(1+I*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && Not[IntegerQ[m]] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^2,x_Symbol] := + -b*(a*c+b*d)^2*(a+b*Tan[e+f*x])^m/(2*a^3*f*m) + + 1/(2*a^2)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[a*c^2-2*b*c*d+a*d^2-2*b*d^2*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m<=-1 && EqQ[a^2+b^2] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])^2/(a_.+b_.*tan[e_.+f_.*x_]),x_Symbol] := + d*(2*b*c-a*d)*x/b^2 + d^2/b*Int[Tan[e+f*x],x] + (b*c-a*d)^2/b^2*Int[1/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^2,x_Symbol] := + (b*c-a*d)^2*(a+b*Tan[e+f*x])^(m+1)/(b*f*(m+1)*(a^2+b^2)) + + 1/(a^2+b^2)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[a*c^2+2*b*c*d-a*d^2-(b*c^2-2*a*c*d-b*d^2)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m<-1 && NeQ[a^2+b^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^2,x_Symbol] := + d^2*(a+b*Tan[e+f*x])^(m+1)/(b*f*(m+1)) + + Int[(a+b*Tan[e+f*x])^m*Simp[c^2-d^2+2*c*d*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && Not[RationalQ[m] && m<=-1] && Not[EqQ[m-2] && EqQ[a]] + + +Int[Sqrt[a_+b_.*tan[e_.+f_.*x_]]/Sqrt[c_.+d_.*tan[e_.+f_.*x_]],x_Symbol] := + -2*a*b/f*Subst[Int[1/(a*c-b*d-2*a^2*x^2),x],x,Sqrt[c+d*Tan[e+f*x]]/Sqrt[a+b*Tan[e+f*x]]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*b*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)/(f*(m-1)*(a*c-b*d)) + + 2*a^2/(a*c-b*d)*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m+n==0 && m>1/2 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n/(2*b*f*m) - + (a*c-b*d)/(2*b^2)*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m+n==0 && m<=-1/2 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(2*f*m*(b*c-a*d)) + + 1/(2*a)*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m+n+1==0 && m<-1 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + -d*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(f*m*(c^2+d^2)) + + a/(a*c-b*d)*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && EqQ[m+n+1] && Not[RationalQ[m] && m<-1] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])^n_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + -(a*c+b*d)*(c+d*Tan[e+f*x])^n/(2*(b*c-a*d)*f*(a+b*Tan[e+f*x])) + + 1/(2*a*(b*c-a*d))*Int[(c+d*Tan[e+f*x])^(n-1)*Simp[a*c*d*(n-1)+b*c^2+b*d^2*n-d*(b*c-a*d)*(n-1)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[n] && 01 + + +Int[1/((a_.+b_.*tan[e_.+f_.*x_])*(c_.+d_.*tan[e_.+f_.*x_])),x_Symbol] := + b/(b*c-a*d)*Int[1/(a+b*Tan[e+f*x]),x] - d/(b*c-a*d)*Int[1/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])^n_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + -a*(c+d*Tan[e+f*x])^(n+1)/(2*f*(b*c-a*d)*(a+b*Tan[e+f*x])) + + 1/(2*a*(b*c -a*d))*Int[(c+d*Tan[e+f*x])^n*Simp[b*c+a*d*(n-1)-b*d*n*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && Not[RationalQ[n] && n>0] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + -a^2*(b*c-a*d)*(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(b*c+a*d)*(n+1)) + + a/(d*(b*c+a*d)*(n+1))*Int[(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)* + Simp[b*(b*c*(m-2)-a*d*(m-2*n-4))+(a*b*c*(m-2)+b^2*d*(n+1)-a^2*d*(m+n-1))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m>1 && n<-1 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^(3/2)/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + 2*a^2/(a*c-b*d)*Int[Sqrt[a+b*Tan[e+f*x]],x] - + (2*b*c*d+a*(c^2-d^2))/(a*(c^2+d^2))*Int[(a-b*Tan[e+f*x])*Sqrt[a+b*Tan[e+f*x]]/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^(3/2)/Sqrt[c_.+d_.*tan[e_.+f_.*x_]],x_Symbol] := + 2*a*Int[Sqrt[a+b*Tan[e+f*x]]/Sqrt[c+d*Tan[e+f*x]],x] + + b/a*Int[(b+a*Tan[e+f*x])*Sqrt[a+b*Tan[e+f*x]]/Sqrt[c+d*Tan[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b^2*(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n-1)) + + a/(d*(m+n-1))*Int[(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^n* + Simp[b*c*(m-2)+a*d*(m+2*n)+(a*c*(m-2)+b*d*(3*m+2*n-4))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && + IntegerQ[2*m] && m>1 && NeQ[m+n-1] && (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*Sqrt[c_.+d_.*tan[e_.+f_.*x_]],x_Symbol] := + -b*(a+b*Tan[e+f*x])^m*Sqrt[c+d*Tan[e+f*x]]/(2*a*f*m) + + 1/(4*a^2*m)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[2*a*c*m+b*d+a*d*(2*m+1)*Tan[e+f*x],x]/Sqrt[c+d*Tan[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m<0 && IntegersQ[2*m] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + -(b*c-a*d)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n-1)/(2*a*f*m) + + 1/(2*a^2*m)*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-2)* + Simp[c*(a*c*m+b*d*(n-1))-d*(b*c*m+a*d*(n-1))-d*(b*d*(m-n+1)-a*c*(m+n-1))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && + RationalQ[m,n] && m<0 && n>1 && (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(2*f*m*(b*c-a*d)) + + 1/(2*a*m*(b*c-a*d))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n* + Simp[b*c*m-a*d*(2*m+n+1)+b*d*(m+n+1)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && + RationalQ[m] && m<0 && (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + d*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n-1)/(f*(m+n-1)) - + 1/(a*(m+n-1))*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n-2)* + Simp[d*(b*c*m+a*d*(-1+n))-a*c^2*(m+n-1)+d*(b*d*m-a*c*(m+2*n-2))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && + RationalQ[n] && n>1 && NeQ[m+n-1] && (IntegerQ[n] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + d*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(f*(n+1)*(c^2+d^2)) - + 1/(a*(c^2+d^2)*(n+1))*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)* + Simp[b*d*m-a*c*(n+1)+a*d*(m+n+1)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] && + RationalQ[n] && n<-1 && (IntegerQ[n] || IntegersQ[2*m,2*n]) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + a/(a*c-b*d)*Int[(a+b*Tan[e+f*x])^m,x] - + d/(a*c-b*d)*Int[(a+b*Tan[e+f*x])^m*(b+a*Tan[e+f*x])/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[Sqrt[a_+b_.*tan[e_.+f_.*x_]]*Sqrt[c_.+d_.*tan[e_.+f_.*x_]],x_Symbol] := + (a*c-b*d)/a*Int[Sqrt[a+b*Tan[e+f*x]]/Sqrt[c+d*Tan[e+f*x]],x] + + d/a*Int[Sqrt[a+b*Tan[e+f*x]]*(b+a*Tan[e+f*x])/Sqrt[c+d*Tan[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + a*b/f*Subst[Int[(a+x)^(m-1)*(c+d/b*x)^n/(b^2+a*x),x],x,b*Tan[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (b*c-a*d)^2*(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(n+1)*(c^2+d^2)) - + 1/(d*(n+1)*(c^2+d^2))*Int[(a+b*Tan[e+f*x])^(m-3)*(c+d*Tan[e+f*x])^(n+1)* + Simp[a^2*d*(b*d*(m-2)-a*c*(n+1))+b*(b*c-2*a*d)*(b*c*(m-2)+a*d*(n+1)) - + d*(n+1)*(3*a^2*b*c-b^3*c-a^3*d+3*a*b^2*d)*Tan[e+f*x] - + b*(a*d*(2*b*c-a*d)*(m+n-1)-b^2*(c^2*(m-2)-d^2*(n+1)))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m>2 && n<-1 && + IntegerQ[2*m] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b^2*(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n-1)) + + 1/(d*(m+n-1))*Int[(a+b*Tan[e+f*x])^(m-3)*(c+d*Tan[e+f*x])^n* + Simp[a^3*d*(m+n-1)-b^2*(b*c*(m-2)+a*d*(1+n))+b*d*(m+n-1)*(3*a^2-b^2)*Tan[e+f*x]- + b^2*(b*c*(m-2)-a*d*(3*m+2*n-4))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && IntegerQ[2*m] && m>2 && + (RationalQ[n] && n>=-1 || IntegerQ[m]) && Not[IntegerQ[n] && n>2 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (b*c-a*d)*(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-1)/(f*(m+1)*(a^2+b^2)) + + 1/((m+1)*(a^2+b^2))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-2)* + Simp[a*c^2*(m+1)+a*d^2*(n-1)+b*c*d*(m-n+2)-(b*c^2-2*a*c*d-b*d^2)*(m+1)*Tan[e+f*x]-d*(b*c-a*d)*(m+n)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m<-1 && 10 && + IntegerQ[2*m] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b^2*(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n+1)/(f*(m+1)*(a^2+b^2)*(b*c-a*d)) + + 1/((m+1)*(a^2+b^2)*(b*c-a*d))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n* + Simp[a*(b*c-a*d)*(m+1)-b^2*d*(m+n+2)-b*(b*c-a*d)*(m+1)*Tan[e+f*x]-b^2*d*(m+n+2)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && IntegerQ[2*m] && m<-1 && + (RationalQ[n] && n<0 || IntegerQ[m]) && Not[IntegerQ[n] && n<-1 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^n/(f*(m+n-1)) + + 1/(m+n-1)*Int[(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n-1)* + Simp[a^2*c*(m+n-1)-b*(b*c*(m-1)+a*d*n)+(2*a*b*c+a^2*d-b^2*d)*(m+n-1)*Tan[e+f*x]+b*(b*c*n+a*d*(2*m+n-2))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m>1 && n>0 && IntegerQ[2*n] + + +Int[1/((a_+b_.*tan[e_.+f_.*x_])*(c_.+d_.*tan[e_.+f_.*x_])),x_Symbol] := + (a*c-b*d)*x/((a^2+b^2)*(c^2+d^2)) + + b^2/((b*c-a*d)*(a^2+b^2))*Int[(b-a*Tan[e+f*x])/(a+b*Tan[e+f*x]),x] - + d^2/((b*c-a*d)*(c^2+d^2))*Int[(d-c*Tan[e+f*x])/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[Sqrt[a_.+b_.*tan[e_.+f_.*x_]]/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(c^2+d^2)*Int[Simp[a*c+b*d+(b*c-a*d)*Tan[e+f*x],x]/Sqrt[a+b*Tan[e+f*x]],x] - + d*(b*c-a*d)/(c^2+d^2)*Int[(1+Tan[e+f*x]^2)/(Sqrt[a+b*Tan[e+f*x]]*(c+d*Tan[e+f*x])),x] /; +FreeQ[{a,b,c,d},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^(3/2)/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(c^2+d^2)*Int[Simp[a^2*c-b^2*c+2*a*b*d+(2*a*b*c-a^2*d+b^2*d)*Tan[e+f*x],x]/Sqrt[a+b*Tan[e+f*x]],x] + + (b*c-a*d)^2/(c^2+d^2)*Int[(1+Tan[e+f*x]^2)/(Sqrt[a+b*Tan[e+f*x]]*(c+d*Tan[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(c^2+d^2)*Int[(a+b*Tan[e+f*x])^m*(c-d*Tan[e+f*x]),x] + + d^2/(c^2+d^2)*Int[(a+b*Tan[e+f*x])^m*(1+Tan[e+f*x]^2)/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && Not[IntegerQ[m]] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b*ff*x)^m*(c+d*ff*x)^n/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_.*(d_./tan[e_.+f_.*x_])^n_,x_Symbol] := + d^m*Int[(b+a*Cot[e+f*x])^m*(d*Cot[e+f*x])^(n-m),x] /; +FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(d_./cot[e_.+f_.*x_])^n_,x_Symbol] := + d^m*Int[(b+a*Tan[e+f*x])^m*(d*Tan[e+f*x])^(n-m),x] /; +FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_.*(d_.*tan[e_.+f_.*x_])^p_)^n_,x_Symbol] := + c^IntPart[n]*(c*(d*Tan[e + f*x])^p)^FracPart[n]/(d*Tan[e + f*x])^(p*FracPart[n])* + Int[(a+b*Tan[e+f*x])^m*(d*Tan[e+f*x])^(n*p),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[n]] && Not[IntegerQ[m]] + + +Int[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.*(d_.*cot[e_.+f_.*x_])^p_)^n_,x_Symbol] := + c^IntPart[n]*(c*(d*Cot[e + f*x])^p)^FracPart[n]/(d*Cot[e + f*x])^(p*FracPart[n])* + Int[(a+b*Cot[e+f*x])^m*(d*Cot[e+f*x])^(n*p),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[n]] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n*) + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*tan[e_.+f_.*x_])^m_*(c_+d_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + Defer[Int][(g*Tan[e+f*x])^p*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +Int[(g_.*cot[e_.+f_.*x_])^p_*(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + g^(m+n)*Int[(g*Cot[e+f*x])^(p-m-n)*(b+a*Cot[e+f*x])^m*(d+c*Cot[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n] + + +Int[(g_.*tan[e_.+f_.*x_])^p_*(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_+d_.*cot[e_.+f_.*x_])^n_.,x_Symbol] := + g^(m+n)*Int[(g*Tan[e+f*x])^(p-m-n)*(b+a*Tan[e+f*x])^m*(d+c*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n] + + +Int[(g_.*tan[e_.+f_.*x_]^q_)^p_*(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + (g*Tan[e+f*x]^q)^p/(g*Tan[e+f*x])^(p*q)*Int[(g*Tan[e+f*x])^(p*q)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p,q},x] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]] + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*cot[e_.+f_.*x_])^n_.,x_Symbol] := + g^n*Int[(g*Tan[e+f*x])^(p-n)*(a+b*Tan[e+f*x])^m*(d+c*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,p},x] && IntegerQ[n] + + +Int[tan[e_.+f_.*x_]^p_.*(a_+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*cot[e_.+f_.*x_])^n_,x_Symbol] := + Int[(b+a*Cot[e+f*x])^m*(c+d*Cot[e+f*x])^n/Cot[e+f*x]^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p] + + +Int[(g_.*tan[e_.+f_.*x_])^p_*(a_+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*cot[e_.+f_.*x_])^n_,x_Symbol] := + Cot[e+f*x]^p*(g*Tan[e+f*x])^p*Int[(b+a*Cot[e+f*x])^m*(c+d*Cot[e+f*x])^n/Cot[e+f*x]^(m+p),x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]] + + +Int[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*tan[e_.+f_.*x_])^m_*(c_+d_.*cot[e_.+f_.*x_])^n_,x_Symbol] := + (g*Tan[e+f*x])^n*(c+d*Cot[e+f*x])^n/(d+c*Tan[e+f*x])^n*Int[(g*Tan[e+f*x])^(p-n)*(a+b*Tan[e+f*x])^m*(d+c*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && Not[IntegerQ[n]] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.3.1 (a+b tan)^m (c+d tan)^n (A+B tan)*) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_.*(c_+d_.*tan[e_.+f_.*x_])^n_.*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + a*c/f*Subst[Int[(a+b*x)^(m-1)*(c+d*x)^(n-1)*(A+B*x),x],x,Tan[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2+b^2] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])*(A_.+B_.*tan[e_.+f_.*x_])/(a_.+b_.*tan[e_.+f_.*x_]),x_Symbol] := + B*d/b*Int[Tan[e+f*x],x] + 1/b*Int[Simp[A*b*c+(A*b*d+B*(b*c-a*d))*Tan[e+f*x],x]/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*(a*c+b*d)*(a+b*Tan[e+f*x])^m/(2*a^2*f*m) + + 1/(2*a*b)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[A*b*c+a*B*c+a*A*d+b*B*d+2*a*B*d*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && RationalQ[m] && m<-1 && EqQ[a^2+b^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (b*c-a*d)*(A*b-a*B)*(a+b*Tan[e+f*x])^(m+1)/(b*f*(m+1)*(a^2+b^2)) + + 1/(a^2+b^2)*Int[(a+b*Tan[e+f*x])^(m+1)*Simp[a*A*c+b*B*c+A*b*d-a*B*d-(A*b*c-a*B*c-a*A*d-b*B*d)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && RationalQ[m] && m<-1 && NeQ[a^2+b^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_.+d_.*tan[e_.+f_.*x_])*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + B*d*(a+b*Tan[e+f*x])^(m+1)/(b*f*(m+1)) + + Int[(a+b*Tan[e+f*x])^m*Simp[A*c-B*d+(B*c+A*d)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && Not[RationalQ[m] && m<=-1] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + -a^2*(B*c-A*d)*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(b*c+a*d)*(n+1)) - + a/(d*(b*c+a*d)*(n+1))*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)* + Simp[A*b*d*(m-n-2)-B*(b*c*(m-1)+a*d*(n+1))+(a*A*d*(m+n)-B*(a*c*(m-1)+b*d*(n+1)))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[m,n] && m>1 && n<-1 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + b*B*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n)) + + 1/(d*(m+n))*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^n* + Simp[a*A*d*(m+n)+B*(a*c*(m-1)-b*d*(n+1))-(B*(b*c-a*d)*(m-1)-d*(A*b+a*B)*(m+n))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[m] && m>1 && Not[RationalQ[n] && n<-1] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n/(2*a*f*m) + + 1/(2*a^2*m)*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-1)* + Simp[A*(a*c*m+b*d*n)-B*(b*c*m+a*d*n)-d*(b*B*(m-n)-a*A*(m+n))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[m,n] && m<0 && n>0 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (a*A+b*B)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(2*f*m*(b*c-a*d)) + + 1/(2*a*m*(b*c-a*d))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n* + Simp[A*(b*c*m-a*d*(2*m+n+1))+B*(a*c*m-b*d*(n+1))+d*(A*b-a*B)*(m+n+1)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[m] && m<0 && Not[RationalQ[n] && n>0] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + B*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n/(f*(m+n)) + + 1/(a*(m+n))*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n-1)* + Simp[a*A*c*(m+n)-B*(b*c*m+a*d*n)+(a*A*d*(m+n)-B*(b*d*m-a*c*n))*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[n] && n>0 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (A*d-B*c)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(f*(n+1)*(c^2+d^2)) - + 1/(a*(n+1)*(c^2+d^2))*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)* + Simp[A*(b*d*m-a*c*(n+1))-B*(b*c*m+a*d*(n+1))-a*(B*c-A*d)*(m+n+1)*Tan[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && RationalQ[n] && n<-1 + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + b*B/f*Subst[Int[(a+b*x)^(m-1)*(c+d*x)^n,x],x,Tan[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && EqQ[A*b+a*B] + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(A_.+B_.*tan[e_.+f_.*x_])/(c_.+d_.*tan[e_.+f_.*x_]),x_Symbol] := + (A*b+a*B)/(b*c+a*d)*Int[(a+b*Tan[e+f*x])^m,x] - + (B*c-A*d)/(b*c+a*d)*Int[(a+b*Tan[e+f*x])^m*(a-b*Tan[e+f*x])/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[A*b+a*B] + + +(* Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)/b*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] + + B/b*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[c^2+d^2] *) + + +Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (A*b+a*B)/b*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] - + B/b*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n*(a-b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2+b^2] && NeQ[A*b+a*B] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_+B_.*tan[e_.+f_.*x_]),x_Symbol] := + A^2/f*Subst[Int[(a+b*x)^m*(c+d*x)^n/(A-B*x),x],x,Tan[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && + Not[IntegersQ[2*m,2*n]] && EqQ[A^2+B^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (A+I*B)/2*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n*(1-I*Tan[e+f*x]),x] + + (A-I*B)/2*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n*(1+I*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && + Not[IntegersQ[2*m,2*n]] && NeQ[A^2+B^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (b*c-a*d)*(B*c-A*d)*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(n+1)*(c^2+d^2)) - + 1/(d*(n+1)*(c^2+d^2))*Int[(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^(n+1)* + Simp[a*A*d*(b*d*(m-1)-a*c*(n+1))+(b*B*c-(A*b+a*B)*d)*(b*c*(m-1)+a*d*(n+1))- + d*((a*A-b*B)*(b*c-a*d)+(A*b+a*B)*(a*c+b*d))*(n+1)*Tan[e+f*x]- + b*(d*(A*b*c+a*B*c-a*A*d)*(m+n)-b*B*(c^2*(m-1)-d^2*(n+1)))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m>1 && n<-1 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + b*B*(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n)) + + 1/(d*(m+n))*Int[(a+b*Tan[e+f*x])^(m-2)*(c+d*Tan[e+f*x])^n* + Simp[a^2*A*d*(m+n)-b*B*(b*c*(m-1)+a*d*(n+1))+ + d*(m+n)*(2*a*A*b+B*(a^2-b^2))*Tan[e+f*x]- + (b*B*(b*c-a*d)*(m-1)-b*(A*b+a*B)*d*(m+n))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m>1 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) && Not[IntegerQ[n] && n>1 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)*(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n/(f*(m+1)*(a^2+b^2)) + + 1/(b*(m+1)*(a^2+b^2))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n-1)* + Simp[b*B*(b*c*(m+1)+a*d*n)+A*b*(a*c*(m+1)-b*d*n)-b*(A*(b*c-a*d)-B*(a*c+b*d))*(m+1)*Tan[e+f*x]-b*d*(A*b-a*B)*(m+n+1)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m<-1 && 00 && n<-1 + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + (A*d^2+c^2*C)*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(d*f*(n+1)*(c^2+d^2)) - + 1/(d*(n+1)*(c^2+d^2))*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^(n+1)* + Simp[A*d*(b*d*m-a*c*(n+1))+c*C*(b*c*m+a*d*(n+1)) - + d*(n+1)*((A-C)*(b*c-a*d))*Tan[e+f*x] + + b*(A*d^2*(m+n+1)+C*(c^2*m-d^2*(n+1)))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m,n] && m>0 && n<-1 + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + C*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n+1)) + + 1/(d*(m+n+1))*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^n* + Simp[a*A*d*(m+n+1)-C*(b*c*m+a*d*(n+1))+d*(A*b+a*B-b*C)*(m+n+1)*Tan[e+f*x]-(C*m*(b*c-a*d)-b*B*d*(m+n+1))*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m>0 && + Not[IntegerQ[n] && n>0 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_.*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + C*(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^(n+1)/(d*f*(m+n+1)) + + 1/(d*(m+n+1))*Int[(a+b*Tan[e+f*x])^(m-1)*(c+d*Tan[e+f*x])^n* + Simp[a*A*d*(m+n+1)-C*(b*c*m+a*d*(n+1))+d*(A*b-b*C)*(m+n+1)*Tan[e+f*x]-C*m*(b*c-a*d)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m>0 && + Not[IntegerQ[n] && n>0 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + (A*b^2-a*(b*B-a*C))*(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n+1)/(f*(m+1)*(b*c-a*d)*(a^2+b^2)) + + 1/((m+1)*(b*c-a*d)*(a^2+b^2))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n* + Simp[A*(a*(b*c -a*d)*(m+1)-b^2*d*(m+n+2))+(b*B-a*C)*(b*c*(m+1)+a*d*(n+1)) - + (m+1)*(b*c-a*d)*(A*b-a*B-b*C)*Tan[e+f*x] - + d*(A*b^2-a*(b*B-a*C))*(m+n+2)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m<-1 && + Not[IntegerQ[n] && n<-1 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + (A*b^2+a^2*C)*(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^(n+1)/(f*(m+1)*(b*c-a*d)*(a^2+b^2)) + + 1/((m+1)*(b*c-a*d)*(a^2+b^2))*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n* + Simp[A*(a*(b*c -a*d)*(m+1)-b^2*d*(m+n+2))-a*C*(b*c*(m+1)+a*d*(n+1)) - + (m+1)*(b*c-a*d)*(A*b-b*C)*Tan[e+f*x] - + d*(A*b^2+a^2*C)*(m+n+2)*Tan[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f,A,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && RationalQ[m] && m<-1 && + Not[IntegerQ[n] && n<-1 && (Not[IntegerQ[m]] || EqQ[c] && NeQ[a])] + + +Int[(A_.+B_.*tan[e_.+f_.*x_]+C_.*tan[e_.+f_.*x_]^2)/((a_+b_.*tan[e_.+f_.*x_])*(c_.+d_.*tan[e_.+f_.*x_])),x_Symbol] := + (a*(A*c-c*C+B*d)+b*(B*c-A*d+C*d))*x/((a^2+b^2)*(c^2+d^2)) + + (A*b^2-a*b*B+a^2*C)/((b*c-a*d)*(a^2+b^2))*Int[(b-a*Tan[e+f*x])/(a+b*Tan[e+f*x]),x] - + (c^2*C-B*c*d+A*d^2)/((b*c-a*d)*(c^2+d^2))*Int[(d-c*Tan[e+f*x])/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(A_.+C_.*tan[e_.+f_.*x_]^2)/((a_+b_.*tan[e_.+f_.*x_])*(c_.+d_.*tan[e_.+f_.*x_])),x_Symbol] := + (a*(A*c-c*C)-b*(A*d-C*d))*x/((a^2+b^2)*(c^2+d^2)) + + (A*b^2+a^2*C)/((b*c-a*d)*(a^2+b^2))*Int[(b-a*Tan[e+f*x])/(a+b*Tan[e+f*x]),x] - + (c^2*C+A*d^2)/((b*c-a*d)*(c^2+d^2))*Int[(d-c*Tan[e+f*x])/(c+d*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,C},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]+C_.*tan[e_.+f_.*x_]^2)/(a_.+b_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(a^2+b^2)*Int[(c+d*Tan[e+f*x])^n*Simp[b*B+a*(A-C)+(a*B-b*(A-C))*Tan[e+f*x],x],x] + + (A*b^2-a*b*B+a^2*C)/(a^2+b^2)*Int[(c+d*Tan[e+f*x])^n*(1+Tan[e+f*x]^2)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && + Not[RationalQ[n] && n>0] && Not[RationalQ[n] && n<=-1] + + +Int[(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+C_.*tan[e_.+f_.*x_]^2)/(a_.+b_.*tan[e_.+f_.*x_]),x_Symbol] := + 1/(a^2+b^2)*Int[(c+d*Tan[e+f*x])^n*Simp[a*(A-C)-(A*b-b*C)*Tan[e+f*x],x],x] + + (A*b^2+a^2*C)/(a^2+b^2)*Int[(c+d*Tan[e+f*x])^n*(1+Tan[e+f*x]^2)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,C,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] && + Not[RationalQ[n] && n>0] && Not[RationalQ[n] && n<=-1] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+B_.*tan[e_.+f_.*x_]+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b*ff*x)^m*(c+d*ff*x)^n*(A+B*ff*x+C*ff^2*x^2)/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + +Int[(a_.+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+C_.*tan[e_.+f_.*x_]^2),x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b*ff*x)^m*(c+d*ff*x)^n*(A+C*ff^2*x^2)/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,d,e,f,A,C,m,n},x] && NeQ[b*c-a*d] && NeQ[a^2+b^2] && NeQ[c^2+d^2] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.7 (d trig)^m (a+b (c tan)^n)^p*) + + +Int[u_.*(a_+b_.*tan[e_.+f_.*x_]^2)^p_,x_Symbol] := + a^p*Int[ActivateTrig[u*sec[e+f*x]^(2*p)],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a-b,0] && IntegerQ[p] + + +Int[u_.*(a_+b_.*tan[e_.+f_.*x_]^2)^p_,x_Symbol] := + Int[ActivateTrig[u*(a*sec[e+f*x]^2)^p],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a-b,0] + + +Int[u_.*(b_.*tan[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + (b*ff^n)^IntPart[p]*(b*Tan[e+f*x]^n)^FracPart[p]/(Tan[e+f*x]/ff)^(n*FracPart[p])* + Int[ActivateTrig[u]*(Tan[e+f*x]/ff)^(n*p),x]] /; +FreeQ[{b,e,f,n,p},x] && Not[IntegerQ[p]] && IntegerQ[n] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[u_.*(b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_,x_Symbol] := + b^IntPart[p]*(b*(c*Tan[e+f*x])^n)^FracPart[p]/(c*Tan[e+f*x])^(n*FracPart[p])* + Int[ActivateTrig[u]*(c*Tan[e+f*x])^(n*p),x] /; +FreeQ[{b,c,e,f,n,p},x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[1/(a_+b_.*tan[e_.+f_.*x_]^2),x_Symbol] := + 1/a*Int[Cos[e+f*x]^2,x] /; +FreeQ[{a,b,e,f},x] && EqQ[a-b] + + +Int[1/(a_+b_.*tan[e_.+f_.*x_]^2),x_Symbol] := + x/(a-b) - b/(a-b)*Int[Sec[e+f*x]^2/(a+b*Tan[e+f*x]^2),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a-b] + + +Int[(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + c*ff/f*Subst[Int[(a+b*(ff*x)^n)^p/(c^2+ff^2*x^2),x],x,c*Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && (IntegersQ[n,p] || IGtQ[p,0] || EqQ[n^2,4] || EqQ[n^2,16]) + + +Int[(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,e,f,n,p},x] + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + c*ff^(m+1)/f*Subst[Int[x^m*(a+b*(ff*x)^n)^p/(c^2+ff^2*x^2)^(m/2+1),x],x,c*Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && IntegerQ[m/2] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_]^2)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sec[e+f*x],x]}, + 1/(f*ff^m)*Subst[Int[(-1+ff^2*x^2)^((m-1)/2)*(a-b+b*ff^2*x^2)^p/x^(m+1),x],x,Sec[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sec[e+f*x],x]}, + 1/(f*ff^m)*Subst[Int[(-1+ff^2*x^2)^((m-1)/2)*(a+b*(-1+ff^2*x^2)^(n/2))^p/x^(m+1),x],x,Sec[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] + + +Int[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*sin[e+f*x])^m*(a+b*(c*tan[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*sin[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(d*Sin[e+f*x])^m*(Sec[e+f*x]^2)^(m/2)/(f*Tan[e+f*x]^m)* + Subst[Int[(ff*x)^m*(a+b*ff^2*x^2)^p/(1+ff^2*x^2)^(m/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,d,e,f,m,p},x] && Not[IntegerQ[m]] + + +Int[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Sin[e+f*x])^m*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*cos[e_.+f_.*x_])^m_*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Cos[e+f*x])^FracPart[m]*(Sec[e+f*x]/d)^FracPart[m]*Int[(Sec[e+f*x]/d)^(-m)*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + c*ff/f*Subst[Int[(d*ff*x/c)^m*(a+b*(ff*x)^n)^p/(c^2+ff^2*x^2),x],x,c*Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && (IGtQ[p,0] || EqQ[n,2] || EqQ[n,4] || IntegerQ[p] && RationalQ[n]) + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*tan[e+f*x])^m*(a+b*(c*tan[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Tan[e+f*x])^m*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*cot[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_]^n_.)^p_.,x_Symbol] := + d^(n*p)*Int[(d*Cot[e+f*x])^(m-n*p)*(b+a*Cot[e+f*x]^n)^p,x] /; +FreeQ[{a,b,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && IntegersQ[n,p] + + +Int[(d_.*cot[e_.+f_.*x_])^m_*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Cot[e+f*x])^FracPart[m]*(Tan[e+f*x]/d)^FracPart[m]*Int[(Tan[e+f*x]/d)^(-m)*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[sec[e_.+f_.*x_]^m_*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/(c^(m-1)*f)*Subst[Int[(c^2+ff^2*x^2)^(m/2-1)*(a+b*(ff*x)^n)^p,x],x,c*Tan[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && IntegerQ[m/2] && (IntegersQ[n,p] || IGtQ[m,0] || IGtQ[p,0] || EqQ[n^2,4] || EqQ[n^2,16]) + + +Int[sec[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff/f*Subst[Int[ExpandToSum[b*(ff*x)^n+a*(1-ff^2*x^2)^(n/2),x]^p/(1-ff^2*x^2)^((m+n*p+1)/2),x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] && IntegerQ[p] + + +Int[sec[e_.+f_.*x_]^m_.*(a_+b_.*tan[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff/f*Subst[Int[1/(1-ff^2*x^2)^((m+1)/2)*((b*(ff*x)^n+a*(1-ff^2*x^2)^(n/2))/(1-ff^2*x^2)^(n/2))^p,x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Int[ExpandTrig[(d*sec[e+f*x])^m*(a+b*(c*tan[e+f*x])^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[p,0] + + +Int[(d_.*sec[e_.+f_.*x_])^m_*(a_+b_.*tan[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff*(d*Sec[e+f*x])^m/(f*(Sec[e+f*x]^2)^(m/2))* + Subst[Int[(1+ff^2*x^2)^(m/2-1)*(a+b*ff^2*x^2)^p,x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,d,e,f,m,p},x] && Not[IntegerQ[m]] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Sec[e+f*x])^m*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*csc[e_.+f_.*x_])^m_*(a_+b_.*(c_.*tan[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Csc[e+f*x])^FracPart[m]*(Sin[e+f*x]/d)^FracPart[m]*Int[(Sin[e+f*x]/d)^(-m)*(a+b*(c*Tan[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.9 trig^m (a+b tan^n+c tan^(2 n))^p*) + + +Int[(a_+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[(a_+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[(a_+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Tan[d+e*x]^n+c*Tan[d+e*x]^(2*n))^p/(b+2*c*Tan[d+e*x]^n)^(2*p)*Int[(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[(a_+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Cot[d+e*x]^n+c*Cot[d+e*x]^(2*n))^p/(b+2*c*Cot[d+e*x]^n)^(2*p)*Int[(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[1/(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/(b-q+2*c*Tan[d+e*x]^n),x] - + 2*c/q*Int[1/(b+q+2*c*Tan[d+e*x]^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[1/(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/(b-q+2*c*Cot[d+e*x]^n),x] - + 2*c/q*Int[1/(b+q+2*c*Cot[d+e*x]^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[sin[d_.+e_.*x_]^m_*(a_.+b_.*(f_.*tan[d_.+e_.*x_])^n_.+c_.*(f_.*tan[d_.+e_.*x_])^n2_.)^p_,x_Symbol] := + f/e*Subst[Int[x^m*(a+b*x^n+c*x^(2*n))^p/(f^2+x^2)^(m/2+1),x],x,f*Tan[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[n2-2*n] && IntegerQ[m/2] + + +Int[cos[d_.+e_.*x_]^m_*(a_.+b_.*(f_.*cot[d_.+e_.*x_])^n_.+c_.*(f_.*cot[d_.+e_.*x_])^n2_.)^p_,x_Symbol] := + -f/e*Subst[Int[x^m*(a+b*x^n+c*x^(2*n))^p/(f^2+x^2)^(m/2+1),x],x,f*Cot[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[n2-2*n] && IntegerQ[m/2] + + +Int[sin[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{g=FreeFactors[Cos[d+e*x],x]}, + -g/e*Subst[Int[(1-g^2*x^2)^((m-1)/2)*ExpandToSum[a*(g*x)^(2*n)+b*(g*x)^n*(1-g^2*x^2)^(n/2)+c*(1-g^2*x^2)^n,x]^p/(g*x)^(2*n*p),x],x,Cos[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && EvenQ[n] && IntegerQ[p] + + +Int[cos[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{g=FreeFactors[Sin[d+e*x],x]}, + g/e*Subst[Int[(1-g^2*x^2)^((m-1)/2)*ExpandToSum[a*(g*x)^(2*n)+b*(g*x)^n*(1-g^2*x^2)^(n/2)+c*(1-g^2*x^2)^n,x]^p/(g*x)^(2*n*p),x],x,Sin[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && EvenQ[n] && IntegerQ[p] + + +Int[cos[d_.+e_.*x_]^m_*(a_.+b_.*(f_.*tan[d_.+e_.*x_])^n_.+c_.*(f_.*tan[d_.+e_.*x_])^n2_.)^p_.,x_Symbol] := + f^(m+1)/e*Subst[Int[(a+b*x^n+c*x^(2*n))^p/(f^2+x^2)^(m/2+1),x],x,f*Tan[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[n2-2*n] && EvenQ[m] + + +Int[sin[d_.+e_.*x_]^m_*(a_.+b_.*(f_.*cot[d_.+e_.*x_])^n_.+c_.*(f_.*cot[d_.+e_.*x_])^n2_.)^p_.,x_Symbol] := + -f^(m+1)/e*Subst[Int[(a+b*x^n+c*x^(2*n))^p/(f^2+x^2)^(m/2+1),x],x,f*Cot[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,n,p},x] && EqQ[n2-2*n] && EvenQ[m] + + +Int[cos[d_.+e_.*x_]^m_*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{g=FreeFactors[Sin[d+e*x],x]}, + g/e*Subst[Int[(1-g^2*x^2)^((m-2*n*p-1)/2)*ExpandToSum[c*x^(2*n)+b*x^n*(1-x^2)^(n/2)+a*(1-x^2)^n,x]^p,x],x,Sin[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && EvenQ[n] && IntegerQ[p] + + +Int[sin[d_.+e_.*x_]^m_*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{g=FreeFactors[Cos[d+e*x],x]}, + -g/e*Subst[Int[(1-g^2*x^2)^((m-2*n*p-1)/2)*ExpandToSum[c*x^(2*n)+b*x^n*(1-x^2)^(n/2)+a*(1-x^2)^n,x]^p,x],x,Cos[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && EvenQ[n] && IntegerQ[p] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + 1/(4^p*c^p)*Int[Tan[d+e*x]^m*(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + 1/(4^p*c^p)*Int[Cot[d+e*x]^m*(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Tan[d+e*x]^n+c*Tan[d+e*x]^(2*n))^p/(b+2*c*Tan[d+e*x]^n)^(2*p)*Int[Tan[d+e*x]^m*(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Cot[d+e*x]^n+c*Cot[d+e*x]^(2*n))^p/(b+2*c*Cot[d+e*x]^n)^(2*p)*Int[Cot[d+e*x]^m*(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*(f_.*tan[d_.+e_.*x_])^n_.+c_.*(f_.*tan[d_.+e_.*x_])^n2_.)^p_,x_Symbol] := + f/e*Subst[Int[(x/f)^m*(a+b*x^n+c*x^(2*n))^p/(f^2+x^2),x],x,f*Tan[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*(f_.*cot[d_.+e_.*x_])^n_.+c_.*(f_.*cot[d_.+e_.*x_])^n2_.)^p_,x_Symbol] := + -f/e*Subst[Int[(x/f)^m*(a+b*x^n+c*x^(2*n))^p/(f^2+x^2),x],x,f*Cot[d+e*x]] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[Cot[d+e*x]^m*(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[Tan[d+e*x]^m*(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_.+c_.*tan[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Tan[d+e*x]^n+c*Tan[d+e*x]^(2*n))^p/(b+2*c*Tan[d+e*x]^n)^(2*p)*Int[Cot[d+e*x]^m*(b+2*c*Tan[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_.+c_.*cot[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Cot[d+e*x]^n+c*Cot[d+e*x]^(2*n))^p/(b+2*c*Cot[d+e*x]^n)^(2*p)*Int[Tan[d+e*x]^m*(b+2*c*Cot[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[cot[d_.+e_.*x_]^m_.*(a_.+b_.*tan[d_.+e_.*x_]^n_+c_.*tan[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{g=FreeFactors[Cot[d+e*x],x]}, + g/e*Subst[Int[(g*x)^(m-2*n*p)*(c+b*(g*x)^n+a*(g*x)^(2*n))^p/(1+g^2*x^2),x],x,Cot[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EvenQ[n] + + +Int[tan[d_.+e_.*x_]^m_.*(a_.+b_.*cot[d_.+e_.*x_]^n_+c_.*cot[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{g=FreeFactors[Tan[d+e*x],x]}, + -g/e*Subst[Int[(g*x)^(m-2*n*p)*(c+b*(g*x)^n+a*(g*x)^(2*n))^p/(1+g^2*x^2),x],x,Tan[d+e*x]/g]] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] && EvenQ[n] + + +Int[(A_+B_.*tan[d_.+e_.*x_])*(a_+b_.*tan[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_]^2)^n_,x_Symbol] := + 1/(4^n*c^n)*Int[(A+B*Tan[d+e*x])*(b+2*c*Tan[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*cot[d_.+e_.*x_])*(a_+b_.*cot[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_]^2)^n_,x_Symbol] := + 1/(4^n*c^n)*Int[(A+B*Cot[d+e*x])*(b+2*c*Cot[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*tan[d_.+e_.*x_])*(a_+b_.*tan[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_]^2)^n_,x_Symbol] := + (a+b*Tan[d+e*x]+c*Tan[d+e*x]^2)^n/(b+2*c*Tan[d+e*x])^(2*n)*Int[(A+B*Tan[d+e*x])*(b+2*c*Tan[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[n]] + + +Int[(A_+B_.*cot[d_.+e_.*x_])*(a_+b_.*cot[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_]^2)^n_,x_Symbol] := + (a+b*Cot[d+e*x]+c*Cot[d+e*x]^2)^n/(b+2*c*Cot[d+e*x])^(2*n)*Int[(A+B*Cot[d+e*x])*(b+2*c*Cot[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[n]] + + +Int[(A_+B_.*tan[d_.+e_.*x_])/(a_.+b_.*tan[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_]^2),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + (B+(b*B-2*A*c)/q)*Int[1/Simp[b+q+2*c*Tan[d+e*x],x],x] + + (B-(b*B-2*A*c)/q)*Int[1/Simp[b-q+2*c*Tan[d+e*x],x],x]] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] + + +Int[(A_+B_.*cot[d_.+e_.*x_])/(a_.+b_.*cot[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_]^2),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + (B+(b*B-2*A*c)/q)*Int[1/Simp[b+q+2*c*Cot[d+e*x],x],x] + + (B-(b*B-2*A*c)/q)*Int[1/Simp[b-q+2*c*Cot[d+e*x],x],x]] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] + + +Int[(A_+B_.*tan[d_.+e_.*x_])*(a_.+b_.*tan[d_.+e_.*x_]+c_.*tan[d_.+e_.*x_]^2)^n_,x_Symbol] := + Int[ExpandTrig[(A+B*tan[d+e*x])*(a+b*tan[d+e*x]+c*tan[d+e*x]^2)^n,x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*cot[d_.+e_.*x_])*(a_.+b_.*cot[d_.+e_.*x_]+c_.*cot[d_.+e_.*x_]^2)^n_,x_Symbol] := + Int[ExpandTrig[(A+B*cot[d+e*x])*(a+b*cot[d+e*x]+c*cot[d+e*x]^2)^n,x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] && IntegerQ[n] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.10 (c+d x)^m (a+b tan)^n*) + + +Int[(c_.+d_.*x_)^m_.*tan[e_.+k_.*Pi+f_.*Complex[0,fz_]*x_],x_Symbol] := + -I*(c+d*x)^(m+1)/(d*(m+1)) + 2*I*Int[(c+d*x)^m*E^(-2*I*k*Pi)*E^(2*(-I*e+f*fz*x))/(1+E^(-2*I*k*Pi)*E^(2*(-I*e+f*fz*x))),x] /; +FreeQ[{c,d,e,f,fz},x] && IntegerQ[4*k] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*tan[e_.+k_.*Pi+f_.*x_],x_Symbol] := + I*(c+d*x)^(m+1)/(d*(m+1)) - 2*I*Int[(c+d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e+f*x))/(1+E^(2*I*k*Pi)*E^(2*I*(e+f*x))),x] /; +FreeQ[{c,d,e,f},x] && IntegerQ[4*k] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*tan[e_.+f_.*Complex[0,fz_]*x_],x_Symbol] := + -I*(c+d*x)^(m+1)/(d*(m+1)) + 2*I*Int[(c+d*x)^m*E^(2*(-I*e+f*fz*x))/(1+E^(2*(-I*e+f*fz*x))),x] /; +FreeQ[{c,d,e,f,fz},x] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*tan[e_.+f_.*x_],x_Symbol] := + I*(c+d*x)^(m+1)/(d*(m+1)) - 2*I*Int[(c+d*x)^m*E^(2*I*(e+f*x))/(1+E^(2*I*(e+f*x))),x] /; +FreeQ[{c,d,e,f},x] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + b*(c+d*x)^m*(b*Tan[e+f*x])^(n-1)/(f*(n-1)) - + b*d*m/(f*(n-1))*Int[(c+d*x)^(m-1)*(b*Tan[e+f*x])^(n-1),x] - + b^2*Int[(c+d*x)^m*(b*Tan[e+f*x])^(n-2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n>1 && m>0 + + +Int[(c_.+d_.*x_)^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + (c+d*x)^m*(b*Tan[e+f*x])^(n+1)/(b*f*(n+1)) - + d*m/(b*f*(n+1))*Int[(c+d*x)^(m-1)*(b*Tan[e+f*x])^(n+1),x] - + 1/b^2*Int[(c+d*x)^m*(b*Tan[e+f*x])^(n+2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n<-1 && m>0 + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(a+b*Tan[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[m,n] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + (c+d*x)^(m+1)/(2*a*d*(m+1)) - + a*(c+d*x)^m/(2*b*f*(a+b*Tan[e+f*x])) + + a*d*m/(2*b*f)*Int[(c+d*x)^(m-1)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m] && m>0 + + +Int[1/((c_.+d_.*x_)^2*(a_+b_.*tan[e_.+f_.*x_])),x_Symbol] := + -1/(d*(c+d*x)*(a+b*Tan[e+f*x])) + + f/(b*d)*Int[Cos[2*e+2*f*x]/(c+d*x),x] - + f/(a*d)*Int[Sin[2*e+2*f*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] + + +Int[(c_.+d_.*x_)^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + f*(c+d*x)^(m+2)/(b*d^2*(m+1)*(m+2)) + + (c+d*x)^(m+1)/(d*(m+1)*(a+b*Tan[e+f*x])) + + 2*b*f/(a*d*(m+1))*Int[(c+d*x)^(m+1)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m] && m<-1 && m!=-2 + + +(* Int[(c_.+d_.*x_)^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + (c+d*x)^(m+1)/(d*(m+1)*(a+b*Tan[e+f*x])) + + f/(b*d*(m+1))*Int[(c+d*x)^(m+1),x] + + 2*b*f/(a*d*(m+1))*Int[(c+d*x)^(m+1)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] && RationalQ[m] && m<-1 *) + + +Int[1/((c_.+d_.*x_)*(a_+b_.*tan[e_.+f_.*x_])),x_Symbol] := + Log[c+d*x]/(2*a*d) + + 1/(2*a)*Int[Cos[2*e+2*f*x]/(c+d*x),x] + + 1/(2*b)*Int[Sin[2*e+2*f*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] + + +Int[(c_.+d_.*x_)^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + (c+d*x)^(m+1)/(2*a*d*(m+1)) + + 1/(2*a)*Int[(c+d*x)^m*E^(2*a/b*(e+f*x)),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2+b^2] && Not[IntegerQ[m]] + + +Int[(c_.+d_.*x_)^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(1/(2*a)+Cos[2*e+2*f*x]/(2*a)+Sin[2*e+2*f*x]/(2*b))^(-n),x],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] && NegativeIntegerQ[m,n] + + +Int[(c_.+d_.*x_)^m_*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(1/(2*a)+E^(2*a/b*(e+f*x))/(2*a))^(-n),x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2+b^2] && NegativeIntegerQ[n] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + With[{u=IntHide[(a+b*Tan[e+f*x])^n,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[Dist[(c+d*x)^(m-1),u,x],x]] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2] && NegativeIntegerQ[n+1] && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*tan[e_.+k_.*Pi+f_.*x_]),x_Symbol] := + (c+d*x)^(m+1)/(d*(m+1)*(a-I*b)) - 2*I*b*Int[(c+d*x)^m/(a^2+b^2+(a-I*b)^2*E^(2*I*k*Pi)*E^(2*I*(e+f*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && IntegerQ[4*k] && NeQ[a^2+b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_./(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := + (c+d*x)^(m+1)/(d*(m+1)*(a-I*b)) - 2*I*b*Int[(c+d*x)^m/(a^2+b^2+(a-I*b)^2*E^(2*I*(e+f*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2+b^2] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)/(a_+b_.*tan[e_.+f_.*x_])^2,x_Symbol] := + -(c+d*x)^2/(2*d*(a^2+b^2)) - + b*(c+d*x)/(f*(a^2+b^2)*(a+b*Tan[e+f*x])) + + 1/(f*(a^2+b^2))*Int[(b*d+2*a*c*f+2*a*d*f*x)/(a+b*Tan[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2+b^2] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(1/(a-I*b)-2*I*b/(a^2+b^2+(a-I*b)^2*E^(2*I*(e+f*x))))^(-n),x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2+b^2] && NegativeIntegerQ[n] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*tan[e_.+f_.*x_]^n_.,x_Symbol] := + If[MatchQ[f,f1_.*Complex[0,j_]], + If[MatchQ[e,e1_.+Pi/2], + I^n*Defer[Int][(c+d*x)^m*Coth[-I*(e-Pi/2)-I*f*x]^n,x], + I^n*Defer[Int][(c+d*x)^m*Tanh[-I*e-I*f*x]^n,x]], + If[MatchQ[e,e1_.+Pi/2], + (-1)^n*Defer[Int][(c+d*x)^m*Cot[e-Pi/2+f*x]^n,x], + Defer[Int][(c+d*x)^m*Tan[e+f*x]^n,x]]] /; +FreeQ[{c,d,e,f,m,n},x] && IntegerQ[n] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*tan[e_.+f_.*x_])^n_.,x_Symbol] := + Defer[Int][(c+d*x)^m*(a+b*Tan[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[u_^m_.*(a_.+b_.*Tan[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Tan[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*(a_.+b_.*Cot[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Cot[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*4.2.11 (e x)^m (a+b tan(c+d x^n))^p*) + + +Int[(a_.+b_.*Tan[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Tan[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Cot[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Cot[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Tan[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Tan[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Cot[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Cot[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Tan[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Tan[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Cot[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Cot[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Tan[u_])^p_.,x_Symbol] := + Int[(a+b*Tan[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Cot[u_])^p_.,x_Symbol] := + Int[(a+b*Cot[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*(a_.+b_.*Tan[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Tan[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Cot[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Cot[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*Tan[c_.+d_.*x_^n_]^2,x_Symbol] := + x^(m-n+1)*Tan[c+d*x^n]/(d*n) - Int[x^m,x] - (m-n+1)/(d*n)*Int[x^(m-n)*Tan[c+d*x^n],x] /; +FreeQ[{c,d,m,n},x] + + +Int[x_^m_.*Cot[c_.+d_.*x_^n_]^2,x_Symbol] := + -x^(m-n+1)*Cot[c+d*x^n]/(d*n) - Int[x^m,x] + (m-n+1)/(d*n)*Int[x^(m-n)*Cot[c+d*x^n],x] /; +FreeQ[{c,d,m,n},x] + + +(* Int[x_^m_.*Tan[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m-n+1)*Tan[a+b*x^n]^(p-1)/(b*n*(p-1)) - + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Tan[a+b*x^n]^(p-1),x] - + Int[x^m*Tan[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b},x] && RationalQ[m,n,p] && 01 *) + + +(* Int[x_^m_.*Cot[a_.+b_.*x_^n_]^p_,x_Symbol] := + -x^(m-n+1)*Cot[a+b*x^n]^(p-1)/(b*n*(p-1)) + + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Cot[a+b*x^n]^(p-1),x] - + Int[x^m*Cot[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b},x] && RationalQ[m,n,p] && 01 *) + + +(* Int[x_^m_.*Tan[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m-n+1)*Tan[a+b*x^n]^(p+1)/(b*n*(p+1)) - + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*Tan[a+b*x^n]^(p+1),x] - + Int[x^m*Tan[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b},x] && RationalQ[m,n,p] && 0=0 && q===1 + + +Int[x_^m_.*Csc[a_.+b_.*x_^n_.]^p_.*Cot[a_.+b_.*x_^n_.]^q_.,x_Symbol] := + -x^(m-n+1)*Csc[a+b*x^n]^p/(b*n*p) + + (m-n+1)/(b*n*p)*Int[x^(m-n)*Csc[a+b*x^n]^p,x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && q===1 + + + + + +(* ::Subsection::Closed:: *) +(*4.2.12 (d+e x)^m tan(a+b x+c x^2)^n*) + + +Int[Tan[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Tan[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[Cot[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Cot[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[(d_+e_.*x_)*Tan[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*Log[Cos[a+b*x+c*x^2]]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e] + + +Int[(d_+e_.*x_)*Cot[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Log[Sin[a+b*x+c*x^2]]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[2*c*d-b*e] + + +Int[(d_.+e_.*x_)*Tan[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -e*Log[Cos[a+b*x+c*x^2]]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Tan[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e] + + +Int[(d_.+e_.*x_)*Cot[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Log[Sin[a+b*x+c*x^2]]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Cot[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[2*c*d-b*e] + + +(* Int[x_^m_*Tan[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + -x^(m-1)*Log[Cos[a+b*x+c*x^2]]/(2*c) - + b/(2*c)*Int[x^(m-1)*Tan[a+b*x+c*x^2],x] + + (m-1)/(2*c)*Int[x^(m-2)*Log[Cos[a+b*x+c*x^2]],x] /; +FreeQ[{a,b,c},x] && RationalQ[m] && m>1 *) + + +(* Int[x_^m_*Cot[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + x^(m-1)*Log[Sin[a+b*x+c*x^2]]/(2*c) - + b/(2*c)*Int[x^(m-1)*Cot[a+b*x+c*x^2],x] - + (m-1)/(2*c)*Int[x^(m-2)*Log[Sin[a+b*x+c*x^2]],x] /; +FreeQ[{a,b,c},x] && RationalQ[m] && m>1*) + + +Int[(d_.+e_.*x_)^m_.*Tan[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Tan[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + +Int[(d_.+e_.*x_)^m_.*Cot[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Cot[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + + +`) + +func resourcesRubi42TangentMBytes() ([]byte, error) { + return _resourcesRubi42TangentM, nil +} + +func resourcesRubi42TangentM() (*asset, error) { + bytes, err := resourcesRubi42TangentMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/4.2 Tangent.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi43SecantM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Secant Function Rules*) + + +(* ::Subsection::Closed:: *) +(*4.3.1.1 (a+b sec)^n*) + + +Int[csc[c_.+d_.*x_]^n_,x_Symbol] := + -1/d*Subst[Int[ExpandIntegrand[(1+x^2)^(n/2-1),x],x],x,Cot[c+d*x]] /; +FreeQ[{c,d},x] && PositiveIntegerQ[n/2] + + +Int[(b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -b*Cos[c+d*x]*(b*Csc[c+d*x])^(n-1)/(d*(n-1)) + + b^2*(n-2)/(n-1)*Int[(b*Csc[c+d*x])^(n-2),x] /; +FreeQ[{b,c,d},x] && n>1 && IntegerQ[2*n] + + +Int[(b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Cos[c+d*x]*(b*Csc[c+d*x])^(n+1)/(b*d*n) + + (n+1)/(b^2*n)*Int[(b*Csc[c+d*x])^(n+2),x] /; +FreeQ[{b,c,d},x] && n<-1 && IntegerQ[2*n] + + +Int[csc[c_.+d_.*x_],x_Symbol] := +(* -ArcCoth[Cos[c+d*x]]/d /; *) + -ArcTanh[Cos[c+d*x]]/d /; +FreeQ[{c,d},x] + + +(* Int[1/csc[c_.+d_.*x_],x_Symbol] := + -Cos[c+d*x]/d /; +FreeQ[{c,d},x] *) + + +Int[(b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + (b*Csc[c+d*x])^n*Sin[c+d*x]^n*Int[1/Sin[c+d*x]^n,x] /; +FreeQ[{b,c,d},x] && EqQ[n^2-1/4] + + +Int[(b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + (b*Csc[c+d*x])^(n-1)*((Sin[c+d*x]/b)^(n-1)*Int[1/(Sin[c+d*x]/b)^n,x]) /; +FreeQ[{b,c,d,n},x] && Not[IntegerQ[n]] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^2,x_Symbol] := + a^2*x + 2*a*b*Int[Csc[c+d*x],x] + b^2*Int[Csc[c+d*x]^2,x] /; +FreeQ[{a,b,c,d},x] + + +Int[Sqrt[a_+b_.*csc[c_.+d_.*x_]],x_Symbol] := + -2*b/d*Subst[Int[1/(a+x^2),x],x,b*Cot[c+d*x]/Sqrt[a+b*Csc[c+d*x]]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -b^2*Cot[c+d*x]*(a+b*Csc[c+d*x])^(n-2)/(d*(n-1)) + + a/(n-1)*Int[(a+b*Csc[c+d*x])^(n-2)*(a*(n-1)+b*(3*n-4)*Csc[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] && RationalQ[n] && n>1 && IntegerQ[2*n] + + +Int[1/Sqrt[a_+b_.*csc[c_.+d_.*x_]],x_Symbol] := + 1/a*Int[Sqrt[a+b*Csc[c+d*x]],x] - + b/a*Int[Csc[c+d*x]/Sqrt[a+b*Csc[c+d*x]],x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -Cot[c+d*x]*(a+b*Csc[c+d*x])^n/(d*(2*n+1)) + + 1/(a^2*(2*n+1))*Int[(a+b*Csc[c+d*x])^(n+1)*(a*(2*n+1)-b*(n+1)*Csc[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] && RationalQ[n] && n<=-1 && IntegerQ[2*n] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + a^n*Cot[c+d*x]/(d*Sqrt[1+Csc[c+d*x]]*Sqrt[1-Csc[c+d*x]])* + Subst[Int[(1+b*x/a)^(n-1/2)/(x*Sqrt[1-b*x/a]),x],x,Csc[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[2*n]] && PositiveQ[a] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + a^IntPart[n]*(a+b*Csc[c+d*x])^FracPart[n]/(1+b/a*Csc[c+d*x])^FracPart[n]*Int[(1+b/a*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[2*n]] && Not[PositiveQ[a]] + + +Int[Sqrt[a_+b_.*csc[c_.+d_.*x_]],x_Symbol] := + 2*(a+b*Csc[c+d*x])/(d*Rt[a+b,2]*Cot[c+d*x])*Sqrt[b*(1+Csc[c+d*x])/(a+b*Csc[c+d*x])]*Sqrt[-b*(1-Csc[c+d*x])/(a+b*Csc[c+d*x])]* + EllipticPi[a/(a+b),ArcSin[Rt[a+b,2]/Sqrt[a+b*Csc[c+d*x]]],(a-b)/(a+b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^(3/2),x_Symbol] := + Int[(a^2+b*(2*a-b)*Csc[c+d*x])/Sqrt[a+b*Csc[c+d*x]],x] + + b^2*Int[Csc[c+d*x]*(1+Csc[c+d*x])/Sqrt[a+b*Csc[c+d*x]],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -b^2*Cot[c+d*x]*(a+b*Csc[c+d*x])^(n-2)/(d*(n-1)) + + 1/(n-1)*Int[(a+b*Csc[c+d*x])^(n-3)* + Simp[a^3*(n-1)+(b*(b^2*(n-2)+3*a^2*(n-1)))*Csc[c+d*x]+(a*b^2*(3*n-4))*Csc[c+d*x]^2,x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && RationalQ[n] && n>2 && IntegerQ[2*n] + + +Int[1/(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + x/a - 1/a*Int[1/(1+a/b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[1/Sqrt[a_+b_.*csc[c_.+d_.*x_]],x_Symbol] := + 2*Rt[a+b,2]/(a*d*Cot[c+d*x])*Sqrt[b*(1-Csc[c+d*x])/(a+b)]*Sqrt[-b*(1+Csc[c+d*x])/(a-b)]* + EllipticPi[(a+b)/a,ArcSin[Sqrt[a+b*Csc[c+d*x]]/Rt[a+b,2]],(a+b)/(a-b)] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + b^2*Cot[c+d*x]*(a+b*Csc[c+d*x])^(n+1)/(a*d*(n+1)*(a^2-b^2)) + + 1/(a*(n+1)*(a^2-b^2))*Int[(a+b*Csc[c+d*x])^(n+1)*Simp[(a^2-b^2)*(n+1)-a*b*(n+1)*Csc[c+d*x]+b^2*(n+2)*Csc[c+d*x]^2,x],x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && RationalQ[n] && n<-1 && IntegerQ[2*n] + + +Int[(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Defer[Int][(a+b*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[a^2-b^2] && Not[IntegerQ[2*n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.1.2 (d sec)^n (a+b sec)^m*) + + +Int[(a_+b_.*csc[e_.+f_.*x_])*(d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + a*Int[(d*Csc[e+f*x])^n,x] + b/d*Int[(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,n},x] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^2*(d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + 2*a*b/d*Int[(d*Csc[e+f*x])^(n+1),x] + Int[(d*Csc[e+f*x])^n*(a^2+b^2*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,n},x] + + +Int[csc[e_.+f_.*x_]^2/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + 1/b*Int[Csc[e+f*x],x] - a/b*Int[Csc[e+f*x]/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] + + +Int[csc[e_.+f_.*x_]^3/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + -Cot[e+f*x]/(b*f) - a/b*Int[Csc[e+f*x]^2/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandTrig[(a+b*csc[e+f*x])^m*(d*csc[e+f*x])^n,x],x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && PositiveIntegerQ[m] && RationalQ[n] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*b*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]) /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -b*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)/(f*m) + a*(2*m-1)/m*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m-1),x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m>1/2 && IntegerQ[2*m] + + +Int[csc[e_.+f_.*x_]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + -Cot[e+f*x]/(f*(b+a*Csc[e+f*x])) /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2/f*Subst[Int[1/(2*a+x^2),x],x,b*Cot[e+f*x]/Sqrt[a+b*Csc[e+f*x]]] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + b*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(a*f*(2*m+1)) + (m+1)/(a*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 && IntegerQ[2*m] + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(2*m+1)) + + m/(b*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(m+1)) + + a*m/(b*(m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m,x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && Not[RationalQ[m] && m<-1/2] + + +Int[csc[e_.+f_.*x_]^3*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + b*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(a*f*(2*m+1)) - + 1/(a^2*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(a*m-b*(2*m+1)*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 + + +Int[csc[e_.+f_.*x_]^3*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m*(b*(m+1)-a*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f,m},x] && EqQ[a^2-b^2] && Not[RationalQ[m] && m<-1/2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*a/(b*f)*Sqrt[a*d/b]*Subst[Int[1/Sqrt[1+x^2/a],x],x,b*Cot[e+f*x]/Sqrt[a+b*Csc[e+f*x]]] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && PositiveQ[a*d/b] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*b*d/f*Subst[Int[1/(b-d*x^2),x],x,b*Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[d*Csc[e+f*x]])] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && Not[PositiveQ[a*d/b]] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -2*b*d*Cot[e+f*x]*(d*Csc[e+f*x])^(n-1)/(f*(2*n-1)*Sqrt[a+b*Csc[e+f*x]]) + + 2*a*d*(n-1)/(b*(2*n-1))*Int[Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n>1 && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*a*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[d*Csc[e+f*x]]) /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a*Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*n*Sqrt[a+b*Csc[e+f*x]]) + + a*(2*n+1)/(2*b*d*n)*Int[Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n<-1/2 && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a^2*d*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]])*Subst[Int[(d*x)^(n-1)/Sqrt[a-b*x],x],x,Csc[e+f*x]] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] + + +Int[Sqrt[d_.*csc[e_.+f_.*x_]]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -Sqrt[2]*Sqrt[a]/(b*f)*Subst[Int[1/Sqrt[1+x^2],x],x,b*Cot[e+f*x]/(a+b*Csc[e+f*x])] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && EqQ[d-a/b] && PositiveQ[a] + + +Int[Sqrt[d_.*csc[e_.+f_.*x_]]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*b*d/(a*f)*Subst[Int[1/(2*b-d*x^2),x],x,b*Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[d*Csc[e+f*x]])] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -a*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n/(f*m) + + b*(2*m-1)/(d*m)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && EqQ[m+n] && RationalQ[m] && m>1/2 && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)/(a*f*(2*m+1)) + + d*(m+1)/(b*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && EqQ[m+n] && RationalQ[m] && m<-1/2 && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*(2*m+1)) + + m/(a*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[m,n] && m+n+1==0 && m<-1/2 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*(m+1)) + + a*m/(b*d*(m+1))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && EqQ[m+n+1] && Not[RationalQ[m] && m<-1/2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + b^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n/(f*n) - + a/(d*n)*Int[(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^(n+1)*(b*(m-2*n-2)-a*(m+2*n-1)*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[m,n] && m>1 && (n<-1 || m==3/2 && n==-1/2) && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n/(f*(m+n-1)) + + b/(m+n-1)*Int[(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n*(b*(m+2*n-1)+a*(3*m+2*n-4)*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && RationalQ[m] && m>1 && NeQ[m+n-1] && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)/(a*f*(2*m+1)) - + d/(a*b*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)*(a*(n-1)-b*(m+n)*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[m,n] && m<-1 && 12 && (IntegersQ[2*m,2*n] || IntegerQ[m]) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*(2*m+1)) + + 1/(a^2*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n*(a*(2*m+n+1)-b*(m+n+1)*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] && RationalQ[m] && m<-1 && (IntegersQ[2*m,2*n] || IntegerQ[m]) + + +Int[(d_.*csc[e_.+f_.*x_])^n_/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + d^2*Cot[e+f*x]*(d*Csc[e+f*x])^(n-2)/(f*(a+b*Csc[e+f*x])) - + d^2/(a*b)*Int[(d*Csc[e+f*x])^(n-2)*(b*(n-2)-a*(n-1)*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n>1 + + +Int[(d_.*csc[e_.+f_.*x_])^n_/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*(a+b*Csc[e+f*x])) - + 1/a^2*Int[(d*Csc[e+f*x])^n*(a*(n-1)-b*n*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n<0 + + +Int[(d_.*csc[e_.+f_.*x_])^n_/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*d*Cot[e+f*x]*(d*Csc[e+f*x])^(n-1)/(a*f*(a+b*Csc[e+f*x])) + + d*(n-1)/(a*b)*Int[(d*Csc[e+f*x])^(n-1)*(a-b*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,n},x] && EqQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^(3/2)/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + d/b*Int[Sqrt[a+b*Csc[e+f*x]]*Sqrt[d*Csc[e+f*x]],x] - + a*d/b*Int[Sqrt[d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*d^2*Cot[e+f*x]*(d*Csc[e+f*x])^(n-2)/(f*(2*n-3)*Sqrt[a+b*Csc[e+f*x]]) + + d^2/(b*(2*n-3))*Int[(d*Csc[e+f*x])^(n-2)*(2*b*(n-2)-a*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n>2 && IntegerQ[2*n] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*n*Sqrt[a+b*Csc[e+f*x]]) + + 1/(2*b*d*n)*Int[(d*Csc[e+f*x])^(n+1)*(a+b*(2*n+1)*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && EqQ[a^2-b^2] && RationalQ[n] && n<0 && IntegerQ[2*n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -d^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-2)/(f*(m+n-1)) + + d^2/(b*(m+n-1))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-2)*(b*(n-2)+a*m*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2-b^2] && RationalQ[n] && n>2 && NeQ[m+n-1] && IntegerQ[n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -(a*d/b)^n*Cot[e+f*x]/(a^(n-2)*f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]])* + Subst[Int[(a-x)^(n-1)*(2*a-x)^(m-1/2)/Sqrt[x],x],x,a-b*Csc[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && PositiveQ[a] && Not[IntegerQ[n]] && PositiveQ[a*d/b] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -(-a*d/b)^n*Cot[e+f*x]/(a^(n-1)*f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]])* + Subst[Int[x^(m-1/2)*(a-x)^(n-1)/Sqrt[2*a-x],x],x,a+b*Csc[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && PositiveQ[a] && Not[IntegerQ[n]] && NegativeQ[a*d/b] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + a^2*d*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]])* + Subst[Int[(d*x)^(n-1)*(a+b*x)^(m-1/2)/Sqrt[a-b*x],x],x,Csc[e+f*x]] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && PositiveQ[a] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + a^IntPart[m]*(a+b*Csc[e+f*x])^FracPart[m]/(1+b/a*Csc[e+f*x])^FracPart[m]*Int[(1+b/a*Csc[e+f*x])^m*(d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[m]] && Not[PositiveQ[a]] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + (a-b)*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] + b*Int[Csc[e+f*x]*(1+Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -b*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)/(f*m) + + 1/m*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m-2)*(b^2*(m-1)+a^2*m+a*b*(2*m-1)*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m>1 && IntegerQ[2*m] + + +(* Int[csc[e_.+f_.*x_]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + -2/f*Subst[Int[1/(a+b-(a-b)*x^2),x],x,Cot[e+f*x]/(1+Csc[e+f*x])] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] *) + + +Int[csc[e_.+f_.*x_]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + 1/b*Int[1/(1+a/b*Sin[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*Rt[a+b,2]/(b*f*Cot[e+f*x])*Sqrt[(b*(1-Csc[e+f*x]))/(a+b)]*Sqrt[-b*(1+Csc[e+f*x])/(a-b)]* + EllipticF[ArcSin[Sqrt[a+b*Csc[e+f*x]]/Rt[a+b,2]],(a+b)/(a-b)] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -b*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(a*(m+1)-b*(m+2)*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegerQ[2*m] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + Cot[e+f*x]/(f*Sqrt[1+Csc[e+f*x]]*Sqrt[1-Csc[e+f*x]])*Subst[Int[(a+b*x)^m/(Sqrt[1+x]*Sqrt[1-x]),x],x,Csc[e+f*x]] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && Not[IntegerQ[2*m]] + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(m+1)) + + m/(m+1)*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(b+a*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m>0 + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + a*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(f*(m+1)*(a^2-b^2)) - + 1/((m+1)*(a^2-b^2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(b*(m+1)-a*(m+2)*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[csc[e_.+f_.*x_]^2/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] + + Int[Csc[e+f*x]*(1+Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -a/b*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m,x] + 1/b*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1),x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]^3*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -a^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(b*f*(m+1)*(a^2-b^2)) + + 1/(b*(m+1)*(a^2-b^2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*Simp[a*b*(m+1)-(a^2+b^2*(m+1))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[csc[e_.+f_.*x_]^3*(a_+b_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + -Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m*(b*(m+1)-a*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2] && Not[RationalQ[m] && m<-1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n/(f*n) - + 1/(d*n)*Int[(a+b*Csc[e+f*x])^(m-3)*(d*Csc[e+f*x])^(n+1)* + Simp[a^2*b*(m-2*n-2)-a*(3*b^2*n+a^2*(n+1))*Csc[e+f*x]-b*(b^2*n+a^2*(m+n-1))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[m,n] && m>2 && (IntegerQ[m] && n<-1 || IntegersQ[m+1/2,2*n] && n<=-1) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n/(f*(m+n-1)) + + 1/(d*(m+n-1))*Int[(a+b*Csc[e+f*x])^(m-3)*(d*Csc[e+f*x])^n* + Simp[a^3*d*(m+n-1)+a*b^2*d*n+b*(b^2*d*(m+n-2)+3*a^2*d*(m+n-1))*Csc[e+f*x]+a*b^2*d*(3*m+2*n-4)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m>2 && + (IntegerQ[m] || IntegersQ[2*m,2*n]) && Not[IntegerQ[n] && n>2 && Not[IntegerQ[m]]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)* + Simp[b*d*(n-1)+a*d*(m+1)*Csc[e+f*x]-b*d*(m+n+1)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[m,n] && m<-1 && 03 || IntegersQ[n+1/2,2*m] && n>2) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*n) - + 1/(a*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1)* + Simp[b*(m+n+1)-a*(n+1)*Csc[e+f*x]-b*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && NegativeIntegerQ[m+1/2,n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + b^2*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*(m+1)*(a^2-b^2)) + + 1/(a*(m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n* + (a^2*(m+1)-b^2*(m+n+1)-a*b*(m+1)*Csc[e+f*x]+b^2*(m+n+2)*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && IntegersQ[2*m,2*n] + + +Int[Sqrt[d_.*csc[e_.+f_.*x_]]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + Sqrt[d*Sin[e+f*x]]*Sqrt[d*Csc[e+f*x]]/d*Int[Sqrt[d*Sin[e+f*x]]/(b+a*Sin[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^(3/2)/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + d*Sqrt[d*Sin[e+f*x]]*Sqrt[d*Csc[e+f*x]]*Int[1/(Sqrt[d*Sin[e+f*x]]*(b+a*Sin[e+f*x])),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^(5/2)/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + d/b*Int[(d*Csc[e+f*x])^(3/2),x] - a*d/b*Int[(d*Csc[e+f*x])^(3/2)/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + -d^3*Cot[e+f*x]*(d*Csc[e+f*x])^(n-3)/(b*f*(n-2)) + + d^3/(b*(n-2))*Int[(d*Csc[e+f*x])^(n-3)*Simp[a*(n-3)+b*(n-3)*Csc[e+f*x]-a*(n-2)*Csc[e+f*x]^2,x]/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n>3 + + +Int[1/(Sqrt[d_.*csc[e_.+f_.*x_]]*(a_+b_.*csc[e_.+f_.*x_])),x_Symbol] := + b^2/(a^2*d^2)*Int[(d*Csc[e+f*x])^(3/2)/(a+b*Csc[e+f*x]),x] + + 1/a^2*Int[(a-b*Csc[e+f*x])/Sqrt[d*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + Cot[e+f*x]*(d*Csc[e+f*x])^n/(a*f*n) - + 1/(a*d*n)*Int[(d*Csc[e+f*x])^(n+1)/(a+b*Csc[e+f*x])* + Simp[b*n-a*(n+1)*Csc[e+f*x]-b*(n+1)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + a*Int[Sqrt[d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] + + b/d*Int[(d*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -2*d*Cos[e+f*x]*Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^(n-1)/(f*(2*n-1)) + + d^2/(2*n-1)*Int[(d*Csc[e+f*x])^(n-2)*Simp[2*a*(n-2)+b*(2*n-3)*Csc[e+f*x]+a*Csc[e+f*x]^2,x]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n>1 && IntegerQ[2*n] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + Sqrt[a+b*Csc[e+f*x]]/(Sqrt[d*Csc[e+f*x]]*Sqrt[b+a*Sin[e+f*x]])*Int[Sqrt[b+a*Sin[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Cot[e+f*x]*Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^n/(f*n) - + 1/(2*d*n)*Int[(d*Csc[e+f*x])^(n+1)*Simp[b-2*a*(n+1)*Csc[e+f*x]-b*(2*n+3)*Csc[e+f*x]^2,x]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 && IntegerQ[2*n] + + +Int[Sqrt[d_.*csc[e_.+f_.*x_]]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + Sqrt[d*Csc[e+f*x]]*Sqrt[b+a*Sin[e+f*x]]/Sqrt[a+b*Csc[e+f*x]]*Int[1/Sqrt[b+a*Sin[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^(3/2)/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + d*Sqrt[d*Csc[e+f*x]]*Sqrt[b+a*Sin[e+f*x]]/Sqrt[a+b*Csc[e+f*x]]*Int[1/(Sin[e+f*x]*Sqrt[b+a*Sin[e+f*x]]),x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*d^2*Cos[e+f*x]*(d*Csc[e+f*x])^(n-2)*Sqrt[a+b*Csc[e+f*x]]/(b*f*(2*n-3)) + + d^3/(b*(2*n-3))*Int[(d*Csc[e+f*x])^(n-3)/Sqrt[a+b*Csc[e+f*x]]* + Simp[2*a*(n-3)+b*(2*n-5)*Csc[e+f*x]-2*a*(n-2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n>2 && IntegerQ[2*n] + + +Int[1/(csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]),x_Symbol] := + -Cos[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(a*f) - b/(2*a)*Int[(1+Csc[e+f*x]^2)/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2] + + +Int[1/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[d_.*csc[e_.+f_.*x_]]),x_Symbol] := + 1/a*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[d*Csc[e+f*x]],x] - + b/(a*d)*Int[Sqrt[d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + Cos[e+f*x]*(d*Csc[e+f*x])^(n+1)*Sqrt[a+b*Csc[e+f*x]]/(a*d*f*n) + + 1/(2*a*d*n)*Int[(d*Csc[e+f*x])^(n+1)/Sqrt[a+b*Csc[e+f*x]]* + Simp[-b*(2*n+1)+2*a*(n+1)*Csc[e+f*x]+b*(2*n+3)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n<-1 && IntegerQ[2*n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a*Cot[e+f*x]*Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^n/(f*n) + + 1/(2*d*n)*Int[(d*Csc[e+f*x])^(n+1)/Sqrt[a+b*Csc[e+f*x]]* + Simp[a*b*(2*n-1)+2*(b^2*n+a^2*(n+1))*Csc[e+f*x]+a*b*(2*n+3)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 && IntegersQ[2*n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -d^3*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-3)/(b*f*(m+n-1)) + + d^3/(b*(m+n-1))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-3)* + Simp[a*(n-3)+b*(m+n-2)*Csc[e+f*x]-a*(n-2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,m},x] && NeQ[a^2-b^2] && RationalQ[n] && n>3 && + (IntegerQ[n] || IntegersQ[2*m,2*n]) && Not[IntegerQ[m] && m>2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n-1)/(f*(m+n-1)) + + d/(m+n-1)*Int[(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^(n-1)* + Simp[a*b*(n-1)+(b^2*(m+n-2)+a^2*(m+n-1))*Csc[e+f*x]+a*b*(2*m+n-2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2] && RationalQ[m,n] && 01 + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + -(e*Cot[c+d*x])^(m+1)*(a+b*Csc[c+d*x])/(d*e*(m+1)) - + 1/(e^2*(m+1))*Int[(e*Cot[c+d*x])^(m+2)*(a*(m+1)+b*(m+2)*Csc[c+d*x]),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m<-1 + + +Int[(a_+b_.*csc[c_.+d_.*x_])/cot[c_.+d_.*x_],x_Symbol] := + Int[(b+a*Sin[c+d*x])/Cos[c+d*x],x] /; +FreeQ[{a,b,c,d},x] + + +Int[(e_.*cot[c_.+d_.*x_])^m_.*(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + a*Int[(e*Cot[c+d*x])^m,x] + b*Int[(e*Cot[c+d*x])^m*Csc[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m},x] + + +Int[cot[c_.+d_.*x_]^m_.*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -(-1)^((m-1)/2)/(d*b^(m-1))*Subst[Int[(b^2-x^2)^((m-1)/2)*(a+x)^n/x,x],x,b*Csc[c+d*x]] /; +FreeQ[{a,b,c,d,n},x] && IntegerQ[(m-1)/2] && NeQ[a^2-b^2] + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(e*Cot[c+d*x])^m,(a+b*Csc[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && PositiveIntegerQ[n] + + +Int[cot[c_.+d_.*x_]^m_.*(a_+b_.*csc[c_.+d_.*x_])^n_.,x_Symbol] := + -2*a^(m/2+n+1/2)/d*Subst[Int[x^m*(2+a*x^2)^(m/2+n-1/2)/(1+a*x^2),x],x,Cot[c+d*x]/Sqrt[a+b*Csc[c+d*x]]] /; +FreeQ[{a,b,c,d},x] && EqQ[a^2-b^2] && IntegerQ[m/2] && IntegerQ[n-1/2] + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + a^(2*n)*e^(-2*n)*Int[(e*Cot[c+d*x])^(m+2*n)/(-a+b*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[a^2-b^2] && NegativeIntegerQ[n] + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + -2^(m+n+1)*(e*Cot[c+d*x])^(m+1)*(a+b*Csc[c+d*x])^n/(d*e*(m+1))*(a/(a+b*Csc[c+d*x]))^(m+n+1)* + AppellF1[(m+1)/2,m+n,1,(m+3)/2,-(a-b*Csc[c+d*x])/(a+b*Csc[c+d*x]),(a-b*Csc[c+d*x])/(a+b*Csc[c+d*x])] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[a^2-b^2] && Not[IntegerQ[n]] + + +Int[Sqrt[e_.*cot[c_.+d_.*x_]]/(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + 1/a*Int[Sqrt[e*Cot[c+d*x]],x] - b/a*Int[Sqrt[e*Cot[c+d*x]]/(b+a*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2] + + +Int[(e_.*cot[c_.+d_.*x_])^m_/(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + -e^2/b^2*Int[(e*Cot[c+d*x])^(m-2)*(a-b*Csc[c+d*x]),x] + + e^2*(a^2-b^2)/b^2*Int[(e*Cot[c+d*x])^(m-2)/(a+b*Csc[c+d*x]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m-1/2] + + +Int[1/(Sqrt[e_.*cot[c_.+d_.*x_]]*(a_+b_.*csc[c_.+d_.*x_])),x_Symbol] := + 1/a*Int[1/Sqrt[e*Cot[c+d*x]],x] - b/a*Int[1/(Sqrt[e*Cot[c+d*x]]*(b+a*Sin[c+d*x])),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2] + + +Int[(e_.*cot[c_.+d_.*x_])^m_/(a_+b_.*csc[c_.+d_.*x_]),x_Symbol] := + 1/(a^2-b^2)*Int[(e*Cot[c+d*x])^m*(a-b*Csc[c+d*x]),x] + + b^2/(e^2*(a^2-b^2))*Int[(e*Cot[c+d*x])^(m+2)/(a+b*Csc[c+d*x]),x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[a^2-b^2] && NegativeIntegerQ[m+1/2] + + +Int[cot[c_.+d_.*x_]^2*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Int[(-1+Csc[c+d*x]^2)*(a+b*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(e*Cot[c+d*x])^m,(a+b*Csc[c+d*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[a^2-b^2] && PositiveIntegerQ[n] + + +Int[cot[c_.+d_.*x_]^m_.*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Int[Cos[c+d*x]^m*(b+a*Sin[c+d*x])^n/Sin[c+d*x]^(m+n),x] /; +FreeQ[{a,b,c,d},x] && NeQ[a^2-b^2] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || m<=1) + + +(* Int[cot[c_.+d_.*x_]^m_*(a_+b_.*csc[c_.+d_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(a+b*Csc[c+d*x])^n,(-1+Csc[c+d*x]^2)^(m/2),x],x] /; +FreeQ[{a,b,c,d,n},x] && NeQ[a^2-b^2] && PositiveIntegerQ[m/2] *) + + +Int[(e_.*cot[c_.+d_.*x_])^m_.*(a_.+b_.*csc[c_.+d_.*x_])^n_.,x_Symbol] := + Defer[Int][(e*Cot[c+d*x])^m*(a+b*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + +Int[(e_.*cot[c_.+d_.*x_])^m_*(a_+b_.*sec[c_.+d_.*x_])^n_.,x_Symbol] := + (e*Cot[c+d*x])^m*Tan[c+d*x]^m*Int[(a+b*Sec[c+d*x])^n/Tan[c+d*x]^m,x] /; +FreeQ[{a,b,c,d,e,m,n},x] && Not[IntegerQ[m]] + + +Int[(e_.*tan[c_.+d_.*x_]^p_)^m_*(a_+b_.*sec[c_.+d_.*x_])^n_.,x_Symbol] := + (e*Tan[c+d*x]^p)^m/(e*Tan[c+d*x])^(m*p)*Int[(e*Tan[c+d*x])^(m*p)*(a+b*Sec[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && Not[IntegerQ[m]] + + +Int[(e_.*cot[c_.+d_.*x_]^p_)^m_*(a_+b_.*csc[c_.+d_.*x_])^n_.,x_Symbol] := + (e*Cot[c+d*x]^p)^m/(e*Cot[c+d*x])^(m*p)*Int[(e*Cot[c+d*x])^(m*p)*(a+b*Csc[c+d*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.2.1 (a+b sec)^m (c+d sec)^n*) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + c^n*Int[ExpandTrig[(1+d/c*csc[e+f*x])^n,(a+b*csc[e+f*x])^m,x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && PositiveIntegerQ[m] && NegativeIntegerQ[n] && m+n<2 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + (-a*c)^m*Int[Cot[e+f*x]^(2*m)*(c+d*Csc[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m] && RationalQ[n] && Not[IntegerQ[n] && m-n>0] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + (-a*c)^(m+1/2)*Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])*Int[Cot[e+f*x]^(2*m),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m+1/2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + 2*a*c*Cot[e+f*x]*(c+d*Csc[e+f*x])^(n-1)/(f*(2*n-1)*Sqrt[a+b*Csc[e+f*x]]) + + c*Int[Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[n] && n>1/2 + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -2*a*Cot[e+f*x]*(c+d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) + + 1/c*Int[Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[n] && n<-1/2 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + -4*a^2*Cot[e+f*x]*(c+d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) + + a/c*Int[Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[n] && n<-1/2 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + -2*a^2*Cot[e+f*x]*(c+d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) + + a*Int[Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && Not[RationalQ[n] && n<=-1/2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(5/2)*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + -8*a^3*Cot[e+f*x]*(c+d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) + + a^2/c^2*Int[Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^(n+2),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && RationalQ[n] && n<-1/2 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -a*c*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])* + Subst[Int[(b+a*x)^(m-1/2)*(d+c*x)^(n-1/2)/x^(m+n),x],x,Sin[e+f*x]] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m-1/2] && EqQ[m+n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + a*c*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])*Subst[Int[(a+b*x)^(m-1/2)*(c+d*x)^(n-1/2)/x,x],x,Csc[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a*c*x + b*d*Int[Csc[e+f*x]^2,x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] + + +Int[(a_+b_.*csc[e_.+f_.*x_])*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a*c*x + (b*c+a*d)*Int[Csc[e+f*x],x] + b*d*Int[Csc[e+f*x]^2,x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[b*c+a*d] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + c*Int[Sqrt[a+b*Csc[e+f*x]],x] + d*Int[Sqrt[a+b*Csc[e+f*x]]*Csc[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a*c*Int[1/Sqrt[a+b*Csc[e+f*x]],x] + + Int[Csc[e+f*x]*(b*c+a*d+b*d*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)/(f*m) + + 1/m*Int[(a+b*Csc[e+f*x])^(m-1)*Simp[a*c*m+(b*c*m+a*d*(2*m-1))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m>1 && EqQ[a^2-b^2] && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)/(f*m) + + 1/m*Int[(a+b*Csc[e+f*x])^(m-2)* + Simp[a^2*c*m+(b^2*d*(m-1)+2*a*b*c*m+a^2*d*m)*Csc[e+f*x]+b*(b*c*m+a*d*(2*m-1))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m>1 && NeQ[a^2-b^2] && IntegerQ[2*m] + + +Int[(c_+d_.*csc[e_.+f_.*x_])/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + c*x/a - (b*c-a*d)/a*Int[Csc[e+f*x]/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[(c_+d_.*csc[e_.+f_.*x_])/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + c/a*Int[Sqrt[a+b*Csc[e+f*x]],x] - (b*c-a*d)/a*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[(c_+d_.*csc[e_.+f_.*x_])/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + c*Int[1/Sqrt[a+b*Csc[e+f*x]],x] + d*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -(b*c-a*d)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(b*f*(2*m+1)) + + 1/(a^2*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*Simp[a*c*(2*m+1)-(b*c-a*d)*(m+1)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m<-1 && EqQ[a^2-b^2] && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + b*(b*c-a*d)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(a*f*(m+1)*(a^2-b^2)) + + 1/(a*(m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)* + Simp[c*(a^2-b^2)*(m+1)-(a*(b*c-a*d)*(m+1))*Csc[e+f*x]+b*(b*c-a*d)*(m+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && RationalQ[m] && m<-1 && NeQ[a^2-b^2] && IntegerQ[2*m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + c*Int[(a+b*Csc[e+f*x])^m,x] + d*Int[(a+b*Csc[e+f*x])^m*Csc[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[b*c-a*d] && Not[IntegerQ[2*m]] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + 1/c*Int[Sqrt[a+b*Csc[e+f*x]],x] - d/c*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a/c*Int[1/Sqrt[a+b*Csc[e+f*x]],x] + (b*c-a*d)/c*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a/c*Int[Sqrt[a+b*Csc[e+f*x]],x] + (b*c-a*d)/c*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +(* Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + b/d*Int[Sqrt[a+b*Csc[e+f*x]],x] - (b*c-a*d)/d*Int[Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] *) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + 1/(c*d)*Int[(a^2*d+b^2*c*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] - + (b*c-a*d)^2/(c*d)*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + 1/(c*(b*c-a*d))*Int[(b*c-a*d-b*d*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] + + d^2/(c*(b*c-a*d))*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[1/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + 1/c*Int[1/Sqrt[a+b*Csc[e+f*x]],x] - d/c*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]/Cot[e+f*x]*Int[Cot[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + c*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] + + d*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + 1/c*Int[Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]],x] - + d/c*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*a/f*Subst[Int[1/(1+a*c*x^2),x],x,Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + a/c*Int[Sqrt[c+d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] + + (b*c-a*d)/c*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + 2*(a+b*Csc[e+f*x])/(c*f*Rt[(a+b)/(c+d),2]*Cot[e+f*x])* + Sqrt[(b*c-a*d)*(1+Csc[e+f*x])/((c-d)*(a+b*Csc[e+f*x]))]* + Sqrt[-(b*c-a*d)*(1-Csc[e+f*x])/((c+d)*(a+b*Csc[e+f*x]))]* + EllipticPi[a*(c+d)/(c*(a+b)),ArcSin[Rt[(a+b)/(c+d),2]*Sqrt[c+d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]]],(a-b)*(c+d)/((a+b)*(c-d))] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[1/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]),x_Symbol] := + Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])*Int[1/Cot[e+f*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[1/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]),x_Symbol] := + 1/a*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] - + b/a*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_])^(3/2),x_Symbol] := + 1/c*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] - + d/c*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x])^(3/2),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[c^2-d^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + a^2*(Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]]))* + Subst[Int[(a+b*x)^(m-1/2)*(c+d*x)^n/(x*Sqrt[a-b*x]),x],x,Csc[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && IntegerQ[m-1/2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^(m+n),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && IntegerQ[m] && IntegerQ[n] && MemberQ[{0,-1,-2},m+n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Sqrt[d+c*Sin[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]/(Sqrt[b+a*Sin[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])* + Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^(m+n),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && IntegerQ[m+1/2] && IntegerQ[n+1/2] && MemberQ[{0,-1,-2},m+n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Sin[e+f*x]^(m+n)*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/((b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n)* + Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^Simplify[m+n],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && EqQ[m+n] && Not[IntegerQ[2*m]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(a+b*csc[e+f*x])^m,(c+d*csc[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && PositiveIntegerQ[n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Defer[Int][(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(d_./sec[e_.+f_.*x_])^n_,x_Symbol] := + d^m*Int[(b+a*Cos[e+f*x])^m*(d*Cos[e+f*x])^(n-m),x] /; +FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_.+b_.*csc[e_.+f_.*x_])^m_.*(d_./csc[e_.+f_.*x_])^n_,x_Symbol] := + d^m*Int[(b+a*Sin[e+f*x])^m*(d*Sin[e+f*x])^(n-m),x] /; +FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(c_.*(d_.*sec[e_.+f_.*x_])^p_)^n_,x_Symbol] := + c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* + Int[(a+b*Sec[e+f*x])^m*(d*Sec[e+f*x])^(n*p),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[n]] + + +Int[(a_.+b_.*csc[e_.+f_.*x_])^m_.*(c_.*(d_.*csc[e_.+f_.*x_])^p_)^n_,x_Symbol] := + c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* + Int[(a+b*Cos[e+f*x])^m*(d*Cos[e+f*x])^(n*p),x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n*) + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + b*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/(a*f*(2*m+1)) /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && EqQ[m+n+1] && NeQ[2*m+1] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + b*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/(a*f*(2*m+1)) + + (m+n+1)/(a*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && NegativeIntegerQ[m+n+1] && NeQ[2*m+1] && + Not[RationalQ[n] && n<0] && Not[PositiveIntegerQ[n+1/2] && n+1/2<-(m+n)] + + +Int[csc[e_.+f_.*x_]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + a*c*Log[1+b/a*Csc[e+f*x]]*Cot[e+f*x]/(b*f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]) /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_.*Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + 2*a*c*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(b*f*(2*m+1)*Sqrt[c+d*Csc[e+f*x]]) /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && NeQ[m+1/2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + 2*a*c*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^(n-1)/(b*f*(2*m+1)) - + d*(2*n-1)/(b*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(c+d*Csc[e+f*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && PositiveIntegerQ[n-1/2] && RationalQ[m] && m<-1/2 + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -d*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^(n-1)/(f*(m+n)) + + c*(2*n-1)/(m+n)*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && PositiveIntegerQ[n-1/2] && Not[RationalQ[m] && m<-1/2] && + Not[PositiveIntegerQ[m-1/2] && m=0 && m*n>0 + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + (-a*c)^(m+1/2)*Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])*Int[Csc[e+f*x]*Cot[e+f*x]^(2*m),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m+1/2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + b*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/(a*f*(2*m+1)) + + (m+n+1)/(a*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,n},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && (NegativeIntegerQ[m,n-1/2] || NegativeIntegerQ[m-1/2,n-1/2] && m=0 && m*n>0 + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^m_,x_Symbol] := + (-a*c)^(m+1/2)*Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])*Int[(g*Csc[e+f*x])^p*Cot[e+f*x]^(2*m),x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m+1/2] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a*c*g*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])* + Subst[Int[(g*x)^(p-1)*(a+b*x)^(m-1/2)*(c+d*x)^(n-1/2),x],x,Csc[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] + + +Int[Sqrt[g_.*csc[e_.+f_.*x_]]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -2*b*g/f*Subst[Int[1/(b*c+a*d-c*g*x^2),x],x,b*Cot[e+f*x]/(Sqrt[g*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[Sqrt[g_.*csc[e_.+f_.*x_]]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + a/c*Int[Sqrt[g*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] + + (b*c-a*d)/(c*g)*Int[(g*Csc[e+f*x])^(3/2)/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -2*b/f*Subst[Int[1/(b*c+a*d+d*x^2),x],x,b*Cot[e+f*x]/Sqrt[a+b*Csc[e+f*x]]] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + -Sqrt[a+b*Csc[e+f*x]]*Sqrt[c/(c+d*Csc[e+f*x])]/(d*f*Sqrt[c*d*(a+b*Csc[e+f*x])/((b*c+a*d)*(c+d*Csc[e+f*x]))])* + EllipticE[ArcSin[c*Cot[e+f*x]/(c+d*Csc[e+f*x])],-(b*c-a*d)/(b*c+a*d)] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + b/d*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] - + (b*c-a*d)/d*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(3/2)*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + g/d*Int[Sqrt[g*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]],x] - + c*g/d*Int[Sqrt[g*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(3/2)*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_]),x_Symbol] := + b/d*Int[(g*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] - + (b*c-a*d)/d*Int[(g*Csc[e+f*x])^(3/2)/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + b/(b*c-a*d)*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] - + d/(b*c-a*d)*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[csc[e_.+f_.*x_]/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + -2*Cot[e+f*x]/(f*(c+d)*Sqrt[a+b*Csc[e+f*x]]*Sqrt[-Cot[e+f*x]^2])*Sqrt[(a+b*Csc[e+f*x])/(a+b)]* + EllipticPi[2*d/(c+d),ArcSin[Sqrt[1-Csc[e+f*x]]/Sqrt[2]],2*b/(a+b)] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(3/2)/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + -a*g/(b*c-a*d)*Int[Sqrt[g*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] + + c*g/(b*c-a*d)*Int[Sqrt[g*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(3/2)/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + g*Sqrt[g*Csc[e+f*x]]*Sqrt[b+a*Sin[e+f*x]]/Sqrt[a+b*Csc[e+f*x]]*Int[1/(Sqrt[b+a*Sin[e+f*x]]*(d+c*Sin[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]^2/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + -a/(b*c-a*d)*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] + + c/(b*c-a*d)*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && (EqQ[a^2-b^2] || EqQ[c^2-d^2]) + + +Int[csc[e_.+f_.*x_]^2/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + 1/d*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] - + c/d*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(5/2)/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + -c^2*g^2/(d*(b*c-a*d))*Int[Sqrt[g*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] + + g^2/(d*(b*c-a*d))*Int[Sqrt[g*Csc[e+f*x]]*(a*c+(b*c-a*d)*Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] + + +Int[(g_.*csc[e_.+f_.*x_])^(5/2)/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])),x_Symbol] := + g/d*Int[(g*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] - + c*g/d*Int[(g*Csc[e+f*x])^(3/2)/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*b/f*Subst[Int[1/(1-b*d*x^2),x],x,Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + -(b*c-a*d)/d*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]),x] + + b/d*Int[Csc[e+f*x]*Sqrt[c+d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && EqQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/Sqrt[c_+d_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*(a+b*Csc[e+f*x])/(d*f*Sqrt[(a+b)/(c+d)]*Cot[e+f*x])* + Sqrt[-(b*c-a*d)*(1-Csc[e+f*x])/((c+d)*(a+b*Csc[e+f*x]))]*Sqrt[(b*c-a*d)*(1+Csc[e+f*x])/((c-d)*(a+b*Csc[e+f*x]))]* + EllipticPi[b*(c+d)/(d*(a+b)),ArcSin[Sqrt[(a+b)/(c+d)]*Sqrt[c+d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]]],(a-b)*(c+d)/((a+b)*(c-d))] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]),x_Symbol] := + -2*a/(b*f)*Subst[Int[1/(2+(a*c-b*d)*x^2),x],x,Cot[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]),x_Symbol] := + -2*(c+d*Csc[e+f*x])/(f*(b*c-a*d)*Rt[(c+d)/(a+b),2]*Cot[e+f*x])* + Sqrt[(b*c-a*d)*(1-Csc[e+f*x])/((a+b)*(c+d*Csc[e+f*x]))]*Sqrt[-(b*c-a*d)*(1+Csc[e+f*x])/((a-b)*(c+d*Csc[e+f*x]))]* + EllipticF[ArcSin[Rt[(c+d)/(a+b),2]*(Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]])],(a+b)*(c-d)/((a-b)*(c+d))] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[csc[e_.+f_.*x_]^2/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*Sqrt[c_+d_.*csc[e_.+f_.*x_]]),x_Symbol] := + -a/b*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]),x] + + 1/b*Int[Csc[e+f*x]*Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] + + +Int[csc[e_.+f_.*x_]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]/(c_+d_.*csc[e_.+f_.*x_])^(3/2),x_Symbol] := + (a-b)/(c-d)*Int[Csc[e+f*x]/(Sqrt[a+b*Csc[e+f*x]]*Sqrt[c+d*Csc[e+f*x]]),x] + + (b*c-a*d)/(c-d)*Int[Csc[e+f*x]*(1+Csc[e+f*x])/(Sqrt[a+b*Csc[e+f*x]]*(c+d*Csc[e+f*x])^(3/2)),x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + a^2*g*Cot[e+f*x]/(f*Sqrt[a+b*Csc[e+f*x]]*Sqrt[a-b*Csc[e+f*x]])* + Subst[Int[(g*x)^(p-1)*(a+b*x)^(m-1/2)*(c+d*x)^n/Sqrt[a-b*x],x],x,Csc[e+f*x]] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && EqQ[a^2-b^2] && NeQ[c^2-d^2] && (EqQ[p-1] || IntegerQ[m-1/2]) + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + 1/g^(m+n)*Int[(g*Csc[e+f*x])^(m+n+p)*(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && NeQ[b*c-a*d] && IntegerQ[m] && IntegerQ[n] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + (g*Csc[e+f*x])^(m+p)*(c+d*Csc[e+f*x])^n/(g^m*(d+c*Sin[e+f*x])^n)*Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && NeQ[b*c-a*d] && EqQ[m+n+p] && IntegerQ[m] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + (g*Csc[e+f*x])^p*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n/((b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n)* + Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && EqQ[m+n+p] && Not[IntegerQ[m]] + + +Int[csc[e_.+f_.*x_]^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Sqrt[d+c*Sin[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]/(Sqrt[b+a*Sin[e+f*x]]*Sqrt[c+d*Csc[e+f*x]])* + Int[(b+a*Sin[e+f*x])^m*(d+c*Sin[e+f*x])^n/Sin[e+f*x]^(m+n+p),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && NeQ[b*c-a*d] && IntegerQ[m-1/2] && IntegerQ[n-1/2] && IntegerQ[p] && MemberQ[{-1,-2},m+n+p] + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*csc[e_.+f_.*x_])^m_*(c_+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Int[ExpandTrig[(g*csc[e+f*x])^p*(a+b*csc[e+f*x])^m*(c+d*csc[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && NeQ[b*c-a*d] && (IntegersQ[m,n] || IntegersQ[m,p] || IntegersQ[n,p]) + + +Int[(g_.*csc[e_.+f_.*x_])^p_.*(a_.+b_.*csc[e_.+f_.*x_])^m_*(c_.+d_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + Defer[Int][(g*Csc[e+f*x])^p*(a+b*Csc[e+f*x])^m*(c+d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +Int[sec[e_.+f_.*x_]*(A_+B_.*sec[e_.+f_.*x_])/(Sqrt[a_+b_.*sec[e_.+f_.*x_]]*(c_+d_.*sec[e_.+f_.*x_])^(3/2)),x_Symbol] := + 2*A*(1+Sec[e+f*x])*Sqrt[(b*c-a*d)*(1-Sec[e+f*x])/((a+b)*(c+d*Sec[e+f*x]))]/ + (f*(b*c-a*d)*Rt[(c+d)/(a+b),2]*Tan[e+f*x]*Sqrt[-(b*c-a*d)*(1+Sec[e+f*x])/((a-b)*(c+d*Sec[e+f*x]))])* + EllipticE[ArcSin[Rt[(c+d)/(a+b),2]*Sqrt[a+b*Sec[e+f*x]]/Sqrt[c+d*Sec[e+f*x]]],(a+b)*(c-d)/((a-b)*(c+d))] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && EqQ[A-B] + + +Int[csc[e_.+f_.*x_]*(A_+B_.*csc[e_.+f_.*x_])/(Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(c_+d_.*csc[e_.+f_.*x_])^(3/2)),x_Symbol] := + -2*A*(1+Csc[e+f*x])*Sqrt[(b*c-a*d)*(1-Csc[e+f*x])/((a+b)*(c+d*Csc[e+f*x]))]/ + (f*(b*c-a*d)*Rt[(c+d)/(a+b),2]*Cot[e+f*x]*Sqrt[-(b*c-a*d)*(1+Csc[e+f*x])/((a-b)*(c+d*Csc[e+f*x]))])* + EllipticE[ArcSin[Rt[(c+d)/(a+b),2]*Sqrt[a+b*Csc[e+f*x]]/Sqrt[c+d*Csc[e+f*x]]],(a+b)*(c-d)/((a-b)*(c+d))] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d] && NeQ[a^2-b^2] && NeQ[c^2-d^2] && EqQ[A-B] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.3.1 (a+b sec)^m (d sec)^n (A+B sec)*) + + +Int[(a_+b_.*csc[e_.+f_.*x_])*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*a*Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*n) + + 1/(d*n)*Int[(d*Csc[e+f*x])^(n+1)*Simp[n*(B*a+A*b)+(B*b*n+A*a*(n+1))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && RationalQ[n] && n<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])*(d_.*csc[e_.+f_.*x_])^n_.*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*B*Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*(n+1)) + + 1/(n+1)*Int[(d*Csc[e+f*x])^n*Simp[A*a*(n+1)+B*b*n+(A*b+B*a)*(n+1)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && Not[RationalQ[n] && n<=-1] + + +Int[csc[e_.+f_.*x_]*(A_+B_.*csc[e_.+f_.*x_])/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + B/b*Int[Csc[e+f*x],x] + (A*b-a*B)/b*Int[Csc[e+f*x]/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,e,f,A,B},x] && NeQ[A*b-a*B] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(m+1)) /; +FreeQ[{a,b,A,B,e,f,m},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && EqQ[a*B*m+A*b*(m+1)] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(a*f*(2*m+1)) + + (a*B*m+A*b*(m+1))/(a*b*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1),x] /; +FreeQ[{a,b,A,B,e,f},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && NeQ[a*B*m+A*b*(m+1)] && RationalQ[m] && m<-1/2 + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(m+1)) + + (a*B*m+A*b*(m+1))/(b*(m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m,x] /; +FreeQ[{a,b,A,B,e,f,m},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && NeQ[a*B*m+A*b*(m+1)] && Not[RationalQ[m] && m<-1/2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(f*(m+1)) + + 1/(m+1)*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*Simp[b*B*m+a*A*(m+1)+(a*B*m+A*b*(m+1))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,A,B,e,f},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m] && m>0 + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*Simp[(a*A-b*B)*(m+1)-(A*b-a*B)*(m+2)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,A,B,e,f},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[csc[e_.+f_.*x_]*(A_+B_.*csc[e_.+f_.*x_])/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + -2*(A*b-a*B)*Rt[a+b*B/A,2]*Sqrt[b*(1-Csc[e+f*x])/(a+b)]*Sqrt[-b*(1+Csc[e+f*x])/(a-b)]/(b^2*f*Cot[e+f*x])* + EllipticE[ArcSin[Sqrt[a+b*Csc[e+f*x]]/Rt[a+b*B/A,2]],(a*A+b*B)/(a*A-b*B)] /; +FreeQ[{a,b,e,f,A,B},x] && NeQ[a^2-b^2] && EqQ[A^2-B^2] + + +Int[csc[e_.+f_.*x_]*(A_+B_.*csc[e_.+f_.*x_])/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + (A-B)*Int[Csc[e+f*x]/Sqrt[a+b*Csc[e+f*x]],x] + + B*Int[Csc[e+f*x]*(1+Csc[e+f*x])/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,e,f,A,B},x] && NeQ[a^2-b^2] && NeQ[A^2-B^2] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + 2*Sqrt[2]*A*(a+b*Csc[e+f*x])^m*(A-B*Csc[e+f*x])*Sqrt[(A+B*Csc[e+f*x])/A]/(B*f*Cot[e+f*x]*(A*(a+b*Csc[e+f*x])/(a*A+b*B))^m)* + AppellF1[1/2,-(1/2),-m,3/2,(A-B*Csc[e+f*x])/(2*A),(b*(A-B*Csc[e+f*x]))/(A*b+a*B)] /; +FreeQ[{a,b,A,B,e,f},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && EqQ[A^2-B^2] && Not[IntegerQ[2*m]] + + +Int[csc[e_.+f_.*x_]*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)/b*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m,x] + B/b*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1),x] /; +FreeQ[{a,b,A,B,e,f,m},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m/(b*f*(2*m+1)) + + 1/(b^2*(2*m+1))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*Simp[A*b*m-a*B*m+b*B*(2*m+1)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,e,f,A,B},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + a*(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(b*f*(m+1)*(a^2-b^2)) - + 1/(b*(m+1)*(a^2-b^2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^(m+1)* + Simp[b*(A*b-a*B)*(m+1)-(a*A*b*(m+2)-B*(a^2+b^2*(m+1)))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 + + +Int[csc[e_.+f_.*x_]^2*(a_+b_.*csc[e_.+f_.*x_])^m_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)/(b*f*(m+2)) + + 1/(b*(m+2))*Int[Csc[e+f*x]*(a+b*Csc[e+f*x])^m*Simp[b*B*(m+1)+(A*b*(m+2)-a*B)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,e,f,A,B,m},x] && NeQ[A*b-a*B] && Not[RationalQ[m] && m<-1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*n) /; +FreeQ[{a,b,d,e,f,A,B,m,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && EqQ[m+n+1] && EqQ[a*A*m-b*B*n] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(b*f*(2*m+1)) + + (a*A*m+b*B*(m+1))/(a^2*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && EqQ[m+n+1] && RationalQ[m] && m<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*n) - + (a*A*m-b*B*n)/(b*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,A,B,m,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && EqQ[m+n+1] && Not[RationalQ[m] && m<=-1] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -2*b*B*Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && EqQ[A*b*(2*n+1)+2*a*B*n] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*b^2*Cot[e+f*x]*(d*Csc[e+f*x])^n/(a*f*n*Sqrt[a+b*Csc[e+f*x]]) + + (A*b*(2*n+1)+2*a*B*n)/(2*a*d*n)*Int[Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^(n+1),x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && NeQ[A*b*(2*n+1)+2*a*B*n] && RationalQ[n] && n<0 + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -2*b*B*Cot[e+f*x]*(d*Csc[e+f*x])^n/(f*(2*n+1)*Sqrt[a+b*Csc[e+f*x]]) + + (A*b*(2*n+1)+2*a*B*n)/(b*(2*n+1))*Int[Sqrt[a+b*Csc[e+f*x]]*(d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && NeQ[A*b*(2*n+1)+2*a*B*n] && Not[RationalQ[n] && n<0] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + a*A*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n/(f*n) - + b/(a*d*n)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n+1)*Simp[a*A*(m-n-1)-b*B*n-(a*B*n+A*b*(m+n))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[m,n] && m>1/2 && n<-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*B*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n/(f*(m+n)) + + 1/(d*(m+n))*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n*Simp[a*A*d*(m+n)+B*(b*d*n)+(A*b*d*(m+n)+a*B*d*(2*m+n-1))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[m] && m>1/2 && Not[RationalQ[n] && n<-1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + d*(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)/(a*f*(2*m+1)) - + 1/(a*b*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)* + Simp[A*(a*d*(n-1))-B*(b*d*(n-1))-d*(a*B*(m-n+1)+A*b*(m+n))*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[m,n] && m<-1/2 && n>0 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(b*f*(2*m+1)) - + 1/(a^2*(2*m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n* + Simp[b*B*n-a*A*(2*m+n+1)+(A*b-a*B)*(m+n+1)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[m] && m<-1/2 && Not[RationalQ[n] && n>0] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)/(f*(m+n)) + + d/(b*(m+n))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)*Simp[b*B*(n-1)+(A*b*(m+n)+a*B*m)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B,m},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[n] && n>1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*n) - + 1/(b*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1)*Simp[a*A*m-b*B*n-A*b*(m+n+1)*Csc[e+f*x],x],x] /; +FreeQ[{a,b,d,e,f,A,B,m},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] && RationalQ[n] && n<0 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + (A*b-a*B)/b*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n,x] + + B/b*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,d,e,f,A,B,m},x] && NeQ[A*b-a*B] && EqQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + a*A*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n/(f*n) + + 1/(d*n)*Int[(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^(n+1)* + Simp[a*(a*B*n-A*b*(m-n-1))+(2*a*b*B*n+A*(b^2*n+a^2*(1+n)))*Csc[e+f*x]+b*(b*B*n+a*A*(m+n))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m,n] && m>1 && n<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -b*B*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n/(f*(m+n)) + + 1/(m+n)*Int[(a+b*Csc[e+f*x])^(m-2)*(d*Csc[e+f*x])^n* + Simp[a^2*A*(m+n)+a*b*B*n+(a*(2*A*b+a*B)*(m+n)+b^2*B*(m+n-1))*Csc[e+f*x]+b*(A*b*(m+n)+a*B*(2*m+n-1))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m] && m>1 && + Not[IntegerQ[n] && n>1 && Not[IntegerQ[m]]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -d*(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(f*(m+1)*(a^2-b^2)) + + 1/((m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)* + Simp[d*(n-1)*(A*b-a*B)+d*(a*A-b*B)*(m+1)*Csc[e+f*x]-d*(A*b-a*B)*(m+n+1)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m,n] && m<-1 && 01 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + b*(A*b-a*B)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*(m+1)*(a^2-b^2)) + + 1/(a*(m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n* + Simp[A*(a^2*(m+1)-b^2*(m+n+1))+a*b*B*n-a*(A*b-a*B)*(m+1)*Csc[e+f*x]+b*(A*b-a*B)*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && Not[NegativeIntegerQ[m+1/2,n]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + -B*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)/(f*(m+n)) + + d/(m+n)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n-1)* + Simp[a*B*(n-1)+(b*B*(m+n-1)+a*A*(m+n))*Csc[e+f*x]+(a*B*m+A*b*(m+n))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m,n] && 00 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*n) - + 1/(d*n)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n+1)* + Simp[A*b*m-a*B*n-(b*B*n+a*A*(n+1))*Csc[e+f*x]-A*b*(m+n+1)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[m,n] && 01 && NeQ[m+n] && + Not[IntegerQ[m] && m>1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*n) + + 1/(a*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1)* + Simp[a*B*n-A*b*(m+n+1)+A*a*(n+1)*Csc[e+f*x]+A*b*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,m},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 + + +Int[(A_+B_.*csc[e_.+f_.*x_])/(Sqrt[d_.*csc[e_.+f_.*x_]]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]),x_Symbol] := + A/a*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[d*Csc[e+f*x]],x] - + (A*b-a*B)/(a*d)*Int[Sqrt[d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +Int[Sqrt[d_.*csc[e_.+f_.*x_]]*(A_+B_.*csc[e_.+f_.*x_])/Sqrt[a_+b_.*csc[e_.+f_.*x_]],x_Symbol] := + A*Int[Sqrt[d*Csc[e+f*x]]/Sqrt[a+b*Csc[e+f*x]],x] + + B/d*Int[(d*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +Int[Sqrt[a_+b_.*csc[e_.+f_.*x_]]*(A_+B_.*csc[e_.+f_.*x_])/Sqrt[d_.*csc[e_.+f_.*x_]],x_Symbol] := + B/d*Int[Sqrt[a+b*Csc[e+f*x]]*Sqrt[d*Csc[e+f*x]],x] + + A*Int[Sqrt[a+b*Csc[e+f*x]]/Sqrt[d*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,A,B},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +Int[(d_.*csc[e_.+f_.*x_])^n_*(A_+B_.*csc[e_.+f_.*x_])/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := + A/a*Int[(d*Csc[e+f*x])^n,x] - (A*b-a*B)/(a*d)*Int[(d*Csc[e+f*x])^(n+1)/(a+b*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,A,B,n},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_.*(A_+B_.*csc[e_.+f_.*x_]),x_Symbol] := + Defer[Int][(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n*(A+B*Csc[e+f*x]),x] /; +FreeQ[{a,b,d,e,f,A,B,m,n},x] && NeQ[A*b-a*B] && NeQ[a^2-b^2] + + +(* Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.*(A_.+B_.*csc[e_.+f_.*x_])^p_.,x_Symbol] := + (-a*c)^m*Int[Cot[e+f*x]^(2*m)*(c+d*Csc[e+f*x])^(n-m)*(A+B*Csc[e+f*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d] && EqQ[a^2-b^2] && IntegerQ[m] && + Not[IntegerQ[n] && (m<0 && n>0 || 00 && n<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*n) - + 1/(d*n)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^(n+1)* + Simp[A*b*m-a*(C*n+A*(n+1))*Csc[e+f*x]-b*(C*n+A*(m+n+1))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C},x] && NeQ[a^2-b^2] && RationalQ[m,n] && m>0 && n<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -C*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*(m+n+1)) + + 1/(m+n+1)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n* + Simp[a*A*(m+n+1)+a*C*n+((A*b+a*B)*(m+n+1)+b*C*(m+n))*Csc[e+f*x]+(b*B*(m+n+1)+a*C*m)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,C,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m>0 && Not[RationalQ[n] && n<=-1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -C*Cot[e+f*x]*(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n/(f*(m+n+1)) + + 1/(m+n+1)*Int[(a+b*Csc[e+f*x])^(m-1)*(d*Csc[e+f*x])^n* + Simp[a*A*(m+n+1)+a*C*n+b*(A*(m+n+1)+C*(m+n))*Csc[e+f*x]+a*C*m*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m>0 && Not[RationalQ[n] && n<=-1] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -d*(A*b^2-a*b*B+a^2*C)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(b*f*(a^2-b^2)*(m+1)) + + d/(b*(a^2-b^2)*(m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)* + Simp[A*b^2*(n-1)-a*(b*B-a*C)*(n-1)+ + b*(a*A-b*B+a*C)*(m+1)*Csc[e+f*x]- + (b*(A*b-a*B)*(m+n+1)+C*(a^2*n+b^2*(m+1)))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,C},x] && NeQ[a^2-b^2] && RationalQ[m,n] && m<-1 && n>0 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -d*(A*b^2+a^2*C)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(b*f*(a^2-b^2)*(m+1)) + + d/(b*(a^2-b^2)*(m+1))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)* + Simp[A*b^2*(n-1)+a^2*C*(n-1)+a*b*(A+C)*(m+1)*Csc[e+f*x]-(A*b^2*(m+n+1)+C*(a^2*n+b^2*(m+1)))*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C},x] && NeQ[a^2-b^2] && RationalQ[m,n] && m<-1 && n>0 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + (A*b^2-a*b*B+a^2*C)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*(m+1)*(a^2-b^2)) + + 1/(a*(m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n* + Simp[a*(a*A-b*B+a*C)*(m+1)-(A*b^2-a*b*B+a^2*C)*(m+n+1)- + a*(A*b-a*B+b*C)*(m+1)*Csc[e+f*x]+ + (A*b^2-a*b*B+a^2*C)*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,C,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && Not[NegativeIntegerQ[m+1/2,n]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + (A*b^2+a^2*C)*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*(m+1)*(a^2-b^2)) + + 1/(a*(m+1)*(a^2-b^2))*Int[(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n* + Simp[a^2*(A+C)*(m+1)-(A*b^2+a^2*C)*(m+n+1)-a*b*(A+C)*(m+1)*Csc[e+f*x]+(A*b^2+a^2*C)*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C,n},x] && NeQ[a^2-b^2] && RationalQ[m] && m<-1 && Not[NegativeIntegerQ[m+1/2,n]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -C*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(b*f*(m+n+1)) + + d/(b*(m+n+1))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)* + Simp[a*C*(n-1)+(A*b*(m+n+1)+b*C*(m+n))*Csc[e+f*x]+(b*B*(m+n+1)-a*C*n)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,C,m},x] && NeQ[a^2-b^2] && RationalQ[n] && n>0 (* && Not[IntegerQ[m] && m>0 && Not[IntegerQ[n]]] *) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + -C*d*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^(n-1)/(b*f*(m+n+1)) + + d/(b*(m+n+1))*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n-1)* + Simp[a*C*(n-1)+(A*b*(m+n+1)+b*C*(m+n))*Csc[e+f*x]-a*C*n*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C,m},x] && NeQ[a^2-b^2] && RationalQ[n] && n>0 (* && Not[IntegerQ[m] && m>0 && Not[IntegerQ[n]]] *) + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*n) + + 1/(a*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1)* + Simp[a*B*n-A*b*(m+n+1)+a*(A+A*n+C*n)*Csc[e+f*x]+A*b*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,B,C,m},x] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_*(d_.*csc[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + A*Cot[e+f*x]*(a+b*Csc[e+f*x])^(m+1)*(d*Csc[e+f*x])^n/(a*f*n) + + 1/(a*d*n)*Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n+1)* + Simp[-A*b*(m+n+1)+a*(A+A*n+C*n)*Csc[e+f*x]+A*b*(m+n+2)*Csc[e+f*x]^2,x],x] /; +FreeQ[{a,b,d,e,f,A,C,m},x] && NeQ[a^2-b^2] && RationalQ[n] && n<=-1 + + +Int[(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2)/(Sqrt[d_.*csc[e_.+f_.*x_]]*(a_+b_.*csc[e_.+f_.*x_])),x_Symbol] := + (A*b^2-a*b*B+a^2*C)/(a^2*d^2)*Int[(d*Csc[e+f*x])^(3/2)/(a+b*Csc[e+f*x]),x] + + 1/a^2*Int[(a*A-(A*b-a*B)*Csc[e+f*x])/Sqrt[d*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,A,B,C},x] && NeQ[a^2-b^2] + + +Int[(A_.+C_.*csc[e_.+f_.*x_]^2)/(Sqrt[d_.*csc[e_.+f_.*x_]]*(a_+b_.*csc[e_.+f_.*x_])),x_Symbol] := + (A*b^2+a^2*C)/(a^2*d^2)*Int[(d*Csc[e+f*x])^(3/2)/(a+b*Csc[e+f*x]),x] + + 1/a^2*Int[(a*A-A*b*Csc[e+f*x])/Sqrt[d*Csc[e+f*x]],x] /; +FreeQ[{a,b,d,e,f,A,C},x] && NeQ[a^2-b^2] + + +Int[(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2)/(Sqrt[d_.*csc[e_.+f_.*x_]]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]),x_Symbol] := + C/d^2*Int[(d*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] + + Int[(A+B*Csc[e+f*x])/(Sqrt[d*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]),x] /; +FreeQ[{a,b,d,e,f,A,B,C},x] && NeQ[a^2-b^2] + + +Int[(A_.+C_.*csc[e_.+f_.*x_]^2)/(Sqrt[d_.*csc[e_.+f_.*x_]]*Sqrt[a_+b_.*csc[e_.+f_.*x_]]),x_Symbol] := + C/d^2*Int[(d*Csc[e+f*x])^(3/2)/Sqrt[a+b*Csc[e+f*x]],x] + + A*Int[1/(Sqrt[d*Csc[e+f*x]]*Sqrt[a+b*Csc[e+f*x]]),x] /; +FreeQ[{a,b,d,e,f,A,C},x] && NeQ[a^2-b^2] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(d_.*csc[e_.+f_.*x_])^n_.*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + Defer[Int][(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n*(A+B*Csc[e+f*x]+C*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,B,C,m,n},x] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(d_.*csc[e_.+f_.*x_])^n_.*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + Defer[Int][(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^n*(A+C*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,C,m,n},x] + + +Int[(a_+b_.*sec[e_.+f_.*x_])^m_.*(d_.*cos[e_.+f_.*x_])^n_*(A_.+B_.*sec[e_.+f_.*x_]+C_.*sec[e_.+f_.*x_]^2),x_Symbol] := + d^(m+2)*Int[(b+a*Cos[e+f*x])^m*(d*Cos[e+f*x])^(n-m-2)*(C+B*Cos[e+f*x]+A*Cos[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,B,C,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + d^(m+2)*Int[(b+a*Sin[e+f*x])^m*(d*Sin[e+f*x])^(n-m-2)*(C+B*Sin[e+f*x]+A*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,B,C,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_+b_.*sec[e_.+f_.*x_])^m_.*(d_.*cos[e_.+f_.*x_])^n_*(A_.+C_.*sec[e_.+f_.*x_]^2),x_Symbol] := + d^(m+2)*Int[(b+a*Cos[e+f*x])^m*(d*Cos[e+f*x])^(n-m-2)*(C+A*Cos[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,C,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + d^(m+2)*Int[(b+a*Sin[e+f*x])^m*(d*Sin[e+f*x])^(n-m-2)*(C+A*Sin[e+f*x]^2),x] /; +FreeQ[{a,b,d,e,f,A,C,n},x] && Not[IntegerQ[n]] && IntegerQ[m] + + +Int[(a_+b_.*sec[e_.+f_.*x_])^m_.*(c_.*(d_.*sec[e_.+f_.*x_])^p_)^n_*(A_.+B_.*sec[e_.+f_.*x_]+C_.*sec[e_.+f_.*x_]^2),x_Symbol] := + c^IntPart[n]*(c*(d*Sec[e+f*x])^p)^FracPart[n]/(d*Sec[e+f*x])^(p*FracPart[n])* + Int[(a+b*Sec[e+f*x])^m*(d*Sec[e+f*x])^(n*p)*(A+B*Sec[e+f*x]+C*Sec[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && Not[IntegerQ[n]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_.*(d_.*csc[e_.+f_.*x_])^p_)^n_*(A_.+B_.*csc[e_.+f_.*x_]+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + c^IntPart[n]*(c*(d*Csc[e+f*x])^p)^FracPart[n]/(d*Csc[e+f*x])^(p*FracPart[n])* + Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n*p)*(A+B*Csc[e+f*x]+C*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && Not[IntegerQ[n]] + + +Int[(a_+b_.*sec[e_.+f_.*x_])^m_.*(c_.*(d_.*sec[e_.+f_.*x_])^p_)^n_*(A_.+C_.*sec[e_.+f_.*x_]^2),x_Symbol] := + c^IntPart[n]*(c*(d*Sec[e+f*x])^p)^FracPart[n]/(d*Sec[e+f*x])^(p*FracPart[n])* + Int[(a+b*Sec[e+f*x])^m*(d*Sec[e+f*x])^(n*p)*(A+C*Sec[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,m,n,p},x] && Not[IntegerQ[n]] + + +Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_.*(d_.*csc[e_.+f_.*x_])^p_)^n_*(A_.+C_.*csc[e_.+f_.*x_]^2),x_Symbol] := + c^IntPart[n]*(c*(d*Csc[e+f*x])^p)^FracPart[n]/(d*Csc[e+f*x])^(p*FracPart[n])* + Int[(a+b*Csc[e+f*x])^m*(d*Csc[e+f*x])^(n*p)*(A+C*Csc[e+f*x]^2),x] /; +FreeQ[{a,b,c,d,e,f,A,C,m,n,p},x] && Not[IntegerQ[n]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.7 (d trig)^m (a+b (c sec)^n)^p*) + + +Int[u_.*(a_+b_.*sec[e_.+f_.*x_]^2)^p_,x_Symbol] := + b^p*Int[ActivateTrig[u*tan[e+f*x]^(2*p)],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a+b,0] && IntegerQ[p] + + +Int[u_.*(a_+b_.*sec[e_.+f_.*x_]^2)^p_,x_Symbol] := + Int[ActivateTrig[u*(b*tan[e+f*x]^2)^p],x] /; +FreeQ[{a,b,e,f,p},x] && EqQ[a+b,0] + + +Int[(b_.*sec[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + b*ff/f*Subst[Int[(b+b*ff^2*x^2)^(p-1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{b,e,f,p},x] && Not[IntegerQ[p]] + + +Int[(b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + b^IntPart[p]*(b*(c*Sec[e+f*x])^n)^FracPart[p]/(c*Sec[e+f*x])^(n*FracPart[p])*Int[(c*Sec[e+f*x])^(n*p),x] /; +FreeQ[{b,c,e,f,n,p},x] && Not[IntegerQ[p]] + + +Int[tan[e_.+f_.*x_]^m_.*(b_.*sec[e_.+f_.*x_]^2)^p_.,x_Symbol] := + b/(2*f)*Subst[Int[(-1+x)^((m-1)/2)*(b*x)^(p-1),x],x,Sec[e+f*x]^2] /; +FreeQ[{b,e,f,p},x] && Not[IntegerQ[p]] && IntegerQ[(m-1)/2] + + +Int[u_.*(b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Sec[e+f*x],x]}, + (b*ff^n)^IntPart[p]*(b*Sec[e+f*x]^n)^FracPart[p]/(Sec[e+f*x]/ff)^(n*FracPart[p])* + Int[ActivateTrig[u]*(Sec[e+f*x]/ff)^(n*p),x]] /; +FreeQ[{b,e,f,n,p},x] && Not[IntegerQ[p]] && IntegerQ[n] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[u_.*(b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + b^IntPart[p]*(b*(c*Sec[e+f*x])^n)^FracPart[p]/(c*Sec[e+f*x])^(n*FracPart[p])* + Int[ActivateTrig[u]*(c*Sec[e+f*x])^(n*p),x] /; +FreeQ[{b,c,e,f,n,p},x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && + (EqQ[u,1] || MatchQ[u,(d_.*trig_[e+f*x])^m_. /; FreeQ[{d,m},x] && MemberQ[{sin,cos,tan,cot,sec,csc},trig]]) + + +Int[1/(a_+b_.*sec[e_.+f_.*x_]^2),x_Symbol] := + x/a - b/a*Int[1/(b+a*Cos[e+f*x]^2),x] /; +FreeQ[{a,b,e,f},x] && NeQ[a+b,0] + + +Int[(a_+b_.*sec[e_.+f_.*x_]^2)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b+b*ff^2*x^2)^p/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && NeQ[a+b,0] && NeQ[p,-1] + + +Int[(a_+b_.*sec[e_.+f_.*x_]^4)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b+2*b*ff^2*x^2+b*ff^4*x^4)^p/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[2*p] + + +Int[(a_+b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(a+b*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[n/2] && IGtQ[p,-2] + + +Int[(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + Defer[Int][(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,e,f,n,p},x] + + +Int[sin[e_.+f_.*x_]^m_*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff^(m+1)/f*Subst[Int[x^m*ExpandToSum[a+b*(1+ff^2*x^2)^(n/2),x]^p/(1+ff^2*x^2)^(m/2+1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[m/2] && IntegerQ[n/2] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Cos[e+f*x],x]}, + -ff/f*Subst[Int[(1-ff^2*x^2)^((m-1)/2)*(b+a*(ff*x)^n)^p/(ff*x)^(n*p),x],x,Cos[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[(m-1)/2] && IntegerQ[n] && IntegerQ[p] + + +Int[sin[e_.+f_.*x_]^m_.*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Cos[e+f*x],x]}, + 1/(f*ff^m)*Subst[Int[(-1+ff^2*x^2)^((m-1)/2)*(a+b*(c*ff*x)^n)^p/x^(m+1),x],x,Sec[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && IntegerQ[(m-1)/2] && (m>0 || EqQ[n,2] || EqQ[n,4]) + + +Int[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Sin[e+f*x])^m*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*cos[e_.+f_.*x_])^m_*(a_+b_.*sec[e_.+f_.*x_]^n_.)^p_.,x_Symbol] := + d^(n*p)*Int[(d*Cos[e+f*x])^(m-n*p)*(b+a*Cos[e+f*x]^n)^p,x] /; +FreeQ[{a,b,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && IntegersQ[n,p] + + +Int[(d_.*cos[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Cos[e+f*x])^FracPart[m]*(Sec[e+f*x]/d)^FracPart[m]*Int[(Sec[e+f*x]/d)^(-m)*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[tan[e_.+f_.*x_]^m_.*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + Module[{ff=FreeFactors[Cos[e+f*x],x]}, + -1/(f*ff^(m+n*p-1))*Subst[Int[(1-ff^2*x^2)^((m-1)/2)*(b+a*(ff*x)^n)^p/x^(m+n*p),x],x,Cos[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,n},x] && IntegerQ[(m-1)/2] && IntegerQ[n] && IntegerQ[p] + + +Int[tan[e_.+f_.*x_]^m_.*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Sec[e+f*x],x]}, + 1/f*Subst[Int[(-1+ff^2*x^2)^((m-1)/2)*(a+b*(c*ff*x)^n)^p/x,x],x,Sec[e+f*x]/ff]] /; +FreeQ[{a,b,c,e,f,n,p},x] && IntegerQ[(m-1)/2] && (m>0 || EqQ[n,2] || EqQ[n,4] || IGtQ[p,0] || IntegersQ[2*n,p]) + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(b_.*sec[e_.+f_.*x_]^2)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + b*ff/f*Subst[Int[(d*ff*x)^m*(b+b*ff^2*x^2)^(p-1),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{b,e,f,m,p},x] + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_.,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(d*ff*x)^m*(a+b*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2),x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,m,p},x] && IntegerQ[n/2] && (IntegerQ[m/2] || EqQ[n,2]) + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + d*(d*Tan[e+f*x])^(m-1)*(b*(c*Sec[e+f*x])^n)^p/(f*(p*n+m-1)) - + d^2*(m-1)/(p*n+m-1)*Int[(d*Tan[e+f*x])^(m-2)*(b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{b,d,c,e,f,p,n},x] && GtQ[m,1] && NeQ[p*n+m-1,0] && IntegersQ[2*p*n,2*m] + + +Int[(d_.*tan[e_.+f_.*x_])^m_*(b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + (d*Tan[e+f*x])^(m+1)*(b*(c*Sec[e+f*x])^n)^p/(d*f*(m+1)) - + (p*n+m+1)/(d^2*(m+1))*Int[(d*Tan[e+f*x])^(m+2)*(b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{b,d,c,e,f,p,n},x] && LtQ[m,-1] && NeQ[p*n+m+1,0] && IntegersQ[2*p*n,2*m] + + +Int[(d_.*tan[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Tan[e+f*x])^m*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*cot[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Cot[e+f*x])^FracPart[m]*(Tan[e+f*x]/d)^FracPart[m]*Int[(Tan[e+f*x]/d)^(-m)*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + +Int[sec[e_.+f_.*x_]^m_*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Tan[e+f*x],x]}, + ff/f*Subst[Int[(1+ff^2*x^2)^(m/2-1)*ExpandToSum[a+b*(1+ff^2*x^2)^(n/2),x]^p,x],x,Tan[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[m/2] && IntegerQ[n/2] + + +Int[sec[e_.+f_.*x_]^m_.*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff/f*Subst[Int[ExpandToSum[b+a*(1-ff^2*x^2)^(n/2),x]^p/(1-ff^2*x^2)^((m+n*p+1)/2),x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,e,f},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] && IntegerQ[p] + + +Int[sec[e_.+f_.*x_]^m_.*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + With[{ff=FreeFactors[Sin[e+f*x],x]}, + ff/f*Subst[Int[(a+b/(1-ff^2*x^2)^(n/2))^p/(1-ff^2*x^2)^((m+1)/2),x],x,Sin[e+f*x]/ff]] /; +FreeQ[{a,b,e,f,p},x] && IntegerQ[(m-1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]] + + +Int[sec[e_.+f_.*x_]^m_.*(a_+b_.*sec[e_.+f_.*x_]^n_)^p_,x_Symbol] := + Int[ExpandTrig[sec[e+f*x]^m*(a+b*sec[e+f*x]^n)^p,x],x] /; +FreeQ[{a,b,e,f},x] && IntegersQ[m,n,p] + + +Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_.,x_Symbol] := + Defer[Int][(d*Sec[e+f*x])^m*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(d_.*csc[e_.+f_.*x_])^m_*(a_+b_.*(c_.*sec[e_.+f_.*x_])^n_)^p_,x_Symbol] := + (d*Csc[e+f*x])^FracPart[m]*(Sin[e+f*x]/d)^FracPart[m]*Int[(Sin[e+f*x]/d)^(-m)*(a+b*(c*Sec[e+f*x])^n)^p,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.9 trig^m (a+b sec^n+c sec^(2 n))^p*) + + +Int[(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[(b+2*c*Sec[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[(b+2*c*Csc[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Sec[d+e*x]^n+c*Sec[d+e*x]^(2*n))^p/(b+2*c*Sec[d+e*x]^n)^(2*p)*Int[u*(b+2*c*Sec[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Csc[d+e*x]^n+c*Csc[d+e*x]^(2*n))^p/(b+2*c*Csc[d+e*x]^n)^(2*p)*Int[u*(b+2*c*Csc[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[1/(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/(b-q+2*c*Sec[d+e*x]^n),x] - + 2*c/q*Int[1/(b+q+2*c*Sec[d+e*x]^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[1/(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + 2*c/q*Int[1/(b-q+2*c*Csc[d+e*x]^n),x] - + 2*c/q*Int[1/(b+q+2*c*Csc[d+e*x]^n),x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && NeQ[b^2-4*a*c] + + +Int[sin[d_.+e_.*x_]^m_.*(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{f=FreeFactors[Cos[d+e*x],x]}, + -f/e*Subst[Int[(1-f^2*x^2)^((m-1)/2)*(b+a*(f*x)^n)^p/(f*x)^(n*p),x],x,Cos[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && IntegersQ[n,p] + + +Int[cos[d_.+e_.*x_]^m_.*(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{f=FreeFactors[Sin[d+e*x],x]}, + f/e*Subst[Int[(1-f^2*x^2)^((m-1)/2)*(b+a*(f*x)^n)^p/(f*x)^(n*p),x],x,Sin[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && OddQ[m] && IntegersQ[n,p] + + +Int[sin[d_.+e_.*x_]^m_*(a_.+b_.*sec[d_.+e_.*x_]^n_+c_.*sec[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{f=FreeFactors[Tan[d+e*x],x]}, + f^(m+1)/e*Subst[Int[x^m*ExpandToSum[a+b*(1+f^2*x^2)^(n/2)+c*(1+f^2*x^2)^n,x]^p/(1+f^2*x^2)^(m/2+1),x],x,Tan[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[n2-2*n] && EvenQ[m] && EvenQ[n] + + +Int[cos[d_.+e_.*x_]^m_*(a_.+b_.*csc[d_.+e_.*x_]^n_+c_.*csc[d_.+e_.*x_]^n2_)^p_.,x_Symbol] := + Module[{f=FreeFactors[Cot[d+e*x],x]}, + -f^(m+1)/e*Subst[Int[x^m*ExpandToSum[a+b*(1+f^2*x^2)^(n/2)+c*(1+f^2*x^2)^n,x]^p/(1+f^2*x^2)^(m/2+1),x],x,Cot[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[n2-2*n] && EvenQ[m] && EvenQ[n] + + +Int[sec[d_.+e_.*x_]^m_.*(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + 1/(4^p*c^p)*Int[Sec[d+e*x]^m*(b+2*c*Sec[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[csc[d_.+e_.*x_]^m_.*(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + 1/(4^p*c^p)*Int[Csc[d+e*x]^m*(b+2*c*Csc[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +Int[sec[d_.+e_.*x_]^m_.*(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Sec[d+e*x]^n+c*Sec[d+e*x]^(2*n))^p/(b+2*c*Sec[d+e*x]^n)^(2*p)*Int[Sec[d+e*x]^m*(b+2*c*Sec[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[csc[d_.+e_.*x_]^m_.*(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + (a+b*Csc[d+e*x]^n+c*Csc[d+e*x]^(2*n))^p/(b+2*c*Csc[d+e*x]^n)^(2*p)*Int[Csc[d+e*x]^m*(b+2*c*Csc[d+e*x]^n)^(2*p),x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] + + +Int[sec[d_.+e_.*x_]^m_.*(a_.+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + Int[ExpandTrig[sec[d+e*x]^m*(a+b*sec[d+e*x]^n+c*sec[d+e*x]^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && IntegersQ[m,n,p] + + +Int[csc[d_.+e_.*x_]^m_.*(a_.+b_.*csc[d_.+e_.*x_]^n_.+c_.*csc[d_.+e_.*x_]^n2_.)^p_,x_Symbol] := + Int[ExpandTrig[csc[d+e*x]^m*(a+b*csc[d+e*x]^n+c*csc[d+e*x]^(2*n))^p,x],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && IntegersQ[m,n,p] + + +Int[tan[d_.+e_.*x_]^m_.*(a_+b_.*sec[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{f=FreeFactors[Cos[d+e*x],x]}, + -1/(e*f^(m+n*p-1))*Subst[Int[(1-f^2*x^2)^((m-1)/2)*(c+b*(f*x)^n+c*(f*x)^(2*n))^p/x^(m+2*n*p),x],x,Cos[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && OddQ[m] && IntegerQ[n] && IntegerQ[p] + + +Int[cot[d_.+e_.*x_]^m_.*(a_+b_.*csc[d_.+e_.*x_]^n_.+c_.*sec[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{f=FreeFactors[Sin[d+e*x],x]}, + 1/(e*f^(m+n*p-1))*Subst[Int[(1-f^2*x^2)^((m-1)/2)*(c+b*(f*x)^n+c*(f*x)^(2*n))^p/x^(m+2*n*p),x],x,Sin[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[n2-2*n] && OddQ[m] && IntegerQ[n] && IntegerQ[p] + + +Int[tan[d_.+e_.*x_]^m_.*(a_+b_.*sec[d_.+e_.*x_]^n_+c_.*sec[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{f=FreeFactors[Tan[d+e*x],x]}, + f^(m+1)/e*Subst[Int[x^m*ExpandToSum[a+b*(1+f^2*x^2)^(n/2)+c*(1+f^2*x^2)^n,x]^p/(1+f^2*x^2),x],x,Tan[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && EvenQ[m] && EvenQ[n] + + +Int[cot[d_.+e_.*x_]^m_.*(a_+b_.*csc[d_.+e_.*x_]^n_+c_.*sec[d_.+e_.*x_]^n2_.)^p_.,x_Symbol] := + Module[{f=FreeFactors[Cot[d+e*x],x]}, + -f^(m+1)/e*Subst[Int[x^m*ExpandToSum[a+b*(1+f^2*x^2)^(n/2)+c*(1+f^2*x^2)^n,x]^p/(1+f^2*x^2),x],x,Cot[d+e*x]/f]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[n2-2*n] && EvenQ[m] && EvenQ[n] + + +Int[(A_+B_.*sec[d_.+e_.*x_])*(a_+b_.*sec[d_.+e_.*x_]+c_.*sec[d_.+e_.*x_]^2)^n_,x_Symbol] := + 1/(4^n*c^n)*Int[(A+B*Sec[d+e*x])*(b+2*c*Sec[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*csc[d_.+e_.*x_])*(a_+b_.*csc[d_.+e_.*x_]+c_.*csc[d_.+e_.*x_]^2)^n_,x_Symbol] := + 1/(4^n*c^n)*Int[(A+B*Csc[d+e*x])*(b+2*c*Csc[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*sec[d_.+e_.*x_])*(a_+b_.*sec[d_.+e_.*x_]+c_.*sec[d_.+e_.*x_]^2)^n_,x_Symbol] := + (a+b*Sec[d+e*x]+c*Sec[d+e*x]^2)^n/(b+2*c*Sec[d+e*x])^(2*n)*Int[(A+B*Sec[d+e*x])*(b+2*c*Sec[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[n]] + + +Int[(A_+B_.*csc[d_.+e_.*x_])*(a_+b_.*csc[d_.+e_.*x_]+c_.*csc[d_.+e_.*x_]^2)^n_,x_Symbol] := + (a+b*Csc[d+e*x]+c*Csc[d+e*x]^2)^n/(b+2*c*Csc[d+e*x])^(2*n)*Int[(A+B*Csc[d+e*x])*(b+2*c*Csc[d+e*x])^(2*n),x] /; +FreeQ[{a,b,c,d,e,A,B},x] && EqQ[b^2-4*a*c] && Not[IntegerQ[n]] + + +Int[(A_+B_.*sec[d_.+e_.*x_])/(a_.+b_.*sec[d_.+e_.*x_]+c_.*sec[d_.+e_.*x_]^2),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + (B+(b*B-2*A*c)/q)*Int[1/(b+q+2*c*Sec[d+e*x]),x] + + (B-(b*B-2*A*c)/q)*Int[1/(b-q+2*c*Sec[d+e*x]),x]] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] + + +Int[(A_+B_.*csc[d_.+e_.*x_])/(a_.+b_.*csc[d_.+e_.*x_]+c_.*csc[d_.+e_.*x_]^2),x_Symbol] := + Module[{q=Rt[b^2-4*a*c,2]}, + (B+(b*B-2*A*c)/q)*Int[1/(b+q+2*c*Csc[d+e*x]),x] + + (B-(b*B-2*A*c)/q)*Int[1/(b-q+2*c*Csc[d+e*x]),x]] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] + + +Int[(A_+B_.*sec[d_.+e_.*x_])*(a_.+b_.*sec[d_.+e_.*x_]+c_.*sec[d_.+e_.*x_]^2)^n_,x_Symbol] := + Int[ExpandTrig[(A+B*sec[d+e*x])*(a+b*sec[d+e*x]+c*sec[d+e*x]^2)^n,x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] && IntegerQ[n] + + +Int[(A_+B_.*csc[d_.+e_.*x_])*(a_.+b_.*csc[d_.+e_.*x_]+c_.*csc[d_.+e_.*x_]^2)^n_,x_Symbol] := + Int[ExpandTrig[(A+B*csc[d+e*x])*(a+b*csc[d+e*x]+c*csc[d+e*x]^2)^n,x],x] /; +FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c] && IntegerQ[n] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.10 (c+d x)^m (a+b sec)^n*) + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+k_.*Pi+f_.*Complex[0,fz_]*x_],x_Symbol] := + -2*(c+d*x)^m*ArcTanh[E^(-I*k*Pi)*E^(-I*e+f*fz*x)]/(f*fz*I) - + d*m/(f*fz*I)*Int[(c+d*x)^(m-1)*Log[1-E^(-I*k*Pi)*E^(-I*e+f*fz*x)],x] + + d*m/(f*fz*I)*Int[(c+d*x)^(m-1)*Log[1+E^(-I*k*Pi)*E^(-I*e+f*fz*x)],x] /; +FreeQ[{c,d,e,f,fz},x] && IntegerQ[2*k] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+k_.*Pi+f_.*x_],x_Symbol] := + -2*(c+d*x)^m*ArcTanh[E^(I*k*Pi)*E^(I*(e+f*x))]/f - + d*m/f*Int[(c+d*x)^(m-1)*Log[1-E^(I*k*Pi)*E^(I*(e+f*x))],x] + + d*m/f*Int[(c+d*x)^(m-1)*Log[1+E^(I*k*Pi)*E^(I*(e+f*x))],x] /; +FreeQ[{c,d,e,f},x] && IntegerQ[2*k] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+f_.*Complex[0,fz_]*x_],x_Symbol] := + -2*(c+d*x)^m*ArcTanh[E^(-I*e+f*fz*x)]/(f*fz*I) - + d*m/(f*fz*I)*Int[(c+d*x)^(m-1)*Log[1-E^(-I*e+f*fz*x)],x] + + d*m/(f*fz*I)*Int[(c+d*x)^(m-1)*Log[1+E^(-I*e+f*fz*x)],x] /; +FreeQ[{c,d,e,f,fz},x] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+f_.*x_],x_Symbol] := + -2*(c+d*x)^m*ArcTanh[E^(I*(e+f*x))]/f - + d*m/f*Int[(c+d*x)^(m-1)*Log[1-E^(I*(e+f*x))],x] + + d*m/f*Int[(c+d*x)^(m-1)*Log[1+E^(I*(e+f*x))],x] /; +FreeQ[{c,d,e,f},x] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+f_.*x_]^2,x_Symbol] := + -(c+d*x)^m*Cot[e+f*x]/f + + d*m/f*Int[(c+d*x)^(m-1)*Cot[e+f*x],x] /; +FreeQ[{c,d,e,f},x] && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*(c+d*x)*Cot[e+f*x]*(b*Csc[e+f*x])^(n-2)/(f*(n-1)) - + b^2*d*(b*Csc[e+f*x])^(n-2)/(f^2*(n-1)*(n-2)) + + b^2*(n-2)/(n-1)*Int[(c+d*x)*(b*Csc[e+f*x])^(n-2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[n] && n>1 && n!=2 + + +Int[(c_.+d_.*x_)^m_*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + -b^2*(c+d*x)^m*Cot[e+f*x]*(b*Csc[e+f*x])^(n-2)/(f*(n-1)) - + b^2*d*m*(c+d*x)^(m-1)*(b*Csc[e+f*x])^(n-2)/(f^2*(n-1)*(n-2)) + + b^2*(n-2)/(n-1)*Int[(c+d*x)^m*(b*Csc[e+f*x])^(n-2),x] + + b^2*d^2*m*(m-1)/(f^2*(n-1)*(n-2))*Int[(c+d*x)^(m-2)*(b*Csc[e+f*x])^(n-2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n>1 && n!=2 && m>1 + + +Int[(c_.+d_.*x_)*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + d*(b*Csc[e+f*x])^n/(f^2*n^2) + + (c+d*x)*Cos[e+f*x]*(b*Csc[e+f*x])^(n+1)/(b*f*n) + + (n+1)/(b^2*n)*Int[(c+d*x)*(b*Csc[e+f*x])^(n+2),x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[n] && n<-1 + + +Int[(c_.+d_.*x_)^m_*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + d*m*(c+d*x)^(m-1)*(b*Csc[e+f*x])^n/(f^2*n^2) + + (c+d*x)^m*Cos[e+f*x]*(b*Csc[e+f*x])^(n+1)/(b*f*n) + + (n+1)/(b^2*n)*Int[(c+d*x)^m*(b*Csc[e+f*x])^(n+2),x] - + d^2*m*(m-1)/(f^2*n^2)*Int[(c+d*x)^(m-2)*(b*Csc[e+f*x])^n,x] /; +FreeQ[{b,c,d,e,f},x] && RationalQ[m,n] && n<-1 && m>1 + + +Int[(c_.+d_.*x_)^m_.*(b_.*csc[e_.+f_.*x_])^n_,x_Symbol] := + (b*Sin[e+f*x])^n*(b*Csc[e+f*x])^n*Int[(c+d*x)^m/(b*Sin[e+f*x])^n,x] /; +FreeQ[{b,c,d,e,f,m,n},x] && Not[IntegerQ[n]] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,(a+b*Csc[e+f*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[m,n] + + +Int[(c_.+d_.*x_)^m_.*(a_+b_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(c+d*x)^m,Sin[e+f*x]^(-n)/(b+a*Sin[e+f*x])^(-n),x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NegativeIntegerQ[n] && PositiveIntegerQ[m] + + +Int[(c_.+d_.*x_)^m_.*csc[e_.+f_.*x_]^n_.,x_Symbol] := + If[MatchQ[f,f1_.*Complex[0,j_]], + If[MatchQ[e,e1_.+Pi/2], + Defer[Int][(c+d*x)^m*Sech[I*(e-Pi/2)+I*f*x]^n,x], + (-I)^n*Defer[Int][(c+d*x)^m*Csch[-I*e-I*f*x]^n,x]], + If[MatchQ[e,e1_.+Pi/2], + Defer[Int][(c+d*x)^m*Sec[e-Pi/2+f*x]^n,x], + Defer[Int][(c+d*x)^m*Csc[e+f*x]^n,x]]] /; +FreeQ[{c,d,e,f,m,n},x] && IntegerQ[n] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*csc[e_.+f_.*x_])^n_.,x_Symbol] := + Defer[Int][(c+d*x)^m*(a+b*Csc[e+f*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[u_^m_.*(a_.+b_.*Sec[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Sec[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*(a_.+b_.*Csc[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Csc[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*4.3.11 (e x)^m (a+b sec(c+d x^n))^p*) + + +Int[(a_.+b_.*Sec[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Sec[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Csc[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Csc[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Sec[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Sec[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Csc[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Csc[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Sec[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Sec[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Csc[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Csc[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Sec[u_])^p_.,x_Symbol] := + Int[(a+b*Sec[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Csc[u_])^p_.,x_Symbol] := + Int[(a+b*Csc[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*(a_.+b_.*Sec[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Sec[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Csc[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Csc[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Sec[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Sec[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[x_^m_.*(a_.+b_.*Csc[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Csc[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Sec[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Sec[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Csc[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Csc[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Sec[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Sec[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(e_*x_)^m_.*(a_.+b_.*Csc[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Csc[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*Sec[a_.+b_.*x_^n_.]^p_*Sin[a_.+b_.*x_^n_.],x_Symbol] := + x^(m-n+1)*Sec[a+b*x^n]^(p-1)/(b*n*(p-1)) - + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Sec[a+b*x^n]^(p-1),x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && NeQ[p-1] + + +Int[x_^m_.*Csc[a_.+b_.*x_^n_.]^p_*Cos[a_.+b_.*x_^n_.],x_Symbol] := + -x^(m-n+1)*Csc[a+b*x^n]^(p-1)/(b*n*(p-1)) + + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Csc[a+b*x^n]^(p-1),x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && NeQ[p-1] + + + +`) + +func resourcesRubi43SecantMBytes() ([]byte, error) { + return _resourcesRubi43SecantM, nil +} + +func resourcesRubi43SecantM() (*asset, error) { + bytes, err := resourcesRubi43SecantMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/4.3 Secant.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi44MiscellaneousTrigFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Miscellaneous Trig Function Rules*) + + +(* ::Subsection::Closed:: *) +(*4.1 Sine normalization rules*) + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.*(d_.*sin[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Tan[a+b*x])^m*(d*Cos[a+b*x])^m/(d*Sin[a+b*x])^m*Int[ActivateTrig[u]*(d*Sin[a+b*x])^(m+n)/(d*Cos[a+b*x])^m,x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.*(d_.*cos[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Tan[a+b*x])^m*(d*Cos[a+b*x])^m/(d*Sin[a+b*x])^m*Int[ActivateTrig[u]*(d*Sin[a+b*x])^m/(d*Cos[a+b*x])^(m-n),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.*(d_.*sin[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cot[a+b*x])^m*(d*Sin[a+b*x])^m/(d*Cos[a+b*x])^m*Int[ActivateTrig[u]*(d*Cos[a+b*x])^m/(d*Sin[a+b*x])^(m-n),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.*(d_.*cos[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cot[a+b*x])^m*(d*Sin[a+b*x])^m/(d*Cos[a+b*x])^m*Int[ActivateTrig[u]*(d*Cos[a+b*x])^(m+n)/(d*Sin[a+b*x])^m,x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*sec[a_.+b_.*x_])^m_.*(d_.*cos[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Sec[a+b*x])^m*(d*Cos[a+b*x])^m*Int[ActivateTrig[u]*(d*Cos[a+b*x])^(n-m),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*sec[a_.+b_.*x_])^m_.*(d_.*cos[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Csc[a+b*x])^m*(d*Sin[a+b*x])^m*Int[ActivateTrig[u]*(d*Sin[a+b*x])^(n-m),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Tan[a+b*x])^m*(c*Cos[a+b*x])^m/(c*Sin[a+b*x])^m*Int[ActivateTrig[u]*(c*Sin[a+b*x])^m/(c*Cos[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Cot[a+b*x])^m*(c*Sin[a+b*x])^m/(c*Cos[a+b*x])^m*Int[ActivateTrig[u]*(c*Cos[a+b*x])^m/(c*Sin[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*sec[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Sec[a+b*x])^m*(c*Cos[a+b*x])^m*Int[ActivateTrig[u]/(c*Cos[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*csc[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Csc[a+b*x])^m*(c*Sin[a+b*x])^m*Int[ActivateTrig[u]/(c*Sin[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*sin[a_.+b_.*x_])^n_.*(A_+B_.*csc[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Sin[a+b*x])^(n-1)*(B+A*Sin[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(c_.*cos[a_.+b_.*x_])^n_.*(A_+B_.*sec[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Cos[a+b*x])^(n-1)*(B+A*Cos[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_+B_.*csc[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Sin[a+b*x])/Sin[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_+B_.*sec[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Cos[a+b*x])/Cos[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownSineIntegrandQ[u,x] + + +Int[u_.*(c_.*sin[a_.+b_.*x_])^n_.*(A_.+B_.*csc[a_.+b_.*x_]+C_.*csc[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Sin[a+b*x])^(n-2)*(C+B*Sin[a+b*x]+A*Sin[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_.*(c_.*cos[a_.+b_.*x_])^n_.*(A_.+B_.*sec[a_.+b_.*x_]+C_.*sec[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Cos[a+b*x])^(n-2)*(C+B*Cos[a+b*x]+A*Cos[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_.*(c_.*sin[a_.+b_.*x_])^n_.*(A_+C_.*csc[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Sin[a+b*x])^(n-2)*(C+A*Sin[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_.*(c_.*cos[a_.+b_.*x_])^n_.*(A_+C_.*sec[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Cos[a+b*x])^(n-2)*(C+A*Cos[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*csc[a_.+b_.*x_]+C_.*csc[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Sin[a+b*x]+A*Sin[a+b*x]^2)/Sin[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*sec[a_.+b_.*x_]+C_.*sec[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Cos[a+b*x]+A*Cos[a+b*x]^2)/Cos[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_+C_.*csc[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Sin[a+b*x]^2)/Sin[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_+C_.*sec[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Cos[a+b*x]^2)/Cos[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownSineIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*sin[a_.+b_.*x_]+C_.*csc[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Sin[a+b*x]+B*Sin[a+b*x]^2)/Sin[a+b*x],x] /; +FreeQ[{a,b,A,B,C},x] + + +Int[u_*(A_.+B_.*cos[a_.+b_.*x_]+C_.*sec[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Cos[a+b*x]+B*Cos[a+b*x]^2)/Cos[a+b*x],x] /; +FreeQ[{a,b,A,B,C},x] + + +Int[u_*(A_.*sin[a_.+b_.*x_]^n_.+B_.*sin[a_.+b_.*x_]^n1_+C_.*sin[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Sin[a+b*x]^n*(A+B*Sin[a+b*x]+C*Sin[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + +Int[u_*(A_.*cos[a_.+b_.*x_]^n_.+B_.*cos[a_.+b_.*x_]^n1_+C_.*cos[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Cos[a+b*x]^n*(A+B*Cos[a+b*x]+C*Cos[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + + + + +(* ::Subsection::Closed:: *) +(*4.2 Tangent normalization rules*) + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.*(d_.*tan[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cot[a+b*x])^m*(d*Tan[a+b*x])^m*Int[ActivateTrig[u]*(d*Tan[a+b*x])^(n-m),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.*(d_.*cos[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Tan[a+b*x])^m*(d*Cos[a+b*x])^m/(d*Sin[a+b*x])^m*Int[ActivateTrig[u]*(d*Sin[a+b*x])^m/(d*Cos[a+b*x])^(m-n),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Cot[a+b*x])^m*(c*Tan[a+b*x])^m*Int[ActivateTrig[u]/(c*Tan[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Cot[a+b*x])^m*(c*Tan[a+b*x])^m*Int[ActivateTrig[u]/(c*Cot[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^n_.*(A_+B_.*cot[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Tan[a+b*x])^(n-1)*(B+A*Tan[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^n_.*(A_+B_.*tan[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Cot[a+b*x])^(n-1)*(B+A*Cot[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(A_+B_.*cot[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Tan[a+b*x])/Tan[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(A_+B_.*tan[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Cot[a+b*x])/Cot[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_.*(c_.*tan[a_.+b_.*x_])^n_.*(A_.+B_.*cot[a_.+b_.*x_]+C_.*cot[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Tan[a+b*x])^(n-2)*(C+B*Tan[a+b*x]+A*Tan[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_.*(c_.*cot[a_.+b_.*x_])^n_.*(A_.+B_.*tan[a_.+b_.*x_]+C_.*tan[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Cot[a+b*x])^(n-2)*(C+B*Cot[a+b*x]+A*Cot[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_.*(c_.*tan[a_.+b_.*x_])^n_.*(A_+C_.*cot[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Tan[a+b*x])^(n-2)*(C+A*Tan[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_.*(c_.*cot[a_.+b_.*x_])^n_.*(A_+C_.*tan[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Cot[a+b*x])^(n-2)*(C+A*Cot[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*cot[a_.+b_.*x_]+C_.*cot[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Tan[a+b*x]+A*Tan[a+b*x]^2)/Tan[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*tan[a_.+b_.*x_]+C_.*tan[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Cot[a+b*x]+A*Cot[a+b*x]^2)/Cot[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(A_+C_.*cot[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Tan[a+b*x]^2)/Tan[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownTangentIntegrandQ[u,x] + + +Int[u_*(A_+C_.*tan[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Cot[a+b*x]^2)/Cot[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownCotangentIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*tan[a_.+b_.*x_]+C_.*cot[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Tan[a+b*x]+B*Tan[a+b*x]^2)/Tan[a+b*x],x] /; +FreeQ[{a,b,A,B,C},x] + + +Int[u_*(A_.*tan[a_.+b_.*x_]^n_.+B_.*tan[a_.+b_.*x_]^n1_+C_.*tan[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Tan[a+b*x]^n*(A+B*Tan[a+b*x]+C*Tan[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + +Int[u_*(A_.*cot[a_.+b_.*x_]^n_.+B_.*cot[a_.+b_.*x_]^n1_+C_.*cot[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Cot[a+b*x]^n*(A+B*Cot[a+b*x]+C*Cot[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + + + + +(* ::Subsection::Closed:: *) +(*4.3 Secant normalization rules*) +(**) + + +Int[u_*(c_.*sin[a_.+b_.*x_])^m_.*(d_.*csc[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Sin[a+b*x])^m*(d*Csc[a+b*x])^m*Int[ActivateTrig[u]*(d*Csc[a+b*x])^(n-m),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*cos[a_.+b_.*x_])^m_.*(d_.*sec[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cos[a+b*x])^m*(d*Sec[a+b*x])^m*Int[ActivateTrig[u]*(d*Sec[a+b*x])^(n-m),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.*(d_.*sec[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Tan[a+b*x])^m*(d*Csc[a+b*x])^m/(d*Sec[a+b*x])^m*Int[ActivateTrig[u]*(d*Sec[a+b*x])^(m+n)/(d*Csc[a+b*x])^m,x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.*(d_.*csc[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Tan[a+b*x])^m*(d*Csc[a+b*x])^m/(d*Sec[a+b*x])^m*Int[ActivateTrig[u]*(d*Sec[a+b*x])^m/(d*Csc[a+b*x])^(m-n),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.*(d_.*sec[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cot[a+b*x])^m*(d*Sec[a+b*x])^m/(d*Csc[a+b*x])^m*Int[ActivateTrig[u]*(d*Csc[a+b*x])^m/(d*Sec[a+b*x])^(m-n),x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.*(d_.*csc[a_.+b_.*x_])^n_.,x_Symbol] := + (c*Cot[a+b*x])^m*(d*Sec[a+b*x])^m/(d*Csc[a+b*x])^m*Int[ActivateTrig[u]*(d*Csc[a+b*x])^(m+n)/(d*Sec[a+b*x])^m,x] /; +FreeQ[{a,b,c,d,m,n},x] && KnownSecantIntegrandQ[u,x] && Not[IntegerQ[m]] + + +Int[u_*(c_.*sin[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Csc[a+b*x])^m*(c*Sin[a+b*x])^m*Int[ActivateTrig[u]/(c*Csc[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*cos[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Cos[a+b*x])^m*(c*Sec[a+b*x])^m*Int[ActivateTrig[u]/(c*Sec[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*tan[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Tan[a+b*x])^m*(c*Csc[a+b*x])^m/(c*Sec[a+b*x])^m*Int[ActivateTrig[u]*(c*Sec[a+b*x])^m/(c*Csc[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*cot[a_.+b_.*x_])^m_.,x_Symbol] := + (c*Cot[a+b*x])^m*(c*Sec[a+b*x])^m/(c*Csc[a+b*x])^m*Int[ActivateTrig[u]*(c*Csc[a+b*x])^m/(c*Sec[a+b*x])^m,x] /; +FreeQ[{a,b,c,m},x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*sec[a_.+b_.*x_])^n_.*(A_+B_.*cos[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Sec[a+b*x])^(n-1)*(B+A*Sec[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(c_.*csc[a_.+b_.*x_])^n_.*(A_+B_.*sin[a_.+b_.*x_]),x_Symbol] := + c*Int[ActivateTrig[u]*(c*Csc[a+b*x])^(n-1)*(B+A*Csc[a+b*x]),x] /; +FreeQ[{a,b,c,A,B,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_+B_.*cos[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Sec[a+b*x])/Sec[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_+B_.*sin[a_.+b_.*x_]),x_Symbol] := + Int[ActivateTrig[u]*(B+A*Csc[a+b*x])/Csc[a+b*x],x] /; +FreeQ[{a,b,A,B},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_.*(c_.*sec[a_.+b_.*x_])^n_.*(A_.+B_.*cos[a_.+b_.*x_]+C_.*cos[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Sec[a+b*x])^(n-2)*(C+B*Sec[a+b*x]+A*Sec[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_.*(c_.*csc[a_.+b_.*x_])^n_.*(A_.+B_.*sin[a_.+b_.*x_]+C_.*sin[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Csc[a+b*x])^(n-2)*(C+B*Csc[a+b*x]+A*Csc[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,B,C,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_.*(c_.*sec[a_.+b_.*x_])^n_.*(A_+C_.*cos[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Sec[a+b*x])^(n-2)*(C+A*Sec[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_.*(c_.*csc[a_.+b_.*x_])^n_.*(A_+C_.*sin[a_.+b_.*x_]^2),x_Symbol] := + c^2*Int[ActivateTrig[u]*(c*Csc[a+b*x])^(n-2)*(C+A*Csc[a+b*x]^2),x] /; +FreeQ[{a,b,c,A,C,n},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*cos[a_.+b_.*x_]+C_.*cos[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Sec[a+b*x]+A*Sec[a+b*x]^2)/Sec[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_.+B_.*sin[a_.+b_.*x_]+C_.*sin[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+B*Csc[a+b*x]+A*Csc[a+b*x]^2)/Csc[a+b*x]^2,x] /; +FreeQ[{a,b,A,B,C},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_+C_.*cos[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Sec[a+b*x]^2)/Sec[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_+C_.*sin[a_.+b_.*x_]^2),x_Symbol] := + Int[ActivateTrig[u]*(C+A*Csc[a+b*x]^2)/Csc[a+b*x]^2,x] /; +FreeQ[{a,b,A,C},x] && KnownSecantIntegrandQ[u,x] + + +Int[u_*(A_.*sec[a_.+b_.*x_]^n_.+B_.*sec[a_.+b_.*x_]^n1_+C_.*sec[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Sec[a+b*x]^n*(A+B*Sec[a+b*x]+C*Sec[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + +Int[u_*(A_.*csc[a_.+b_.*x_]^n_.+B_.*csc[a_.+b_.*x_]^n1_+C_.*csc[a_.+b_.*x_]^n2_),x_Symbol] := + Int[ActivateTrig[u]*Csc[a+b*x]^n*(A+B*Csc[a+b*x]+C*Csc[a+b*x]^2),x] /; +FreeQ[{a,b,A,B,C,n},x] && EqQ[n1-n-1] && EqQ[n2-n-2] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.1 (c trig)^m (d trig)^n*) + + +Int[sin[a_.+b_.*x_]*sin[c_.+d_.*x_],x_Symbol] := + Sin[a-c+(b-d)*x]/(2*(b-d)) - Sin[a+c+(b+d)*x]/(2*(b+d)) /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-d^2] + + +Int[cos[a_.+b_.*x_]*cos[c_.+d_.*x_],x_Symbol] := + Sin[a-c+(b-d)*x]/(2*(b-d)) + Sin[a+c+(b+d)*x]/(2*(b+d)) /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-d^2] + + +Int[sin[a_.+b_.*x_]*cos[c_.+d_.*x_],x_Symbol] := + -Cos[a-c+(b-d)*x]/(2*(b-d)) - Cos[a+c+(b+d)*x]/(2*(b+d)) /; +FreeQ[{a,b,c,d},x] && NeQ[b^2-d^2] + + +Int[cos[a_.+b_.*x_]^2*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + 1/2*Int[(g*Sin[c+d*x])^p,x] + + 1/2*Int[Cos[c+d*x]*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && PositiveIntegerQ[p/2] + + +Int[sin[a_.+b_.*x_]^2*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + 1/2*Int[(g*Sin[c+d*x])^p,x] - + 1/2*Int[Cos[c+d*x]*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && PositiveIntegerQ[p/2] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*sin[c_.+d_.*x_]^p_.,x_Symbol] := + 2^p/e^p*Int[(e*Cos[a+b*x])^(m+p)*Sin[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && IntegerQ[p] + + +Int[(f_.*sin[a_.+b_.*x_])^n_.*sin[c_.+d_.*x_]^p_.,x_Symbol] := + 2^p/f^p*Int[Cos[a+b*x]^p*(f*Sin[a+b*x])^(n+p),x] /; +FreeQ[{a,b,c,d,f,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && IntegerQ[p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e^2*(e*Cos[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+p-1] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e^2*(e*Sin[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+p-1] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(b*g*m) /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+2*p+2] + + +Int[(e_.*sin[a_.+b_.*x_])^m_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Sin[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(b*g*m) /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+2*p+2] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e^2*(e*Cos[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + e^4*(m+p-1)/(4*g^2*(p+1))*Int[(e*Cos[a+b*x])^(m-4)*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>2 && p<-1 && + (m>3 || p==-3/2) && IntegersQ[2*m,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e^2*(e*Sin[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + e^4*(m+p-1)/(4*g^2*(p+1))*Int[(e*Sin[a+b*x])^(m-4)*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>2 && p<-1 && + (m>3 || p==-3/2) && IntegersQ[2*m,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Cos[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + e^2*(m+2*p+2)/(4*g^2*(p+1))*Int[(e*Cos[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>1 && p<-1 && + NeQ[m+2*p+2] && (p<-2 || m==2) && IntegersQ[2*m,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Sin[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + e^2*(m+2*p+2)/(4*g^2*(p+1))*Int[(e*Sin[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>1 && p<-1 && + NeQ[m+2*p+2] && (p<-2 || m==2) && IntegersQ[2*m,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e^2*(e*Cos[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(m+2*p)) + + e^2*(m+p-1)/(m+2*p)*Int[(e*Cos[a+b*x])^(m-2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m>1 && NeQ[m+2*p] && + IntegersQ[2*m,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e^2*(e*Sin[a+b*x])^(m-2)*(g*Sin[c+d*x])^(p+1)/(2*b*g*(m+2*p)) + + e^2*(m+p-1)/(m+2*p)*Int[(e*Sin[a+b*x])^(m-2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m>1 && NeQ[m+2*p] && + IntegersQ[2*m,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(2*b*g*(m+p+1)) + + (m+2*p+2)/(e^2*(m+p+1))*Int[(e*Cos[a+b*x])^(m+2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m<-1 && + NeQ[m+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Sin[a+b*x])^m*(g*Sin[c+d*x])^(p+1)/(2*b*g*(m+p+1)) + + (m+2*p+2)/(e^2*(m+p+1))*Int[(e*Sin[a+b*x])^(m+2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m<-1 && + NeQ[m+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*p] + + +Int[cos[a_.+b_.*x_]*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + 2*Sin[a+b*x]*(g*Sin[c+d*x])^p/(d*(2*p+1)) + 2*p*g/(2*p+1)*Int[Sin[a+b*x]*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[p] && p>0 && IntegerQ[2*p] + + +Int[sin[a_.+b_.*x_]*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -2*Cos[a+b*x]*(g*Sin[c+d*x])^p/(d*(2*p+1)) + 2*p*g/(2*p+1)*Int[Cos[a+b*x]*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[p] && p>0 && IntegerQ[2*p] + + +Int[cos[a_.+b_.*x_]*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + Cos[a+b*x]*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + (2*p+3)/(2*g*(p+1))*Int[Sin[a+b*x]*(g*Sin[c+d*x])^(p+1),x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[sin[a_.+b_.*x_]*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -Sin[a+b*x]*(g*Sin[c+d*x])^(p+1)/(2*b*g*(p+1)) + + (2*p+3)/(2*g*(p+1))*Int[Cos[a+b*x]*(g*Sin[c+d*x])^(p+1),x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[p] && p<-1 && IntegerQ[2*p] + + +Int[cos[a_.+b_.*x_]/Sqrt[sin[c_.+d_.*x_]],x_Symbol] := + -ArcSin[Cos[a+b*x]-Sin[a+b*x]]/d + Log[Cos[a+b*x]+Sin[a+b*x]+Sqrt[Sin[c+d*x]]]/d /; +FreeQ[{a,b,c,d},x] && EqQ[b*c-a*d] && EqQ[d/b-2] + + +Int[sin[a_.+b_.*x_]/Sqrt[sin[c_.+d_.*x_]],x_Symbol] := + -ArcSin[Cos[a+b*x]-Sin[a+b*x]]/d - Log[Cos[a+b*x]+Sin[a+b*x]+Sqrt[Sin[c+d*x]]]/d /; +FreeQ[{a,b,c,d},x] && EqQ[b*c-a*d] && EqQ[d/b-2] + + +Int[(g_.*sin[c_.+d_.*x_])^p_/cos[a_.+b_.*x_],x_Symbol] := + 2*g*Int[Sin[a+b*x]*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && IntegerQ[2*p] + + +Int[(g_.*sin[c_.+d_.*x_])^p_/sin[a_.+b_.*x_],x_Symbol] := + 2*g*Int[Cos[a+b*x]*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && IntegerQ[2*p] + + +(* Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^(m+1)*Sin[a+b*x]*(g*Sin[c+d*x])^p/(b*e*(m+p+1)*(Sin[a+b*x]^2)^((p+1)/2))* + Hypergeometric2F1[-(p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2] /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && Not[IntegerQ[m+p]] *) + + +(* Int[(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -Cos[a+b*x]*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(p+1)*(Sin[a+b*x]^2)^((n+p+1)/2))* + Hypergeometric2F1[-(n+p-1)/2,(p+1)/2,(p+3)/2,Cos[a+b*x]^2] /; +FreeQ[{a,b,c,d,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && Not[IntegerQ[n+p]] *) + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (g*Sin[c+d*x])^p/((e*Cos[a+b*x])^p*Sin[a+b*x]^p)*Int[(e*Cos[a+b*x])^(m+p)*Sin[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] + + +Int[(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (g*Sin[c+d*x])^p/(Cos[a+b*x]^p*(f*Sin[a+b*x])^p)*Int[Cos[a+b*x]^p*(f*Sin[a+b*x])^(n+p),x] /; +FreeQ[{a,b,c,d,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] + + +Int[cos[a_.+b_.*x_]^2*sin[a_.+b_.*x_]^2*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + 1/4*Int[(g*Sin[c+d*x])^p,x] - + 1/4*Int[Cos[c+d*x]^2*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && PositiveIntegerQ[p/2] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(f_.*sin[a_.+b_.*x_])^n_.*sin[c_.+d_.*x_]^p_.,x_Symbol] := + 2^p/(e^p*f^p)*Int[(e*Cos[a+b*x])^(m+p)*(f*Sin[a+b*x])^(n+p),x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && IntegerQ[p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e*(e*Cos[a+b*x])^(m-1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(n+p+1)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+p+1] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e*(e*Sin[a+b*x])^(m-1)*(f*Cos[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(n+p+1)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+p+1] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)) /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && EqQ[m+n+2*p+2] && NeQ[m+p+1] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e^2*(e*Cos[a+b*x])^(m-2)*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^(p+1)/(2*b*g*(n+p+1)) + + e^4*(m+p-1)/(4*g^2*(n+p+1))*Int[(e*Cos[a+b*x])^(m-4)*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>3 && p<-1 && + NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e^2*(e*Sin[a+b*x])^(m-2)*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^(p+1)/(2*b*g*(n+p+1)) + + e^4*(m+p-1)/(4*g^2*(n+p+1))*Int[(e*Sin[a+b*x])^(m-4)*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>3 && p<-1 && + NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Cos[a+b*x])^m*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^(p+1)/(2*b*g*(n+p+1)) + + e^2*(m+n+2*p+2)/(4*g^2*(n+p+1))*Int[(e*Cos[a+b*x])^(m-2)*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>1 && p<-1 && + NeQ[m+n+2*p+2] && NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] && (p<-2 || m==2 || m==3) + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Sin[a+b*x])^m*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^(p+1)/(2*b*g*(n+p+1)) + + e^2*(m+n+2*p+2)/(4*g^2*(n+p+1))*Int[(e*Sin[a+b*x])^(m-2)*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^(p+2),x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,p] && m>1 && p<-1 && + NeQ[m+n+2*p+2] && NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] && (p<-2 || m==2 || m==3) + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e*(e*Cos[a+b*x])^(m-1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(n+p+1)) + + e^2*(m+p-1)/(f^2*(n+p+1))*Int[(e*Cos[a+b*x])^(m-2)*(f*Sin[a+b*x])^(n+2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n] && m>1 && n<-1 && + NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e*(e*Sin[a+b*x])^(m-1)*(f*Cos[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(n+p+1)) + + e^2*(m+p-1)/(f^2*(n+p+1))*Int[(e*Sin[a+b*x])^(m-2)*(f*Cos[a+b*x])^(n+2)*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n] && m>1 && n<-1 && + NeQ[n+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + e*(e*Cos[a+b*x])^(m-1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(m+n+2*p)) + + e^2*(m+p-1)/(m+n+2*p)*Int[(e*Cos[a+b*x])^(m-2)*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m>1 && NeQ[m+n+2*p] && + IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -e*(e*Sin[a+b*x])^(m-1)*(f*Cos[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(m+n+2*p)) + + e^2*(m+p-1)/(m+n+2*p)*Int[(e*Sin[a+b*x])^(m-2)*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m>1 && NeQ[m+n+2*p] && + IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -f*(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n-1)*(g*Sin[c+d*x])^p/(b*e*(m+n+2*p)) + + 2*f*g*(n+p-1)/(e*(m+n+2*p))*Int[(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n-1)*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n,p] && m<-1 && n>0 && p>0 && + NeQ[m+n+2*p] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + f*(e*Sin[a+b*x])^(m+1)*(f*Cos[a+b*x])^(n-1)*(g*Sin[c+d*x])^p/(b*e*(m+n+2*p)) + + 2*f*g*(n+p-1)/(e*(m+n+2*p))*Int[(e*Sin[a+b*x])^(m+1)*(f*Cos[a+b*x])^(n-1)*(g*Sin[c+d*x])^(p-1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n,p] && m<-1 && n>0 && p>0 && + NeQ[m+n+2*p] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)) + + f*(m+n+2*p+2)/(2*e*g*(m+p+1))*Int[(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n-1)*(g*Sin[c+d*x])^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n,p] && m<-1 && n>0 && p<-1 && + NeQ[m+n+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Sin[a+b*x])^(m+1)*(f*Cos[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)) + + f*(m+n+2*p+2)/(2*e*g*(m+p+1))*Int[(e*Sin[a+b*x])^(m+1)*(f*Cos[a+b*x])^(n-1)*(g*Sin[c+d*x])^(p+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m,n,p] && m<-1 && n>0 && p<-1 && + NeQ[m+n+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)) + + (m+n+2*p+2)/(e^2*(m+p+1))*Int[(e*Cos[a+b*x])^(m+2)*(f*Sin[a+b*x])^n*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m<-1 && + NeQ[m+n+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*n,2*p] + + +Int[(e_.*sin[a_.+b_.*x_])^m_*(f_.*cos[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (e*Sin[a+b*x])^(m+1)*(f*Cos[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)) + + (m+n+2*p+2)/(e^2*(m+p+1))*Int[(e*Sin[a+b*x])^(m+2)*(f*Cos[a+b*x])^n*(g*Sin[c+d*x])^p,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && RationalQ[m] && m<-1 && + NeQ[m+n+2*p+2] && NeQ[m+p+1] && IntegersQ[2*m,2*n,2*p] + + +(* Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+p+1)*(Sin[a+b*x]^2)^((n+p+1)/2))* + Hypergeometric2F1[-(n+p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] && Not[IntegerQ[m+p]] && Not[IntegerQ[n+p]] *) + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := + (g*Sin[c+d*x])^p/((e*Cos[a+b*x])^p*(f*Sin[a+b*x])^p)*Int[(e*Cos[a+b*x])^(m+p)*(f*Sin[a+b*x])^(n+p),x] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d] && EqQ[d/b-2] && Not[IntegerQ[p]] + + +Int[(e_.*cos[a_.+b_.*x_])^m_.*sin[c_.+d_.*x_],x_Symbol] := + -(m+2)*(e*Cos[a+b*x])^(m+1)*Cos[(m+1)*(a+b*x)]/(d*e*(m+1)) /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b*c-a*d] && EqQ[d/b-Abs[m+2]] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.3 Inert trig functions*) + + +Int[(a_.*F_[c_.+d_.*x_]^p_)^n_,x_Symbol] := + With[{v=ActivateTrig[F[c+d*x]]}, + a^IntPart[n]*(v/NonfreeFactors[v,x])^(p*IntPart[n])*(a*v^p)^FracPart[n]/NonfreeFactors[v,x]^(p*FracPart[n])* + Int[NonfreeFactors[v,x]^(n*p),x]] /; +FreeQ[{a,c,d,n,p},x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p] + + +Int[(a_.*(b_.*F_[c_.+d_.*x_])^p_)^n_.,x_Symbol] := + With[{v=ActivateTrig[F[c+d*x]]}, + a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])*Int[(b*v)^(n*p),x]] /; +FreeQ[{a,b,c,d,n,p},x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]] + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[1,Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && (F===Cos || F===cos) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[1,Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && (F===Sin || F===sin) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + 1/(b*c)*Subst[Int[SubstFor[1/x,Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && (F===Cot || F===cot) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -1/(b*c)*Subst[Int[SubstFor[1/x,Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && (F===Tan || F===tan) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^2,x_Symbol] := + With[{d=FreeFactors[Tan[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[1,Tan[c*(a+b*x)]/d,u,x],x],x,Tan[c*(a+b*x)]/d] /; + FunctionOfQ[Tan[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && NonsumQ[u] && (F===Sec || F===sec) + + +Int[u_/cos[c_.*(a_.+b_.*x_)]^2,x_Symbol] := + With[{d=FreeFactors[Tan[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[1,Tan[c*(a+b*x)]/d,u,x],x],x,Tan[c*(a+b*x)]/d] /; + FunctionOfQ[Tan[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && NonsumQ[u] + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^2,x_Symbol] := + With[{d=FreeFactors[Cot[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[1,Cot[c*(a+b*x)]/d,u,x],x],x,Cot[c*(a+b*x)]/d] /; + FunctionOfQ[Cot[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && NonsumQ[u] && (F===Csc || F===csc) + + +Int[u_/sin[c_.*(a_.+b_.*x_)]^2,x_Symbol] := + With[{d=FreeFactors[Cot[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[1,Cot[c*(a+b*x)]/d,u,x],x],x,Cot[c*(a+b*x)]/d] /; + FunctionOfQ[Cot[c*(a+b*x)]/d,u,x,True]] /; +FreeQ[{a,b,c},x] && NonsumQ[u] + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_.,x_Symbol] := + With[{d=FreeFactors[Tan[c*(a+b*x)],x]}, + 1/(b*c*d^(n-1))*Subst[Int[SubstFor[1/(x^n*(1+d^2*x^2)),Tan[c*(a+b*x)]/d,u,x],x],x,Tan[c*(a+b*x)]/d] /; + FunctionOfQ[Tan[c*(a+b*x)]/d,u,x,True] && TryPureTanSubst[ActivateTrig[u]*Cot[c*(a+b*x)]^n,x]] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && (F===Cot || F===cot) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_.,x_Symbol] := + With[{d=FreeFactors[Cot[c*(a+b*x)],x]}, + -1/(b*c*d^(n-1))*Subst[Int[SubstFor[1/(x^n*(1+d^2*x^2)),Cot[c*(a+b*x)]/d,u,x],x],x,Cot[c*(a+b*x)]/d] /; + FunctionOfQ[Cot[c*(a+b*x)]/d,u,x,True] && TryPureTanSubst[ActivateTrig[u]*Tan[c*(a+b*x)]^n,x]] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && (F===Tan || F===tan) + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Cot[a+b*x]],x]","-1/b*Subst[Int[F[x]/(1+x^2),x],x,Cot[a+b*x]]",Hold[ + With[{d=FreeFactors[Cot[v],x]}, + Dist[-d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Cot[v]/d,u,x],x],x,Cot[v]/d],x]]]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v],x],u,x,True] && TryPureTanSubst[ActivateTrig[u],x]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Cot[v],x]}, + Dist[-d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Cot[v]/d,u,x],x],x,Cot[v]/d],x]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v],x],u,x,True] && TryPureTanSubst[ActivateTrig[u],x]]] + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Tan[a+b*x]],x]","1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]",Hold[ + With[{d=FreeFactors[Tan[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v]/d,u,x],x],x,Tan[v]/d],x]]]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v],x],u,x,True] && TryPureTanSubst[ActivateTrig[u],x]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Tan[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v]/d,u,x],x],x,Tan[v]/d],x]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v],x],u,x,True] && TryPureTanSubst[ActivateTrig[u],x]]] + + +Int[F_[a_.+b_.*x_]^p_.*G_[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[ActivateTrig[F[a+b*x]^p*G[c+d*x]^q],x],x] /; +FreeQ[{a,b,c,d},x] && (F===sin || F===cos) && (G===sin || G===cos) && PositiveIntegerQ[p,q] + + +Int[F_[a_.+b_.*x_]^p_.*G_[c_.+d_.*x_]^q_.*H_[e_.+f_.*x_]^r_.,x_Symbol] := + Int[ExpandTrigReduce[ActivateTrig[F[a+b*x]^p*G[c+d*x]^q*H[e+f*x]^r],x],x] /; +FreeQ[{a,b,c,d,e,f},x] && (F===sin || F===cos) && (G===sin || G===cos) && (H===sin || H===cos) && PositiveIntegerQ[p,q,r] + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[1,Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && (F===Cos || F===cos) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[1,Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && (F===Sin || F===sin) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + 1/(b*c)*Subst[Int[SubstFor[1/x,Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && (F===Cot || F===cot) + + +Int[u_*F_[c_.*(a_.+b_.*x_)],x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -1/(b*c)*Subst[Int[SubstFor[1/x,Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && (F===Tan || F===tan) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[(1-d^2*x^2)^((n-1)/2),Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Cos || F===cos) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + d/(b*c)*Subst[Int[SubstFor[(1-d^2*x^2)^((-n-1)/2),Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Sec || F===sec) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[(1-d^2*x^2)^((n-1)/2),Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Sin || F===sin) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -d/(b*c)*Subst[Int[SubstFor[(1-d^2*x^2)^((-n-1)/2),Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Csc || F===csc) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Sin[c*(a+b*x)],x]}, + 1/(b*c*d^(n-1))*Subst[Int[SubstFor[(1-d^2*x^2)^((n-1)/2)/x^n,Sin[c*(a+b*x)]/d,u,x],x],x,Sin[c*(a+b*x)]/d] /; + FunctionOfQ[Sin[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Cot || F===cot) + + +Int[u_*F_[c_.*(a_.+b_.*x_)]^n_,x_Symbol] := + With[{d=FreeFactors[Cos[c*(a+b*x)],x]}, + -1/(b*c*d^(n-1))*Subst[Int[SubstFor[(1-d^2*x^2)^((n-1)/2)/x^n,Cos[c*(a+b*x)]/d,u,x],x],x,Cos[c*(a+b*x)]/d] /; + FunctionOfQ[Cos[c*(a+b*x)]/d,u,x]] /; +FreeQ[{a,b,c},x] && OddQ[n] && NonsumQ[u] && (F===Tan || F===tan) + + +Int[u_*(v_+d_.*F_[c_.*(a_.+b_.*x_)]^n_.),x_Symbol] := + With[{e=FreeFactors[Sin[c*(a+b*x)],x]}, + Int[ActivateTrig[u*v],x] + d*Int[ActivateTrig[u]*Cos[c*(a+b*x)]^n,x] /; + FunctionOfQ[Sin[c*(a+b*x)]/e,u,x]] /; +FreeQ[{a,b,c,d},x] && Not[FreeQ[v,x]] && OddQ[n] && NonsumQ[u] && (F===Cos || F===cos) + + +Int[u_*(v_+d_.*F_[c_.*(a_.+b_.*x_)]^n_.),x_Symbol] := + With[{e=FreeFactors[Cos[c*(a+b*x)],x]}, + Int[ActivateTrig[u*v],x] + d*Int[ActivateTrig[u]*Sin[c*(a+b*x)]^n,x] /; + FunctionOfQ[Cos[c*(a+b*x)]/e,u,x]] /; +FreeQ[{a,b,c,d},x] && Not[FreeQ[v,x]] && OddQ[n] && NonsumQ[u] && (F===Sin || F===sin) + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Sin[a+b*x]]*Cos[a+b*x],x]","Subst[Int[F[x],x],x,Sin[a+b*x]]/b",Hold[ + With[{d=FreeFactors[Sin[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1,Sin[v]/d,u/Cos[v],x],x],x,Sin[v]/d],x]]]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v],x],u/Cos[v],x]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Sin[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1,Sin[v]/d,u/Cos[v],x],x],x,Sin[v]/d],x]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v],x],u/Cos[v],x]]] + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Cos[a+b*x]]*Sin[a+b*x],x]","-Subst[Int[F[x],x],x,Cos[a+b*x]]/b",Hold[ + With[{d=FreeFactors[Cos[v],x]}, + Dist[-d/Coefficient[v,x,1],Subst[Int[SubstFor[1,Cos[v]/d,u/Sin[v],x],x],x,Cos[v]/d],x]]]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v],x],u/Sin[v],x]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Cos[v],x]}, + Dist[-d/Coefficient[v,x,1],Subst[Int[SubstFor[1,Cos[v]/d,u/Sin[v],x],x],x,Cos[v]/d],x]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v],x],u/Sin[v],x]]] + + +Int[u_.*(a_.+b_.*cos[d_.+e_.*x_]^2+c_.*sin[d_.+e_.*x_]^2)^p_.,x_Symbol] := + (a+c)^p*Int[ActivateTrig[u],x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[b-c] + + +Int[u_.*(a_.+b_.*tan[d_.+e_.*x_]^2+c_.*sec[d_.+e_.*x_]^2)^p_.,x_Symbol] := + (a+c)^p*Int[ActivateTrig[u],x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[b+c] + + +Int[u_.*(a_.+b_.*cot[d_.+e_.*x_]^2+c_.*csc[d_.+e_.*x_]^2)^p_.,x_Symbol] := + (a+c)^p*Int[ActivateTrig[u],x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[b+c] + + +Int[u_/y_,x_Symbol] := + With[{q=DerivativeDivides[ActivateTrig[y],ActivateTrig[u],x]}, + q*Log[RemoveContent[ActivateTrig[y],x]] /; + Not[FalseQ[q]]] /; +Not[InertTrigFreeQ[u]] + + +Int[u_/(y_*w_),x_Symbol] := + With[{q=DerivativeDivides[ActivateTrig[y*w],ActivateTrig[u],x]}, + q*Log[RemoveContent[ActivateTrig[y*w],x]] /; + Not[FalseQ[q]]] /; +Not[InertTrigFreeQ[u]] + + +Int[u_*y_^m_.,x_Symbol] := + With[{q=DerivativeDivides[ActivateTrig[y],ActivateTrig[u],x]}, + q*ActivateTrig[y^(m+1)]/(m+1) /; + Not[FalseQ[q]]] /; +FreeQ[m,x] && NeQ[m+1] && Not[InertTrigFreeQ[u]] + + +Int[u_*y_^m_.*z_^n_.,x_Symbol] := + With[{q=DerivativeDivides[ActivateTrig[y*z],ActivateTrig[u*z^(n-m)],x]}, + q*ActivateTrig[y^(m+1)*z^(m+1)]/(m+1) /; + Not[FalseQ[q]]] /; +FreeQ[{m,n},x] && NeQ[m+1] && Not[InertTrigFreeQ[u]] + + +Int[u_.*(a_.*F_[c_.+d_.*x_]^p_)^n_,x_Symbol] := + With[{v=ActivateTrig[F[c+d*x]]}, + a^IntPart[n]*(v/NonfreeFactors[v,x])^(p*IntPart[n])*(a*v^p)^FracPart[n]/NonfreeFactors[v,x]^(p*FracPart[n])* + Int[ActivateTrig[u]*NonfreeFactors[v,x]^(n*p),x]] /; +FreeQ[{a,c,d,n,p},x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p] + + +Int[u_.*(a_.*(b_.*F_[c_.+d_.*x_])^p_)^n_.,x_Symbol] := + With[{v=ActivateTrig[F[c+d*x]]}, + a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])*Int[ActivateTrig[u]*(b*v)^(n*p),x]] /; +FreeQ[{a,b,c,d,n,p},x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]] + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Tan[a+b*x]],x]","1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]",Hold[ + With[{d=FreeFactors[Tan[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v]/d,u,x],x],x,Tan[v]/d],x]]]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v],x],u,x]] /; +SimplifyFlag && InverseFunctionFreeQ[u,x], + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Tan[v],x]}, + Dist[d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v]/d,u,x],x],x,Tan[v]/d],x]] /; + Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v],x],u,x]] /; +InverseFunctionFreeQ[u,x]] + + +Int[u_.*(a_.*tan[c_.+d_.*x_]^n_.+b_.*sec[c_.+d_.*x_]^n_.)^p_,x_Symbol] := + Int[ActivateTrig[u]*Sec[c+d*x]^(n*p)*(b+a*Sin[c+d*x]^n)^p,x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] + + +Int[u_.*(a_.*cot[c_.+d_.*x_]^n_.+b_.*csc[c_.+d_.*x_]^n_.)^p_,x_Symbol] := + Int[ActivateTrig[u]*Csc[c+d*x]^(n*p)*(b+a*Cos[c+d*x]^n)^p,x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] + + +Int[u_*(a_*F_[c_.+d_.*x_]^p_.+b_.*F_[c_.+d_.*x_]^q_.)^n_.,x_Symbol] := + Int[ActivateTrig[u*F[c+d*x]^(n*p)*(a+b*F[c+d*x]^(q-p))^n],x] /; +FreeQ[{a,b,c,d,p,q},x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q-p] + + +Int[u_*(a_*F_[d_.+e_.*x_]^p_.+b_.*F_[d_.+e_.*x_]^q_.+c_.*F_[d_.+e_.*x_]^r_.)^n_.,x_Symbol] := + Int[ActivateTrig[u*F[d+e*x]^(n*p)*(a+b*F[d+e*x]^(q-p)+c*F[d+e*x]^(r-p))^n],x] /; +FreeQ[{a,b,c,d,e,p,q,r},x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q-p] && PosQ[r-p] + + +Int[u_*(a_+b_.*F_[d_.+e_.*x_]^p_.+c_.*F_[d_.+e_.*x_]^q_.)^n_.,x_Symbol] := + Int[ActivateTrig[u*F[d+e*x]^(n*p)*(b+a*F[d+e*x]^(-p)+c*F[d+e*x]^(q-p))^n],x] /; +FreeQ[{a,b,c,d,e,p,q},x] && InertTrigQ[F] && IntegerQ[n] && NegQ[p] + + +Int[u_.*(a_.*cos[c_.+d_.*x_]+b_.*sin[c_.+d_.*x_])^n_.,x_Symbol] := + Int[ActivateTrig[u]*(a*E^(-a/b*(c+d*x)))^n,x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[a^2+b^2] + + +Int[u_,x_Symbol] := + Int[TrigSimplify[u],x] /; +TrigSimplifyQ[u] + + +Int[u_.*(a_*v_)^p_,x_Symbol] := + With[{uu=ActivateTrig[u],vv=ActivateTrig[v]}, + a^IntPart[p]*(a*vv)^FracPart[p]/(vv^FracPart[p])*Int[uu*vv^p,x]] /; +FreeQ[{a,p},x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]] + + +Int[u_.*(v_^m_)^p_,x_Symbol] := + With[{uu=ActivateTrig[u],vv=ActivateTrig[v]}, + (vv^m)^FracPart[p]/(vv^(m*FracPart[p]))*Int[uu*vv^(m*p),x]] /; +FreeQ[{m,p},x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]] + + +Int[u_.*(v_^m_.*w_^n_.)^p_,x_Symbol] := + With[{uu=ActivateTrig[u],vv=ActivateTrig[v],ww=ActivateTrig[w]}, + (vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))*Int[uu*vv^(m*p)*ww^(n*p),x]] /; +FreeQ[{m,n,p},x] && Not[IntegerQ[p]] && (Not[InertTrigFreeQ[v]] || Not[InertTrigFreeQ[w]]) + + +Int[u_,x_Symbol] := + With[{v=ExpandTrig[u,x]}, + Int[v,x] /; + SumQ[v]] /; +Not[InertTrigFreeQ[u]] + + +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{w=Block[{ShowSteps=False,$StepCounter=Null}, + Int[SubstFor[1/(1+FreeFactors[Tan[FunctionOfTrig[u,x]/2],x]^2*x^2),Tan[FunctionOfTrig[u,x]/2]/FreeFactors[Tan[FunctionOfTrig[u,x]/2],x],u,x],x]]}, + ShowStep["","Int[F[Sin[a+b*x],Cos[a+b*x]],x]","2/b*Subst[Int[1/(1+x^2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x,Tan[(a+b*x)/2]]",Hold[ + Module[{v=FunctionOfTrig[u,x],d}, + d=FreeFactors[Tan[v/2],x]; + Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/2]/d,u,x],x],x,Tan[v/2]/d],x]]]] /; + FreeQ[w,Int]] /; +SimplifyFlag && InverseFunctionFreeQ[u,x] && Not[FalseQ[FunctionOfTrig[u,x]]], + +Int[u_,x_Symbol] := + With[{w=Block[{ShowSteps=False,$StepCounter=Null}, + Int[SubstFor[1/(1+FreeFactors[Tan[FunctionOfTrig[u,x]/2],x]^2*x^2),Tan[FunctionOfTrig[u,x]/2]/FreeFactors[Tan[FunctionOfTrig[u,x]/2],x],u,x],x]]}, + Module[{v=FunctionOfTrig[u,x],d}, + d=FreeFactors[Tan[v/2],x]; + Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/2]/d,u,x],x],x,Tan[v/2]/d],x]] /; + FreeQ[w,Int]] /; +InverseFunctionFreeQ[u,x] && Not[FalseQ[FunctionOfTrig[u,x]]]] + + +(* If[ShowSteps, + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + ShowStep["","Int[F[Sin[a+b*x],Cos[a+b*x]],x]","2/b*Subst[Int[1/(1+x^2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x,Tan[(a+b*x)/2]]",Hold[ + With[{d=FreeFactors[Tan[v/2],x]}, + Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/2]/d,u,x],x],x,Tan[v/2]/d],x]]]] /; + Not[FalseQ[v]]] /; +SimplifyFlag && InverseFunctionFreeQ[u,x], + +Int[u_,x_Symbol] := + With[{v=FunctionOfTrig[u,x]}, + With[{d=FreeFactors[Tan[v/2],x]}, + Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/2]/d,u,x],x],x,Tan[v/2]/d],x]] /; + Not[FalseQ[v]]] /; +InverseFunctionFreeQ[u,x]] *) + + +Int[u_,x_Symbol] := + With[{v=ActivateTrig[u]}, + Defer[Int][v,x]] /; +Not[InertTrigFreeQ[u]] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.5 (c+d x)^m trig(a+b x)^n trig(a+b x)^p*) + + +Int[(c_.+d_.*x_)^m_.*Sin[a_.+b_.*x_]^n_.*Cos[a_.+b_.*x_],x_Symbol] := + (c+d*x)^m*Sin[a+b*x]^(n+1)/(b*(n+1)) - + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Sin[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sin[a_.+b_.*x_]*Cos[a_.+b_.*x_]^n_.,x_Symbol] := + -(c+d*x)^m*Cos[a+b*x]^(n+1)/(b*(n+1)) + + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Cos[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sin[a_.+b_.*x_]^n_.*Cos[a_.+b_.*x_]^p_.,x_Symbol] := + Int[ExpandTrigReduce[(c+d*x)^m,Sin[a+b*x]^n*Cos[a+b*x]^p,x],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Sin[a_.+b_.*x_]^n_.*Tan[a_.+b_.*x_]^p_.,x_Symbol] := + -Int[(c+d*x)^m*Sin[a+b*x]^n*Tan[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Sin[a+b*x]^(n-2)*Tan[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Cos[a_.+b_.*x_]^n_.*Cot[a_.+b_.*x_]^p_.,x_Symbol] := + -Int[(c+d*x)^m*Cos[a+b*x]^n*Cot[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Cos[a+b*x]^(n-2)*Cot[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Sec[a_.+b_.*x_]^n_.*Tan[a_.+b_.*x_]^p_.,x_Symbol] := + (c+d*x)^m*Sec[a+b*x]^n/(b*n) - + d*m/(b*n)*Int[(c+d*x)^(m-1)*Sec[a+b*x]^n,x] /; +FreeQ[{a,b,c,d,n},x] && p===1 && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^n_.*Cot[a_.+b_.*x_]^p_.,x_Symbol] := + -(c+d*x)^m*Csc[a+b*x]^n/(b*n) + + d*m/(b*n)*Int[(c+d*x)^(m-1)*Csc[a+b*x]^n,x] /; +FreeQ[{a,b,c,d,n},x] && p===1 && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*Sec[a_.+b_.*x_]^2*Tan[a_.+b_.*x_]^n_.,x_Symbol] := + (c+d*x)^m*Tan[a+b*x]^(n+1)/(b*(n+1)) - + d*m/(b*(n +1))*Int[(c+d*x)^(m-1)*Tan[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^2*Cot[a_.+b_.*x_]^n_.,x_Symbol] := + -(c+d*x)^m*Cot[a+b*x]^(n+1)/(b*(n+1)) + + d*m/(b*(n +1))*Int[(c+d*x)^(m-1)*Cot[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sec[a_.+b_.*x_]*Tan[a_.+b_.*x_]^p_,x_Symbol] := + -Int[(c+d*x)^m*Sec[a+b*x]*Tan[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Sec[a+b*x]^3*Tan[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Sec[a_.+b_.*x_]^n_.*Tan[a_.+b_.*x_]^p_,x_Symbol] := + -Int[(c+d*x)^m*Sec[a+b*x]^n*Tan[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Sec[a+b*x]^(n+2)*Tan[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m,n},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]*Cot[a_.+b_.*x_]^p_,x_Symbol] := + -Int[(c+d*x)^m*Csc[a+b*x]*Cot[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Csc[a+b*x]^3*Cot[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^n_.*Cot[a_.+b_.*x_]^p_,x_Symbol] := + -Int[(c+d*x)^m*Csc[a+b*x]^n*Cot[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Csc[a+b*x]^(n+2)*Cot[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m,n},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Sec[a_.+b_.*x_]^n_.*Tan[a_.+b_.*x_]^p_.,x_Symbol] := + Module[{u=IntHide[Sec[a+b*x]^n*Tan[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d,n,p},x] && PositiveIntegerQ[m] && (EvenQ[n] || OddQ[p]) + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^n_.*Cot[a_.+b_.*x_]^p_.,x_Symbol] := + Module[{u=IntHide[Csc[a+b*x]^n*Cot[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d,n,p},x] && PositiveIntegerQ[m] && (EvenQ[n] || OddQ[p]) + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^n_.*Sec[a_.+b_.*x_]^n_., x_Symbol] := + 2^n*Int[(c+d*x)^m*Csc[2*a+2*b*x]^n,x] /; +FreeQ[{a,b,c,d},x] && RationalQ[m] && IntegerQ[n] + + +Int[(c_.+d_.*x_)^m_.*Csc[a_.+b_.*x_]^n_.*Sec[a_.+b_.*x_]^p_., x_Symbol] := + Module[{u=IntHide[Csc[a+b*x]^n*Sec[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && RationalQ[m] && m>0 && n!=p + + +Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*F[ExpandToSum[v,x]]^n*G[ExpandToSum[v,x]]^p,x] /; +FreeQ[{m,n,p},x] && TrigQ[F] && TrigQ[G] && EqQ[v-w] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]] + + +Int[(e_.+f_.*x_)^m_.*Cos[c_.+d_.*x_]*(a_+b_.*Sin[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Sin[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Sin[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sin[c_.+d_.*x_]*(a_+b_.*Cos[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Cos[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Cos[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sec[c_.+d_.*x_]^2*(a_+b_.*Tan[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Tan[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Tan[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Csc[c_.+d_.*x_]^2*(a_+b_.*Cot[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Cot[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Cot[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sec[c_.+d_.*x_]*Tan[c_.+d_.*x_]*(a_+b_.*Sec[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Sec[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Sec[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Csc[c_.+d_.*x_]*Cot[c_.+d_.*x_]*(a_+b_.*Csc[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Csc[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Csc[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sin[a_.+b_.*x_]^p_.*Sin[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Sin[a+b*x]^p*Sin[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[p,q] && IntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*Cos[a_.+b_.*x_]^p_.*Cos[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Cos[a+b*x]^p*Cos[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[p,q] && IntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*Sin[a_.+b_.*x_]^p_.*Cos[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Sin[a+b*x]^p*Cos[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[p,q] + + +Int[(e_.+f_.*x_)^m_.*F_[a_.+b_.*x_]^p_.*G_[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigExpand[(e+f*x)^m*G[c+d*x]^q,F,c+d*x,p,b/d,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && MemberQ[{Sin,Cos},F] && MemberQ[{Sec,Csc},G] && + PositiveIntegerQ[p,q] && EqQ[b*c-a*d] && PositiveIntegerQ[b/d-1] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.6 F^(c (a+b x)) trig(d+e x)^n*) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_],x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Sin[d+e*x]/(e^2+b^2*c^2*Log[F]^2) - + e*F^(c*(a+b*x))*Cos[d+e*x]/(e^2+b^2*c^2*Log[F]^2) /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2+b^2*c^2*Log[F]^2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_],x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Cos[d+e*x]/(e^2+b^2*c^2*Log[F]^2) + + e*F^(c*(a+b*x))*Sin[d+e*x]/(e^2+b^2*c^2*Log[F]^2) /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2+b^2*c^2*Log[F]^2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Sin[d+e*x]^n/(e^2*n^2+b^2*c^2*Log[F]^2) - + e*n*F^(c*(a+b*x))*Cos[d+e*x]*Sin[d+e*x]^(n-1)/(e^2*n^2+b^2*c^2*Log[F]^2) + + (n*(n-1)*e^2)/(e^2*n^2+b^2*c^2*Log[F]^2)*Int[F^(c*(a+b*x))*Sin[d+e*x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n>1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^m_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Cos[d+e*x]^m/(e^2*m^2+b^2*c^2*Log[F]^2) + + e*m*F^(c*(a+b*x))*Sin[d+e*x]*Cos[d+e*x]^(m-1)/(e^2*m^2+b^2*c^2*Log[F]^2) + + (m*(m-1)*e^2)/(e^2*m^2+b^2*c^2*Log[F]^2)*Int[F^(c*(a+b*x))*Cos[d+e*x]^(m-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*m^2+b^2*c^2*Log[F]^2] && RationalQ[m] && m>1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sin[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) + + F^(c*(a+b*x))*Cos[d+e*x]*Sin[d+e*x]^(n+1)/(e*(n+1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n+2)^2+b^2*c^2*Log[F]^2] && NeQ[n+1] && NeQ[n+2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Cos[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) - + F^(c*(a+b*x))*Sin[d+e*x]*Cos[d+e*x]^(n+1)/(e*(n+1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n+2)^2+b^2*c^2*Log[F]^2] && NeQ[n+1] && NeQ[n+2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sin[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) + + F^(c*(a+b*x))*Cos[d+e*x]*Sin[d+e*x]^(n+1)/(e*(n+1)) + + (e^2*(n+2)^2+b^2*c^2*Log[F]^2)/(e^2*(n+1)*(n+2))*Int[F^(c*(a+b*x))*Sin[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n+2)^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 && n!=-2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Cos[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) - + F^(c*(a+b*x))*Sin[d+e*x]*Cos[d+e*x]^(n+1)/(e*(n+1)) + + (e^2*(n+2)^2+b^2*c^2*Log[F]^2)/(e^2*(n+1)*(n+2))*Int[F^(c*(a+b*x))*Cos[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n+2)^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 && n!=-2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_,x_Symbol] := + E^(I*n*(d+e*x))*Sin[d+e*x]^n/(-1+E^(2*I*(d+e*x)))^n*Int[F^(c*(a+b*x))*(-1+E^(2*I*(d+e*x)))^n/E^(I*n*(d+e*x)),x] /; +FreeQ[{F,a,b,c,d,e,n},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_,x_Symbol] := + E^(I*n*(d+e*x))*Cos[d+e*x]^n/(1+E^(2*I*(d+e*x)))^n*Int[F^(c*(a+b*x))*(1+E^(2*I*(d+e*x)))^n/E^(I*n*(d+e*x)),x] /; +FreeQ[{F,a,b,c,d,e,n},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Tan[d_.+e_.*x_]^n_.,x_Symbol] := + I^n*Int[ExpandIntegrand[F^(c*(a+b*x))*(1-E^(2*I*(d+e*x)))^n/(1+E^(2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cot[d_.+e_.*x_]^n_.,x_Symbol] := + (-I)^n*Int[ExpandIntegrand[F^(c*(a+b*x))*(1+E^(2*I*(d+e*x)))^n/(1-E^(2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*(Sec[d+e x]^n/(e^2*n^2+b^2*c^2*Log[F]^2)) - + e*n*F^(c*(a+b*x))*Sec[d+e x]^(n+1)*(Sin[d+e x]/(e^2*n^2+b^2*c^2*Log[F]^2)) + + e^2*n*((n+1)/(e^2*n^2+b^2*c^2*Log[F]^2))*Int[F^(c*(a+b*x))*Sec[d+e x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*(Csc[d+e x]^n/(e^2*n^2+b^2*c^2*Log[F]^2)) + + e*n*F^(c*(a+b*x))*Csc[d+e x]^(n+1)*(Cos[d+e x]/(e^2*n^2+b^2*c^2*Log[F]^2)) + + e^2*n*((n+1)/(e^2*n^2+b^2*c^2*Log[F]^2))*Int[F^(c*(a+b*x))*Csc[d+e x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sec[d+e x]^(n-2)/(e^2*(n-1)*(n-2)) + + F^(c*(a+b*x))*Sec[d+e x]^(n-1)*Sin[d+e x]/(e*(n-1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[b^2*c^2*Log[F]^2+e^2*(n-2)^2] && NeQ[n-1] && NeQ[n-2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Csc[d+e x]^(n-2)/(e^2*(n-1)*(n-2)) + + F^(c*(a+b*x))*Csc[d+e x]^(n-1)*Cos[d+e x]/(e*(n-1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[b^2*c^2*Log[F]^2+e^2*(n-2)^2] && NeQ[n-1] && NeQ[n-2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sec[d+e x]^(n-2)/(e^2*(n-1)*(n-2)) + + F^(c*(a+b*x))*Sec[d+e x]^(n-1)*Sin[d+e x]/(e*(n-1)) + + (e^2*(n-2)^2+b^2*c^2*Log[F]^2)/(e^2*(n-1)*(n-2))*Int[F^(c*(a+b*x))*Sec[d+e x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[b^2*c^2*Log[F]^2+e^2*(n-2)^2] && RationalQ[n] && n>1 && n!=2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Csc[d+e x]^(n-2)/(e^2*(n-1)*(n-2)) - + F^(c*(a+b*x))*Csc[d+e x]^(n-1)*Cos[d+e x]/(e*(n-1)) + + (e^2*(n-2)^2+b^2*c^2*Log[F]^2)/(e^2*(n-1)*(n-2))*Int[F^(c*(a+b*x))*Csc[d+e x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[b^2*c^2*Log[F]^2+e^2*(n-2)^2] && RationalQ[n] && n>1 && n!=2 + + +(* Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_.,x_Symbol] := + 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(I*n*(d+e*x))/(1+E^(2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *) + + +(* Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_.,x_Symbol] := + (2*I)^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-I*n*(d+e*x))/(1-E^(-2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+k_.*Pi+e_.*x_]^n_.,x_Symbol] := + 2^n*E^(I*k*n*Pi)*E^(I*n*(d+e*x))*F^(c*(a+b*x))/(I*e*n+b*c*Log[F])* + Hypergeometric2F1[n,n/2-I*b*c*Log[F]/(2*e),1+n/2-I*b*c*Log[F]/(2*e),-E^(2*I*k*Pi)*E^(2*I*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[4*k] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_.,x_Symbol] := + 2^n*E^(I*n*(d+e*x))*F^(c*(a+b*x))/(I*e*n+b*c*Log[F])* + Hypergeometric2F1[n,n/2-I*b*c*Log[F]/(2*e),1+n/2-I*b*c*Log[F]/(2*e),-E^(2*I*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+k_.*Pi+e_.*x_]^n_.,x_Symbol] := + (-2*I)^n*E^(I*k*n*Pi)*E^(I*n*(d+e*x))*(F^(c*(a+b*x))/(I*e*n+b*c*Log[F]))* + Hypergeometric2F1[n,n/2-I*b*c*Log[F]/(2*e),1+n/2-I*b*c*Log[F]/(2*e),E^(2*I*k*Pi)*E^(2*I*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[4*k] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_.,x_Symbol] := + (-2*I)^n*E^(I*n*(d+e*x))*(F^(c*(a+b*x))/(I*e*n+b*c*Log[F]))* + Hypergeometric2F1[n,n/2-I*b*c*Log[F]/(2*e),1+n/2-I*b*c*Log[F]/(2*e),E^(2*I*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_.,x_Symbol] := + (1+E^(2*I*(d+e*x)))^n*Sec[d+e*x]^n/E^(I*n*(d+e*x))*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(I*n*(d+e*x))/(1+E^(2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_.,x_Symbol] := + (1-E^(-2*I*(d+e*x)))^n*Csc[d+e*x]^n/E^(-I*n*(d+e*x))*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-I*n*(d+e*x))/(1-E^(-2*I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Sin[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*f^n*Int[F^(c*(a+b*x))*Cos[d/2+e*x/2-f*Pi/(4*g)]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f^2-g^2] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Cos[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*f^n*Int[F^(c*(a+b*x))*Cos[d/2+e*x/2]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f-g] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Cos[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*f^n*Int[F^(c*(a+b*x))*Sin[d/2+e*x/2]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f+g] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^m_.*(f_+g_.*Sin[d_.+e_.*x_])^n_.,x_Symbol] := + g^n*Int[F^(c*(a+b*x))*Tan[f*Pi/(4*g)-d/2-e*x/2]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f^2-g^2] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^m_.*(f_+g_.*Cos[d_.+e_.*x_])^n_.,x_Symbol] := + f^n*Int[F^(c*(a+b*x))*Tan[d/2+e*x/2]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f-g] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^m_.*(f_+g_.*Cos[d_.+e_.*x_])^n_.,x_Symbol] := + f^n*Int[F^(c*(a+b*x))*Cot[d/2+e*x/2]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f+g] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*(h_+i_.*Cos[d_.+e_.*x_])/(f_+g_.*Sin[d_.+e_.*x_]),x_Symbol] := + 2*i*Int[F^(c*(a+b*x))*(Cos[d+e*x]/(f+g*Sin[d+e*x])),x] + + Int[F^(c*(a+b*x))*((h-i*Cos[d+e*x])/(f+g*Sin[d+e*x])),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,i},x] && EqQ[f^2-g^2] && EqQ[h^2-i^2] && EqQ[g*h-f*i] + + +Int[F_^(c_.*(a_.+b_.*x_))*(h_+i_.*Sin[d_.+e_.*x_])/(f_+g_.*Cos[d_.+e_.*x_]),x_Symbol] := + 2*i*Int[F^(c*(a+b*x))*(Sin[d+e*x]/(f+g*Cos[d+e*x])),x] + + Int[F^(c*(a+b*x))*((h-i*Sin[d+e*x])/(f+g*Cos[d+e*x])),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,i},x] && EqQ[f^2-g^2] && EqQ[h^2-i^2] && EqQ[g*h+f*i] + + +Int[F_^(c_.*u_)*G_[v_]^n_.,x_Symbol] := + Int[F^(c*ExpandToSum[u,x])*G[ExpandToSum[v,x]]^n,x] /; +FreeQ[{F,c,n},x] && TrigQ[G] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_.,x_Symbol] := + Module[{u=IntHide[F^(c*(a+b*x))*Sin[d+e*x]^n,x]}, + Dist[(f*x)^m,u,x] - f*m*Int[(f*x)^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] && GtQ[m,0] + + +Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_.,x_Symbol] := + Module[{u=IntHide[F^(c*(a+b*x))*Cos[d+e*x]^n,x]}, + Dist[(f*x)^m,u,x] - f*m*Int[(f*x)^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] && GtQ[m,0] + + +Int[(f_.*x_)^m_*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_],x_Symbol] := + (f*x)^(m+1)/(f*(m+1))*F^(c*(a+b*x))*Sin[d+e*x] - + e/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Cos[d+e*x],x] - + b*c*Log[F]/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Sin[d+e*x],x] /; +FreeQ[{F,a,b,c,d,e,f,m},x] && (LtQ[m,-1] || SumSimplerQ[m,1]) + + +Int[(f_.*x_)^m_*F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_],x_Symbol] := + (f*x)^(m+1)/(f*(m+1))*F^(c*(a+b*x))*Cos[d+e*x] + + e/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Sin[d+e*x],x] - + b*c*Log[F]/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Cos[d+e*x],x] /; +FreeQ[{F,a,b,c,d,e,f,m},x] && (LtQ[m,-1] || SumSimplerQ[m,1]) + + +(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_.,x_Symbol] := + I^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))-E^(I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *) + + +(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_.,x_Symbol] := + 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))+E^(I*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^m_.*Cos[f_.+g_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[F^(c*(a+b*x)),Sin[d+e*x]^m*Cos[f+g*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && PositiveIntegerQ[m,n] + + +Int[x_^p_.*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^m_.*Cos[f_.+g_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[x^p*F^(c*(a+b*x)),Sin[d+e*x]^m*Cos[f+g*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && PositiveIntegerQ[m,n,p] + + +Int[F_^(c_.*(a_.+b_.*x_))*G_[d_.+e_.*x_]^m_.*H_[d_.+e_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^(c*(a+b*x)),G[d+e*x]^m*H[d+e*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && PositiveIntegerQ[m,n] && TrigQ[G] && TrigQ[H] + + +Int[F_^u_*Sin[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Sin[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[n] + + +Int[F_^u_*Cos[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Cos[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[n] + + +Int[F_^u_*Sin[v_]^m_.*Cos[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Sin[v]^m*Cos[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[m,n] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.7 x^m trig(a+b log(c x^n))^p*) + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(p+2)*Sin[a+b*Log[c*x^n]]^(p+2)/(p+1) + + x*Cot[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p+2)^2+1] && NeQ[p+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(p+2)*Cos[a+b*Log[c*x^n]]^(p+2)/(p+1) - + x*Tan[a+b*Log[c*x^n]]*Cos[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p+2)^2+1] && NeQ[p+1] + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[ExpandIntegrand[(E^(a*b*n*p)/(2*b*n*p)*(c*x^n)^(-1/(n*p))-E^(-a*b*n*p)/(2*b*n*p)*(c*x^n)^(1/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[ExpandIntegrand[(E^(a*b*n*p)/2*(c*x^n)^(-1/(n*p))-E^(-a*b*n*p)/2*(c*x^n)^(1/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2+1] + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + x*Sin[a+b*Log[c*x^n]]/(b^2*n^2+1) - + b*n*x*Cos[a+b*Log[c*x^n]]/(b^2*n^2+1) /; +FreeQ[{a,b,c,n},x] && NeQ[b^2*n^2+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + x*Cos[a+b*Log[c*x^n]]/(1+b^2*n^2) + + b*n*x*Sin[a+b*Log[c*x^n]]/(b^2*n^2+1) /; +FreeQ[{a,b,c,n},x] && NeQ[b^2*n^2+1] + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Sin[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+1) - + b*n*p*x*Cos[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2+1) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2+1)*Int[Sin[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Cos[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+1) + + b*n*p*x*Cos[a+b*Log[c*x^n]]^(p-1)*Sin[a+b*Log[c*x^n]]/(b^2*n^2*p^2+1) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2+1)*Int[Cos[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2+1] + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Cot[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + x*Sin[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2+1)/(b^2*n^2*(p+1)*(p+2))*Int[Sin[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Tan[a+b*Log[c*x^n]]*Cos[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + x*Cos[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2+1)/(b^2*n^2*(p+1)*(p+2))*Int[Cos[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2+1] + + +Int[Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(I/(E^(I*a)*(c*x^n)^(I*b))-I*E^(I*a)*(c*x^n)^(I*b))^p/((1-I*b*n*p)*(2-2*E^(2*I*a)*(c*x^n)^(2*I*b))^p)* + Hypergeometric2F1[-p,(1-I*b*n*p)/(2*I*b*n),1+(1-I*b*n*p)/(2*I*b*n),E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2+1] + + +Int[Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(1/(E^(I*a)*(c*x^n)^(I*b))+E^(I*a)*(c*x^n)^(I*b))^p/((1-I*b*n*p)*(2+2*E^(2*I*a)*(c*x^n)^(2*I*b))^p)* + Hypergeometric2F1[-p,(1-I*b*n*p)/(2*I*b*n),1+(1-I*b*n*p)/(2*I*b*n),-E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2+1] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p+2)*x^(m+1)*Sin[a+b*Log[c*x^n]]^(p+2)/((m+1)*(p+1)) + + x^(m+1)*Cot[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p+2)^2+(m+1)^2] && NeQ[p+1] && NeQ[m+1] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p+2)*x^(m+1)*Cos[a+b*Log[c*x^n]]^(p+2)/((m+1)*(p+1)) - + x^(m+1)*Tan[a+b*Log[c*x^n]]*Cos[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p+2)^2+(m+1)^2] && NeQ[p+1] && NeQ[m+1] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 1/2^p*Int[ExpandIntegrand[x^m*((m+1)/(b*n*p)*E^(a*b*n*p/(m+1))*(c*x^n)^(-(m+1)/(n*p)) - + (m+1)/(b*n*p)*E^(-a*b*n*p/(m+1))*(c*x^n)^((m+1)/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,m,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 1/2^p*Int[ExpandIntegrand[x^m*(E^(a*b*n*p/(m+1))*(c*x^n)^(-(m+1)/(n*p)) - + E^(-a*b*n*p/(m+1))*(c*x^n)^((m+1)/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,m,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + (m+1)*x^(m+1)*Sin[a+b*Log[c*x^n]]/(b^2*n^2+(m+1)^2) - + b*n*x^(m+1)*Cos[a+b*Log[c*x^n]]/(b^2*n^2+(m+1)^2) /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2+(m+1)^2] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + (m+1)*x^(m+1)*Cos[a+b*Log[c*x^n]]/(b^2*n^2+(m+1)^2) + + b*n*x^(m+1)*Sin[a+b*Log[c*x^n]]/(b^2*n^2+(m+1)^2) /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2+(m+1)^2] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (m+1)*x^(m+1)*Sin[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+(m+1)^2) - + b*n*p*x^(m+1)*Cos[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2+(m+1)^2) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2+(m+1)^2)*Int[x^m*Sin[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (m+1)*x^(m+1)*Cos[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+(m+1)^2) + + b*n*p*x^(m+1)*Sin[a+b*Log[c*x^n]]*Cos[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2+(m+1)^2) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2+(m+1)^2)*Int[x^m*Cos[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*Cot[a+b*Log[c*x^n]]*Sin[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + (m+1)*x^(m+1)*Sin[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2+(m+1)^2)/(b^2*n^2*(p+1)*(p+2))*Int[x^m*Sin[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2+(m+1)^2] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x^(m+1)*Tan[a+b*Log[c*x^n]]*Cos[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + (m+1)*x^(m+1)*Cos[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2+(m+1)^2)/(b^2*n^2*(p+1)*(p+2))*Int[x^m*Cos[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2+(m+1)^2] + + +Int[x_^m_.*Sin[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*(I*E^(-I*a)*(c*x^n)^(-I*b)-I*E^(I*a)*(c*x^n)^(I*b))^p/((m+1-I*b*n*p)*(2-2*E^(2*I*a)*(c*x^n)^(2*I*b))^p)* + Hypergeometric2F1[-p,(m+1-I*b*n*p)/(2*I*b*n),1+(m+1-I*b*n*p)/(2*I*b*n),E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Cos[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*(E^(-I*a)*(c*x^n)^(-I*b)+E^(I*a)*(c*x^n)^(I*b))^p/((m+1-I*b*n*p)*(2+2*E^(2*I*a)*(c*x^n)^(2*I*b))^p)* + Hypergeometric2F1[-p,(m+1-I*b*n*p)/(2*I*b*n),1+(m+1-I*b*n*p)/(2*I*b*n),-E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[Sec[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*E^(a*b*n)*Int[(c*x^n)^(1/n)/(E^(2*a*b*n)+(c*x^n)^(2/n)),x] /; +FreeQ[{a,b,c,n},x] && EqQ[b^2*n^2+1] + + +Int[Csc[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*b*n*E^(a*b*n)*Int[(c*x^n)^(1/n)/(E^(2*a*b*n)-(c*x^n)^(2/n)),x] /; +FreeQ[{a,b,c,n},x] && EqQ[b^2*n^2+1] + + +Int[Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x*Sec[a+b*Log[c*x^n]]^(p-2)/(p-1) + + x*Tan[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p-2)^2+1] && NeQ[p-1] + + +Int[Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x*Csc[a+b*Log[c*x^n]]^(p-2)/(p-1) - + x*Cot[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p-2)^2+1] && NeQ[p-1] + + +Int[Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Tan[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + x*Sec[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2+1)/(b^2*n^2*(p-1)*(p-2))*Int[Sec[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2+1] + + +Int[Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Cot[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + x*Csc[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2+1)/(b^2*n^2*(p-1)*(p-2))*Int[Csc[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2+1] + + +Int[Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -b*n*p*x*Sin[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2+1) + + x*Sec[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+1) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2+1)*Int[Sec[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2+1] + + +Int[Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + b*n*p*x*Cos[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2+1) + + x*Csc[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+1) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2+1)*Int[Csc[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2+1] + + +Int[Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + x*(2+2*E^(2*I*a)*(c*x^n)^(2*I*b))^p/(1+I*b*n*p)* + (E^(I*a)*(c*x^n)^(I*b)/(1+E^(2*I*a)*(c*x^n)^(2*I*b)))^p* + Hypergeometric2F1[p,(1+I*b*n*p)/(2*I*b*n),1+(1+I*b*n*p)/(2*I*b*n),-E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2+1] + + +Int[Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + x*(2-2*E^(2*I*a)*(c*x^n)^(2*I*b))^p/(1+I*b*n*p)* + (-I*E^(I*a)*(c*x^n)^(I*b)/(1-E^(2*I*a)*(c*x^n)^(2*I*b)))^p* + Hypergeometric2F1[p,(1+I*b*n*p)/(2*I*b*n),1+(1+I*b*n*p)/(2*I*b*n),E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2+1] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*E^(a*b*n/(m+1))*Int[x^m*(c*x^n)^((m+1)/n)/(E^(2*a*b*n/(m+1))+(c*x^n)^(2*(m+1)/n)),x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[b^2*n^2+(m+1)^2] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*b*n/(m+1)*E^(a*b*n/(m+1))*Int[x^m*(c*x^n)^((m+1)/n)/(E^(2*a*b*n/(m+1))-(c*x^n)^(2*(m+1)/n)),x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[b^2*n^2+(m+1)^2] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x^(m+1)*Sec[a+b*Log[c*x^n]]^(p-2)/((m+1)*(p-1)) + + x^(m+1)*Tan[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p-2)^2+(m+1)^2] && NeQ[m+1] && NeQ[p-1] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x^(m+1)*Csc[a+b*Log[c*x^n]]^(p-2)/((m+1)*(p-1)) - + x^(m+1)*Cot[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p-2)^2+(m+1)^2] && NeQ[m+1] && NeQ[p-1] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*Tan[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + (m+1)*x^(m+1)*Sec[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2+(m+1)^2)/(b^2*n^2*(p-1)*(p-2))*Int[x^m*Sec[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2+(m+1)^2] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x^(m+1)*Cot[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + (m+1)*x^(m+1)*Csc[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2+(m+1)^2)/(b^2*n^2*(p-1)*(p-2))*Int[x^m*Csc[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2+(m+1)^2] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -b*n*p*x^(m+1)*Sin[a+b*Log[c*x^n]]*Sec[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2+(m+1)^2) + + (m+1)*x^(m+1)*Sec[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+(m+1)^2) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2+(m+1)^2)*Int[x^m*Sec[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + b*n*p*x^(m+1)*Cos[a+b*Log[c*x^n]]*Csc[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2+(m+1)^2) + + (m+1)*x^(m+1)*Csc[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2+(m+1)^2) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2+(m+1)^2)*Int[x^m*Csc[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + x^(m+1)*(2+2*E^(2*I*a)*(c*x^n)^(2*I*b))^p/(m+1+I*b*n*p)* + (E^(I*a)*(c*x^n)^(I*b)/(1+E^(2*I*a)*(c*x^n)^(2*I*b)))^p* + Hypergeometric2F1[p,(m+1+I*b*n*p)/(2*I*b*n),1+(m+1+I*b*n*p)/(2*I*b*n),-E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + x^(m+1)*(2-2*E^(2*I*a)*(c*x^n)^(2*I*b))^p/(m+1+I*b*n*p)* + (-I*E^(I*a)*(c*x^n)^(I*b)/(1-E^(2*I*a)*(c*x^n)^(2*I*b)))^p* + Hypergeometric2F1[p,(m+1+I*b*n*p)/(2*I*b*n),1+(m+1+I*b*n*p)/(2*I*b*n),E^(2*I*a)*(c*x^n)^(2*I*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2+(m+1)^2] + + +Int[Sin[a_.*x_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + -Cos[a*x*Log[b*x]^p]/a - + p*Int[Sin[a*x*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b},x] && RationalQ[p] && p>0 + + +Int[Cos[a_.*x_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sin[a*x*Log[b*x]^p]/a - + p*Int[Cos[a*x*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b},x] && RationalQ[p] && p>0 + + +Int[Sin[a_.*x_^n_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + -Cos[a*x^n*Log[b*x]^p]/(a*n*x^(n-1)) - + p/n*Int[Sin[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] - + (n-1)/(a*n)*Int[Cos[a*x^n*Log[b*x]^p]/x^n,x] /; +FreeQ[{a,b},x] && RationalQ[n,p] && p>0 + + +Int[Cos[a_.*x_^n_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sin[a*x^n*Log[b*x]^p]/(a*n*x^(n-1)) - + p/n*Int[Cos[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] + + (n-1)/(a*n)*Int[Sin[a*x^n*Log[b*x]^p]/x^n,x] /; +FreeQ[{a,b},x] && RationalQ[n,p] && p>0 + + +Int[x_^m_.*Sin[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + -Cos[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Sin[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-n+1] && RationalQ[p] && p>0 + + +Int[x_^m_.*Cos[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sin[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Cos[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-n+1] && RationalQ[p] && p>0 + + +Int[x_^m_.*Sin[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + -x^(m-n+1)*Cos[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Sin[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] + + (m-n+1)/(a*n)*Int[x^(m-n)*Cos[a*x^n*Log[b*x]^p],x] /; +FreeQ[{a,b,m,n},x] && RationalQ[p] && p>0 && NeQ[m-n+1] + + +Int[x_^m_*Cos[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + x^(m-n+1)*Sin[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Cos[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] - + (m-n+1)/(a*n)*Int[x^(m-n)*Sin[a*x^n*Log[b*x]^p],x] /; +FreeQ[{a,b,m,n},x] && RationalQ[p] && p>0 && NeQ[m-n+1] + + + + + +(* ::Subsection::Closed:: *) +(*4.4.8 Active trig functions*) + + +Int[Sin[a_./(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Sin[a*x]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,c,d},x] && PositiveIntegerQ[n] + + +Int[Cos[a_./(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Cos[a*x]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,c,d},x] && PositiveIntegerQ[n] + + +Int[Sin[e_.*(a_.+b_.*x_)/(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Sin[b*e/d-e*(b*c-a*d)*x/d]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[n] && NeQ[b*c-a*d] + + +Int[Cos[e_.*(a_.+b_.*x_)/(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Cos[b*e/d-e*(b*c-a*d)*x/d]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[n] && NeQ[b*c-a*d] + + +Int[Sin[u_]^n_.,x_Symbol] := + Module[{lst=QuotientOfLinearsParts[u,x]}, + Int[Sin[(lst[[1]]+lst[[2]]*x)/(lst[[3]]+lst[[4]]*x)]^n,x]] /; +PositiveIntegerQ[n] && QuotientOfLinearsQ[u,x] + + +Int[Cos[u_]^n_.,x_Symbol] := + Module[{lst=QuotientOfLinearsParts[u,x]}, + Int[Cos[(lst[[1]]+lst[[2]]*x)/(lst[[3]]+lst[[4]]*x)]^n,x]] /; +PositiveIntegerQ[n] && QuotientOfLinearsQ[u,x] + + +Int[u_.*Sin[v_]^p_.*Sin[w_]^q_.,x_Symbol] := + Int[u*Sin[v]^(p+q),x] /; +EqQ[v-w] + + +Int[u_.*Cos[v_]^p_.*Cos[w_]^q_.,x_Symbol] := + Int[u*Cos[v]^(p+q),x] /; +EqQ[v-w] + + +Int[Sin[v_]^p_.*Sin[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q,x],x] /; +(PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) && PositiveIntegerQ[p,q] + + +Int[Cos[v_]^p_.*Cos[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Cos[v]^p*Cos[w]^q,x],x] /; +(PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) && PositiveIntegerQ[p,q] + + +Int[x_^m_.*Sin[v_]^p_.*Sin[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Sin[v]^p*Sin[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[x_^m_.*Cos[v_]^p_.*Cos[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Cos[v]^p*Cos[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[u_.*Sin[v_]^p_.*Cos[w_]^p_.,x_Symbol] := + 1/2^p*Int[u*Sin[2*v]^p,x] /; +EqQ[v-w] && IntegerQ[p] + + +Int[Sin[v_]^p_.*Cos[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q,x],x] /; +PositiveIntegerQ[p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[x_^m_.*Sin[v_]^p_.*Cos[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Sin[v]^p*Cos[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[Sin[v_]*Tan[w_]^n_.,x_Symbol] := + -Int[Cos[v]*Tan[w]^(n-1),x] + Cos[v-w]*Int[Sec[w]*Tan[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cos[v_]*Cot[w_]^n_.,x_Symbol] := + -Int[Sin[v]*Cot[w]^(n-1),x] + Cos[v-w]*Int[Csc[w]*Cot[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sin[v_]*Cot[w_]^n_.,x_Symbol] := + Int[Cos[v]*Cot[w]^(n-1),x] + Sin[v-w]*Int[Csc[w]*Cot[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cos[v_]*Tan[w_]^n_.,x_Symbol] := + Int[Sin[v]*Tan[w]^(n-1),x] - Sin[v-w]*Int[Sec[w]*Tan[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sin[v_]*Sec[w_]^n_.,x_Symbol] := + Cos[v-w]*Int[Tan[w]*Sec[w]^(n-1),x] + Sin[v-w]*Int[Sec[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cos[v_]*Csc[w_]^n_.,x_Symbol] := + Cos[v-w]*Int[Cot[w]*Csc[w]^(n-1),x] - Sin[v-w]*Int[Csc[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sin[v_]*Csc[w_]^n_.,x_Symbol] := + Sin[v-w]*Int[Cot[w]*Csc[w]^(n-1),x] + Cos[v-w]*Int[Csc[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cos[v_]*Sec[w_]^n_.,x_Symbol] := + -Sin[v-w]*Int[Tan[w]*Sec[w]^(n-1),x] + Cos[v-w]*Int[Sec[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[(e_.+f_.*x_)^m_.*(a_+b_.*Sin[c_.+d_.*x_]*Cos[c_.+d_.*x_])^n_.,x_Symbol] := + Int[(e+f*x)^m*(a+b*Sin[2*c+2*d*x]/2)^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[x_^m_.*(a_+b_.*Sin[c_.+d_.*x_]^2)^n_,x_Symbol] := + 1/2^n*Int[x^m*(2*a+b-b*Cos[2*c+2*d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a+b] && IntegersQ[m,n] && m>0 && n<0 && (n==-1 || m==1 && n==-2) + + +Int[x_^m_.*(a_+b_.*Cos[c_.+d_.*x_]^2)^n_,x_Symbol] := + 1/2^n*Int[x^m*(2*a+b+b*Cos[2*c+2*d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a+b] && IntegersQ[m,n] && m>0 && n<0 && (n==-1 || m==1 && n==-2) + + +Int[(e_.+f_.*x_)^m_.*Sin[a_.+b_.*(c_+d_.*x_)^n_]^p_.,x_Symbol] := + 1/d^(m+1)*Subst[Int[(d*e-c*f+f*x)^m*Sin[a+b*x^n]^p,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && RationalQ[p] + + +Int[(e_.+f_.*x_)^m_.*Cos[a_.+b_.*(c_+d_.*x_)^n_]^p_.,x_Symbol] := + 1/d^(m+1)*Subst[Int[(d*e-c*f+f*x)^m*Cos[a+b*x^n]^p,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && RationalQ[p] + + +Int[(f_.+g_.*x_)^m_./(a_.+b_.*Cos[d_.+e_.*x_]^2+c_.*Sin[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b+c+(b-c)*Cos[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(f_.+g_.*x_)^m_.*Sec[d_.+e_.*x_]^2/(b_+c_.*Tan[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(b+c+(b-c)*Cos[2*d+2*e*x]),x] /; +FreeQ[{b,c,d,e,f,g},x] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*Sec[d_.+e_.*x_]^2/(b_.+a_.*Sec[d_.+e_.*x_]^2+c_.*Tan[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b+c+(b-c)*Cos[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(f_.+g_.*x_)^m_.*Csc[d_.+e_.*x_]^2/(c_+b_.*Cot[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(b+c+(b-c)*Cos[2*d+2*e*x]),x] /; +FreeQ[{b,c,d,e,f,g},x] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*Csc[d_.+e_.*x_]^2/(c_.+b_.*Cot[d_.+e_.*x_]^2+a_.*Csc[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b+c+(b-c)*Cos[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(e_.+f_.*x_)^m_.*Cos[c_.+d_.*x_]/(a_+b_.*Sin[c_.+d_.*x_]),x_Symbol] := + -I*(e+f*x)^(m+1)/(b*f*(m+1)) + + Int[(e+f*x)^m*E^(I*(c+d*x))/(a-Rt[a^2-b^2,2]-I*b*E^(I*(c+d*x))),x] + + Int[(e+f*x)^m*E^(I*(c+d*x))/(a+Rt[a^2-b^2,2]-I*b*E^(I*(c+d*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && PosQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Sin[c_.+d_.*x_]/(a_+b_.*Cos[c_.+d_.*x_]),x_Symbol] := + I*(e+f*x)^(m+1)/(b*f*(m+1)) - + I*Int[(e+f*x)^m*E^(I*(c+d*x))/(a-Rt[a^2-b^2,2]+b*E^(I*(c+d*x))),x] - + I*Int[(e+f*x)^m*E^(I*(c+d*x))/(a+Rt[a^2-b^2,2]+b*E^(I*(c+d*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && PosQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Cos[c_.+d_.*x_]/(a_+b_.*Sin[c_.+d_.*x_]),x_Symbol] := + -I*(e+f*x)^(m+1)/(b*f*(m+1)) + + I*Int[(e+f*x)^m*E^(I*(c+d*x))/(I*a-Rt[-a^2+b^2,2]+b*E^(I*(c+d*x))),x] + + I*Int[(e+f*x)^m*E^(I*(c+d*x))/(I*a+Rt[-a^2+b^2,2]+b*E^(I*(c+d*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NegQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Sin[c_.+d_.*x_]/(a_+b_.*Cos[c_.+d_.*x_]),x_Symbol] := + I*(e+f*x)^(m+1)/(b*f*(m+1)) + + Int[(e+f*x)^m*E^(I*(c+d*x))/(I*a-Rt[-a^2+b^2,2]+I*b*E^(I*(c+d*x))),x] + + Int[(e+f*x)^m*E^(I*(c+d*x))/(I*a+Rt[-a^2+b^2,2]+I*b*E^(I*(c+d*x))),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NegQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Cos[c_.+d_.*x_]^n_/(a_+b_.*Sin[c_.+d_.*x_]),x_Symbol] := + 1/a*Int[(e+f*x)^m*Cos[c+d*x]^(n-2),x] - + 1/b*Int[(e+f*x)^m*Cos[c+d*x]^(n-2)*Sin[c+d*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && EqQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Sin[c_.+d_.*x_]^n_/(a_+b_.*Cos[c_.+d_.*x_]),x_Symbol] := + 1/a*Int[(e+f*x)^m*Sin[c+d*x]^(n-2),x] - + 1/b*Int[(e+f*x)^m*Sin[c+d*x]^(n-2)*Cos[c+d*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && EqQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Cos[c_.+d_.*x_]^n_/(a_+b_.*Sin[c_.+d_.*x_]),x_Symbol] := + a/b^2*Int[(e+f*x)^m*Cos[c+d*x]^(n-2),x] - + 1/b*Int[(e+f*x)^m*Cos[c+d*x]^(n-2)*Sin[c+d*x],x] - + (a^2-b^2)/b^2*Int[(e+f*x)^m*Cos[c+d*x]^(n-2)/(a+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && NeQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Sin[c_.+d_.*x_]^n_/(a_+b_.*Cos[c_.+d_.*x_]),x_Symbol] := + a/b^2*Int[(e+f*x)^m*Sin[c+d*x]^(n-2),x] - + 1/b*Int[(e+f*x)^m*Sin[c+d*x]^(n-2)*Cos[c+d*x],x] - + (a^2-b^2)/b^2*Int[(e+f*x)^m*Sin[c+d*x]^(n-2)/(a+b*Cos[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && NeQ[a^2-b^2] + + +Int[(e_.+f_.*x_)*(A_+B_.*Sin[c_.+d_.*x_])/(a_+b_.*Sin[c_.+d_.*x_])^2,x_Symbol] := + -B*(e+f*x)*Cos[c+d*x]/(a*d*(a+b*Sin[c+d*x])) + + B*f/(a*d)*Int[Cos[c+d*x]/(a+b*Sin[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && EqQ[a*A-b*B] + + +Int[(e_.+f_.*x_)*(A_+B_.*Cos[c_.+d_.*x_])/(a_+b_.*Cos[c_.+d_.*x_])^2,x_Symbol] := + B*(e+f*x)*Sin[c+d*x]/(a*d*(a+b*Cos[c+d*x])) - + B*f/(a*d)*Int[Sin[c+d*x]/(a+b*Cos[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && EqQ[a*A-b*B] + + +Int[(g_.+h_.*x_)^p_.*(a_+b_.*Sin[e_.+f_.*x_])^m_.*(c_+d_.*Sin[e_.+f_.*x_])^n_.,x_Symbol] := + a^m*c^m*Int[(g+h*x)^p*Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] && IntegerQ[p] && IntegerQ[m] && IGeQ[n-m,0] + + +Int[(g_.+h_.*x_)^p_.*(a_+b_.*Cos[e_.+f_.*x_])^m_.*(c_+d_.*Cos[e_.+f_.*x_])^n_.,x_Symbol] := + a^m*c^m*Int[(g+h*x)^p*Sin[e+f*x]^(2*m)*(c+d*Cos[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] && IntegerQ[p] && IntegerQ[m] && IGeQ[n-m,0] + + +Int[(g_.+h_.*x_)^p_.*(a_+b_.*Sin[e_.+f_.*x_])^m_*(c_+d_.*Sin[e_.+f_.*x_])^n_,x_Symbol] := + a^IntPart[m]*c^IntPart[m]*(a+b*Sin[e+f*x])^FracPart[m]*(c+d*Sin[e+f*x])^FracPart[m]/Cos[e+f*x]^(2*FracPart[m])* + Int[(g+h*x)^p*Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n-m,0] + + +Int[(g_.+h_.*x_)^p_.*(a_+b_.*Cos[e_.+f_.*x_])^m_*(c_+d_.*Cos[e_.+f_.*x_])^n_,x_Symbol] := + a^IntPart[m]*c^IntPart[m]*(a+b*Cos[e+f*x])^FracPart[m]*(c+d*Cos[e+f*x])^FracPart[m]/Sin[e+f*x]^(2*FracPart[m])* + Int[(g+h*x)^p*Sin[e+f*x]^(2*m)*(c+d*Cos[e+f*x])^(n-m),x] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n-m,0] + + +Int[Sec[v_]^m_.*(a_+b_.*Tan[v_])^n_., x_Symbol] := + Int[(a*Cos[v]+b*Sin[v])^n,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && m+n==0 && OddQ[m] + + +Int[Csc[v_]^m_.*(a_+b_.*Cot[v_])^n_., x_Symbol] := + Int[(b*Cos[v]+a*Sin[v])^n,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && m+n==0 && OddQ[m] + + +Int[u_.*Sin[a_.+b_.*x_]^m_.*Sin[c_.+d_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[u,Sin[a+b*x]^m*Sin[c+d*x]^n,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m,n] + + +Int[u_.*Cos[a_.+b_.*x_]^m_.*Cos[c_.+d_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[u,Cos[a+b*x]^m*Cos[c+d*x]^n,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m,n] + + +Int[Sec[a_.+b_.*x_]*Sec[c_+d_.*x_],x_Symbol] := + -Csc[(b*c-a*d)/d]*Int[Tan[a+b*x],x] + Csc[(b*c-a*d)/b]*Int[Tan[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Csc[a_.+b_.*x_]*Csc[c_+d_.*x_],x_Symbol] := + Csc[(b*c-a*d)/b]*Int[Cot[a+b*x],x] - Csc[(b*c-a*d)/d]*Int[Cot[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Tan[a_.+b_.*x_]*Tan[c_+d_.*x_],x_Symbol] := + -b*x/d + b/d*Cos[(b*c-a*d)/d]*Int[Sec[a+b*x]*Sec[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Cot[a_.+b_.*x_]*Cot[c_+d_.*x_],x_Symbol] := + -b*x/d + Cos[(b*c-a*d)/d]*Int[Csc[a+b*x]*Csc[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[u_.*(a_.*Cos[v_]+b_.*Sin[v_])^n_.,x_Symbol] := + Int[u*(a*E^(-a/b*v))^n,x] /; +FreeQ[{a,b,n},x] && EqQ[a^2+b^2] + + + +`) + +func resourcesRubi44MiscellaneousTrigFunctionsMBytes() ([]byte, error) { + return _resourcesRubi44MiscellaneousTrigFunctionsM, nil +} + +func resourcesRubi44MiscellaneousTrigFunctionsM() (*asset, error) { + bytes, err := resourcesRubi44MiscellaneousTrigFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/4.4 Miscellaneous trig functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi5InverseTrigFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Inverse Trig Function Rules*) + + +(* ::Subsection::Closed:: *) +(*5.1.1 (a+b arcsin(c x))^n*) + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcSin[c*x])^n - + b*c*n*Int[x*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcCos[c*x])^n + + b*c*n*Int[x*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + Sqrt[1-c^2*x^2]*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) + + c/(b*(n+1))*Int[x*(a+b*ArcSin[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -Sqrt[1-c^2*x^2]*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - + c/(b*(n+1))*Int[x*(a+b*ArcCos[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + 1/(b*c)*Subst[Int[x^n*Cos[a/b-x/b],x],x,a+b*ArcSin[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + 1/(b*c)*Subst[Int[x^n*Sin[a/b-x/b],x],x,a+b*ArcCos[c*x]] /; +FreeQ[{a,b,c,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*5.1.2 (d x)^m (a+b arcsin(c x))^n*) + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./x_,x_Symbol] := + Subst[Int[(a+b*x)^n/Tan[x],x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./x_,x_Symbol] := + -Subst[Int[(a+b*x)^n/Cot[x],x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d*x)^(m+1)*(a+b*ArcSin[c*x])^n/(d*(m+1)) - + b*c*n/(d*(m+1))*Int[(d*x)^(m+1)*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d*x)^(m+1)*(a+b*ArcCos[c*x])^n/(d*(m+1)) + + b*c*n/(d*(m+1))*Int[(d*x)^(m+1)*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcSin[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n>0 + + +Int[x_^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcCos[c*x])^n/(m+1) + + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n>0 + + +Int[x_^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[1-c^2*x^2]*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + 1/(b*c^(m+1)*(n+1))*Subst[Int[ExpandTrigReduce[(a+b*x)^(n+1),Sin[x]^(m-1)*(m-(m+1)*Sin[x]^2),x],x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && -2<=n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -x^m*Sqrt[1-c^2*x^2]*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - + 1/(b*c^(m+1)*(n+1))*Subst[Int[ExpandTrigReduce[(a+b*x)^(n+1),Cos[x]^(m-1)*(m-(m+1)*Cos[x]^2),x],x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && -2<=n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[1-c^2*x^2]*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(a+b*ArcSin[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] + + c*(m+1)/(b*(n+1))*Int[x^(m+1)*(a+b*ArcSin[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-2 + + +Int[x_^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -x^m*Sqrt[1-c^2*x^2]*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) + + m/(b*c*(n+1))*Int[x^(m-1)*(a+b*ArcCos[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] - + c*(m+1)/(b*(n+1))*Int[x^(m+1)*(a+b*ArcCos[c*x])^(n+1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-2 + + +Int[x_^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Sin[x]^m*Cos[x],x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -1/c^(m+1)*Subst[Int[(a+b*x)^n*Cos[x]^m*Sin[x],x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[m] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d*x)^m*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,m,n},x] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d*x)^m*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,m,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n*) + + +(* Int[(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n,x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveQ[d] *) + + +(* Int[(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n,x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveQ[d] *) + + +Int[1/(Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSin[c_.*x_])),x_Symbol] := + Log[a+b*ArcSin[c*x]]/(b*c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[1/(Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCos[c_.*x_])),x_Symbol] := + -Log[a+b*ArcCos[c*x]]/(b*c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + (a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveQ[d] && NeQ[n+1] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveQ[d] && NeQ[n+1] + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcSin[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcCos[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && Not[PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^p/(2*p+1)*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^p/(2*p+1)*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + x*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/2 - + b*c*n*Sqrt[d+e*x^2]/(2*Sqrt[1-c^2*x^2])*Int[x*(a+b*ArcSin[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/(2*Sqrt[1-c^2*x^2])*Int[(a+b*ArcSin[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + x*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/2 + + b*c*n*Sqrt[d+e*x^2]/(2*Sqrt[1-c^2*x^2])*Int[x*(a+b*ArcCos[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/(2*Sqrt[1-c^2*x^2])*Int[(a+b*ArcCos[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/((2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/((2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcSin[c*x])^n/(d*Sqrt[d+e*x^2]) - + b*c*n/Sqrt[d]*Int[x*(a+b*ArcSin[c*x])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && PositiveQ[d] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcCos[c*x])^n/(d*Sqrt[d+e*x^2]) + + b*c*n/Sqrt[d]*Int[x*(a+b*ArcCos[c*x])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && PositiveQ[d] + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcSin[c*x])^n/(d*Sqrt[d+e*x^2]) - + b*c*n*Sqrt[1-c^2*x^2]/(d*Sqrt[d+e*x^2])*Int[x*(a+b*ArcSin[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcCos[c*x])^n/(d*Sqrt[d+e*x^2]) + + b*c*n*Sqrt[1-c^2*x^2]/(d*Sqrt[d+e*x^2])*Int[x*(a+b*ArcCos[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +(* Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*c*n*d^p/(2*(p+1))*Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*c*n*d^p/(2*(p+1))*Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/(c*d)*Subst[Int[(a+b*x)^n*Sec[x],x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -1/(c*d)*Subst[Int[(a+b*x)^n*Csc[x],x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) + + c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - + c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) + + c*(2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - + c*(2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + d^p/c*Subst[Int[(a+b*x)^n*Cos[x]^(2*p+1),x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[2*p] && (IntegerQ[p] || PositiveQ[d]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -d^p/c*Subst[Int[(a+b*x)^n*Sin[x]^(2*p+1),x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[2*p] && (IntegerQ[p] || PositiveQ[d]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1-c^2*x^2]*Int[(1-c^2*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[2*p] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1-c^2*x^2]*Int[(1-c^2*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[2*p] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[c^2*d+e] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[c^2*d+e] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSin[c*x])^n,(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && NeQ[c^2*d+e] && IntegerQ[p] && (p>0 || PositiveIntegerQ[n]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCos[c*x])^n,(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && NeQ[c^2*d+e] && IntegerQ[p] && (p>0 || PositiveIntegerQ[n]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(d*f+e*g*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2-g^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(d*f+e*g*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2-g^2] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n*) + + +Int[x_*(a_.+b_.*ArcSin[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -1/e*Subst[Int[(a+b*x)^n*Tan[x],x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcCos[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Subst[Int[(a+b*x)^n*Cot[x],x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) + + b*n*d^p/(2*c*(p+1))*Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - + b*n*d^p/(2*c*(p+1))*Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) + + b*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - + b*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + 1/d*Subst[Int[(a+b*x)^n/(Cos[x]*Sin[x]),x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + -1/d*Subst[Int[(a+b*x)^n/(Cos[x]*Sin[x]),x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] && + (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] && + (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])/x_,x_Symbol] := + (d+e*x^2)^p*(a+b*ArcSin[c*x])/(2*p) - + b*c*d^p/(2*p)*Int[(1-c^2*x^2)^(p-1/2),x] + + d*Int[(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])/x,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])/x_,x_Symbol] := + (d+e*x^2)^p*(a+b*ArcCos[c*x])/(2*p) + + b*c*d^p/(2*p)*Int[(1-c^2*x^2)^(p-1/2),x] + + d*Int[(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])/x,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])/(f*(m+1)) - + b*c*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2),x] - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] && NegativeIntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])/(f*(m+1)) + + b*c*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2),x] - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] && NegativeIntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1-c^2*x^2)^p,x]}, + Dist[d^p*(a+b*ArcSin[c*x]),u,x] - b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) && + p!=-1/2 && PositiveQ[d] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1-c^2*x^2)^p,x]}, + Dist[d^p*(a+b*ArcCos[c*x]),u,x] + b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) && + p!=-1/2 && PositiveQ[d] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1-c^2*x^2)^p,x]}, + (a+b*ArcSin[c*x])*Int[x^m*(d+e*x^2)^p,x] - + b*c*d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1-c^2*x^2]*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p+1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1-c^2*x^2)^p,x]}, + (a+b*ArcCos[c*x])*Int[x^m*(d+e*x^2)^p,x] + + b*c*d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1-c^2*x^2]*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p+1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/(f*(m+1)) - + b*c*n*Sqrt[d+e*x^2]/(f*(m+1)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcSin[c*x])^(n-1),x] + + c^2*Sqrt[d+e*x^2]/(f^2*(m+1)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+2)*(a+b*ArcSin[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/(f*(m+1)) + + b*c*n*Sqrt[d+e*x^2]/(f*(m+1)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcCos[c*x])^(n-1),x] + + c^2*Sqrt[d+e*x^2]/(f^2*(m+1)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+2)*(a+b*ArcCos[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (IntegerQ[p] || PositiveQ[d]) && (RationalQ[m] || EqQ[n-1]) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (IntegerQ[p] || PositiveQ[d]) && (RationalQ[m] || EqQ[n-1]) *) + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/(f*(m+2)) - + b*c*n*Sqrt[d+e*x^2]/(f*(m+2)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcSin[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/((m+2)*Sqrt[1-c^2*x^2])*Int[(f*x)^m*(a+b*ArcSin[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && Not[RationalQ[m] && m<-1] && (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/(f*(m+2)) + + b*c*n*Sqrt[d+e*x^2]/(f*(m+2)*Sqrt[1-c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcCos[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/((m+2)*Sqrt[1-c^2*x^2])*Int[(f*x)^m*(a+b*ArcCos[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && Not[RationalQ[m] && m<-1] && (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (RationalQ[m] || EqQ[n-1]) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[p] || PositiveQ[d]) && (IntegerQ[m] || IntegerQ[p] || n==1) *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[p] || PositiveQ[d]) && (IntegerQ[m] || IntegerQ[p] || n==1) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*f*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[m] || IntegerQ[p] || n==1) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*f*(p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[m] || IntegerQ[p] || n==1) + + +(* Int[(f_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/(e*m) + + b*f*n/(c*m*Sqrt[d])*Int[(f*x)^(m-1)*(a+b*ArcSin[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSin[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && PositiveQ[d] && IntegerQ[m] *) + + +(* Int[(f_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/(e*m) - + b*f*n*Sqrt[1-c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b*ArcCos[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcCos[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && PositiveQ[d] && IntegerQ[m] *) + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/(e*m) + + b*f*n*Sqrt[1-c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b*ArcSin[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSin[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && IntegerQ[m] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/(e*m) - + b*f*n*Sqrt[1-c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b*ArcCos[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcCos[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && IntegerQ[m] + + +Int[x_^m_*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*Sin[x]^m,x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[x_^m_*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*Cos[x]^m,x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f*x)^(m+1)*(a+b*ArcSin[c*x])*Hypergeometric2F1[1/2,(1+m)/2,(3+m)/2,c^2*x^2]/(Sqrt[d]*f*(m+1)) - + b*c*(f*x)^(m+2)*HypergeometricPFQ[{1,1+m/2,1+m/2},{3/2+m/2,2+m/2},c^2*x^2]/(Sqrt[d]*f^2*(m+1)*(m+2)) /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && PositiveQ[d] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f*x)^(m+1)*(a+b*ArcCos[c*x])*Hypergeometric2F1[1/2,(1+m)/2,(3+m)/2,c^2*x^2]/(Sqrt[d]*f*(m+1)) + + b*c*(f*x)^(m+2)*HypergeometricPFQ[{1,1+m/2,1+m/2},{3/2+m/2,2+m/2},c^2*x^2]/(Sqrt[d]*f^2*(m+1)*(m+2)) /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && PositiveQ[d] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(f*x)^m*(a+b*ArcSin[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && Not[PositiveQ[d]] && (IntegerQ[m] || n==1) + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(f*x)^m*(a+b*ArcCos[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && Not[PositiveQ[d]] && (IntegerQ[m] || n==1) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(e*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] + + b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && + (IntegerQ[p] || PositiveQ[d]) && IntegerQ[m] *) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(e*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] - + b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && + (IntegerQ[p] || PositiveQ[d]) && IntegerQ[m] *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(e*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] + + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(c*(m+2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(e*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] - + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(c*(m+2*p+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && IntegerQ[m] + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) + + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) + + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f*x)^m*(a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + f*m/(b*c*Sqrt[d]*(n+1))*Int[(f*x)^(m-1)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && PositiveQ[d] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + -(f*x)^m*(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) + + f*m/(b*c*Sqrt[d]*(n+1))*Int[(f*x)^(m-1)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && PositiveQ[d] + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] + + c*(m+2*p+1)*d^p/(b*f*(n+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] && + (IntegerQ[p] || PositiveQ[d]) *) + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) + + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] - + c*(m+2*p+1)*d^p/(b*f*(n+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] && + (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] + + c*(m+2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*f*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) + + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] - + c*(m+2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*f*(n+1)*(1-c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Sin[x]^m*Cos[x]^(2*p+1),x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Cos[x]^m*Sin[x]^(2*p+1),x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[x^m*(1-c^2*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && Not[(IntegerQ[p] || PositiveQ[d])] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[x^m*(1-c^2*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && Not[(IntegerQ[p] || PositiveQ[d])] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSin[c*x])^n/Sqrt[d+e*x^2],(f*x)^m*(d+e*x^2)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[c^2*d+e] && PositiveQ[d] && PositiveIntegerQ[p+1/2] && Not[PositiveIntegerQ[(m+1)/2]] && + IntegerQ[m] && -30 || PositiveIntegerQ[(m-1)/2] && m+p<=0) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[c^2*d+e] && IntegerQ[p] && (p>0 || PositiveIntegerQ[(m-1)/2] && m+p<=0) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSin[c*x])^n,(f*x)^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[c^2*d+e] && PositiveIntegerQ[n] && IntegerQ[p] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCos[c*x])^n,(f*x)^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[c^2*d+e] && PositiveIntegerQ[n] && IntegerQ[p] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(h_.*x_)^m_.*(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(h*x)^m*(d*f+e*g*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2-g^2] && Not[IntegerQ[p]] + + +Int[(h_.*x_)^m_.*(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(h*x)^m*(d*f+e*g*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2-g^2] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*5.1.5 u (a+b arcsin(c x))^n*) + + +Int[(a_.+b_.*ArcSin[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + Subst[Int[(a+b*x)^n*Cos[x]/(c*d+e*Sin[x]),x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCos[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + -Subst[Int[(a+b*x)^n*Sin[x]/(c*d+e*Cos[x]),x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n] + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*ArcSin[c*x])^n/(e*(m+1)) - + b*c*n/(e*(m+1))*Int[(d+e*x)^(m+1)*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*ArcCos[c*x])^n/(e*(m+1)) + + b*c*n/(e*(m+1))*Int[(d+e*x)^(m+1)*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-1 + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-1 + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Cos[x]*(c*d+e*Sin[x])^m,x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && PositiveIntegerQ[m] + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + -1/c^(m+1)*Subst[Int[(a+b*x)^n*Sin[x]*(c*d+e*Cos[x])^m,x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && PositiveIntegerQ[m] + + +Int[Px_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[ExpandExpression[Px,x],x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] + + +Int[Px_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[ExpandExpression[Px,x],x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] + + +(* Int[Px_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + With[{u=IntHide[Px,x]}, + Dist[(a+b*ArcSin[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] *) + + +(* Int[Px_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + With[{u=IntHide[Px,x]}, + Dist[(a+b*ArcCos[c*x])^n,u,x] + b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] *) + + +Int[Px_*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c,n},x] && PolynomialQ[Px,x] + + +Int[Px_*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c,n},x] && PolynomialQ[Px,x] + + +Int[Px_*(d_.+e_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[Px*(d+e*x)^m,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,m},x] && PolynomialQ[Px,x] + + +Int[Px_*(d_.+e_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[Px*(d+e*x)^m,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,m},x] && PolynomialQ[Px,x] + + +Int[(f_.+g_.*x_)^p_.*(d_+e_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + With[{u=IntHide[(f+g*x)^p*(d+e*x)^m,x]}, + Dist[(a+b*ArcSin[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[n,p] && NegativeIntegerQ[m] && m+p+1<0 + + +Int[(f_.+g_.*x_)^p_.*(d_+e_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + With[{u=IntHide[(f+g*x)^p*(d+e*x)^m,x]}, + Dist[(a+b*ArcCos[c*x])^n,u,x] + b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[n,p] && NegativeIntegerQ[m] && m+p+1<0 + + +Int[(f_.+g_.*x_+h_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_/(d_+e_.*x_)^2,x_Symbol] := + With[{u=IntHide[(f+g*x+h*x^2)^p/(d+e*x)^2,x]}, + Dist[(a+b*ArcSin[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[n,p] && EqQ[e*g-2*d*h] + + +Int[(f_.+g_.*x_+h_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_/(d_+e_.*x_)^2,x_Symbol] := + With[{u=IntHide[(f+g*x+h*x^2)^p/(d+e*x)^2,x]}, + Dist[(a+b*ArcCos[c*x])^n,u,x] + b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[n,p] && EqQ[e*g-2*d*h] + + +Int[Px_*(d_+e_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(d+e*x)^m*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[Px_*(d_+e_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(d+e*x)^m*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f+g*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[Dist[1/Sqrt[1-c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && m>0 && + (m<-2*p-1 || m>3) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f+g*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[Dist[1/Sqrt[1-c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && m>0 && + (m<-2*p-1 || m>3) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m>0 && + (n==1 && p>-1 || p>0 || m==1 || m==2 && p<-2) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m>0 && + (n==1 && p>-1 || p>0 || m==1 || m==2 && p<-2) + + +Int[(f_+g_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d+e*x^2)*(a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + 1/(b*c*Sqrt[d]*(n+1))*Int[(d*g*m+2*e*f*x+e*g*(m+2)*x^2)*(f+g*x)^(m-1)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -(f+g*x)^m*(d+e*x^2)*(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) + + 1/(b*c*Sqrt[d]*(n+1))*Int[(d*g*m+2*e*f*x+e*g*(m+2)*x^2)*(f+g*x)^(m-1)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n,(f+g*x)^m*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n,(f+g*x)^m*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d+e*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + 1/(b*c*Sqrt[d]*(n+1))* + Int[ExpandIntegrand[(f+g*x)^(m-1)*(a+b*ArcSin[c*x])^(n+1),(d*g*m+e*f*(2*p+1)*x+e*g*(m+2*p+1)*x^2)*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveIntegerQ[p-1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + -(f+g*x)^m*(d+e*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) + + 1/(b*c*Sqrt[d]*(n+1))* + Int[ExpandIntegrand[(f+g*x)^(m-1)*(a+b*ArcCos[c*x])^(n+1),(d*g*m+e*f*(2*p+1)*x+e*g*(m+2*p+1)*x^2)*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveIntegerQ[p-1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f+g*x)^m*(a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + g*m/(b*c*Sqrt[d]*(n+1))*Int[(f+g*x)^(m-1)*(a+b*ArcSin[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && m>0 && RationalQ[n] && n<-1 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + -(f+g*x)^m*(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) + + g*m/(b*c*Sqrt[d]*(n+1))*Int[(f+g*x)^(m-1)*(a+b*ArcCos[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && m>0 && RationalQ[n] && n<-1 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*(c*f+g*Sin[x])^m,x],x,ArcSin[c*x]] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && (m>0 || PositiveIntegerQ[n]) + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*(c*f+g*Cos[x])^m,x],x,ArcCos[c*x]] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && PositiveQ[d] && (m>0 || PositiveIntegerQ[n]) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSin[c*x])^n/Sqrt[d+e*x^2],(f+g*x)^m*(d+e*x^2)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCos[c*x])^n/Sqrt[d+e*x^2],(f+g*x)^m*(d+e*x^2)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[c^2*d+e] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[(f+g*x)^m*(1-c^2*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[(f+g*x)^m*(1-c^2*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Log[h*(f+g*x)^m]*(a+b*ArcSin[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + g*m/(b*c*Sqrt[d]*(n+1))*Int[(a+b*ArcSin[c*x])^(n+1)/(f+g*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m},x] && EqQ[c^2*d+e] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -Log[h*(f+g*x)^m]*(a+b*ArcCos[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) + + g*m/(b*c*Sqrt[d]*(n+1))*Int[(a+b*ArcCos[c*x])^(n+1)/(f+g*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m},x] && EqQ[c^2*d+e] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[Log[h*(f+g*x)^m]*(1-c^2*x^2)^p*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1-c^2*x^2)^FracPart[p]*Int[Log[h*(f+g*x)^m]*(1-c^2*x^2)^p*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x)^m*(f+g*x)^m,x]}, + Dist[a+b*ArcSin[c*x],u,x] - b*c*Int[Dist[1/Sqrt[1-c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NegativeIntegerQ[m+1/2] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x)^m*(f+g*x)^m,x]}, + Dist[a+b*ArcCos[c*x],u,x] + b*c*Int[Dist[1/Sqrt[1-c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NegativeIntegerQ[m+1/2] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^m*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && IntegerQ[m] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(f+g*x)^m*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && IntegerQ[m] + + +Int[u_*(a_.+b_.*ArcSin[c_.*x_]),x_Symbol] := + With[{v=IntHide[u,x]}, + Dist[a+b*ArcSin[c*x],v,x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1-c^2*x^2],x],x] /; + InverseFunctionFreeQ[v,x]] /; +FreeQ[{a,b,c},x] + + +Int[u_*(a_.+b_.*ArcCos[c_.*x_]),x_Symbol] := + With[{v=IntHide[u,x]}, + Dist[a+b*ArcCos[c*x],v,x] + b*c*Int[SimplifyIntegrand[v/Sqrt[1-c^2*x^2],x],x] /; + InverseFunctionFreeQ[v,x]] /; +FreeQ[{a,b,c},x] + + +Int[Px_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,n},x] && PolynomialQ[Px,x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[Px_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,n},x] && PolynomialQ[Px,x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[Px_.*(f_+g_.*(d_+e_.*x_^2)^p_)^m_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(f+g*(d+e*x^2)^p)^m*(a+b*ArcSin[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PolynomialQ[Px,x] && EqQ[c^2*d+e] && PositiveIntegerQ[p+1/2] && IntegersQ[m,n] + + +Int[Px_.*(f_+g_.*(d_+e_.*x_^2)^p_)^m_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(f+g*(d+e*x^2)^p)^m*(a+b*ArcCos[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PolynomialQ[Px,x] && EqQ[c^2*d+e] && PositiveIntegerQ[p+1/2] && IntegersQ[m,n] + + +Int[RFx_*ArcSin[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[ArcSin[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[c,x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*ArcCos[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[ArcCos[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[c,x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[RFx*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[RFx*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(d_+e_.*x_^2)^p_*ArcSin[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(d+e*x^2)^p*ArcSin[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[RFx_*(d_+e_.*x_^2)^p_*ArcCos[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(d+e*x^2)^p*ArcCos[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[RFx_*(d_+e_.*x_^2)^p_*(a_+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p,RFx*(a+b*ArcSin[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[RFx_*(d_+e_.*x_^2)^p_*(a_+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p,RFx*(a+b*ArcCos[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[u_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := + Defer[Int][u*(a+b*ArcSin[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[u_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := + Defer[Int][u*(a+b*ArcCos[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*5.1.6 Miscellaneous inverse sine*) +(**) + + +Int[(a_.+b_.*ArcSin[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(a+b*ArcSin[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,n},x] + + +Int[(a_.+b_.*ArcCos[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(a+b*ArcCos[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,n},x] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*ArcSin[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(a+b*ArcSin[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*ArcCos[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(a+b*ArcCos[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(-C/d^2+C/d^2*x^2)^p*(a+b*ArcSin[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,A,B,C,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(-C/d^2+C/d^2*x^2)^p*(a+b*ArcCos[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,A,B,C,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(e_.+f_.*x_)^m_.*(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(-C/d^2+C/d^2*x^2)^p*(a+b*ArcSin[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(e_.+f_.*x_)^m_.*(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(-C/d^2+C/d^2*x^2)^p*(a+b*ArcCos[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[Sqrt[a_.+b_.*ArcSin[c_+d_.*x_^2]],x_Symbol] := + x*Sqrt[a+b*ArcSin[c+d*x^2]] - + Sqrt[Pi]*x*(Cos[a/(2*b)]+c*Sin[a/(2*b)])*FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Sqrt[c/b]*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) + + Sqrt[Pi]*x*(Cos[a/(2*b)]-c*Sin[a/(2*b)])*FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Sqrt[c/b]*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] + + +Int[Sqrt[a_.+b_.*ArcCos[1+d_.*x_^2]],x_Symbol] := + -2*Sqrt[a+b*ArcCos[1+d*x^2]]*Sin[ArcCos[1+d*x^2]/2]^2/(d*x) - + 2*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(Sqrt[1/b]*d*x) + + 2*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(Sqrt[1/b]*d*x) /; +FreeQ[{a,b,d},x] + + +Int[Sqrt[a_.+b_.*ArcCos[-1+d_.*x_^2]],x_Symbol] := + 2*Sqrt[a+b*ArcCos[-1+d*x^2]]*Cos[(1/2)*ArcCos[-1+d*x^2]]^2/(d*x) - + 2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(Sqrt[1/b]*d*x) - + 2*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(Sqrt[1/b]*d*x) /; +FreeQ[{a,b,d},x] + + +Int[(a_.+b_.*ArcSin[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcSin[c+d*x^2])^n + + 2*b*n*Sqrt[-2*c*d*x^2-d^2*x^4]*(a+b*ArcSin[c+d*x^2])^(n-1)/(d*x) - + 4*b^2*n*(n-1)*Int[(a+b*ArcSin[c+d*x^2])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n>1 + + +Int[(a_.+b_.*ArcCos[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcCos[c+d*x^2])^n - + 2*b*n*Sqrt[-2*c*d*x^2-d^2*x^4]*(a+b*ArcCos[c+d*x^2])^(n-1)/(d*x) - + 4*b^2*n*(n-1)*Int[(a+b*ArcCos[c+d*x^2])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n>1 + + +Int[1/(a_.+b_.*ArcSin[c_+d_.*x_^2]),x_Symbol] := + -x*(c*Cos[a/(2*b)]-Sin[a/(2*b)])*CosIntegral[(c/(2*b))*(a+b*ArcSin[c+d*x^2])]/ + (2*b*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) - + x*(c*Cos[a/(2*b)]+Sin[a/(2*b)])*SinIntegral[(c/(2*b))*(a+b*ArcSin[c+d*x^2])]/ + (2*b*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] + + +Int[1/(a_.+b_.*ArcCos[1+d_.*x_^2]),x_Symbol] := + x*Cos[a/(2*b)]*CosIntegral[(a+b*ArcCos[1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[-d*x^2]) + + x*Sin[a/(2*b)]*SinIntegral[(a+b*ArcCos[1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[-d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCos[-1+d_.*x_^2]),x_Symbol] := + x*Sin[a/(2*b)]*CosIntegral[(a+b*ArcCos[-1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) - + x*Cos[a/(2*b)]*SinIntegral[(a+b*ArcCos[-1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/Sqrt[a_.+b_.*ArcSin[c_+d_.*x_^2]],x_Symbol] := + -Sqrt[Pi]*x*(Cos[a/(2*b)]-c*Sin[a/(2*b)])*FresnelC[1/(Sqrt[b*c]*Sqrt[Pi])*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Sqrt[b*c]*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) - + Sqrt[Pi]*x*(Cos[a/(2*b)]+c*Sin[a/(2*b)])*FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Sqrt[b*c]*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] + + +Int[1/Sqrt[a_.+b_.*ArcCos[1+d_.*x_^2]],x_Symbol] := + -2*Sqrt[Pi/b]*Cos[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(d*x) - + 2*Sqrt[Pi/b]*Sin[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/Sqrt[a_.+b_.*ArcCos[-1+d_.*x_^2]],x_Symbol] := + 2*Sqrt[Pi/b]*Sin[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(d*x) - + 2*Sqrt[Pi/b]*Cos[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcSin[c_+d_.*x_^2])^(3/2),x_Symbol] := + -Sqrt[-2*c*d*x^2-d^2*x^4]/(b*d*x*Sqrt[a+b*ArcSin[c+d*x^2]]) - + (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)]+c*Sin[a/(2*b)])*FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Cos[(1/2)*ArcSin[c+d*x^2]]-c*Sin[ArcSin[c+d*x^2]/2]) + + (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)]-c*Sin[a/(2*b)])*FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a+b*ArcSin[c+d*x^2]]]/ + (Cos[(1/2)*ArcSin[c+d*x^2]]-c*Sin[ArcSin[c+d*x^2]/2]) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] + + +Int[1/(a_.+b_.*ArcCos[1+d_.*x_^2])^(3/2),x_Symbol] := + Sqrt[-2*d*x^2-d^2*x^4]/(b*d*x*Sqrt[a+b*ArcCos[1+d*x^2]]) - + 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(d*x) + + 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCos[-1+d_.*x_^2])^(3/2),x_Symbol] := + Sqrt[2*d*x^2-d^2*x^4]/(b*d*x*Sqrt[a+b*ArcCos[-1+d*x^2]]) - + 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(d*x) - + 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1+d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a+b*ArcCos[-1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcSin[c_+d_.*x_^2])^2,x_Symbol] := + -Sqrt[-2*c*d*x^2-d^2*x^4]/(2*b*d*x*(a+b*ArcSin[c+d*x^2])) - + x*(Cos[a/(2*b)]+c*Sin[a/(2*b)])*CosIntegral[(c/(2*b))*(a+b*ArcSin[c+d*x^2])]/ + (4*b^2*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) + + x*(Cos[a/(2*b)]-c*Sin[a/(2*b)])*SinIntegral[(c/(2*b))*(a+b*ArcSin[c+d*x^2])]/ + (4*b^2*(Cos[ArcSin[c+d*x^2]/2]-c*Sin[ArcSin[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] + + +Int[1/(a_.+b_.*ArcCos[1+d_.*x_^2])^2,x_Symbol] := + Sqrt[-2*d*x^2-d^2*x^4]/(2*b*d*x*(a+b*ArcCos[1+d*x^2])) + + x*Sin[a/(2*b)]*CosIntegral[(a+b*ArcCos[1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2]) - + x*Cos[a/(2*b)]*SinIntegral[(a+b*ArcCos[1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCos[-1+d_.*x_^2])^2,x_Symbol] := + Sqrt[2*d*x^2-d^2*x^4]/(2*b*d*x*(a+b*ArcCos[-1+d*x^2])) - + x*Cos[a/(2*b)]*CosIntegral[(a+b*ArcCos[-1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) - + x*Sin[a/(2*b)]*SinIntegral[(a+b*ArcCos[-1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[(a_.+b_.*ArcSin[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcSin[c+d*x^2])^(n+2)/(4*b^2*(n+1)*(n+2)) + + Sqrt[-2*c*d*x^2-d^2*x^4]*(a+b*ArcSin[c+d*x^2])^(n+1)/(2*b*d*(n+1)*x) - + 1/(4*b^2*(n+1)*(n+2))*Int[(a+b*ArcSin[c+d*x^2])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n<-1 && n!=-2 + + +Int[(a_.+b_.*ArcCos[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcCos[c+d*x^2])^(n+2)/(4*b^2*(n+1)*(n+2)) - + Sqrt[-2*c*d*x^2-d^2*x^4]*(a+b*ArcCos[c+d*x^2])^(n+1)/(2*b*d*(n+1)*x) - + 1/(4*b^2*(n+1)*(n+2))*Int[(a+b*ArcCos[c+d*x^2])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n<-1 && n!=-2 + + +Int[ArcSin[a_.*x_^p_]^n_./x_,x_Symbol] := + 1/p*Subst[Int[x^n*Cot[x],x],x,ArcSin[a*x^p]] /; +FreeQ[{a,p},x] && PositiveIntegerQ[n] + + +Int[ArcCos[a_.*x_^p_]^n_./x_,x_Symbol] := + -1/p*Subst[Int[x^n*Tan[x],x],x,ArcCos[a*x^p]] /; +FreeQ[{a,p},x] && PositiveIntegerQ[n] + + +Int[u_.*ArcSin[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCsc[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCos[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcSec[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[ArcSin[Sqrt[1+b_.*x_^2]]^n_./Sqrt[1+b_.*x_^2],x_Symbol] := + Sqrt[-b*x^2]/(b*x)*Subst[Int[ArcSin[x]^n/Sqrt[1-x^2],x],x,Sqrt[1+b*x^2]] /; +FreeQ[{b,n},x] + + +Int[ArcCos[Sqrt[1+b_.*x_^2]]^n_./Sqrt[1+b_.*x_^2],x_Symbol] := + Sqrt[-b*x^2]/(b*x)*Subst[Int[ArcCos[x]^n/Sqrt[1-x^2],x],x,Sqrt[1+b*x^2]] /; +FreeQ[{b,n},x] + + +Int[u_.*f_^(c_.*ArcSin[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[ReplaceAll[u,x->-a/b+Sin[x]/b]*f^(c*x^n)*Cos[x],x],x,ArcSin[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[u_.*f_^(c_.*ArcCos[a_.+b_.*x_]^n_.),x_Symbol] := + -1/b*Subst[Int[ReplaceAll[u,x->-a/b+Cos[x]/b]*f^(c*x^n)*Sin[x],x],x,ArcCos[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[ArcSin[a_.*x_^2+b_.*Sqrt[c_+d_.*x_^2]],x_Symbol] := + x*ArcSin[a*x^2+b*Sqrt[c+d*x^2]] - + x*Sqrt[b^2*d+a^2*x^2+2*a*b*Sqrt[c+d*x^2]]/Sqrt[(-x^2)*(b^2*d+a^2*x^2+2*a*b*Sqrt[c+d*x^2])]* + Int[x*(b*d+2*a*Sqrt[c+d*x^2])/(Sqrt[c+d*x^2]*Sqrt[b^2*d +a^2*x^2+2*a*b*Sqrt[c+d*x^2]]),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2*c,1] + + +Int[ArcCos[a_.*x_^2+b_.*Sqrt[c_+d_.*x_^2]],x_Symbol] := + x*ArcCos[a*x^2+b*Sqrt[c+d*x^2]] + + x*Sqrt[b^2*d+a^2*x^2+2*a*b*Sqrt[c+d*x^2]]/Sqrt[(-x^2)*(b^2*d+a^2*x^2+2*a*b*Sqrt[c+d*x^2])]* + Int[x*(b*d+2*a*Sqrt[c+d*x^2])/(Sqrt[c+d*x^2]*Sqrt[b^2*d+a^2*x^2+2*a*b*Sqrt[c+d*x^2]]),x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2*c,1] + + +Int[ArcSin[u_],x_Symbol] := + x*ArcSin[u] - + Int[SimplifyIntegrand[x*D[u,x]/Sqrt[1-u^2],x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[ArcCos[u_],x_Symbol] := + x*ArcCos[u] + + Int[SimplifyIntegrand[x*D[u,x]/Sqrt[1-u^2],x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcSin[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcSin[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/Sqrt[1-u^2],x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCos[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCos[u])/(d*(m+1)) + + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/Sqrt[1-u^2],x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[v_*(a_.+b_.*ArcSin[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcSin[u]),w,x] - + b*Int[SimplifyIntegrand[w*D[u,x]/Sqrt[1-u^2],x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + +Int[v_*(a_.+b_.*ArcCos[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCos[u]),w,x] + + b*Int[SimplifyIntegrand[w*D[u,x]/Sqrt[1-u^2],x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + + + + +(* ::Subsection::Closed:: *) +(*5.2.1 u (a+b arctan(c x))^n*) + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcTan[c*x])^n - + b*c*n*Int[x*(a+b*ArcTan[c*x])^(n-1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcCot[c*x])^n + + b*c*n*Int[x*(a+b*ArcCot[c*x])^(n-1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + Defer[Int][(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] && Not[PositiveIntegerQ[n]] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + Defer[Int][(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] && Not[PositiveIntegerQ[n]] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + -(a+b*ArcTan[c*x])^n*Log[2*d/(d+e*x)]/e + + b*c*n/e*Int[(a+b*ArcTan[c*x])^(n-1)*Log[2*d/(d+e*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + -(a+b*ArcCot[c*x])^n*Log[2*d/(d+e*x)]/e - + b*c*n/e*Int[(a+b*ArcCot[c*x])^(n-1)*Log[2*d/(d+e*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] + + +Int[ArcTan[c_.*x_]/(d_+e_.*x_),x_Symbol] := + -ArcTan[c*d/e]*Log[d+e*x]/e + I*PolyLog[2,Simp[I*c*(d+e*x)/(I*c*d-e),x]]/(2*e) - I*PolyLog[2,Simp[I*c*(d+e*x)/(I*c*d+e),x]]/(2*e) /; +FreeQ[{c,d,e},x] && PositiveQ[I*c*d/e+1] && NegativeQ[I*c*d/e-1] + + +(* Int[ArcCot[c_.*x_]/(d_+e_.*x_),x_Symbol] := + ? /; +FreeQ[{c,d,e},x] && PositiveQ[I*c*d/e+1] && NegativeQ[I*c*d/e-1] *) + + +Int[ArcTan[c_.*x_]/(d_.+e_.*x_),x_Symbol] := + I/2*Int[Log[1-I*c*x]/(d+e*x),x] - I/2*Int[Log[1+I*c*x]/(d+e*x),x] /; +FreeQ[{c,d,e},x] + + +Int[ArcCot[c_.*x_]/(d_.+e_.*x_),x_Symbol] := + I/2*Int[Log[1-I/(c*x)]/(d+e*x),x] - I/2*Int[Log[1+I/(c*x)]/(d+e*x),x] /; +FreeQ[{c,d,e},x] + + +Int[(a_+b_.*ArcTan[c_.*x_])/(d_.+e_.*x_),x_Symbol] := + a/e*Log[RemoveContent[d+e*x,x]] + b*Int[ArcTan[c*x]/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(a_+b_.*ArcCot[c_.*x_])/(d_.+e_.*x_),x_Symbol] := + a/e*Log[RemoveContent[d+e*x,x]] + b*Int[ArcCot[c*x]/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + (d+e*x)^(p+1)*(a+b*ArcTan[c*x])/(e*(p+1)) - + b*c/(e*(p+1))*Int[(d+e*x)^(p+1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + (d+e*x)^(p+1)*(a+b*ArcCot[c*x])/(e*(p+1)) + + b*c/(e*(p+1))*Int[(d+e*x)^(p+1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_/x_,x_Symbol] := + 2*(a+b*ArcTan[c*x])^n*ArcTanh[1-2*I/(I-c*x)] - + 2*b*c*n*Int[(a+b*ArcTan[c*x])^(n-1)*ArcTanh[1-2*I/(I-c*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_/x_,x_Symbol] := + 2*(a+b*ArcCot[c*x])^n*ArcCoth[1-2*I/(I-c*x)] + + 2*b*c*n*Int[(a+b*ArcCot[c*x])^(n-1)*ArcCoth[1-2*I/(I-c*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[x_^m_.*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcTan[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcTan[c*x])^(n-1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,m},x] && IntegerQ[n] && n>1 && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcCot[c*x])^n/(m+1) + + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCot[c*x])^(n-1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,m},x] && IntegerQ[n] && n>1 && NeQ[m+1] + + +Int[(d_+e_.*x_)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^p*(a+b*ArcTan[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n,p] + + +Int[(d_+e_.*x_)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^p*(a+b*ArcCot[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n,p] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[x_^m_.*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/e*Int[x^(m-1)*(a+b*ArcTan[c*x])^n,x] - + d/e*Int[x^(m-1)*(a+b*ArcTan[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] && RationalQ[m] && m>0 + + +Int[x_^m_.*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/e*Int[x^(m-1)*(a+b*ArcCot[c*x])^n,x] - + d/e*Int[x^(m-1)*(a+b*ArcCot[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] && RationalQ[m] && m>0 + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(x_*(d_+e_.*x_)),x_Symbol] := + (a+b*ArcTan[c*x])^n*Log[2*e*x/(d+e*x)]/d - + b*c*n/d*Int[(a+b*ArcTan[c*x])^(n-1)*Log[2*e*x/(d+e*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(x_*(d_+e_.*x_)),x_Symbol] := + (a+b*ArcCot[c*x])^n*Log[2*e*x/(d+e*x)]/d + + b*c*n/d*Int[(a+b*ArcCot[c*x])^(n-1)*Log[2*e*x/(d+e*x)]/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTan[c*x])^n,x] - + e/d*Int[x^(m+1)*(a+b*ArcTan[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCot[c*x])^n,x] - + e/d*Int[x^(m+1)*(a+b*ArcCot[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2+e^2] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[x_^m_.*(d_+e_.*x_)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x)^p*(a+b*ArcTan[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || NeQ[a] || IntegerQ[m]) + + +Int[x_^m_.*(d_+e_.*x_)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x)^p*(a+b*ArcCot[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || NeQ[a] || IntegerQ[m]) + + +Int[x_^m_.*(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^p/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcTan[c*x])/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcTan[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + b*(d+e*x^2)^p/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcCot[c*x])/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCot[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + -b*n*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n-1)/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcTan[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcTan[c*x])^n,x] + + b^2*d*n*(n-1)/(2*p*(2*p+1))*Int[(d+e*x^2)^(p-1)*(a+b*ArcTan[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p>0 && n>1 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + b*n*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n-1)/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcCot[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCot[c*x])^n,x] + + b^2*d*n*(n-1)/(2*p*(2*p+1))*Int[(d+e*x^2)^(p-1)*(a+b*ArcCot[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p>0 && n>1 + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcTan[c_.*x_])),x_Symbol] := + Log[RemoveContent[a+b*ArcTan[c*x],x]]/(b*c*d) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcCot[c_.*x_])),x_Symbol] := + -Log[RemoveContent[a+b*ArcCot[c*x],x]]/(b*c*d) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NeQ[n+1] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NeQ[n+1] + + +Int[(a_.+b_.*ArcTan[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + -2*I*(a+b*ArcTan[c*x])*ArcTan[Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) + + I*b*PolyLog[2,-I*Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) - + I*b*PolyLog[2,I*Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + -2*I*(a+b*ArcCot[c*x])*ArcTan[Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) - + I*b*PolyLog[2,-I*Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) + + I*b*PolyLog[2,I*Sqrt[1+I*c*x]/Sqrt[1-I*c*x]]/(c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n*Sec[x],x],x,ArcTan[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -x*Sqrt[1+1/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n*Csc[x],x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcTan[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcCot[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcTan[c*x])^n/(2*d*(d+e*x^2)) + + (a+b*ArcTan[c*x])^(n+1)/(2*b*c*d^2*(n+1)) - + b*c*n/2*Int[x*(a+b*ArcTan[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcCot[c*x])^n/(2*d*(d+e*x^2)) - + (a+b*ArcCot[c*x])^(n+1)/(2*b*c*d^2*(n+1)) + + b*c*n/2*Int[x*(a+b*ArcCot[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcTan[c_.*x_])/(d_+e_.*x_^2)^(3/2),x_Symbol] := + b/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcTan[c*x])/(d*Sqrt[d+e*x^2]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcCot[c*x])/(d*Sqrt[d+e*x^2]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + b*(d+e*x^2)^(p+1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p<-1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p<-1 && p!=-3/2 + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_/(d_+e_.*x_^2)^(3/2),x_Symbol] := + b*n*(a+b*ArcTan[c*x])^(n-1)/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcTan[c*x])^n/(d*Sqrt[d+e*x^2]) - + b^2*n*(n-1)*Int[(a+b*ArcTan[c*x])^(n-2)/(d+e*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>1 + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b*n*(a+b*ArcCot[c*x])^(n-1)/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcCot[c*x])^n/(d*Sqrt[d+e*x^2]) - + b^2*n*(n-1)*Int[(a+b*ArcCot[c*x])^(n-2)/(d+e*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + b*n*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^(n-1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n,x] - + b^2*n*(n-1)/(4*(p+1)^2)*Int[(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p<-1 && n>1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + -b*n*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^(n-1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n,x] - + b^2*n*(n-1)/(4*(p+1)^2)*Int[(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p<-1 && n>1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) - + 2*c*(p+1)/(b*(n+1))*Int[x*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p<-1 && n<-1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + -(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) + + 2*c*(p+1)/(b*(n+1))*Int[x*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && p<-1 && n<-1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + d^p/c*Subst[Int[(a+b*x)^n/Cos[x]^(2*(p+1)),x],x,ArcTan[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NegativeIntegerQ[2*(p+1)] && (IntegerQ[p] || PositiveQ[d]) + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + d^(p+1/2)*Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(1+c^2*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NegativeIntegerQ[2*(p+1)] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + -d^p/c*Subst[Int[(a+b*x)^n/Sin[x]^(2*(p+1)),x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NegativeIntegerQ[2*(p+1)] && IntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + -d^(p+1/2)*x*Sqrt[(1+c^2*x^2)/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n/Sin[x]^(2*(p+1)),x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && NegativeIntegerQ[2*(p+1)] && Not[IntegerQ[p]] + + +Int[ArcTan[c_.*x_]/(d_.+e_.*x_^2),x_Symbol] := + I/2*Int[Log[1-I*c*x]/(d+e*x^2),x] - I/2*Int[Log[1+I*c*x]/(d+e*x^2),x] /; +FreeQ[{c,d,e},x] + + +Int[ArcCot[c_.*x_]/(d_.+e_.*x_^2),x_Symbol] := + I/2*Int[Log[1-I/(c*x)]/(d+e*x^2),x] - I/2*Int[Log[1+I/(c*x)]/(d+e*x^2),x] /; +FreeQ[{c,d,e},x] + + +Int[(a_+b_.*ArcTan[c_.*x_])/(d_.+e_.*x_^2),x_Symbol] := + a*Int[1/(d+e*x^2),x] + b*Int[ArcTan[c*x]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(a_+b_.*ArcCot[c_.*x_])/(d_.+e_.*x_^2),x_Symbol] := + a*Int[1/(d+e*x^2),x] + b*Int[ArcCot[c*x]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcTan[c*x],u,x] - b*c*Int[ExpandIntegrand[u/(1+c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (IntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCot[c*x],u,x] + b*c*Int[ExpandIntegrand[u/(1+c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (IntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcTan[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcCot[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTan[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCot[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -I*(a+b*ArcTan[c*x])^(n+1)/(b*e*(n+1)) - + 1/(c*d)*Int[(a+b*ArcTan[c*x])^n/(I-c*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + I*(a+b*ArcCot[c*x])^(n+1)/(b*e*(n+1)) - + 1/(c*d)*Int[(a+b*ArcCot[c*x])^n/(I-c*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcTan[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + x*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) - + 1/(b*c*d*(n+1))*Int[(a+b*ArcTan[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && Not[PositiveIntegerQ[n]] && NeQ[n+1] + + +Int[x_*(a_.+b_.*ArcCot[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + -x*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) + + 1/(b*c*d*(n+1))*Int[(a+b*ArcCot[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && Not[PositiveIntegerQ[n]] && NeQ[n+1] + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcTan[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcCot[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + -I*(a+b*ArcTan[c*x])^(n+1)/(b*d*(n+1)) + + I/d*Int[(a+b*ArcTan[c*x])^n/(x*(I+c*x)),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + I*(a+b*ArcCot[c*x])^(n+1)/(b*d*(n+1)) + + I/d*Int[(a+b*ArcCot[c*x])^n/(x*(I+c*x)),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTan[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCot[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + x^m*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*d*(n+1))*Int[x^(m-1)*(a+b*ArcTan[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + -x^m*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) + + m/(b*c*d*(n+1))*Int[x^(m-1)*(a+b*ArcCot[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcTan[c_.*x_])/(d_+e_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcTan[c*x]),x^m/(d+e*x^2),x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[m] && Not[m==1 && NeQ[a]] + + +Int[x_^m_.*(a_.+b_.*ArcCot[c_.*x_])/(d_+e_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCot[c*x]),x^m/(d+e*x^2),x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[m] && Not[m==1 && NeQ[a]] + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n/(2*e*(p+1)) - + b*n/(2*c*(p+1))*Int[(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n/(2*e*(p+1)) + + b*n/(2*c*(p+1))*Int[(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(a_.+b_.*ArcTan[c_.*x_])^n_/(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)*(d+e*x^2)) - + (1-c^2*x^2)*(a+b*ArcTan[c*x])^(n+2)/(b^2*e*(n+1)*(n+2)*(d+e*x^2)) - + 4/(b^2*(n+1)*(n+2))*Int[x*(a+b*ArcTan[c*x])^(n+2)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && n!=-2 + + +Int[x_*(a_.+b_.*ArcCot[c_.*x_])^n_/(d_+e_.*x_^2)^2,x_Symbol] := + -x*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)*(d+e*x^2)) - + (1-c^2*x^2)*(a+b*ArcCot[c*x])^(n+2)/(b^2*e*(n+1)*(n+2)*(d+e*x^2)) - + 4/(b^2*(n+1)*(n+2))*Int[x*(a+b*ArcCot[c*x])^(n+2)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && n!=-2 + + +Int[x_^2*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c^3*d*(p+1)^2) + + x*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])/(2*c^2*d*(p+1)) - + 1/(2*c^2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p<-1 && p!=-5/2 + + +Int[x_^2*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + b*(d+e*x^2)^(p+1)/(4*c^3*d*(p+1)^2) + + x*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])/(2*c^2*d*(p+1)) - + 1/(2*c^2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[p] && p<-1 && p!=-5/2 + + +Int[x_^2*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + (a+b*ArcTan[c*x])^(n+1)/(2*b*c^3*d^2*(n+1)) - + x*(a+b*ArcTan[c*x])^n/(2*c^2*d*(d+e*x^2)) + + b*n/(2*c)*Int[x*(a+b*ArcTan[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[x_^2*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + -(a+b*ArcCot[c*x])^(n+1)/(2*b*c^3*d^2*(n+1)) - + x*(a+b*ArcCot[c*x])^n/(2*c^2*d*(d+e*x^2)) - + b*n/(2*c)*Int[x*(a+b*ArcCot[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + b*x^m*(d+e*x^2)^(p+1)/(c*d*m^2) - + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])/(c^2*d*m) + + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[p] && p<-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + -b*x^m*(d+e*x^2)^(p+1)/(c*d*m^2) - + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])/(c^2*d*m) + + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[p] && p<-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + b*n*x^m*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^(n-1)/(c*d*m^2) - + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n/(c^2*d*m) - + b^2*n*(n-1)/m^2*Int[x^m*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n-2),x] + + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[n,p] && p<-1 && n>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + -b*n*x^m*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^(n-1)/(c*d*m^2) - + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n/(c^2*d*m) - + b^2*n*(n-1)/m^2*Int[x^m*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n-2),x] + + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[n,p] && p<-1 && n>1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_,x_Symbol] := + -x^m*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) + + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[e-c^2*d] && EqQ[m+2*p+2] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + x^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[e-c^2*d] && EqQ[m+2*p+3] && RationalQ[n] && n>0 && NeQ[m+1] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + x^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n/(d*(m+1)) + + b*c*n/(m+1)*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[e-c^2*d] && EqQ[m+2*p+3] && RationalQ[n] && n>0 && NeQ[m+1] + + +Int[x_^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcTan[c*x])/(m+2) - + b*c*d/(m+2)*Int[x^(m+1)/Sqrt[d+e*x^2],x] + + d/(m+2)*Int[x^m*(a+b*ArcTan[c*x])/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && NeQ[m+2] + + +Int[x_^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCot[c*x])/(m+2) + + b*c*d/(m+2)*Int[x^(m+1)/Sqrt[d+e*x^2],x] + + d/(m+2)*Int[x^m*(a+b*ArcCot[c*x])/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && NeQ[m+2] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && IntegerQ[p] && p>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && IntegerQ[p] && p>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + d*Int[x^m*(d+e*x^2)^(p-1)*(a+b*ArcTan[c*x])^n,x] + + c^2*d*Int[x^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && RationalQ[p] && p>0 && PositiveIntegerQ[n] && (RationalQ[m] || EqQ[n,1] && IntegerQ[p]) + + +Int[x_^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + d*Int[x^m*(d+e*x^2)^(p-1)*(a+b*ArcCot[c*x])^n,x] + + c^2*d*Int[x^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[e-c^2*d] && RationalQ[p] && p>0 && PositiveIntegerQ[n] && (RationalQ[m] || EqQ[n,1] && IntegerQ[p]) + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcTan[c*x])^n/(c^2*d*m) - + b*n/(c*m)*Int[x^(m-1)*(a+b*ArcTan[c*x])^(n-1)/Sqrt[d+e*x^2],x] - + (m-1)/(c^2*m)*Int[x^(m-2)*(a+b*ArcTan[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcCot[c*x])^n/(c^2*d*m) + + b*n/(c*m)*Int[x^(m-1)*(a+b*ArcCot[c*x])^(n-1)/Sqrt[d+e*x^2],x] - + (m-1)/(c^2*m)*Int[x^(m-2)*(a+b*ArcCot[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 + + +Int[(a_.+b_.*ArcTan[c_.*x_])/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -2/Sqrt[d]*(a+b*ArcTan[c*x])*ArcTanh[Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] + + I*b/Sqrt[d]*PolyLog[2,-Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] - + I*b/Sqrt[d]*PolyLog[2,Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -2/Sqrt[d]*(a+b*ArcCot[c*x])*ArcTanh[Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] - + I*b/Sqrt[d]*PolyLog[2,-Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] + + I*b/Sqrt[d]*PolyLog[2,Sqrt[1+I*c*x]/Sqrt[1-I*c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + 1/Sqrt[d]*Subst[Int[(a+b*x)^n*Csc[x],x],x,ArcTan[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -c*x*Sqrt[1+1/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n*Sec[x],x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcTan[c*x])^n/(x*Sqrt[1+c^2*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcCot[c*x])^n/(x*Sqrt[1+c^2*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_./(x_^2*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -Sqrt[d+e*x^2]*(a+b*ArcTan[c*x])^n/(d*x) + + b*c*n*Int[(a+b*ArcTan[c*x])^(n-1)/(x*Sqrt[d+e*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_./(x_^2*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -Sqrt[d+e*x^2]*(a+b*ArcCot[c*x])^n/(d*x) - + b*c*n*Int[(a+b*ArcCot[c*x])^(n-1)/(x*Sqrt[d+e*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[x_^m_*(a_.+b_.*ArcTan[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcTan[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcTan[c*x])^(n-1)/Sqrt[d+e*x^2],x] - + c^2*(m+2)/(m+1)*Int[x^(m+2)*(a+b*ArcTan[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 && m!=-2 + + +Int[x_^m_*(a_.+b_.*ArcCot[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCot[c*x])^n/(d*(m+1)) + + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCot[c*x])^(n-1)/Sqrt[d+e*x^2],x] - + c^2*(m+2)/(m+1)*Int[x^(m+2)*(a+b*ArcCot[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 && m!=-2 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + 1/e*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n,x] - + d/e*Int[x^(m-2)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegersQ[m,n,2*p] && p<-1 && m>1 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + 1/e*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n,x] - + d/e*Int[x^(m-2)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegersQ[m,n,2*p] && p<-1 && m>1 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + 1/d*Int[x^m*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^n,x] - + e/d*Int[x^(m+2)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegersQ[m,n,2*p] && p<-1 && m<0 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + 1/d*Int[x^m*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^n,x] - + e/d*Int[x^(m+2)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegersQ[m,n,2*p] && p<-1 && m<0 && n!=-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n+1),x] - + c*(m+2*p+2)/(b*(n+1))*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcTan[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n,p] && p<-1 && n<-1 && NeQ[m+2*p+2] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + -x^m*(d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])^(n+1)/(b*c*d*(n+1)) + + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n+1),x] + + c*(m+2*p+2)/(b*(n+1))*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcCot[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[m,n,p] && p<-1 && n<-1 && NeQ[m+2*p+2] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Sin[x]^m/Cos[x]^(m+2*(p+1)),x],x,ArcTan[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && (IntegerQ[p] || PositiveQ[d]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + d^(p+1/2)*Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[x^m*(1+c^2*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + -d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Cos[x]^m/Sin[x]^(m+2*(p+1)),x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && IntegerQ[p] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + -d^(p+1/2)*x*Sqrt[(1+c^2*x^2)/(c^2*x^2)]/(c^m*Sqrt[d+e*x^2])*Subst[Int[(a+b*x)^n*Cos[x]^m/Sin[x]^(m+2*(p+1)),x],x,ArcCot[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && Not[IntegerQ[p]] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTan[c*x])/(2*e*(p+1)) - + b*c/(2*e*(p+1))*Int[(d+e*x^2)^(p+1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCot[c*x])/(2*e*(p+1)) + + b*c/(2*e*(p+1))*Int[(d+e*x^2)^(p+1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcTan[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(1+c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCot[c*x],u,x] + b*c*Int[SimplifyIntegrand[u/(1+c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcTan[c*x])^n,x^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || p<-1 && IntegerQ[m] && m!=1) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCot[c*x])^n,x^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || p<-1 && IntegerQ[m] && m!=1) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_+b_.*ArcTan[c_.*x_]),x_Symbol] := + a*Int[x^m*(d+e*x^2)^p,x] + b*Int[x^m*(d+e*x^2)^p*ArcTan[c*x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_+b_.*ArcCot[c_.*x_]),x_Symbol] := + a*Int[x^m*(d+e*x^2)^p,x] + b*Int[x^m*(d+e*x^2)^p*ArcCot[c*x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTan[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcTan[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCot[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcCot[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[ArcTanh[u_]*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+u]*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[1-u]*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I+c*x))^2] + + +Int[ArcCoth[u_]*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[SimplifyIntegrand[1+1/u,x]]*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[SimplifyIntegrand[1-1/u,x]]*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I+c*x))^2] + + +Int[ArcTanh[u_]*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+u]*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[1-u]*(a+b*ArcTan[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I-c*x))^2] + + +Int[ArcCoth[u_]*(a_.+b_.*ArcCot[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[SimplifyIntegrand[1+1/u,x]]*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[SimplifyIntegrand[1-1/u,x]]*(a+b*ArcCot[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I-c*x))^2] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + I*(a+b*ArcTan[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) - + b*n*I/2*Int[(a+b*ArcTan[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2*I/(I+c*x))^2] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + I*(a+b*ArcCot[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) + + b*n*I/2*Int[(a+b*ArcCot[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2*I/(I+c*x))^2] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + -I*(a+b*ArcTan[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) + + b*n*I/2*Int[(a+b*ArcTan[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2*I/(I-c*x))^2] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + -I*(a+b*ArcCot[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) - + b*n*I/2*Int[(a+b*ArcCot[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2*I/(I-c*x))^2] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + -I*(a+b*ArcTan[c*x])^n*PolyLog[p+1,u]/(2*c*d) + + b*n*I/2*Int[(a+b*ArcTan[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I+c*x))^2] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + -I*(a+b*ArcCot[c*x])^n*PolyLog[p+1,u]/(2*c*d) - + b*n*I/2*Int[(a+b*ArcCot[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I+c*x))^2] + + +Int[(a_.+b_.*ArcTan[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + I*(a+b*ArcTan[c*x])^n*PolyLog[p+1,u]/(2*c*d) - + b*n*I/2*Int[(a+b*ArcTan[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I-c*x))^2] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + I*(a+b*ArcCot[c*x])^n*PolyLog[p+1,u]/(2*c*d) + + b*n*I/2*Int[(a+b*ArcCot[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[u^2-(1-2*I/(I-c*x))^2] + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcCot[c_.*x_])*(a_.+b_.*ArcTan[c_.*x_])),x_Symbol] := + (-Log[a+b*ArcCot[c*x]]+Log[a+b*ArcTan[c*x]])/(b*c*d*(2*a+b*ArcCot[c*x]+b*ArcTan[c*x])) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] + + +Int[(a_.+b_.*ArcCot[c_.*x_])^m_.*(a_.+b_.*ArcTan[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcCot[c*x])^(m+1)*(a+b*ArcTan[c*x])^n/(b*c*d*(m+1)) + + n/(m+1)*Int[(a+b*ArcCot[c*x])^(m+1)*(a+b*ArcTan[c*x])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegersQ[m,n] && 00 + + +Int[x_^m_.*ArcCot[a_.+b_.*f_^(c_.+d_.*x_)],x_Symbol] := + I/2*Int[x^m*Log[1-I/(a+b*f^(c+d*x))],x] - + I/2*Int[x^m*Log[1+I/(a+b*f^(c+d*x))],x] /; +FreeQ[{a,b,c,d,f},x] && IntegerQ[m] && m>0 + + +Int[u_.*ArcTan[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCot[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCot[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcTan[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[1/(Sqrt[a_.+b_.*x_^2]*ArcTan[c_.*x_/Sqrt[a_.+b_.*x_^2]]),x_Symbol] := + 1/c*Log[ArcTan[c*x/Sqrt[a+b*x^2]]] /; +FreeQ[{a,b,c},x] && EqQ[b+c^2] + + +Int[1/(Sqrt[a_.+b_.*x_^2]*ArcCot[c_.*x_/Sqrt[a_.+b_.*x_^2]]),x_Symbol] := + -1/c*Log[ArcCot[c*x/Sqrt[a+b*x^2]]] /; +FreeQ[{a,b,c},x] && EqQ[b+c^2] + + +Int[ArcTan[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[a_.+b_.*x_^2],x_Symbol] := + ArcTan[c*x/Sqrt[a+b*x^2]]^(m+1)/(c*(m+1)) /; +FreeQ[{a,b,c,m},x] && EqQ[b+c^2] && NeQ[m+1] + + +Int[ArcCot[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[a_.+b_.*x_^2],x_Symbol] := + -ArcCot[c*x/Sqrt[a+b*x^2]]^(m+1)/(c*(m+1)) /; +FreeQ[{a,b,c,m},x] && EqQ[b+c^2] && NeQ[m+1] + + +Int[ArcTan[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[d_.+e_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2]/Sqrt[d+e*x^2]*Int[ArcTan[c*x/Sqrt[a+b*x^2]]^m/Sqrt[a+b*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b+c^2] && EqQ[b*d-a*e] + + +Int[ArcCot[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[d_.+e_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2]/Sqrt[d+e*x^2]*Int[ArcCot[c*x/Sqrt[a+b*x^2]]^m/Sqrt[a+b*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b+c^2] && EqQ[b*d-a*e] + + +Int[u_.*ArcTan[v_+s_.*Sqrt[w_]],x_Symbol] := + Pi*s/4*Int[u,x] + 1/2*Int[u*ArcTan[v],x] /; +EqQ[s^2-1] && EqQ[w-(v^2+1)] + + +Int[u_.*ArcCot[v_+s_.*Sqrt[w_]],x_Symbol] := + Pi*s/4*Int[u,x] - 1/2*Int[u*ArcTan[v],x] /; +EqQ[s^2-1] && EqQ[w-(v^2+1)] + + +If[ShowSteps, + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + ShowStep["","Int[f[x,ArcTan[a+b*x]]/(1+(a+b*x)^2),x]", + "Subst[Int[f[-a/b+Tan[x]/b,x],x],x,ArcTan[a+b*x]]/b",Hold[ + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Sec[x]^(2*(n+1)),x],x], x, tmp]]] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcTan && EqQ[Discriminant[v,x]*tmp[[1]]^2+D[v,x]^2]] /; +SimplifyFlag && QuadraticQ[v,x] && IntegerQ[n] && n<0 && NegQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]], + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Sec[x]^(2*(n+1)),x],x], x, tmp] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcTan && EqQ[Discriminant[v,x]*tmp[[1]]^2+D[v,x]^2]] /; +QuadraticQ[v,x] && IntegerQ[n] && n<0 && NegQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]]] + + +If[ShowSteps, + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + ShowStep["","Int[f[x,ArcCot[a+b*x]]/(1+(a+b*x)^2),x]", + "-Subst[Int[f[-a/b+Cot[x]/b,x],x],x,ArcCot[a+b*x]]/b",Hold[ + -(-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Csc[x]^(2*(n+1)),x],x], x, tmp]]] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcCot && EqQ[Discriminant[v,x]*tmp[[1]]^2+D[v,x]^2]] /; +SimplifyFlag && QuadraticQ[v,x] && IntegerQ[n] && n<0 && NegQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]], + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + -(-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Csc[x]^(2*(n+1)),x],x], x, tmp] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcCot && EqQ[Discriminant[v,x]*tmp[[1]]^2+D[v,x]^2]] /; +QuadraticQ[v,x] && IntegerQ[n] && n<0 && NegQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]]] + + +Int[ArcTan[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Tan[a+b*x]] - + I*b*Int[x/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c+I*d)^2+1] + + +Int[ArcCot[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Tan[a+b*x]] + + I*b*Int[x/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c+I*d)^2+1] + + +Int[ArcTan[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Cot[a+b*x]] - + I*b*Int[x/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-I*d)^2+1] + + +Int[ArcCot[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Cot[a+b*x]] + + I*b*Int[x/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-I*d)^2+1] + + +Int[ArcTan[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Tan[a+b*x]] - + b*(1+I*c+d)*Int[x*E^(2*I*a+2*I*b*x)/(1+I*c-d+(1+I*c+d)*E^(2*I*a+2*I*b*x)),x] + + b*(1-I*c-d)*Int[x*E^(2*I*a+2*I*b*x)/(1-I*c+d+(1-I*c-d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c+I*d)^2+1] + + +Int[ArcCot[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Tan[a+b*x]] + + b*(1+I*c+d)*Int[x*E^(2*I*a+2*I*b*x)/(1+I*c-d+(1+I*c+d)*E^(2*I*a+2*I*b*x)),x] - + b*(1-I*c-d)*Int[x*E^(2*I*a+2*I*b*x)/(1-I*c+d+(1-I*c-d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c+I*d)^2+1] + + +Int[ArcTan[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Cot[a+b*x]] + + b*(1+I*c-d)*Int[x*E^(2*I*a+2*I*b*x)/(1+I*c+d-(1+I*c-d)*E^(2*I*a+2*I*b*x)),x] - + b*(1-I*c+d)*Int[x*E^(2*I*a+2*I*b*x)/(1-I*c-d-(1-I*c+d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c+I*d)^2+1] + + +Int[ArcCot[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Cot[a+b*x]] - + b*(1+I*c-d)*Int[x*E^(2*I*a+2*I*b*x)/(1+I*c+d-(1+I*c-d)*E^(2*I*a+2*I*b*x)),x] + + b*(1-I*c+d)*Int[x*E^(2*I*a+2*I*b*x)/(1-I*c-d-(1-I*c+d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Tan[a+b*x]]/(f*(m+1)) - + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c+I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Tan[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c+I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Cot[a+b*x]]/(f*(m+1)) - + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Cot[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Tan[a+b*x]]/(f*(m+1)) - + b*(1+I*c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+I*c-d+(1+I*c+d)*E^(2*I*a+2*I*b*x)),x] + + b*(1-I*c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-I*c+d+(1-I*c-d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c+I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Tan[a+b*x]]/(f*(m+1)) + + b*(1+I*c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+I*c-d+(1+I*c+d)*E^(2*I*a+2*I*b*x)),x] - + b*(1-I*c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-I*c+d+(1-I*c-d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c+I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Cot[a+b*x]]/(f*(m+1)) + + b*(1+I*c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+I*c+d-(1+I*c-d)*E^(2*I*a+2*I*b*x)),x] - + b*(1-I*c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-I*c-d-(1-I*c+d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-I*d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Cot[a+b*x]]/(f*(m+1)) - + b*(1+I*c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+I*c+d-(1+I*c-d)*E^(2*I*a+2*I*b*x)),x] + + b*(1-I*c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-I*c-d-(1-I*c+d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-I*d)^2+1] + + +Int[ArcTan[Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[Tanh[a+b*x]] - b*Int[x*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcCot[Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[Tanh[a+b*x]] + b*Int[x*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcTan[Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[Coth[a+b*x]] + b*Int[x*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcCot[Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[Coth[a+b*x]] - b*Int[x*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[Tanh[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[Tanh[a+b*x]]/(f*(m+1)) + b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[Coth[a+b*x]]/(f*(m+1)) + b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[Coth[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sech[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[ArcTan[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Tanh[a+b*x]] - + b*Int[x/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2+1] + + +Int[ArcCot[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Tanh[a+b*x]] + + b*Int[x/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2+1] + + +Int[ArcTan[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Coth[a+b*x]] - + b*Int[x/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2+1] + + +Int[ArcCot[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Coth[a+b*x]] + + b*Int[x/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2+1] + + +Int[ArcTan[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Tanh[a+b*x]] + + I*b*(I-c-d)*Int[x*E^(2*a+2*b*x)/(I-c+d+(I-c-d)*E^(2*a+2*b*x)),x] - + I*b*(I+c+d)*Int[x*E^(2*a+2*b*x)/(I+c-d+(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2+1] + + +Int[ArcCot[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Tanh[a+b*x]] - + I*b*(I-c-d)*Int[x*E^(2*a+2*b*x)/(I-c+d+(I-c-d)*E^(2*a+2*b*x)),x] + + I*b*(I+c+d)*Int[x*E^(2*a+2*b*x)/(I+c-d+(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2+1] + + +Int[ArcTan[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcTan[c+d*Coth[a+b*x]] - + I*b*(I-c-d)*Int[x*E^(2*a+2*b*x)/(I-c+d-(I-c-d)*E^(2*a+2*b*x)),x] + + I*b*(I+c+d)*Int[x*E^(2*a+2*b*x)/(I+c-d-(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2+1] + + +Int[ArcCot[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcCot[c+d*Coth[a+b*x]] + + I*b*(I-c-d)*Int[x*E^(2*a+2*b*x)/(I-c+d-(I-c-d)*E^(2*a+2*b*x)),x] - + I*b*(I+c+d)*Int[x*E^(2*a+2*b*x)/(I+c-d-(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Tanh[a+b*x]]/(f*(m+1)) - + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Tanh[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Coth[a+b*x]]/(f*(m+1)) - + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Coth[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Tanh[a+b*x]]/(f*(m+1)) + + I*b*(I-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I-c+d+(I-c-d)*E^(2*a+2*b*x)),x] - + I*b*(I+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I+c-d+(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Tanh[a+b*x]]/(f*(m+1)) - + I*b*(I-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I-c+d+(I-c-d)*E^(2*a+2*b*x)),x] + + I*b*(I+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I+c-d+(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcTan[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTan[c+d*Coth[a+b*x]]/(f*(m+1)) - + I*b*(I-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I-c+d-(I-c-d)*E^(2*a+2*b*x)),x] + + I*b*(I+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I+c-d-(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2+1] + + +Int[(e_.+f_.*x_)^m_.*ArcCot[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCot[c+d*Coth[a+b*x]]/(f*(m+1)) + + I*b*(I-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I-c+d-(I-c-d)*E^(2*a+2*b*x)),x] - + I*b*(I+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(I+c-d-(I+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2+1] + + +Int[ArcTan[u_],x_Symbol] := + x*ArcTan[u] - + Int[SimplifyIntegrand[x*D[u,x]/(1+u^2),x],x] /; +InverseFunctionFreeQ[u,x] + + +Int[ArcCot[u_],x_Symbol] := + x*ArcCot[u] + + Int[SimplifyIntegrand[x*D[u,x]/(1+u^2),x],x] /; +InverseFunctionFreeQ[u,x] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcTan[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcTan[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(1+u^2),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCot[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCot[u])/(d*(m+1)) + + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(1+u^2),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1,x]] + + +Int[v_*(a_.+b_.*ArcTan[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcTan[u]),w,x] - b*Int[SimplifyIntegrand[w*D[u,x]/(1+u^2),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] && FalseQ[FunctionOfLinear[v*(a+b*ArcTan[u]),x]] + + +Int[v_*(a_.+b_.*ArcCot[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCot[u]),w,x] + b*Int[SimplifyIntegrand[w*D[u,x]/(1+u^2),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] && FalseQ[FunctionOfLinear[v*(a+b*ArcCot[u]),x]] + + +Int[ArcTan[v_]*Log[w_]/(a_.+b_.*x_),x_Symbol] := + I/2*Int[Log[1-I*v]*Log[w]/(a+b*x),x] - I/2*Int[Log[1+I*v]*Log[w]/(a+b*x),x] /; +FreeQ[{a,b},x] && LinearQ[v,x] && LinearQ[w,x] && EqQ[Simplify[D[v/(a+b*x),x]]] && EqQ[Simplify[D[w/(a+b*x),x]]] + + +Int[ArcTan[v_]*Log[w_],x_Symbol] := + x*ArcTan[v]*Log[w] - + Int[SimplifyIntegrand[x*Log[w]*D[v,x]/(1+v^2),x],x] - + Int[SimplifyIntegrand[x*ArcTan[v]*D[w,x]/w,x],x] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + +Int[ArcCot[v_]*Log[w_],x_Symbol] := + x*ArcCot[v]*Log[w] + + Int[SimplifyIntegrand[x*Log[w]*D[v,x]/(1+v^2),x],x] - + Int[SimplifyIntegrand[x*ArcCot[v]*D[w,x]/w,x],x] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + +Int[u_*ArcTan[v_]*Log[w_],x_Symbol] := + With[{z=IntHide[u,x]}, + Dist[ArcTan[v]*Log[w],z,x] - + Int[SimplifyIntegrand[z*Log[w]*D[v,x]/(1+v^2),x],x] - + Int[SimplifyIntegrand[z*ArcTan[v]*D[w,x]/w,x],x] /; + InverseFunctionFreeQ[z,x]] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + +Int[u_*ArcCot[v_]*Log[w_],x_Symbol] := + With[{z=IntHide[u,x]}, + Dist[ArcCot[v]*Log[w],z,x] + + Int[SimplifyIntegrand[z*Log[w]*D[v,x]/(1+v^2),x],x] - + Int[SimplifyIntegrand[z*ArcCot[v]*D[w,x]/w,x],x] /; + InverseFunctionFreeQ[z,x]] /; +InverseFunctionFreeQ[v,x] && InverseFunctionFreeQ[w,x] + + + + + +(* ::Subsection::Closed:: *) +(*5.3/1 Miscellaneous inverse secant*) + + +Int[ArcSec[c_.*x_],x_Symbol] := + x*ArcSec[c*x] - 1/c*Int[1/(x*Sqrt[1-1/(c^2*x^2)]),x] /; +FreeQ[c,x] + + +Int[ArcCsc[c_.*x_],x_Symbol] := + x*ArcCsc[c*x] + 1/c*Int[1/(x*Sqrt[1-1/(c^2*x^2)]),x] /; +FreeQ[c,x] + + +Int[(a_.+b_.*ArcSec[c_.*x_])^n_,x_Symbol] := + 1/c*Subst[Int[(a+b*x)^n*Sec[x]*Tan[x],x],x,ArcSec[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcCsc[c_.*x_])^n_,x_Symbol] := + -1/c*Subst[Int[(a+b*x)^n*Csc[x]*Cot[x],x],x,ArcCsc[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcSec[c_.*x_])/x_,x_Symbol] := + -Subst[Int[(a+b*ArcCos[x/c])/x,x],x,1/x] /; +FreeQ[{a,b,c},x] + + +Int[(a_.+b_.*ArcCsc[c_.*x_])/x_,x_Symbol] := + -Subst[Int[(a+b*ArcSin[x/c])/x,x],x,1/x] /; +FreeQ[{a,b,c},x] + + +Int[x_^m_.*(a_.+b_.*ArcSec[c_.*x_]),x_Symbol] := + x^(m+1)*(a+b*ArcSec[c*x])/(m+1) - + b/(c*(m+1))*Int[x^(m-1)/Sqrt[1-1/(c^2*x^2)],x] /; +FreeQ[{a,b,c,m},x] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcCsc[c_.*x_]),x_Symbol] := + x^(m+1)*(a+b*ArcCsc[c*x])/(m+1) + + b/(c*(m+1))*Int[x^(m-1)/Sqrt[1-1/(c^2*x^2)],x] /; +FreeQ[{a,b,c,m},x] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcSec[c_.*x_])^n_,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Sec[x]^(m+1)*Tan[x],x],x,ArcSec[c*x]] /; +FreeQ[{a,b,c,n},x] && IntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcCsc[c_.*x_])^n_,x_Symbol] := + -1/c^(m+1)*Subst[Int[(a+b*x)^n*Csc[x]^(m+1)*Cot[x],x],x,ArcCsc[c*x]] /; +FreeQ[{a,b,c,n},x] && IntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(a+b*ArcSec[c*x])^n,x] /; +FreeQ[{a,b,c,m,n},x] + + +Int[x_^m_.*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(a+b*ArcCsc[c*x])^n,x] /; +FreeQ[{a,b,c,m,n},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[(a+b*ArcSec[c*x]),u,x] - b*c*x/Sqrt[c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2-1]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[(a+b*ArcCsc[c*x]),u,x] + b*c*x/Sqrt[c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2-1]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegerQ[p] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegerQ[p] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcSec[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCsc[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSec[c*x])/(2*e*(p+1)) - + b*c*x/(2*e*(p+1)*Sqrt[c^2*x^2])*Int[(d+e*x^2)^(p+1)/(x*Sqrt[c^2*x^2-1]),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCsc[c*x])/(2*e*(p+1)) + + b*c*x/(2*e*(p+1)*Sqrt[c^2*x^2])*Int[(d+e*x^2)^(p+1)/(x*Sqrt[c^2*x^2-1]),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[(a+b*ArcSec[c*x]),u,x] - b*c*x/Sqrt[c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2-1]),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[(a+b*ArcCsc[c*x]),u,x] + b*c*x/Sqrt[c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2-1]),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegersQ[m,p] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegersQ[m,p] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcCos[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcSin[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSec[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcSec[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsc[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcCsc[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[ArcSec[a_+b_.*x_],x_Symbol] := + (a+b*x)*ArcSec[a+b*x]/b - + Int[1/((a+b*x)*Sqrt[1-1/(a+b*x)^2]),x] /; +FreeQ[{a,b},x] + + +Int[ArcCsc[a_+b_.*x_],x_Symbol] := + (a+b*x)*ArcCsc[a+b*x]/b + + Int[1/((a+b*x)*Sqrt[1-1/(a+b*x)^2]),x] /; +FreeQ[{a,b},x] + + +Int[ArcSec[a_+b_.*x_]^n_,x_Symbol] := + 1/b*Subst[Int[x^n*Sec[x]*Tan[x],x],x,ArcSec[a+b*x]] /; +FreeQ[{a,b,n},x] + + +Int[ArcCsc[a_+b_.*x_]^n_,x_Symbol] := + -1/b*Subst[Int[x^n*Csc[x]*Cot[x],x],x,ArcCsc[a+b*x]] /; +FreeQ[{a,b,n},x] + + +Int[ArcSec[a_+b_.*x_]/x_,x_Symbol] := + ArcSec[a+b*x]*Log[1-(1-Sqrt[1-a^2])*E^(I*ArcSec[a+b*x])/a] + + ArcSec[a+b*x]*Log[1-(1+Sqrt[1-a^2])*E^(I*ArcSec[a+b*x])/a] - + ArcSec[a+b*x]*Log[1+E^(2*I*ArcSec[a+b*x])] - + I*PolyLog[2,(1-Sqrt[1-a^2])*E^(I*ArcSec[a+b*x])/a] - + I*PolyLog[2,(1+Sqrt[1-a^2])*E^(I*ArcSec[a+b*x])/a] + + 1/2*I*PolyLog[2,-E^(2*I*ArcSec[a+b*x])] /; +FreeQ[{a,b},x] + + +Int[ArcCsc[a_+b_.*x_]/x_,x_Symbol] := + I*ArcCsc[a+b*x]^2 + + ArcCsc[a+b*x]*Log[1-I*(1-Sqrt[1-a^2])/(E^(I*ArcCsc[a+b*x])*a)] + + ArcCsc[a+b*x]*Log[1-I*(1+Sqrt[1-a^2])/(E^(I*ArcCsc[a+b*x])*a)] - + ArcCsc[a+b*x]*Log[1-E^(2*I*ArcCsc[a+b*x])] + + I*PolyLog[2,I*(1-Sqrt[1-a^2])/(E^(I*ArcCsc[a+b*x])*a)] + + I*PolyLog[2,I*(1+Sqrt[1-a^2])/(E^(I*ArcCsc[a+b*x])*a)] + + 1/2*I*PolyLog[2,E^(2*I*ArcCsc[a+b*x])] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*ArcSec[a_+b_.*x_],x_Symbol] := + -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcSec[a+b*x]/(b^(m+1)*(m+1)) - + 1/(b^(m+1)*(m+1))*Subst[Int[(((-a)*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)*Sqrt[1-x^2]),x],x,1/(a+b*x)] /; +FreeQ[{a,b,m},x] && IntegerQ[m] && NeQ[m+1] + + +Int[x_^m_.*ArcCsc[a_+b_.*x_],x_Symbol] := + -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcCsc[a+b*x]/(b^(m+1)*(m+1)) + + 1/(b^(m+1)*(m+1))*Subst[Int[(((-a)*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)*Sqrt[1-x^2]),x],x,1/(a+b*x)] /; +FreeQ[{a,b,m},x] && IntegerQ[m] && NeQ[m+1] + + +Int[x_^m_.*ArcSec[a_+b_.*x_]^n_,x_Symbol] := + 1/b^(m+1)*Subst[Int[x^n*(-a+Sec[x])^m*Sec[x]*Tan[x],x],x,ArcSec[a+b*x]] /; +FreeQ[{a,b,n},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*ArcCsc[a_+b_.*x_]^n_,x_Symbol] := + -1/b^(m+1)*Subst[Int[x^n*(-a+Csc[x])^m*Csc[x]*Cot[x],x],x,ArcCsc[a+b*x]] /; +FreeQ[{a,b,n},x] && PositiveIntegerQ[m] + + +Int[u_.*ArcSec[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCos[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCsc[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcSin[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*f_^(c_.*ArcSec[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[ReplaceAll[u,x->-a/b+Sec[x]/b]*f^(c*x^n)*Sec[x]*Tan[x],x],x,ArcSec[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[u_.*f_^(c_.*ArcCsc[a_.+b_.*x_]^n_.),x_Symbol] := + -1/b*Subst[Int[ReplaceAll[u,x->-a/b+Csc[x]/b]*f^(c*x^n)*Csc[x]*Cot[x],x],x,ArcCsc[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[ArcSec[u_],x_Symbol] := + x*ArcSec[u] - + u/Sqrt[u^2]*Int[SimplifyIntegrand[x*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[ArcCsc[u_],x_Symbol] := + x*ArcCsc[u] + + u/Sqrt[u^2]*Int[SimplifyIntegrand[x*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcSec[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcSec[u])/(d*(m+1)) - + b*u/(d*(m+1)*Sqrt[u^2])*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCsc[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCsc[u])/(d*(m+1)) + + b*u/(d*(m+1)*Sqrt[u^2])*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[v_*(a_.+b_.*ArcSec[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcSec[u]),w,x] - b*u/Sqrt[u^2]*Int[SimplifyIntegrand[w*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + +Int[v_*(a_.+b_.*ArcCsc[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCsc[u]),w,x] + b*u/Sqrt[u^2]*Int[SimplifyIntegrand[w*D[u,x]/(u*Sqrt[u^2-1]),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + + +`) + +func resourcesRubi5InverseTrigFunctionsMBytes() ([]byte, error) { + return _resourcesRubi5InverseTrigFunctionsM, nil +} + +func resourcesRubi5InverseTrigFunctionsM() (*asset, error) { + bytes, err := resourcesRubi5InverseTrigFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/5 Inverse trig functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi6HyperbolicFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Hyperbolic Function Rules*) + + +(* ::Subsection::Closed:: *) +(*6.1.10 (c+d x)^m (a+b sinh)^n*) + + +Int[u_^m_.*(a_.+b_.*Sinh[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Sinh[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*(a_.+b_.*Cosh[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Cosh[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*6.1.11 (e x)^m (a+b x^n)^p sinh*) + + +Int[(a_+b_.*x_^n_)^p_.*Sinh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sinh[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_.*Cosh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cosh[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + x^(-n+1)*(a+b*x^n)^(p+1)*Sinh[c+d*x]/(b*n*(p+1)) - + (-n+1)/(b*n*(p+1))*Int[x^(-n)*(a+b*x^n)^(p+1)*Sinh[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(-n+1)*(a+b*x^n)^(p+1)*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && p<-1 && n>2 + + +Int[(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + x^(-n+1)*(a+b*x^n)^(p+1)*Cosh[c+d*x]/(b*n*(p+1)) - + (-n+1)/(b*n*(p+1))*Int[x^(-n)*(a+b*x^n)^(p+1)*Cosh[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(-n+1)*(a+b*x^n)^(p+1)*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && p<-1 && n>2 + + +Int[(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sinh[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cosh[c+d*x],(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + Int[x^(n*p)*(b+a*x^(-n))^p*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + Int[x^(n*p)*(b+a*x^(-n))^p*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + Defer[Int][(a+b*x^n)^p*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + Defer[Int][(a+b*x^n)^p*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Sinh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sinh[c+d*x],(e*x)^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && PositiveIntegerQ[p] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Cosh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cosh[c+d*x],(e*x)^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && PositiveIntegerQ[p] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + e^m*(a+b*x^n)^(p+1)*Sinh[c+d*x]/(b*n*(p+1)) - + d*e^m/(b*n*(p+1))*Int[(a+b*x^n)^(p+1)*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && IntegerQ[p] && EqQ[m-n+1] && RationalQ[p] && p<-1 && (IntegerQ[n] || PositiveQ[e]) + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + e^m*(a+b*x^n)^(p+1)*Cosh[c+d*x]/(b*n*(p+1)) - + d*e^m/(b*n*(p+1))*Int[(a+b*x^n)^(p+1)*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n},x] && IntegerQ[p] && EqQ[m-n+1] && RationalQ[p] && p<-1 && (IntegerQ[n] || PositiveQ[e]) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + x^(m-n+1)*(a+b*x^n)^(p+1)*Sinh[c+d*x]/(b*n*(p+1)) - + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*(a+b*x^n)^(p+1)*Sinh[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(m-n+1)*(a+b*x^n)^(p+1)*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && RationalQ[m] && p<-1 && (m-n+1>0 || n>2) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + x^(m-n+1)*(a+b*x^n)^(p+1)*Cosh[c+d*x]/(b*n*(p+1)) - + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*(a+b*x^n)^(p+1)*Cosh[c+d*x],x] - + d/(b*n*(p+1))*Int[x^(m-n+1)*(a+b*x^n)^(p+1)*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && PositiveIntegerQ[n] && RationalQ[m] && p<-1 && (m-n+1>0 || n>2) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Sinh[c+d*x],x^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && IntegerQ[m] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + Int[ExpandIntegrand[Cosh[c+d*x],x^m*(a+b*x^n)^p,x],x] /; +FreeQ[{a,b,c,d},x] && NegativeIntegerQ[p] && IntegerQ[m] && PositiveIntegerQ[n] && (n==2 || p==-1) + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Sinh[c_.+d_.*x_],x_Symbol] := + Int[x^(m+n*p)*(b+a*x^(-n))^p*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d,m},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[x_^m_.*(a_+b_.*x_^n_)^p_*Cosh[c_.+d_.*x_],x_Symbol] := + Int[x^(m+n*p)*(b+a*x^(-n))^p*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d,m},x] && NegativeIntegerQ[p] && NegativeIntegerQ[n] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Sinh[c_.+d_.*x_],x_Symbol] := + Defer[Int][(e*x)^m*(a+b*x^n)^p*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_.*x_)^m_.*(a_+b_.*x_^n_)^p_.*Cosh[c_.+d_.*x_],x_Symbol] := + Defer[Int][(e*x)^m*(a+b*x^n)^p*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + + + + +(* ::Subsection::Closed:: *) +(*6.1.12 (e x)^m (a+b sinh(c+d x^n))^p*) + + +Int[Sinh[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(c+d*x^n),x] - 1/2*Int[E^(-c-d*x^n),x] /; +FreeQ[{c,d},x] && IntegerQ[n] && n>1 + + +Int[Cosh[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(c+d*x^n),x] + 1/2*Int[E^(-c-d*x^n),x] /; +FreeQ[{c,d},x] && IntegerQ[n] && n>1 + + +Int[(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Sinh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && n>1 && p>1 + + +Int[(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Cosh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && n>1 && p>1 + + +Int[(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(a+b*Sinh[c+d*x^(-n)])^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(a+b*Cosh[c+d*x^(-n)])^p/x^2,x],x,1/x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && NegativeIntegerQ[n] + + +Int[(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Module[{k=Denominator[n]}, + k*Subst[Int[x^(k-1)*(a+b*Sinh[c+d*x^(k*n)])^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && FractionQ[n] + + +Int[(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Module[{k=Denominator[n]}, + k*Subst[Int[x^(k-1)*(a+b*Cosh[c+d*x^(k*n)])^p,x],x,x^(1/k)]] /; +FreeQ[{a,b,c,d},x] && IntegerQ[p] && FractionQ[n] + + +Int[Sinh[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(c+d*x^n),x] - 1/2*Int[E^(-c-d*x^n),x] /; +FreeQ[{c,d,n},x] + + +Int[Cosh[c_.+d_.*x_^n_],x_Symbol] := + 1/2*Int[E^(c+d*x^n),x] + 1/2*Int[E^(-c-d*x^n),x] /; +FreeQ[{c,d,n},x] + + +Int[(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Sinh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(a+b*Cosh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[p] + + +Int[(a_.+b_.*Sinh[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Sinh[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n},x] && IntegerQ[p] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Cosh[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Cosh[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n},x] && IntegerQ[p] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Sinh[c_.+d_.*u_^n_])^p_,x_Symbol] := + Defer[Int][(a+b*Sinh[c+d*u^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] + + +Int[(a_.+b_.*Cosh[c_.+d_.*u_^n_])^p_,x_Symbol] := + Defer[Int][(a+b*Cosh[c+d*u^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] + + +Int[(a_.+b_.*Sinh[u_])^p_.,x_Symbol] := + Int[(a+b*Sinh[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Cosh[u_])^p_.,x_Symbol] := + Int[(a+b*Cosh[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[Sinh[d_.*x_^n_]/x_,x_Symbol] := + SinhIntegral[d*x^n]/n /; +FreeQ[{d,n},x] + + +Int[Cosh[d_.*x_^n_]/x_,x_Symbol] := + CoshIntegral[d*x^n]/n /; +FreeQ[{d,n},x] + + +Int[Sinh[c_+d_.*x_^n_]/x_,x_Symbol] := + Sinh[c]*Int[Cosh[d*x^n]/x,x] + Cosh[c]*Int[Sinh[d*x^n]/x,x] /; +FreeQ[{c,d,n},x] + + +Int[Cosh[c_+d_.*x_^n_]/x_,x_Symbol] := + Cosh[c]*Int[Cosh[d*x^n]/x,x] + Sinh[c]*Int[Sinh[d*x^n]/x,x] /; +FreeQ[{c,d,n},x] + + +Int[x_^m_.*(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Sinh[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] && (EqQ[p,1] || EqQ[m,n-1] || IntegerQ[p] && Simplify[(m+1)/n]>0) + + +Int[x_^m_.*(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Cosh[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] && (EqQ[p,1] || EqQ[m,n-1] || IntegerQ[p] && Simplify[(m+1)/n]>0) + + +Int[(e_*x_)^m_*(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Sinh[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_*x_)^m_*(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Cosh[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && IntegerQ[Simplify[(m+1)/n]] + + +Int[(e_.*x_)^m_.*Sinh[c_.+d_.*x_^n_],x_Symbol] := + e^(n-1)*(e*x)^(m-n+1)*Cosh[c+d*x^n]/(d*n) - + e^n*(m-n+1)/(d*n)*Int[(e*x)^(m-n)*Cosh[c+d*x^n],x] /; +FreeQ[{c,d,e},x] && PositiveIntegerQ[n] && RationalQ[m] && 01 && NeQ[n-1] + + +Int[x_^m_.*Cosh[a_.+b_.*x_^n_]^p_,x_Symbol] := + -Cosh[a+b*x^n]^p/((n-1)*x^(n-1)) + + b*n*p/(n-1)*Int[Cosh[a+b*x^n]^(p-1)*Sinh[a+b*x^n],x] /; +FreeQ[{a,b},x] && IntegersQ[n,p] && EqQ[m+n] && p>1 && NeQ[n-1] + + +Int[x_^m_.*Sinh[a_.+b_.*x_^n_]^p_,x_Symbol] := + -n*Sinh[a+b*x^n]^p/(b^2*n^2*p^2) + + x^n*Cosh[a+b*x^n]*Sinh[a+b*x^n]^(p-1)/(b*n*p) - + (p-1)/p*Int[x^m*Sinh[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p>1 + + +Int[x_^m_.*Cosh[a_.+b_.*x_^n_]^p_,x_Symbol] := + -n*Cosh[a+b*x^n]^p/(b^2*n^2*p^2) + + x^n*Sinh[a+b*x^n]*Cosh[a+b*x^n]^(p-1)/(b*n*p) + + (p-1)/p*Int[x^m*Cosh[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p>1 + + +Int[x_^m_.*Sinh[a_.+b_.*x_^n_]^p_,x_Symbol] := + -(m-n+1)*x^(m-2*n+1)*Sinh[a+b*x^n]^p/(b^2*n^2*p^2) + + x^(m-n+1)*Cosh[a+b*x^n]*Sinh[a+b*x^n]^(p-1)/(b*n*p) - + (p-1)/p*Int[x^m*Sinh[a+b*x^n]^(p-2),x] + + (m-n+1)*(m-2*n+1)/(b^2*n^2*p^2)*Int[x^(m-2*n)*Sinh[a+b*x^n]^p,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p>1 && 0<2*n1 && 0<2*n1 && 0<2*n<1-m && NeQ[m+n+1] + + +Int[x_^m_.*Cosh[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m+1)*Cosh[a+b*x^n]^p/(m+1) - + b*n*p*x^(m+n+1)*Sinh[a+b*x^n]*Cosh[a+b*x^n]^(p-1)/((m+1)*(m+n+1)) + + b^2*n^2*p^2/((m+1)*(m+n+1))*Int[x^(m+2*n)*Cosh[a+b*x^n]^p,x] - + b^2*n^2*p*(p-1)/((m+1)*(m+n+1))*Int[x^(m+2*n)*Cosh[a+b*x^n]^(p-2),x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p>1 && 0<2*n<1-m && NeQ[m+n+1] + + +Int[(e_.*x_)^m_*(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + With[{k=Denominator[m]}, + k/e*Subst[Int[x^(k*(m+1)-1)*(a+b*Sinh[c+d*x^(k*n)/e^n])^p,x],x,(e*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] && FractionQ[m] + + +Int[(e_.*x_)^m_*(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + With[{k=Denominator[m]}, + k/e*Subst[Int[x^(k*(m+1)-1)*(a+b*Cosh[c+d*x^(k*n)/e^n])^p,x],x,(e*x)^(1/k)]] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] && FractionQ[m] + + +Int[(e_.*x_)^m_.*(a_.+b_.*Sinh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(e*x)^m,(a+b*Sinh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && p>1 + + +Int[(e_.*x_)^m_.*(a_.+b_.*Cosh[c_.+d_.*x_^n_])^p_,x_Symbol] := + Int[ExpandTrigReduce[(e*x)^m,(a+b*Cosh[c+d*x^n])^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && p>1 + + +Int[x_^m_.*Sinh[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^n*Cosh[a+b*x^n]*Sinh[a+b*x^n]^(p+1)/(b*n*(p+1)) - + n*Sinh[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) - + (p+2)/(p+1)*Int[x^m*Sinh[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p<-1 && p!=-2 + + +Int[x_^m_.*Cosh[a_.+b_.*x_^n_]^p_,x_Symbol] := + -x^n*Sinh[a+b*x^n]*Cosh[a+b*x^n]^(p+1)/(b*n*(p+1)) + + n*Cosh[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (p+2)/(p+1)*Int[x^m*Cosh[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-2*n+1] && RationalQ[p] && p<-1 && p!=-2 + + +Int[x_^m_.*Sinh[a_.+b_.*x_^n_]^p_,x_Symbol] := + x^(m-n+1)*Cosh[a+b*x^n]*Sinh[a+b*x^n]^(p+1)/(b*n*(p+1)) - + (m-n+1)*x^(m-2*n+1)*Sinh[a+b*x^n]^(p+2)/(b^2*n^2*(p+1)*(p+2)) - + (p+2)/(p+1)*Int[x^m*Sinh[a+b*x^n]^(p+2),x] + + (m-n+1)*(m-2*n+1)/(b^2*n^2*(p+1)*(p+2))*Int[x^(m-2*n)*Sinh[a+b*x^n]^(p+2),x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && RationalQ[p] && p<-1 && p!=-2 && 0<2*n1 + + +Int[Cosh[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[Cosh[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[Sinh[v_]^n_.,x_Symbol] := + Int[Sinh[ExpandToSum[v,x]]^n,x] /; +PositiveIntegerQ[n] && QuadraticQ[v,x] && Not[QuadraticMatchQ[v,x]] + + +Int[Cosh[v_]^n_.,x_Symbol] := + Int[Cosh[ExpandToSum[v,x]]^n,x] /; +PositiveIntegerQ[n] && QuadraticQ[v,x] && Not[QuadraticMatchQ[v,x]] + + +Int[(d_.+e_.*x_)*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Cosh[a+b*x+c*x^2]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Sinh[a+b*x+c*x^2]/(2*c) /; +FreeQ[{a,b,c,d,e},x] && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Cosh[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Sinh[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Cosh[a+b*x+c*x^2]/(2*c) - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m>1 && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Sinh[a+b*x+c*x^2]/(2*c) - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m>1 && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Cosh[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[(d+e*x)^(m-1)*Sinh[a+b*x+c*x^2],x] - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m>1 && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*(d+e*x)^(m-1)*Sinh[a+b*x+c*x^2]/(2*c) - + (b*e-2*c*d)/(2*c)*Int[(d+e*x)^(m-1)*Cosh[a+b*x+c*x^2],x] - + e^2*(m-1)/(2*c)*Int[(d+e*x)^(m-2)*Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m>1 && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Sinh[a+b*x+c*x^2]/(e*(m+1)) - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m<-1 && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Cosh[a+b*x+c*x^2]/(e*(m+1)) - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m<-1 && EqQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Sinh[a+b*x+c*x^2]/(e*(m+1)) - + (b*e-2*c*d)/(e^2*(m+1))*Int[(d+e*x)^(m+1)*Cosh[a+b*x+c*x^2],x] - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m<-1 && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + (d+e*x)^(m+1)*Cosh[a+b*x+c*x^2]/(e*(m+1)) - + (b*e-2*c*d)/(e^2*(m+1))*Int[(d+e*x)^(m+1)*Sinh[a+b*x+c*x^2],x] - + 2*c/(e^2*(m+1))*Int[(d+e*x)^(m+2)*Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m] && m<-1 && NeQ[b*e-2*c*d] + + +Int[(d_.+e_.*x_)^m_.*Sinh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + Defer[Int][(d+e*x)^m*Sinh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] + + +Int[(d_.+e_.*x_)^m_.*Cosh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + Defer[Int][(d+e*x)^m*Cosh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] + + +Int[(d_.+e_.*x_)^m_.*Sinh[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[(d+e*x)^m,Sinh[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[n] && n>1 + + +Int[(d_.+e_.*x_)^m_.*Cosh[a_.+b_.*x_+c_.*x_^2]^n_,x_Symbol] := + Int[ExpandTrigReduce[(d+e*x)^m,Cosh[a+b*x+c*x^2]^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[n] && n>1 + + +Int[u_^m_.*Sinh[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Sinh[ExpandToSum[v,x]]^n,x] /; +FreeQ[m,x] && PositiveIntegerQ[n] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + +Int[u_^m_.*Cosh[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Cosh[ExpandToSum[v,x]]^n,x] /; +FreeQ[m,x] && PositiveIntegerQ[n] && LinearQ[u,x] && QuadraticQ[v,x] && Not[LinearMatchQ[u,x] && QuadraticMatchQ[v,x]] + + + + + +(* ::Subsection::Closed:: *) +(*6.2.10 (c+d x)^m (a+b tanh)^n*) + + +Int[u_^m_.*(a_.+b_.*Tanh[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Tanh[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*(a_.+b_.*Coth[v_])^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*(a+b*Coth[ExpandToSum[v,x]])^n,x] /; +FreeQ[{a,b,m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*6.2.11 (e x)^m (a+b tanh(c+d x^n))^p*) + + +Int[(a_.+b_.*Tanh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Tanh[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Coth[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Coth[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Tanh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Tanh[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Coth[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Coth[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Tanh[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Tanh[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Coth[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Coth[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Tanh[u_])^p_.,x_Symbol] := + Int[(a+b*Tanh[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Coth[u_])^p_.,x_Symbol] := + Int[(a+b*Coth[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*(a_.+b_.*Tanh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Tanh[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Coth[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Coth[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*Tanh[c_.+d_.*x_^n_]^2,x_Symbol] := + -x^(m-n+1)*Tanh[c+d*x^n]/(d*n) + Int[x^m,x] + (m-n+1)/(d*n)*Int[x^(m-n)*Tanh[c+d*x^n],x] /; +FreeQ[{c,d,m,n},x] + + +Int[x_^m_.*Coth[c_.+d_.*x_^n_]^2,x_Symbol] := + -x^(m-n+1)*Coth[c+d*x^n]/(d*n) + Int[x^m,x] + (m-n+1)/(d*n)*Int[x^(m-n)*Coth[c+d*x^n],x] /; +FreeQ[{c,d,m,n},x] + + +Int[x_^m_.*(a_.+b_.*Tanh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Tanh[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[x_^m_.*(a_.+b_.*Coth[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Coth[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Tanh[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Tanh[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Coth[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Coth[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Tanh[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Tanh[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(e_*x_)^m_.*(a_.+b_.*Coth[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Coth[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*Sech[a_.+b_.*x_^n_.]^p_.*Tanh[a_.+b_.*x_^n_.]^q_.,x_Symbol] := + -x^(m-n+1)*Sech[a+b*x^n]^p/(b*n*p) + + (m-n+1)/(b*n*p)*Int[x^(m-n)*Sech[a+b*x^n]^p,x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && q===1 + + +Int[x_^m_.*Csch[a_.+b_.*x_^n_.]^p_.*Coth[a_.+b_.*x_^n_.]^q_.,x_Symbol] := + -x^(m-n+1)*Csch[a+b*x^n]^p/(b*n*p) + + (m-n+1)/(b*n*p)*Int[x^(m-n)*Csch[a+b*x^n]^p,x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && q===1 + + + + + +(* ::Subsection::Closed:: *) +(*6.2.12 (d+e x)^m tanh(a+b x+c x^2)^n*) +(**) + + +Int[Tanh[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Tanh[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[Coth[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][Coth[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[(d_.+e_.*x_)*Tanh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Log[Cosh[a+b*x+c*x^2]]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Tanh[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_)*Coth[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + e*Log[Sinh[a+b*x+c*x^2]]/(2*c) + + (2*c*d-b*e)/(2*c)*Int[Coth[a+b*x+c*x^2],x] /; +FreeQ[{a,b,c,d,e},x] + + +(* Int[x_^m_*Tanh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + x^(m-1)*Log[Cosh[a+b*x+c*x^2]]/(2*c) - + b/(2*c)*Int[x^(m-1)*Tanh[a+b*x+c*x^2],x] - + (m-1)/(2*c)*Int[x^(m-2)*Log[Cosh[a+b*x+c*x^2]],x] /; +FreeQ[{a,b,c},x] && RationalQ[m] && m>1 *) + + +(* Int[x_^m_*Coth[a_.+b_.*x_+c_.*x_^2],x_Symbol] := + x^(m-1)*Log[Sinh[a+b*x+c*x^2]]/(2*c) - + b/(2*c)*Int[x^(m-1)*Coth[a+b*x+c*x^2],x] - + (m-1)/(2*c)*Int[x^(m-2)*Log[Sinh[a+b*x+c*x^2]],x] /; +FreeQ[{a,b,c},x] && RationalQ[m] && m>1 *) + + +Int[(d_.+e_.*x_)^m_.*Tanh[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Tanh[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + +Int[(d_.+e_.*x_)^m_.*Coth[a_.+b_.*x_+c_.*x_^2]^n_.,x_Symbol] := + Defer[Int][(d+e*x)^m*Coth[a+b*x+c*x^2]^n,x] /; +FreeQ[{a,b,c,d,e,m,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*6.3.10 (c+d x)^m (a+b sech)^n*) + + +Int[u_^m_.*Sech[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Sech[ExpandToSum[v,x]]^n,x] /; +FreeQ[{m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[u_^m_.*Csch[v_]^n_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*Csch[ExpandToSum[v,x]]^n,x] /; +FreeQ[{m,n},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + + + + +(* ::Subsection::Closed:: *) +(*6.3.11 (e x)^m (a+b sech(c+d x^n))^p*) +(**) + + +Int[(a_.+b_.*Sech[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Sech[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Csch[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(1/n-1)*(a+b*Csch[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[1/n] && IntegerQ[p] + + +Int[(a_.+b_.*Sech[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Sech[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Csch[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][(a+b*Csch[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,n,p},x] + + +Int[(a_.+b_.*Sech[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Sech[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Csch[c_.+d_.*u_^n_])^p_.,x_Symbol] := + 1/Coefficient[u,x,1]*Subst[Int[(a+b*Csch[c+d*x^n])^p,x],x,u] /; +FreeQ[{a,b,c,d,n,p},x] && LinearQ[u,x] && NeQ[u-x] + + +Int[(a_.+b_.*Sech[u_])^p_.,x_Symbol] := + Int[(a+b*Sech[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(a_.+b_.*Csch[u_])^p_.,x_Symbol] := + Int[(a+b*Csch[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*(a_.+b_.*Sech[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Sech[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Csch[c_.+d_.*x_^n_])^p_.,x_Symbol] := + 1/n*Subst[Int[x^(Simplify[(m+1)/n]-1)*(a+b*Csch[c+d*x])^p,x],x,x^n] /; +FreeQ[{a,b,c,d,m,n,p},x] && PositiveIntegerQ[Simplify[(m+1)/n]] && IntegerQ[p] + + +Int[x_^m_.*(a_.+b_.*Sech[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Sech[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[x_^m_.*(a_.+b_.*Csch[c_.+d_.*x_^n_])^p_.,x_Symbol] := + Defer[Int][x^m*(a+b*Csch[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Sech[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Sech[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Csch[c_.+d_.*x_^n_])^p_.,x_Symbol] := + e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]*Int[x^m*(a+b*Csch[c+d*x^n])^p,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(e_*x_)^m_.*(a_.+b_.*Sech[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Sech[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[(e_*x_)^m_.*(a_.+b_.*Csch[u_])^p_.,x_Symbol] := + Int[(e*x)^m*(a+b*Csch[ExpandToSum[u,x]])^p,x] /; +FreeQ[{a,b,e,m,p},x] && BinomialQ[u,x] && Not[BinomialMatchQ[u,x]] + + +Int[x_^m_.*Sech[a_.+b_.*x_^n_.]^p_*Sinh[a_.+b_.*x_^n_.],x_Symbol] := + -x^(m-n+1)*Sech[a+b*x^n]^(p-1)/(b*n*(p-1)) + + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Sech[a+b*x^n]^(p-1),x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && NeQ[p-1] + + +Int[x_^m_.*Csch[a_.+b_.*x_^n_.]^p_*Cosh[a_.+b_.*x_^n_.],x_Symbol] := + -x^(m-n+1)*Csch[a+b*x^n]^(p-1)/(b*n*(p-1)) + + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Csch[a+b*x^n]^(p-1),x] /; +FreeQ[{a,b,p},x] && RationalQ[m] && IntegerQ[n] && m-n>=0 && NeQ[p-1] + + + + + +(* ::Subsection::Closed:: *) +(*6.4.5 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p*) + + +Int[(c_.+d_.*x_)^m_.*Sinh[a_.+b_.*x_]^n_.*Cosh[a_.+b_.*x_],x_Symbol] := + (c+d*x)^m*Sinh[a+b*x]^(n+1)/(b*(n+1)) - + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Sinh[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sinh[a_.+b_.*x_]*Cosh[a_.+b_.*x_]^n_.,x_Symbol] := + (c+d*x)^m*Cosh[a+b*x]^(n+1)/(b*(n+1)) - + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Cosh[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sinh[a_.+b_.*x_]^n_.*Cosh[a_.+b_.*x_]^p_.,x_Symbol] := + Int[ExpandTrigReduce[(c+d*x)^m,Sinh[a+b*x]^n*Cosh[a+b*x]^p,x],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Sinh[a_.+b_.*x_]^n_.*Tanh[a_.+b_.*x_]^p_.,x_Symbol] := + Int[(c+d*x)^m*Sinh[a+b*x]^n*Tanh[a+b*x]^(p-2),x] - Int[(c+d*x)^m*Sinh[a+b*x]^(n-2)*Tanh[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Cosh[a_.+b_.*x_]^n_.*Coth[a_.+b_.*x_]^p_.,x_Symbol] := + Int[(c+d*x)^m*Cosh[a+b*x]^n*Coth[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Cosh[a+b*x]^(n-2)*Coth[a+b*x]^p,x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n,p] + + +Int[(c_.+d_.*x_)^m_.*Sech[a_.+b_.*x_]^n_.*Tanh[a_.+b_.*x_]^p_.,x_Symbol] := + -(c+d*x)^m*Sech[a+b*x]^n/(b*n) + + d*m/(b*n)*Int[(c+d*x)^(m-1)*Sech[a+b*x]^n,x] /; +FreeQ[{a,b,c,d,n},x] && p===1 && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^n_.*Coth[a_.+b_.*x_]^p_.,x_Symbol] := + -(c+d*x)^m*Csch[a+b*x]^n/(b*n) + + d*m/(b*n)*Int[(c+d*x)^(m-1)*Csch[a+b*x]^n,x] /; +FreeQ[{a,b,c,d,n},x] && p===1 && RationalQ[m] && m>0 + + +Int[(c_.+d_.*x_)^m_.*Sech[a_.+b_.*x_]^2*Tanh[a_.+b_.*x_]^n_.,x_Symbol] := + (c+d*x)^m*Tanh[a+b*x]^(n+1)/(b*(n+1)) - + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Tanh[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^2*Coth[a_.+b_.*x_]^n_.,x_Symbol] := + -(c+d*x)^m*Coth[a+b*x]^(n+1)/(b*(n+1)) + + d*m/(b*(n+1))*Int[(c+d*x)^(m-1)*Coth[a+b*x]^(n+1),x] /; +FreeQ[{a,b,c,d,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(c_.+d_.*x_)^m_.*Sech[a_.+b_.*x_]*Tanh[a_.+b_.*x_]^p_,x_Symbol] := + Int[(c+d*x)^m*Sech[a+b*x]*Tanh[a+b*x]^(p-2),x] - Int[(c+d*x)^m*Sech[a+b*x]^3*Tanh[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Sech[a_.+b_.*x_]^n_.*Tanh[a_.+b_.*x_]^p_,x_Symbol] := + Int[(c+d*x)^m*Sech[a+b*x]^n*Tanh[a+b*x]^(p-2),x] - Int[(c+d*x)^m*Sech[a+b*x]^(n+2)*Tanh[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m,n},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]*Coth[a_.+b_.*x_]^p_,x_Symbol] := + Int[(c+d*x)^m*Csch[a+b*x]*Coth[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Csch[a+b*x]^3*Coth[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^n_.*Coth[a_.+b_.*x_]^p_,x_Symbol] := + Int[(c+d*x)^m*Csch[a+b*x]^n*Coth[a+b*x]^(p-2),x] + Int[(c+d*x)^m*Csch[a+b*x]^(n+2)*Coth[a+b*x]^(p-2),x] /; +FreeQ[{a,b,c,d,m,n},x] && PositiveIntegerQ[p/2] + + +Int[(c_.+d_.*x_)^m_.*Sech[a_.+b_.*x_]^n_.*Tanh[a_.+b_.*x_]^p_.,x_Symbol] := + With[{u=IntHide[Sech[a+b*x]^n*Tanh[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d,n,p},x] && PositiveIntegerQ[m] && (EvenQ[n] || OddQ[p]) + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^n_.*Coth[a_.+b_.*x_]^p_.,x_Symbol] := + With[{u=IntHide[Csch[a+b*x]^n*Coth[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d,n,p},x] && PositiveIntegerQ[m] && (EvenQ[n] || OddQ[p]) + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^n_.*Sech[a_.+b_.*x_]^n_., x_Symbol] := + 2^n*Int[(c+d*x)^m*Csch[2*a+2*b*x]^n,x] /; +FreeQ[{a,b,c,d},x] && RationalQ[m] && IntegerQ[n] + + +Int[(c_.+d_.*x_)^m_.*Csch[a_.+b_.*x_]^n_.*Sech[a_.+b_.*x_]^p_., x_Symbol] := + With[{u=IntHide[Csch[a+b*x]^n*Sech[a+b*x]^p,x]}, + Dist[(c+d*x)^m,u,x] - d*m*Int[(c+d*x)^(m-1)*u,x]] /; +FreeQ[{a,b,c,d},x] && IntegersQ[n,p] && RationalQ[m] && m>0 && n!=p + + +Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_.,x_Symbol] := + Int[ExpandToSum[u,x]^m*F[ExpandToSum[v,x]]^n*G[ExpandToSum[v,x]]^p,x] /; +FreeQ[{m,n,p},x] && HyperbolicQ[F] && HyperbolicQ[G] && EqQ[v-w] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]] + + +Int[(e_.+f_.*x_)^m_.*Cosh[c_.+d_.*x_]*(a_+b_.*Sinh[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Sinh[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Sinh[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sinh[c_.+d_.*x_]*(a_+b_.*Cosh[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Cosh[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Cosh[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sech[c_.+d_.*x_]^2*(a_+b_.*Tanh[c_.+d_.*x_])^n_.,x_Symbol] := + (e+f*x)^m*(a+b*Tanh[c+d*x])^(n+1)/(b*d*(n+1)) - + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Tanh[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Csch[c_.+d_.*x_]^2*(a_+b_.*Coth[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Coth[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Coth[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sech[c_.+d_.*x_]*Tanh[c_.+d_.*x_]*(a_+b_.*Sech[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Sech[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Sech[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Csch[c_.+d_.*x_]*Coth[c_.+d_.*x_]*(a_+b_.*Csch[c_.+d_.*x_])^n_.,x_Symbol] := + -(e+f*x)^m*(a+b*Csch[c+d*x])^(n+1)/(b*d*(n+1)) + + f*m/(b*d*(n+1))*Int[(e+f*x)^(m-1)*(a+b*Csch[c+d*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && NeQ[n+1] + + +Int[(e_.+f_.*x_)^m_.*Sinh[a_.+b_.*x_]^p_.*Sinh[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Sinh[a+b*x]^p*Sinh[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[p,q] && IntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*Cosh[a_.+b_.*x_]^p_.*Cosh[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Cosh[a+b*x]^p*Cosh[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[p,q] && IntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*Sinh[a_.+b_.*x_]^p_.*Cosh[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[(e+f*x)^m,Sinh[a+b*x]^p*Cosh[c+d*x]^q,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && PositiveIntegerQ[p,q] + + +Int[(e_.+f_.*x_)^m_.*F_[a_.+b_.*x_]^p_.*G_[c_.+d_.*x_]^q_.,x_Symbol] := + Int[ExpandTrigExpand[(e+f*x)^m*G[c+d*x]^q,F,c+d*x,p,b/d,x],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && MemberQ[{Sinh,Cosh},F] && MemberQ[{Sech,Csch},G] && + PositiveIntegerQ[p,q] && EqQ[b*c-a*d] && PositiveIntegerQ[b/d-1] + + + + + +(* ::Subsection::Closed:: *) +(*6.4.6 F^(c (a+b x)) hyper(d+e x)^n*) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_],x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sinh[d+e*x]/(e^2-b^2*c^2*Log[F]^2) + + e*F^(c*(a+b*x))*Cosh[d+e*x]/(e^2-b^2*c^2*Log[F]^2) /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2-b^2*c^2*Log[F]^2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_],x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Cosh[d+e*x]/(e^2-b^2*c^2*Log[F]^2) + + e*F^(c*(a+b*x))*Sinh[d+e*x]/(e^2-b^2*c^2*Log[F]^2) /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2-b^2*c^2*Log[F]^2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sinh[d+e*x]^n/(e^2*n^2-b^2*c^2*Log[F]^2) + + e*n*F^(c*(a+b*x))*Cosh[d+e*x]*Sinh[d+e*x]^(n-1)/(e^2*n^2-b^2*c^2*Log[F]^2) - + n*(n-1)*e^2/(e^2*n^2-b^2*c^2*Log[F]^2)*Int[F^(c*(a+b*x))*Sinh[d+e*x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n>1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Cosh[d+e*x]^n/(e^2*n^2-b^2*c^2*Log[F]^2) + + e*n*F^(c*(a+b*x))*Sinh[d+e*x]*Cosh[d+e*x]^(n-1)/(e^2*n^2-b^2*c^2*Log[F]^2) + + n*(n-1)*e^2/(e^2*n^2-b^2*c^2*Log[F]^2)*Int[F^(c*(a+b*x))*Cosh[d+e*x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n>1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sinh[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) + + F^(c*(a+b*x))*Cosh[d+e*x]*Sinh[d+e*x]^(n+1)/(e*(n+1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n+2)^2-b^2*c^2*Log[F]^2] && NeQ[n+1] && NeQ[n+2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Cosh[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) - + F^(c*(a+b*x))*Sinh[d+e*x]*Cosh[d+e*x]^(n+1)/(e*(n+1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n+2)^2-b^2*c^2*Log[F]^2] && NeQ[n+1] && NeQ[n+2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Sinh[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) + + F^(c*(a+b*x))*Cosh[d+e*x]*Sinh[d+e*x]^(n+1)/(e*(n+1)) - + (e^2*(n+2)^2-b^2*c^2*Log[F]^2)/(e^2*(n+1)*(n+2))*Int[F^(c*(a+b*x))*Sinh[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n+2)^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 && n!=-2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Cosh[d+e*x]^(n+2)/(e^2*(n+1)*(n+2)) - + F^(c*(a+b*x))*Sinh[d+e*x]*Cosh[d+e*x]^(n+1)/(e*(n+1)) + + (e^2*(n+2)^2-b^2*c^2*Log[F]^2)/(e^2*(n+1)*(n+2))*Int[F^(c*(a+b*x))*Cosh[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n+2)^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 && n!=-2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_,x_Symbol] := + E^(n*(d+e*x))*Sinh[d+e*x]^n/(-1+E^(2*(d+e*x)))^n*Int[F^(c*(a+b*x))*(-1+E^(2*(d+e*x)))^n/E^(n*(d+e*x)),x] /; +FreeQ[{F,a,b,c,d,e,n},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_,x_Symbol] := + E^(n*(d+e*x))*Cosh[d+e*x]^n/(1+E^(2*(d+e*x)))^n*Int[F^(c*(a+b*x))*(1+E^(2*(d+e*x)))^n/E^(n*(d+e*x)),x] /; +FreeQ[{F,a,b,c,d,e,n},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Tanh[d_.+e_.*x_]^n_.,x_Symbol] := + Int[ExpandIntegrand[F^(c*(a+b*x))*(-1+E^(2*(d+e*x)))^n/(1+E^(2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Coth[d_.+e_.*x_]^n_.,x_Symbol] := + Int[ExpandIntegrand[F^(c*(a+b*x))*(1+E^(2*(d+e*x)))^n/(-1+E^(2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*(Sech[d+e*x]^n/(e^2*n^2-b^2*c^2*Log[F]^2)) - + e*n*F^(c*(a+b*x))*Sech[d+e*x]^(n+1)*(Sinh[d+e*x]/(e^2*n^2-b^2*c^2*Log[F]^2)) + + e^2*n*((n+1)/(e^2*n^2-b^2*c^2*Log[F]^2))*Int[F^(c*(a+b*x))*Sech[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*(Csch[d+e*x]^n/(e^2*n^2-b^2*c^2*Log[F]^2)) - + e*n*F^(c*(a+b*x))*Csch[d+e*x]^(n+1)*(Cosh[d+e*x]/(e^2*n^2-b^2*c^2*Log[F]^2)) - + e^2*n*((n+1)/(e^2*n^2-b^2*c^2*Log[F]^2))*Int[F^(c*(a+b*x))*Csch[d+e*x]^(n+2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*n^2+b^2*c^2*Log[F]^2] && RationalQ[n] && n<-1 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Sech[d+e*x]^(n-2)/(e^2*(n-1)*(n-2)) + + F^(c*(a+b*x))*Sech[d+e*x]^(n-1)*Sinh[d+e*x]/(e*(n-1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n-2)^2-b^2*c^2*Log[F]^2] && NeQ[n-1] && NeQ[n-2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Csch[d+e*x]^(n-2)/(e^2*(n-1)*(n-2)) - + F^(c*(a+b*x))*Csch[d+e*x]^(n-1)*Cosh[d+e*x]/(e*(n-1)) /; +FreeQ[{F,a,b,c,d,e,n},x] && EqQ[e^2*(n-2)^2-b^2*c^2*Log[F]^2] && NeQ[n-1] && NeQ[n-2] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_,x_Symbol] := + b*c*Log[F]*F^(c*(a+b*x))*Sech[d+e*x]^(n-2)/(e^2*(n-1)*(n-2)) + + F^(c*(a+b*x))*Sech[d+e*x]^(n-1)*Sinh[d+e*x]/(e*(n-1)) + + (e^2*(n-2)^2-b^2*c^2*Log[F]^2)/(e^2*(n-1)*(n-2))*Int[F^(c*(a+b*x))*Sech[d+e*x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n-2)^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n>1 && n!=2 + + +Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_,x_Symbol] := + -b*c*Log[F]*F^(c*(a+b*x))*Csch[d+e*x]^(n-2)/(e^2*(n-1)*(n-2)) - + F^(c*(a+b*x))*Csch[d+e*x]^(n-1)*Cosh[d+e*x]/(e*(n-1)) - + (e^2*(n-2)^2-b^2*c^2*Log[F]^2)/(e^2*(n-1)*(n-2))*Int[F^(c*(a+b*x))*Csch[d+e*x]^(n-2),x] /; +FreeQ[{F,a,b,c,d,e},x] && NeQ[e^2*(n-2)^2-b^2*c^2*Log[F]^2] && RationalQ[n] && n>1 && n!=2 + + +(* Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_.,x_Symbol] := + 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(n*(d+e*x))/(1+E^(2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *) + + +(* Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_.,x_Symbol] := + 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-n*(d+e*x))/(1-E^(-2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_.,x_Symbol] := + 2^n*E^(n*(d+e*x))*F^(c*(a+b*x))/(e*n+b*c*Log[F])*Hypergeometric2F1[n,n/2+b*c*Log[F]/(2*e),1+n/2+b*c*Log[F]/(2*e),-E^(2*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_.,x_Symbol] := + (-2)^n*E^(n*(d+e*x))*F^(c*(a+b*x))/(e*n+b*c*Log[F])*Hypergeometric2F1[n,n/2+b*c*Log[F]/(2*e),1+n/2+b*c*Log[F]/(2*e),E^(2*(d+e*x))] /; +FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_.,x_Symbol] := + (1+E^(2*(d+e*x)))^n*Sech[d+e*x]^n/E^(n*(d+e*x))*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(n*(d+e*x))/(1+E^(2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_.,x_Symbol] := + (1-E^(-2*(d+e*x)))^n*Csch[d+e*x]^n/E^(-n*(d+e*x))*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-n*(d+e*x))/(1-E^(-2*(d+e*x)))^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && Not[IntegerQ[n]] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Sinh[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*f^n*Int[F^(c*(a+b*x))*Cosh[d/2+e*x/2-f*Pi/(4*g)]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f^2+g^2] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Cosh[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*g^n*Int[F^(c*(a+b*x))*Cosh[d/2+e*x/2]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f-g] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*(f_+g_.*Cosh[d_.+e_.*x_])^n_.,x_Symbol] := + 2^n*g^n*Int[F^(c*(a+b*x))*Sinh[d/2+e*x/2]^(2*n),x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f+g] && NegativeIntegerQ[n] + + +Int[F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^m_.*(f_+g_.*Sinh[d_.+e_.*x_])^n_.,x_Symbol] := + g^n*Int[F^(c*(a+b*x))*Tanh[d/2+e*x/2-f*Pi/(4*g)]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f^2+g^2] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^m_.*(f_+g_.*Cosh[d_.+e_.*x_])^n_.,x_Symbol] := + g^n*Int[F^(c*(a+b*x))*Tanh[d/2+e*x/2]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f-g] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^m_.*(f_+g_.*Cosh[d_.+e_.*x_])^n_.,x_Symbol] := + g^n*Int[F^(c*(a+b*x))*Coth[d/2+e*x/2]^m,x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && EqQ[f+g] && IntegersQ[m,n] && m+n==0 + + +Int[F_^(c_.*(a_.+b_.*x_))*(h_+i_.*Cosh[d_.+e_.*x_])/(f_+g_.*Sinh[d_.+e_.*x_]),x_Symbol] := + 2*i*Int[F^(c*(a+b*x))*(Cosh[d+e*x]/(f+g*Sinh[d+e*x])),x] + + Int[F^(c*(a+b*x))*((h-i*Cosh[d+e*x])/(f+g*Sinh[d+e*x])),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,i},x] && EqQ[f^2+g^2] && EqQ[h^2-i^2] && EqQ[g*h-f*i] + + +Int[F_^(c_.*(a_.+b_.*x_))*(h_+i_.*Sinh[d_.+e_.*x_])/(f_+g_.*Cosh[d_.+e_.*x_]),x_Symbol] := + 2*i*Int[F^(c*(a+b*x))*(Sinh[d+e*x]/(f+g*Cosh[d+e*x])),x] + + Int[F^(c*(a+b*x))*((h-i*Sinh[d+e*x])/(f+g*Cosh[d+e*x])),x] /; +FreeQ[{F,a,b,c,d,e,f,g,h,i},x] && EqQ[f^2-g^2] && EqQ[h^2+i^2] && EqQ[g*h+f*i] + + +Int[F_^(c_.*u_)*G_[v_]^n_.,x_Symbol] := + Int[F^(c*ExpandToSum[u,x])*G[ExpandToSum[v,x]]^n,x] /; +FreeQ[{F,c,n},x] && HyperbolicQ[G] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]] + + +Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_.,x_Symbol] := + Module[{u=IntHide[F^(c*(a+b*x))*Sinh[d+e*x]^n,x]}, + Dist[(f*x)^m,u,x] - f*m*Int[(f*x)^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] && GtQ[m,0] + + +Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_.,x_Symbol] := + Module[{u=IntHide[F^(c*(a+b*x))*Cosh[d+e*x]^n,x]}, + Dist[(f*x)^m,u,x] - f*m*Int[(f*x)^(m-1)*u,x]] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] && GtQ[m,0] + + +Int[(f_.*x_)^m_*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_],x_Symbol] := + (f*x)^(m+1)/(f*(m+1))*F^(c*(a+b*x))*Sinh[d+e*x] - + e/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Cosh[d+e*x],x] - + b*c*Log[F]/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Sinh[d+e*x],x] /; +FreeQ[{F,a,b,c,d,e,f,m},x] && (LtQ[m,-1] || SumSimplerQ[m,1]) + + +Int[(f_.*x_)^m_*F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_],x_Symbol] := + (f*x)^(m+1)/(f*(m+1))*F^(c*(a+b*x))*Cosh[d+e*x] - + e/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Sinh[d+e*x],x] - + b*c*Log[F]/(f*(m+1))*Int[(f*x)^(m+1)*F^(c*(a+b*x))*Cosh[d+e*x],x] /; +FreeQ[{F,a,b,c,d,e,f,m},x] && (LtQ[m,-1] || SumSimplerQ[m,1]) + + +(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_.,x_Symbol] := + (-1)^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))-E^(d+e*x))^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *) + + +(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_.,x_Symbol] := + 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))+E^(d+e*x))^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *) + + +Int[F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^m_.*Cosh[f_.+g_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[F^(c*(a+b*x)),Sinh[d+e*x]^m*Cosh[f+g*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && PositiveIntegerQ[m,n] + + +Int[x_^p_.*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^m_.*Cosh[f_.+g_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[x^p*F^(c*(a+b*x)),Sinh[d+e*x]^m*Cosh[f+g*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e,f,g},x] && PositiveIntegerQ[m,n,p] + + +Int[F_^(c_.*(a_.+b_.*x_))*G_[d_.+e_.*x_]^m_.*H_[d_.+e_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^(c*(a+b*x)),G[d+e*x]^m*H[d+e*x]^n,x],x] /; +FreeQ[{F,a,b,c,d,e},x] && PositiveIntegerQ[m,n] && HyperbolicQ[G] && HyperbolicQ[H] + + +Int[F_^u_*Sinh[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Sinh[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[n] + + +Int[F_^u_*Cosh[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Cosh[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[n] + + +Int[F_^u_*Sinh[v_]^m_.*Cosh[v_]^n_.,x_Symbol] := + Int[ExpandTrigToExp[F^u,Sinh[v]^m*Cosh[v]^n,x],x] /; +FreeQ[F,x] && (LinearQ[u,x] || PolyQ[u,x,2]) && (LinearQ[v,x] || PolyQ[v,x,2]) && PositiveIntegerQ[m,n] + + + + + +(* ::Subsection::Closed:: *) +(*6.4.7 x^m hyper(a+b log(c x^n))^p*) + + +Int[Sinh[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[((c*x^n)^b/2 - 1/(2*(c*x^n)^b))^p,x] /; +FreeQ[c,x] && RationalQ[b,n,p] + + +Int[Cosh[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[((c*x^n)^b/2 + 1/(2*(c*x^n)^b))^p,x] /; +FreeQ[c,x] && RationalQ[b,n,p] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*(p+2)*Sinh[a+b*Log[c*x^n]]^(p+2)/(p+1) + + x*Coth[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p+2)^2-1] && NeQ[p+1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(p+2)*Cosh[a+b*Log[c*x^n]]^(p+2)/(p+1) - + x*Tanh[a+b*Log[c*x^n]]*Cosh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p+2)^2-1] && NeQ[p+1] + + +Int[Sqrt[Sinh[a_.+b_.*Log[c_.*x_^n_.]]],x_Symbol] := + x*Sqrt[Sinh[a+b*Log[c*x^n]]]/Sqrt[-1+E^(2*a)*(c*x^n)^(4/n)]* + Int[Sqrt[-1+E^(2*a)*(c*x^n)^(4/n)]/x,x] /; +FreeQ[{a,b,c,n},x] && EqQ[b*n-2] + + +Int[Sqrt[Cosh[a_.+b_.*Log[c_.*x_^n_.]]],x_Symbol] := + x*Sqrt[Cosh[a+b*Log[c*x^n]]]/Sqrt[1+E^(2*a)*(c*x^n)^(4/n)]* + Int[Sqrt[1+E^(2*a)*(c*x^n)^(4/n)]/x,x] /; +FreeQ[{a,b,c,n},x] && EqQ[b*n-2] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[ExpandIntegrand[(-E^(-a*b*n*p)/(2*b*n*p)*(c*x^n)^(-1/(n*p)) + E^(a*b*n*p)/(2*b*n*p)*(c*x^n)^(1/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2-1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + Int[ExpandIntegrand[(E^(-a*b*n*p)/2*(c*x^n)^(-1/(n*p)) + E^(a*b*n*p)/2*(c*x^n)^(1/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2-1] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -x*Sinh[a+b*Log[c*x^n]]/(b^2*n^2-1) + + b*n*x*Cosh[a+b*Log[c*x^n]]/(b^2*n^2-1) /; +FreeQ[{a,b,c,n},x] && NeQ[b^2*n^2-1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -x*Cosh[a+b*Log[c*x^n]]/(b^2*n^2-1) + + b*n*x*Sinh[a+b*Log[c*x^n]]/(b^2*n^2-1) /; +FreeQ[{a,b,c,n},x] && NeQ[b^2*n^2-1] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Sinh[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-1) + + b*n*p*x*Cosh[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2-1) - + b^2*n^2*p*(p-1)/(b^2*n^2*p^2-1)*Int[Sinh[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2-1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Cosh[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-1) + + b*n*p*x*Sinh[a+b*Log[c*x^n]]*Cosh[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2-1) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2-1)*Int[Cosh[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && NeQ[b^2*n^2*p^2-1] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Coth[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + x*Sinh[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) - + (b^2*n^2*(p+2)^2-1)/(b^2*n^2*(p+1)*(p+2))*Int[Sinh[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2-1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Tanh[a+b*Log[c*x^n]]*Cosh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) + + x*Cosh[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2-1)/(b^2*n^2*(p+1)*(p+2))*Int[Cosh[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && p!=-2 && NeQ[b^2*n^2*(p+2)^2-1] + + +Int[Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(-E^(-a)*(c*x^n)^(-b)+E^a*(c*x^n)^b)^p/((b*n*p+1)*(2-2*E^(-2*a)*(c*x^n)^(-2*b))^p)* + Hypergeometric2F1[-p,-(1+b*n*p)/(2*b*n),1-(1+b*n*p)/(2*b*n),E^(-2*a)*(c*x^n)^(-2*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2-1] + + +Int[Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*(E^(-a)*(c*x^n)^(-b)+E^a*(c*x^n)^b)^p/((b*n*p+1)*(2+2*E^(-2*a)*(c*x^n)^(-2*b))^p)* + Hypergeometric2F1[-p,-(1+b*n*p)/(2*b*n),1-(1+b*n*p)/(2*b*n),-E^(-2*a)*(c*x^n)^(-2*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2-1] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(p+2)*x^(m+1)*Sinh[a+b*Log[c*x^n]]^(p+2)/((m+1)*(p+1)) + + x^(m+1)*Coth[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p+2)^2-(m+1)^2] && NeQ[p+1] && NeQ[m+1] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p+2)*x^(m+1)*Cosh[a+b*Log[c*x^n]]^(p+2)/((m+1)*(p+1)) - + x^(m+1)*Tanh[a+b*Log[c*x^n]]*Cosh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p+2)^2-(m+1)^2] && NeQ[p+1] && NeQ[m+1] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 1/2^p*Int[ExpandIntegrand[x^m*(-(m+1)/(b*n*p)*E^(-a*b*n*p/(m+1))*(c*x^n)^(-(m+1)/(n*p)) + + (m+1)/(b*n*p)*E^((a*b*n*p)/(m+1))*(c*x^n)^((m+1)/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,m,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 1/2^p*Int[ExpandIntegrand[x^m*(E^((a*b*n*p)/(m+1))*(c*x^n)^((m+1)/(n*p)) + + E^(-a*b*n*p/(m+1))*(c*x^n)^(-(m+1)/(n*p)))^p,x],x] /; +FreeQ[{a,b,c,m,n},x] && PositiveIntegerQ[p] && EqQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -(m+1)*x^(m+1)*Sinh[a+b*Log[c*x^n]]/(b^2*n^2-(m+1)^2) + + b*n*x^(m+1)*Cosh[a+b*Log[c*x^n]]/(b^2*n^2-(m+1)^2) /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2-(m+1)^2] && NeQ[m+1] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -(m+1)*x^(m+1)*Cosh[a+b*Log[c*x^n]]/(b^2*n^2-(m+1)^2) + + b*n*x^(m+1)*Sinh[a+b*Log[c*x^n]]/(b^2*n^2-(m+1)^2) /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2-(m+1)^2] && NeQ[m+1] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(m+1)*x^(m+1)*Sinh[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-(m+1)^2) + + b*n*p*x^(m+1)*Cosh[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p-1)/(b^2*n^2*p^2-(m+1)^2) - + b^2*n^2*p*(p-1)/(b^2*n^2*p^2-(m+1)^2)*Int[x^m*Sinh[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2*p^2-(m+1)^2] && RationalQ[p] && p>1 && NeQ[m+1] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(m+1)*x^(m+1)*Cosh[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-(m+1)^2) + + b*n*p*x^(m+1)*Cosh[a+b*Log[c*x^n]]^(p-1)*Sinh[a+b*Log[c*x^n]]/(b^2*n^2*p^2-(m+1)^2) + + b^2*n^2*p*(p-1)/(b^2*n^2*p^2-(m+1)^2)*Int[x^m*Cosh[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2*p^2-(m+1)^2] && RationalQ[p] && p>1 && NeQ[m+1] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*Coth[a+b*Log[c*x^n]]*Sinh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) - + (m+1)*x^(m+1)*Sinh[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) - + (b^2*n^2*(p+2)^2-(m+1)^2)/(b^2*n^2*(p+1)*(p+2))*Int[x^m*Sinh[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2*(p+2)^2-(m+1)^2] && RationalQ[p] && p<-1 && p!=-2 && NeQ[m+1] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x^(m+1)*Tanh[a+b*Log[c*x^n]]*Cosh[a+b*Log[c*x^n]]^(p+2)/(b*n*(p+1)) + + (m+1)*x^(m+1)*Cosh[a+b*Log[c*x^n]]^(p+2)/(b^2*n^2*(p+1)*(p+2)) + + (b^2*n^2*(p+2)^2-(m+1)^2)/(b^2*n^2*(p+1)*(p+2))*Int[x^m*Cosh[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && NeQ[b^2*n^2*(p+2)^2-(m+1)^2] && RationalQ[p] && p<-1 && p!=-2 && NeQ[m+1] + + +Int[x_^m_.*Sinh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*(-E^(-a)*(c*x^n)^(-b)+E^a*(c*x^n)^b)^p/((m+b*n*p+1)*(2-2*E^(-2*a)*(c*x^n)^(-2*b))^p)* + Hypergeometric2F1[-p,-(m+b*n*p+1)/(2*b*n),1-(m+b*n*p+1)/(2*b*n),E^(-2*a)*(c*x^n)^(-2*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Cosh[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*(E^(-a)*(c*x^n)^(-b)+E^a*(c*x^n)^b)^p/((m+b*n*p+1)*(2+2*E^(-2*a)*(c*x^n)^(-2*b))^p)* + Hypergeometric2F1[-p,-(m+b*n*p+1)/(2*b*n),1-(m+b*n*p+1)/(2*b*n),-E^(-2*a)*(c*x^n)^(-2*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[Sech[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*Int[((c*x^n)^b/(1+(c*x^n)^(2*b)))^p,x] /; +FreeQ[c,x] && RationalQ[b,n,p] + + +Int[Csch[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*Int[((c*x^n)^b/(-1+(c*x^n)^(2*b)))^p,x] /; +FreeQ[c,x] && RationalQ[b,n,p] + + +Int[Sech[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*E^(-a*b*n)*Int[(c*x^n)^(1/n)/(E^(-2*a*b*n)+(c*x^n)^(2/n)),x] /; +FreeQ[{a,b,c,n},x] && EqQ[b^2*n^2-1] + + +Int[Csch[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -2*b*n*E^(-a*b*n)*Int[(c*x^n)^(1/n)/(E^(-2*a*b*n)-(c*x^n)^(2/n)),x] /; +FreeQ[{a,b,c,n},x] && EqQ[b^2*n^2-1] + + +Int[Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x*Sech[a+b*Log[c*x^n]]^(p-2)/(p-1) + + x*Tanh[a+b*Log[c*x^n]]*Sech[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p-2)^2-1] && NeQ[p-1] + + +Int[Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(p-2)*x*Csch[a+b*Log[c*x^n]]^(p-2)/(p-1) - + x*Coth[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,n,p},x] && EqQ[b^2*n^2*(p-2)^2-1] && NeQ[p-1] + + +Int[Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x*Tanh[a+b*Log[c*x^n]]*Sech[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) + + x*Sech[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2-1)/(b^2*n^2*(p-1)*(p-2))*Int[Sech[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2-1] + + +Int[Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x*Coth[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + x*Csch[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) - + (b^2*n^2*(p-2)^2-1)/(b^2*n^2*(p-1)*(p-2))*Int[Csch[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2-1] + + +Int[Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -b*n*p*x*Sech[a+b*Log[c*x^n]]^(p+1)*Sinh[a+b*Log[c*x^n]]/(b^2*n^2*p^2-1) - + x*Sech[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-1) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2-1)*Int[Sech[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2-1] + + +Int[Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -b*n*p*x*Cosh[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2-1) - + x*Csch[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-1) - + b^2*n^2*p*(p+1)/(b^2*n^2*p^2-1)*Int[Csch[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2-1] + + +Int[Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*x*(E^(2*a)*(c*x^n)^(2*b)+1)^p/(b*n*p+1)*(E^a*(c*x^n)^b/(E^(2*a)*(c*x^n)^(2*b)+1))^p* + Hypergeometric2F1[p,(b*n*p+1)/(2*b*n),1+(b*n*p+1)/(2*b*n),-E^(2*a)*(c*x^n)^(2*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2-1] + + +Int[Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + x*(2-2*E^(2*a)*(c*x^n)^(2*b))^p/(b*n*p+1)*(E^a*(c*x^n)^b/(E^(2*a)*(c*x^n)^(2*b)-1))^p* + Hypergeometric2F1[p,(b*n*p+1)/(2*b*n),1+(b*n*p+1)/(2*b*n),E^(2*a)*(c*x^n)^(2*b)] /; +FreeQ[{a,b,c,n,p},x] && NeQ[b^2*n^2*p^2-1] + + +Int[x_^m_.*Sech[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*Int[x^m*((c*x^n)^b/(1+(c*x^n)^(2*b)))^p,x] /; +FreeQ[c,x] && RationalQ[b,m,n,p] + + +Int[x_^m_.*Csch[b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*Int[x^m*((c*x^n)^b/(-1+(c*x^n)^(2*b)))^p,x] /; +FreeQ[c,x] && RationalQ[b,m,n,p] + + +Int[x_^m_.*Sec[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + 2*E^(-a*b*n/(m+1))*Int[x^m*(c*x^n)^((m+1)/n)/(E^(-2*a*b*n/(m+1))+(c*x^n)^(2*(m+1)/n)),x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[b^2*n^2-(m+1)^2] + + +Int[x_^m_.*Csc[a_.+b_.*Log[c_.*x_^n_.]],x_Symbol] := + -2*b*n/(m+1)*E^(-a*b*n/(m+1))*Int[x^m*(c*x^n)^((m+1)/n)/(E^(-2*a*b*n/(m+1))-(c*x^n)^(2*(m+1)/n)),x] /; +FreeQ[{a,b,c,m,n},x] && EqQ[b^2*n^2-(m+1)^2] + + +Int[x_^m_.*Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + (p-2)*x^(m+1)*Sech[a+b*Log[c*x^n]]^(p-2)/((m+1)*(p-1)) + + x^(m+1)*Tanh[a+b*Log[c*x^n]]*Sech[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p-2)^2+(m+1)^2] && NeQ[m+1] && NeQ[p-1] + + +Int[x_^m_.*Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(p-2)*x^(m+1)*Csch[a+b*Log[c*x^n]]^(p-2)/((m+1)*(p-1)) - + x^(m+1)*Coth[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[b^2*n^2*(p-2)^2+(m+1)^2] && NeQ[m+1] && NeQ[p-1] + + +Int[x_^m_.*Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + x^(m+1)*Tanh[a+b*Log[c*x^n]]*Sech[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) + + (m+1)*x^(m+1)*Sech[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) + + (b^2*n^2*(p-2)^2-(m+1)^2)/(b^2*n^2*(p-1)*(p-2))*Int[x^m*Sech[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2-(m+1)^2] + + +Int[x_^m_.*Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -x^(m+1)*Coth[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p-2)/(b*n*(p-1)) - + (m+1)*x^(m+1)*Csch[a+b*Log[c*x^n]]^(p-2)/(b^2*n^2*(p-1)*(p-2)) - + (b^2*n^2*(p-2)^2-(m+1)^2)/(b^2*n^2*(p-1)*(p-2))*Int[x^m*Csch[a+b*Log[c*x^n]]^(p-2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p>1 && p!=2 && NeQ[b^2*n^2*(p-2)^2-(m+1)^2] + + +Int[x_^m_.*Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(m+1)*x^(m+1)*Sech[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-(m+1)^2) - + b*n*p*x^(m+1)*Sech[a+b*Log[c*x^n]]^(p+1)*Sinh[a+b*Log[c*x^n]]/(b^2*n^2*p^2-(m+1)^2) + + b^2*n^2*p*(p+1)/(b^2*n^2*p^2-(m+1)^2)*Int[x^m*Sech[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_,x_Symbol] := + -(m+1)*x^(m+1)*Csch[a+b*Log[c*x^n]]^p/(b^2*n^2*p^2-(m+1)^2) - + b*n*p*x^(m+1)*Cosh[a+b*Log[c*x^n]]*Csch[a+b*Log[c*x^n]]^(p+1)/(b^2*n^2*p^2-(m+1)^2) - + b^2*n^2*p*(p+1)/(b^2*n^2*p^2-(m+1)^2)*Int[x^m*Csch[a+b*Log[c*x^n]]^(p+2),x] /; +FreeQ[{a,b,c,m,n},x] && RationalQ[p] && p<-1 && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Sech[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*x^(m+1)*(E^(2*a)*(c*x^n)^(2*b)+1)^p/(m+b*n*p+1)*(E^a*(c*x^n)^b/(E^(2*a)*(c*x^n)^(2*b)+1))^p* + Hypergeometric2F1[p,(m+b*n*p+1)/(2*b*n),1+(m+b*n*p+1)/(2*b*n),-E^(2*a)*(c*x^n)^(2*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[x_^m_.*Csch[a_.+b_.*Log[c_.*x_^n_.]]^p_.,x_Symbol] := + 2^p*x^(m+1)*(-E^(2*a)*(c*x^n)^(2*b)+1)^p/(m+b*n*p+1)*(E^a*(c*x^n)^b/(E^(2*a)*(c*x^n)^(2*b)-1))^p* + Hypergeometric2F1[p,(m+b*n*p+1)/(2*b*n),1+(m+b*n*p+1)/(2*b*n),E^(2*a)*(c*x^n)^(2*b)] /; +FreeQ[{a,b,c,m,n,p},x] && NeQ[b^2*n^2*p^2-(m+1)^2] + + +Int[Sinh[a_.*x_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Cosh[a*x*Log[b*x]^p]/a - + p*Int[Sinh[a*x*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b},x] && RationalQ[p] && p>0 + + +Int[Cosh[a_.*x_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sinh[a*x*Log[b*x]^p]/a - + p*Int[Cosh[a*x*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b},x] && RationalQ[p] && p>0 + + +Int[Sinh[a_.*x_^n_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Cosh[a*x^n*Log[b*x]^p]/(a*n*x^(n-1)) - + p/n*Int[Sinh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] + + (n-1)/(a*n)*Int[Cosh[a*x^n*Log[b*x]^p]/x^n,x] /; +FreeQ[{a,b},x] && RationalQ[n,p] && p>0 + + +Int[Cosh[a_.*x_^n_*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sinh[a*x^n*Log[b*x]^p]/(a*n*x^(n-1)) - + p/n*Int[Cosh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] + + (n-1)/(a*n)*Int[Sinh[a*x^n*Log[b*x]^p]/x^n,x] /; +FreeQ[{a,b},x] && RationalQ[n,p] && p>0 + + +Int[x_^m_.*Sinh[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + -Cosh[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^(n-1)*Sinh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-n+1] && RationalQ[p] && p>0 + + +Int[x_^m_.*Cosh[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + Sinh[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^(n-1)*Cosh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] /; +FreeQ[{a,b,m,n},x] && EqQ[m-n+1] && RationalQ[p] && p>0 + + +Int[x_^m_*Sinh[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + x^(m-n+1)*Cosh[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Sinh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] - + (m-n+1)/(a*n)*Int[x^(m-n)*Cosh[a*x^n*Log[b*x]^p],x] /; +FreeQ[{a,b},x] && RationalQ[m,n,p] && p>0 && NeQ[m-n+1] + + +Int[x_^m_*Cosh[a_.*x_^n_.*Log[b_.*x_]^p_.]*Log[b_.*x_]^p_.,x_Symbol] := + x^(m-n+1)*Sinh[a*x^n*Log[b*x]^p]/(a*n) - + p/n*Int[x^m*Cosh[a*x^n*Log[b*x]^p]*Log[b*x]^(p-1),x] - + (m-n+1)/(a*n)*Int[x^(m-n)*Sinh[a*x^n*Log[b*x]^p],x] /; +FreeQ[{a,b},x] && RationalQ[m,n,p] && p>0 && NeQ[m-n+1] + + + + + +(* ::Subsection::Closed:: *) +(*6.4.8 Active hyperbolic functions*) + + +Int[Sinh[a_./(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Sinh[a*x]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,c,d},x] && PositiveIntegerQ[n] + + +Int[Cosh[a_./(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Cosh[a*x]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,c,d},x] && PositiveIntegerQ[n] + + +Int[Sinh[e_.*(a_.+b_.*x_)/(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Sinh[b*e/d-e*(b*c-a*d)*x/d]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[n] && NeQ[b*c-a*d] + + +Int[Cosh[e_.*(a_.+b_.*x_)/(c_.+d_.*x_)]^n_.,x_Symbol] := + -1/d*Subst[Int[Cosh[b*e/d-e*(b*c-a*d)*x/d]^n/x^2,x],x,1/(c+d*x)] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[n] && NeQ[b*c-a*d] + + +Int[Sinh[u_]^n_.,x_Symbol] := + With[{lst=QuotientOfLinearsParts[u,x]}, + Int[Sinh[(lst[[1]]+lst[[2]]*x)/(lst[[3]]+lst[[4]]*x)]^n,x]] /; +PositiveIntegerQ[n] && QuotientOfLinearsQ[u,x] + + +Int[Cosh[u_]^n_.,x_Symbol] := + With[{lst=QuotientOfLinearsParts[u,x]}, + Int[Cosh[(lst[[1]]+lst[[2]]*x)/(lst[[3]]+lst[[4]]*x)]^n,x]] /; +PositiveIntegerQ[n] && QuotientOfLinearsQ[u,x] + + +Int[u_.*Sinh[v_]^p_.*Sinh[w_]^q_.,x_Symbol] := + Int[u*Sinh[v]^(p+q),x] /; +EqQ[v-w] + + +Int[u_.*Cosh[v_]^p_.*Cosh[w_]^q_.,x_Symbol] := + Int[u*Cosh[v]^(p+q),x] /; +EqQ[v-w] + + +Int[Sinh[v_]^p_.*Sinh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Sinh[v]^p*Sinh[w]^q,x],x] /; +PositiveIntegerQ[p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[Cosh[v_]^p_.*Cosh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Cosh[v]^p*Cosh[w]^q,x],x] /; +PositiveIntegerQ[p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[x_^m_.*Sinh[v_]^p_.*Sinh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Sinh[v]^p*Sinh[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[x_^m_.*Cosh[v_]^p_.*Cosh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Cosh[v]^p*Cosh[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[u_.*Sinh[v_]^p_.*Cosh[w_]^p_.,x_Symbol] := + 1/2^p*Int[u*Sinh[2*v]^p,x] /; +EqQ[v-w] && IntegerQ[p] + + +Int[Sinh[v_]^p_.*Cosh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q,x],x] /; +PositiveIntegerQ[p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[x_^m_.*Sinh[v_]^p_.*Cosh[w_]^q_.,x_Symbol] := + Int[ExpandTrigReduce[x^m,Sinh[v]^p*Cosh[w]^q,x],x] /; +PositiveIntegerQ[m,p,q] && (PolynomialQ[v,x] && PolynomialQ[w,x] || BinomialQ[{v,w},x] && IndependentQ[Cancel[v/w],x]) + + +Int[Sinh[v_]*Tanh[w_]^n_.,x_Symbol] := + Int[Cosh[v]*Tanh[w]^(n-1),x] - Cosh[v-w]*Int[Sech[w]*Tanh[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cosh[v_]*Coth[w_]^n_.,x_Symbol] := + Int[Sinh[v]*Coth[w]^(n-1),x] + Cosh[v-w]*Int[Csch[w]*Coth[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sinh[v_]*Coth[w_]^n_.,x_Symbol] := + Int[Cosh[v]*Coth[w]^(n-1),x] + Sinh[v-w]*Int[Csch[w]*Coth[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cosh[v_]*Tanh[w_]^n_.,x_Symbol] := + Int[Sinh[v]*Tanh[w]^(n-1),x] - Sinh[v-w]*Int[Sech[w]*Tanh[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sinh[v_]*Sech[w_]^n_.,x_Symbol] := + Cosh[v-w]*Int[Tanh[w]*Sech[w]^(n-1),x] + Sinh[v-w]*Int[Sech[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cosh[v_]*Csch[w_]^n_.,x_Symbol] := + Cosh[v-w]*Int[Coth[w]*Csch[w]^(n-1),x] + Sinh[v-w]*Int[Csch[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Sinh[v_]*Csch[w_]^n_.,x_Symbol] := + Sinh[v-w]*Int[Coth[w]*Csch[w]^(n-1),x] + Cosh[v-w]*Int[Csch[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[Cosh[v_]*Sech[w_]^n_.,x_Symbol] := + Sinh[v-w]*Int[Tanh[w]*Sech[w]^(n-1),x] + Cosh[v-w]*Int[Sech[w]^(n-1),x] /; +RationalQ[n] && n>0 && FreeQ[v-w,x] && NeQ[v-w] + + +Int[(e_.+f_.*x_)^m_.*(a_+b_.*Sinh[c_.+d_.*x_]*Cosh[c_.+d_.*x_])^n_.,x_Symbol] := + Int[(e+f*x)^m*(a+b*Sinh[2*c+2*d*x]/2)^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[x_^m_.*(a_+b_.*Sinh[c_.+d_.*x_]^2)^n_,x_Symbol] := + 1/2^n*Int[x^m*(2*a-b+b*Cosh[2*c+2*d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a-b] && IntegersQ[m,n] && m>0 && n<0 && (n==-1 || m==1 && n==-2) + + +Int[x_^m_.*(a_+b_.*Cosh[c_.+d_.*x_]^2)^n_,x_Symbol] := + 1/2^n*Int[x^m*(2*a+b+b*Cosh[2*c+2*d*x])^n,x] /; +FreeQ[{a,b,c,d},x] && NeQ[a+b] && IntegersQ[m,n] && m>0 && n<0 && (n==-1 || m==1 && n==-2) + + +Int[(e_.+f_.*x_)^m_.*Sinh[a_.+b_.*(c_+d_.*x_)^n_]^p_.,x_Symbol] := + 1/d^(m+1)*Subst[Int[(d*e-c*f+f*x)^m*Sinh[a+b*x^n]^p,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && RationalQ[p] + + +Int[(e_.+f_.*x_)^m_.*Cosh[a_.+b_.*(c_+d_.*x_)^n_]^p_.,x_Symbol] := + 1/d^(m+1)*Subst[Int[(d*e-c*f+f*x)^m*Cosh[a+b*x^n]^p,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,n},x] && PositiveIntegerQ[m] && RationalQ[p] + + +Int[(f_.+g_.*x_)^m_./(a_.+b_.*Cosh[d_.+e_.*x_]^2+c_.*Sinh[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b-c+(b+c)*Cosh[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(f_.+g_.*x_)^m_.*Sech[d_.+e_.*x_]^2/(b_+c_.*Tanh[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(b-c+(b+c)*Cosh[2*d+2*e*x]),x] /; +FreeQ[{b,c,d,e,f,g},x] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*Sech[d_.+e_.*x_]^2/(b_.+a_.*Sech[d_.+e_.*x_]^2+c_.*Tanh[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b-c+(b+c)*Cosh[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(f_.+g_.*x_)^m_.*Csch[d_.+e_.*x_]^2/(c_+b_.*Coth[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(b-c+(b+c)*Cosh[2*d+2*e*x]),x] /; +FreeQ[{b,c,d,e,f,g},x] && PositiveIntegerQ[m] + + +Int[(f_.+g_.*x_)^m_.*Csch[d_.+e_.*x_]^2/(c_.+b_.*Coth[d_.+e_.*x_]^2+a_.*Csch[d_.+e_.*x_]^2),x_Symbol] := + 2*Int[(f+g*x)^m/(2*a+b-c+(b+c)*Cosh[2*d+2*e*x]),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[m] && NeQ[a+b] && NeQ[a+c] + + +Int[(e_.+f_.*x_)^m_.*Cosh[c_.+d_.*x_]/(a_+b_.*Sinh[c_.+d_.*x_]),x_Symbol] := + -(e+f*x)^(m+1)/(b*f*(m+1)) + + Int[(e+f*x)^m*E^(c+d*x)/(a-Rt[a^2+b^2,2]+b*E^(c+d*x)),x] + + Int[(e+f*x)^m*E^(c+d*x)/(a+Rt[a^2+b^2,2]+b*E^(c+d*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] (* && PosQ[a^2+b^2] *) + + +Int[(e_.+f_.*x_)^m_.*Sinh[c_.+d_.*x_]/(a_+b_.*Cosh[c_.+d_.*x_]),x_Symbol] := + -(e+f*x)^(m+1)/(b*f*(m+1)) + + Int[(e+f*x)^m*E^(c+d*x)/(a-Rt[a^2-b^2,2]+b*E^(c+d*x)),x] + + Int[(e+f*x)^m*E^(c+d*x)/(a+Rt[a^2-b^2,2]+b*E^(c+d*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] (* && PosQ[a^2+b^2] *) + + +Int[(e_.+f_.*x_)^m_.*Cosh[c_.+d_.*x_]^n_/(a_+b_.*Sinh[c_.+d_.*x_]),x_Symbol] := + 1/a*Int[(e+f*x)^m*Cosh[c+d*x]^(n-2),x] + + 1/b*Int[(e+f*x)^m*Cosh[c+d*x]^(n-2)*Sinh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && EqQ[a^2+b^2] + + +Int[(e_.+f_.*x_)^m_.*Sinh[c_.+d_.*x_]^n_/(a_+b_.*Cosh[c_.+d_.*x_]),x_Symbol] := + 1/a*Int[(e+f*x)^m*Sinh[c+d*x]^(n-2),x] + + 1/b*Int[(e+f*x)^m*Sinh[c+d*x]^(n-2)*Cosh[c+d*x],x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && EqQ[a^2-b^2] + + +Int[(e_.+f_.*x_)^m_.*Cosh[c_.+d_.*x_]^n_/(a_+b_.*Sinh[c_.+d_.*x_]),x_Symbol] := + -a/b^2*Int[(e+f*x)^m*Cosh[c+d*x]^(n-2),x] + + 1/b*Int[(e+f*x)^m*Cosh[c+d*x]^(n-2)*Sinh[c+d*x],x] + + (a^2+b^2)/b^2*Int[(e+f*x)^m*Cosh[c+d*x]^(n-2)/(a+b*Sinh[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && NeQ[a^2+b^2] + + +Int[(e_.+f_.*x_)^m_.*Sinh[c_.+d_.*x_]^n_/(a_+b_.*Cosh[c_.+d_.*x_]),x_Symbol] := + -a/b^2*Int[(e+f*x)^m*Sinh[c+d*x]^(n-2),x] + + 1/b*Int[(e+f*x)^m*Sinh[c+d*x]^(n-2)*Cosh[c+d*x],x] + + (a^2-b^2)/b^2*Int[(e+f*x)^m*Sinh[c+d*x]^(n-2)/(a+b*Cosh[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && IntegerQ[n] && n>1 && NeQ[a^2-b^2] + + +Int[(e_.+f_.*x_)*(A_+B_.*Sinh[c_.+d_.*x_])/(a_+b_.*Sinh[c_.+d_.*x_])^2,x_Symbol] := + B*(e+f*x)*Cosh[c+d*x]/(a*d*(a+b*Sinh[c+d*x])) - + B*f/(a*d)*Int[Cosh[c+d*x]/(a+b*Sinh[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && EqQ[a*A+b*B] + + +Int[(e_.+f_.*x_)*(A_+B_.*Cosh[c_.+d_.*x_])/(a_+b_.*Cosh[c_.+d_.*x_])^2,x_Symbol] := + B*(e+f*x)*Sinh[c+d*x]/(a*d*(a+b*Cosh[c+d*x])) - + B*f/(a*d)*Int[Sinh[c+d*x]/(a+b*Cosh[c+d*x]),x] /; +FreeQ[{a,b,c,d,e,f,A,B},x] && EqQ[a*A-b*B] + + +Int[Sech[v_]^m_.*(a_+b_.*Tanh[v_])^n_.,x_Symbol] := + Int[(a*Cosh[v]+b*Sinh[v])^n,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && m+n==0 && OddQ[m] + + +Int[Csch[v_]^m_.*(a_+b_.*Coth[v_])^n_.,x_Symbol] := + Int[(b*Cosh[v]+a*Sinh[v])^n,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && m+n==0 && OddQ[m] + + +Int[u_.*Sinh[a_.+b_.*x_]^m_.*Sinh[c_.+d_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[u,Sinh[a+b*x]^m*Sinh[c+d*x]^n,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m,n] + + +Int[u_.*Cosh[a_.+b_.*x_]^m_.*Cosh[c_.+d_.*x_]^n_.,x_Symbol] := + Int[ExpandTrigReduce[u,Cosh[a+b*x]^m*Cosh[c+d*x]^n,x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m,n] + + +Int[Sech[a_.+b_.*x_]*Sech[c_+d_.*x_],x_Symbol] := + -Csch[(b*c-a*d)/d]*Int[Tanh[a+b*x],x] + Csch[(b*c-a*d)/b]*Int[Tanh[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Csch[a_.+b_.*x_]*Csch[c_+d_.*x_],x_Symbol] := + Csch[(b*c-a*d)/b]*Int[Coth[a+b*x],x] - Csch[(b*c-a*d)/d]*Int[Coth[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Tanh[a_.+b_.*x_]*Tanh[c_+d_.*x_],x_Symbol] := + b*x/d - b/d*Cosh[(b*c-a*d)/d]*Int[Sech[a+b*x]*Sech[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[Coth[a_.+b_.*x_]*Coth[c_+d_.*x_],x_Symbol] := + b*x/d + Cosh[(b*c-a*d)/d]*Int[Csch[a+b*x]*Csch[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && EqQ[b^2-d^2] && NeQ[b*c-a*d] + + +Int[u_.*(a_.*Cosh[v_]+b_.*Sinh[v_])^n_.,x_Symbol] := + Int[u*(a*E^(a/b*v))^n,x] /; +FreeQ[{a,b,n},x] && EqQ[a^2-b^2] + + + +`) + +func resourcesRubi6HyperbolicFunctionsMBytes() ([]byte, error) { + return _resourcesRubi6HyperbolicFunctionsM, nil +} + +func resourcesRubi6HyperbolicFunctionsM() (*asset, error) { + bytes, err := resourcesRubi6HyperbolicFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/6 Hyperbolic functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi7InverseHyperbolicFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Inverse Hyperbolic Function Rules*) + + +(* ::Subsection::Closed:: *) +(*7.1.1 (a+b arcsinh(c x))^n*) + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcSinh[c*x])^n - + b*c*n*Int[x*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcCosh[c*x])^n - + b*c*n*Int[x*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + Sqrt[1+c^2*x^2]*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + c/(b*(n+1))*Int[x*(a+b*ArcSinh[c*x])^(n+1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + Sqrt[-1+c*x]*Sqrt[1+c*x]*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - + c/(b*(n+1))*Int[x*(a+b*ArcCosh[c*x])^(n+1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + 1/(b*c)*Subst[Int[x^n*Cosh[a/b-x/b],x],x,a+b*ArcSinh[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + -1/(b*c)*Subst[Int[x^n*Sinh[a/b-x/b],x],x,a+b*ArcCosh[c*x]] /; +FreeQ[{a,b,c,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*7.1.2 (d x)^m (a+b arcsinh(c x))^n*) + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./x_,x_Symbol] := + Subst[Int[(a+b*x)^n/Tanh[x],x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./x_,x_Symbol] := + Subst[Int[(a+b*x)^n/Coth[x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d*x)^(m+1)*(a+b*ArcSinh[c*x])^n/(d*(m+1)) - + b*c*n/(d*(m+1))*Int[(d*x)^(m+1)*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d*x)^(m+1)*(a+b*ArcCosh[c*x])^n/(d*(m+1)) - + b*c*n/(d*(m+1))*Int[(d*x)^(m+1)*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c,d,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcSinh[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n>0 + + +Int[x_^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcCosh[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n>0 + + +Int[x_^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[1+c^2*x^2]*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + 1/(b*c^(m+1)*(n+1))*Subst[Int[ExpandTrigReduce[(a+b*x)^(n+1),Sinh[x]^(m-1)*(m+(m+1)*Sinh[x]^2),x],x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && -2<=n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[-1+c*x]*Sqrt[1+c*x]*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + 1/(b*c^(m+1)*(n+1))*Subst[Int[ExpandTrigReduce[(a+b*x)^(n+1)*Cosh[x]^(m-1)*(m-(m+1)*Cosh[x]^2),x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && -2<=n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[1+c^2*x^2]*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(a+b*ArcSinh[c*x])^(n+1)/Sqrt[1+c^2*x^2],x] - + c*(m+1)/(b*(n+1))*Int[x^(m+1)*(a+b*ArcSinh[c*x])^(n+1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-2 + + +Int[x_^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + x^m*Sqrt[-1+c*x]*Sqrt[1+c*x]*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + m/(b*c*(n+1))*Int[x^(m-1)*(a+b*ArcCosh[c*x])^(n+1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] - + c*(m+1)/(b*(n+1))*Int[x^(m+1)*(a+b*ArcCosh[c*x])^(n+1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-2 + + +Int[x_^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Sinh[x]^m*Cosh[x],x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Cosh[x]^m*Sinh[x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,n},x] && PositiveIntegerQ[m] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d*x)^m*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,m,n},x] + + +Int[(d_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d*x)^m*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,m,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n*) + + +Int[1/(Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSinh[c_.*x_])),x_Symbol] := + Log[a+b*ArcSinh[c*x]]/(b*c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] + + +Int[1/(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]*(a_.+b_.*ArcCosh[c_.*x_])),x_Symbol] := + Log[a+b*ArcCosh[c*x]]/(b*c*Sqrt[-d1*d2]) /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveQ[d1] && NegativeQ[d2] + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + (a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveQ[d] && NeQ[n+1] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + (a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveQ[d1] && NegativeQ[d2] && NeQ[n+1] + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + Sqrt[1+c*x]*Sqrt[-1+c*x]/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x])*Int[(a+b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && Not[PositiveQ[d1] && NegativeQ[d2]] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^p/(2*p+1)*Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p>0 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d)^p/((2*p+1))*Int[x*(-1+c*x)^(p-1/2)*(1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && IntegerQ[p] + + +(* Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(2*p+1) + + 2*d1*d2*p/(2*p+1)*Int[(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^p/((2*p+1))*Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p>0 && IntegerQ[p-1/2] && + (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + x*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/2 - + b*c*n*Sqrt[d+e*x^2]/(2*Sqrt[1+c^2*x^2])*Int[x*(a+b*ArcSinh[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/(2*Sqrt[1+c^2*x^2])*Int[(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/2 - + b*c*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(2*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[x*(a+b*ArcCosh[c*x])^(n-1),x] - + Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(2*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(a+b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + x*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/((2*p+1)*(1+c^2*x^2)^FracPart[p])* + Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p>0 + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(2*p+1) + + 2*d1*d2*p/(2*p+1)*Int[(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/((2*p+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p>0 && IntegerQ[p-1/2] + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + x*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(2*p+1) + + 2*d1*d2*p/(2*p+1)*Int[(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/((2*p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[x*(-1+c*x)^(p-1/2)*(1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p>0 + + +(* Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcSinh[c*x])^n/(d*Sqrt[d+e*x^2]) - + b*c*n/Sqrt[d]*Int[x*(a+b*ArcSinh[c*x])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && PositiveQ[d] *) + + +(* Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./((d1_+e1_.*x_)^(3/2)*(d2_+e2_.*x_)^(3/2)),x_Symbol] := + x*(a+b*ArcCosh[c*x])^n/(d1*d2*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]) + + b*c*n/Sqrt[-d1*d2]*Int[x*(a+b*ArcCosh[c*x])^(n-1)/(d1*d2+e1*e2*x^2),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol] := + x*(a+b*ArcSinh[c*x])^n/(d*Sqrt[d+e*x^2]) - + b*c*n*Sqrt[1+c^2*x^2]/(d*Sqrt[d+e*x^2])*Int[x*(a+b*ArcSinh[c*x])^(n-1)/(1+c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./((d1_+e1_.*x_)^(3/2)*(d2_+e2_.*x_)^(3/2)),x_Symbol] := + x*(a+b*ArcCosh[c*x])^n/(d1*d2*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]) + + b*c*n*Sqrt[1+c*x]*Sqrt[-1+c*x]/(d1*d2*Sqrt[d1+e1*x]*Sqrt[d2+e2*x])*Int[x*(a+b*ArcCosh[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 + + +(* Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] + + b*c*n*d^p/(2*(p+1))*Int[x*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d)^p/(2*(p+1))*Int[x*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && IntegerQ[p] + + +(* Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -x*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*(p+1)) + + (2*p+3)/(2*d1*d2*(p+1))*Int[(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^p/(2*(p+1))*Int[x*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 && + IntegerQ[p+1/2] && (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + -x*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*(p+1)*(1+c^2*x^2)^FracPart[p])* + Int[x*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 + + +Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -x*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*(p+1)) + + (2*p+3)/(2*d1*d2*(p+1))*Int[(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^(p+1/2)*Sqrt[1+c*x]*Sqrt[-1+c*x]/(2*(p+1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x])* + Int[x*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 && IntegerQ[p+1/2] + + +Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -x*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*(p+1)) + + (2*p+3)/(2*d1*d2*(p+1))*Int[(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[x*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p<-1 && p!=-3/2 + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/(c*d)*Subst[Int[(a+b*x)^n*Sech[x],x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + -1/(c*d)*Subst[Int[(a+b*x)^n*Csch[x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + d^p*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (-d)^p*(-1+c*x)^(p+1/2)*(1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - + c*(-d)^p*(2*p+1)/(b*(n+1))*Int[x*(-1+c*x)^(p-1/2)*(1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[p] + + +(* Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (-d1*d2)^p*(-1+c*x)^(p+1/2)*(1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - + c*(-d1*d2)^p*(2*p+1)/(b*(n+1))*Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && IntegerQ[p-1/2] && + (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + Sqrt[1+c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + c*(2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*(n+1)*(1+c^2*x^2)^FracPart[p])* + Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - + c*(2*p+1)*(-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(b*(n+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && IntegerQ[p-1/2] + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - + c*(2*p+1)*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(b*(n+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[x*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + d^p/c*Subst[Int[(a+b*x)^n*Cosh[x]^(2*p+1),x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[2*p] && (IntegerQ[p] || PositiveQ[d]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^p/c*Subst[Int[(a+b*x)^n*Sinh[x]^(2*p+1),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^p/c*Subst[Int[(a+b*x)^n*Sinh[x]^(2*p+1),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveIntegerQ[p+1/2] && + (PositiveQ[d1] && NegativeQ[d2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1+c^2*x^2]*Int[(1+c^2*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && PositiveIntegerQ[2*p] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveIntegerQ[2*p] && Not[PositiveQ[d1] && NegativeQ[d2]] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[e-c^2*d] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && NeQ[c^2*d+e] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Sqrt[1-c^2*x^2]/(Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) *) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSinh[c*x])^n,(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && NeQ[e-c^2*d] && IntegerQ[p] && (p>0 || PositiveIntegerQ[n]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCosh[c*x])^n,(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,n},x] && NeQ[c^2*d+e] && IntegerQ[p] && (p>0 || PositiveIntegerQ[n]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && IntegerQ[p] + + +Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n,p},x] + + +Int[(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(d*f+e*g*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2+g^2] && Not[IntegerQ[p]] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^IntPart[p]*(d+e*x^2)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[c^2*d+e] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n*) + + +Int[x_*(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Subst[Int[(a+b*x)^n*Tanh[x],x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Subst[Int[(a+b*x)^n*Coth[x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - + b*n*d^p/(2*c*(p+1))*Int[(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && NeQ[p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e*(p+1)) - + b*n*(-d)^p/(2*c*(p+1))*Int[(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] && IntegerQ[p] + + +(* Int[x_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - + b*n*(-d1*d2)^(p-1/2)/(2*c*(p+1))*Int[(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && IntegerQ[p+1/2] && + (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - + b*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1+c^2*x^2)^FracPart[p])* + Int[(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - + b*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*c*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && NeQ[p+1] && IntegerQ[p+1/2] + + +Int[x_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - + b*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*c*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + 1/d*Subst[Int[(a+b*x)^n/(Cosh[x]*Sinh[x]),x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + -1/d*Subst[Int[(a+b*x)^n/(Cosh[x]*Sinh[x]),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] && + (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(d*f*(m+1)) + + b*c*n*(-d)^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] && IntegerQ[p] + + +(* Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1*d2*f*(m+1)) + + b*c*n*(-d1*d2)^p/(f*(m+1))*Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && + NeQ[m+1] && IntegerQ[p+1/2] && (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1*d2*f*(m+1)) + + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(f*(m+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && + NeQ[m+1] && IntegerQ[p+1/2] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1*d2*f*(m+1)) + + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(f*(m+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && EqQ[m+2*p+3] && NeQ[m+1] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])/x_,x_Symbol] := + (d+e*x^2)^p*(a+b*ArcSinh[c*x])/(2*p) - + b*c*d^p/(2*p)*Int[(1+c^2*x^2)^(p-1/2),x] + + d*Int[(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])/x,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])/x_,x_Symbol] := + (d+e*x^2)^p*(a+b*ArcCosh[c*x])/(2*p) - + b*c*(-d)^p/(2*p)*Int[(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2),x] + + d*Int[(d+e*x^2)^(p-1)*(a+b*ArcCosh[c*x])/x,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])/(f*(m+1)) - + b*c*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2),x] - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && PositiveIntegerQ[p] && NegativeIntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCosh[c*x])/(f*(m+1)) - + b*c*(-d)^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2),x] - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCosh[c*x]),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] && NegativeIntegerQ[(m+1)/2] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1+c^2*x^2)^p,x]}, + Dist[d^p*(a+b*ArcSinh[c*x]),u,x] - b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && IntegerQ[p-1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) && + p!=-1/2 && PositiveQ[d] + + +Int[x_^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1+c*x)^p*(-1+c*x)^p,x]}, + Dist[(-d1*d2)^p*(a+b*ArcCosh[c*x]),u,x] - b*c*(-d1*d2)^p*Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[p-1/2] && + (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) && p!=-1/2 && PositiveQ[d1] && NegativeQ[d2] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1+c^2*x^2)^p,x]}, + (a+b*ArcSinh[c*x])*Int[x^m*(d+e*x^2)^p,x] - + b*c*d^(p-1/2)*Sqrt[d+e*x^2]/Sqrt[1+c^2*x^2]*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveIntegerQ[p+1/2] && (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) + + +Int[x_^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(1+c*x)^p*(-1+c*x)^p,x]}, + (a+b*ArcCosh[c*x])*Int[x^m*(d1+e1*x)^p*(d2+e2*x)^p,x] - + b*c*(-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveIntegerQ[p+1/2] && + (PositiveIntegerQ[(m+1)/2] || NegativeIntegerQ[(m+2*p+3)/2]) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d)^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && IntegerQ[p] + + +(* Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(f*(m+1)) - + 2*e1*e2*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^p/(f*(m+1))*Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && + IntegerQ[p-1/2] && (PositiveQ[d1] && NegativeQ[d2]) *) + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/(f*(m+1)) - + b*c*n*Sqrt[d+e*x^2]/(f*(m+1)*Sqrt[1+c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcSinh[c*x])^(n-1),x] - + c^2*Sqrt[d+e*x^2]/(f^2*(m+1)*Sqrt[1+c^2*x^2])*Int[(f*x)^(m+2)*(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 + + +Int[(f_.*x_)^m_*Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/(f*(m+1)) - + b*c*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(f*(m+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[(f*x)^(m+1)*(a+b*ArcCosh[c*x])^(n-1),x] - + c^2*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(f^2*(m+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[((f*x)^(m+2)*(a+b*ArcCosh[c*x])^n)/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m<-1 + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+1)) - + 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(f*(m+1)) - + 2*e1*e2*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(f*(m+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n,p] && n>0 && p>0 && m<-1 && IntegerQ[p-1/2] + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (IntegerQ[p] || PositiveQ[d]) && (RationalQ[m] || EqQ[n-1]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d)^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + IntegerQ[p] && (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/(f*(m+2)) - + b*c*n*Sqrt[d+e*x^2]/(f*(m+2)*Sqrt[1+c^2*x^2])*Int[(f*x)^(m+1)*(a+b*ArcSinh[c*x])^(n-1),x] + + Sqrt[d+e*x^2]/((m+2)*Sqrt[1+c^2*x^2])*Int[(f*x)^m*(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && Not[RationalQ[m] && m<-1] && (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/(f*(m+2)) - + b*c*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(f*(m+2)*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(f*x)^(m+1)*(a+b*ArcCosh[c*x])^(n-1),x] - + Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/((m+2)*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(f*x)^m*(a+b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && Not[RationalQ[m] && m<-1] && + (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+2*p+1)) + + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+2*p+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + (RationalQ[m] || EqQ[n-1]) + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(f*(m+2*p+1)) + + 2*d1*d2*p/(m+2*p+1)*Int[(f*x)^m*(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^(p-1/2)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(f*(m+2*p+1)*Sqrt[1+c*x]*Sqrt[-1+c*x])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p>0 && Not[RationalQ[m] && m<-1] && + IntegerQ[p-1/2] && (RationalQ[m] || EqQ[n-1]) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(d*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n,x] + + b*c*n*(-d)^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] - + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(f*(m+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1*d2*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] + + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(f*(m+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] && + IntegerQ[p+1/2] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1*d2*f*(m+1)) + + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] + + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(f*(m+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m<-1 && IntegerQ[m] + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] - + b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && p<-1 && m>1 && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d)^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && p<-1 && m>1 && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - + f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] - + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*c*(p+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - + f^2*(m-1)/(2*e1*e2*(p+1))*Int[(f*x)^(m-2)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*c*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n,p] && n>0 && p<-1 && m>1 && IntegerQ[p+1/2] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - + f^2*(m-1)/(2*e1*e2*(p+1))*Int[(f*x)^(m-2)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*c*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n,p] && n>0 && p<-1 && Not[IntegerQ[p]] && m>1 + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] + + b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[p] || PositiveQ[d]) && (IntegerQ[m] || IntegerQ[p] || n==1) *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d)^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && IntegerQ[p] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*f*(p+1)) + + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] + + b*c*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(2*f*(p+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n,p] && n>0 && p<-1 && Not[RationalQ[m] && m>1] && + (IntegerQ[m] || IntegerQ[p] || n==1) + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*f*(p+1)) + + (m+2*p+3)/(2*d1*d2*(p+1))*Int[(f*x)^m*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*f*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p<-1 && + Not[RationalQ[m] && m>1] && (IntegerQ[m] || n==1) && IntegerQ[p+1/2] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + -(f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*f*(p+1)) + + (m+2*p+3)/(2*d1*d2*(p+1))*Int[(f*x)^m*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n,x] - + b*c*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(2*f*(p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n,p] && n>0 && p<-1 && + Not[RationalQ[m] && m>1] && (IntegerQ[m] || IntegerQ[p] || n==1) + + +(* Int[(f_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/(e*m) - + b*f*n*Sqrt[1+c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b*ArcSinh[c*x])^(n-1),x] - + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSinh[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 && PositiveQ[d] && IntegerQ[m] *) + + +(* Int[(f_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/(e1*e2*m) + + b*f*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(c*d1*d2*m*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(f*x)^(m-1)*(a+b*ArcCosh[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[(f*x)^(m-2)*(a+b*ArcCosh[c*x])^n/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m>1 && IntegerQ[m] *) + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/(e*m) - + b*f*n*Sqrt[1+c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b*ArcSinh[c*x])^(n-1),x] - + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSinh[c*x])^n)/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 && IntegerQ[m] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + f*(f*x)^(m-1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/(e1*e2*m) + + b*f*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(c*d1*d2*m*Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[(f*x)^(m-1)*(a+b*ArcCosh[c*x])^(n-1),x] + + f^2*(m-1)/(c^2*m)*Int[(f*x)^(m-2)*(a+b*ArcCosh[c*x])^n/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m>1 && IntegerQ[m] + + +Int[x_^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*Sinh[x]^m,x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[e-c^2*d] && PositiveQ[d] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[x_^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + 1/(c^(m+1)*Sqrt[-d1*d2])*Subst[Int[(a+b*x)^n*Cosh[x]^m,x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveIntegerQ[n] && + PositiveQ[d1] && NegativeQ[d2] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f*x)^(m+1)*(a+b*ArcSinh[c*x])*Hypergeometric2F1[1/2,(1+m)/2,(3+m)/2,-c^2*x^2]/(Sqrt[d]*f*(m+1)) - + b*c*(f*x)^(m+2)*HypergeometricPFQ[{1,1+m/2,1+m/2},{3/2+m/2,2+m/2},-c^2*x^2]/(Sqrt[d]*f^2*(m+1)*(m+2)) /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && PositiveQ[d] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])/(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + (f*x)^(m+1)*Sqrt[1-c^2*x^2]*(a+b*ArcCosh[c*x])*Hypergeometric2F1[1/2,(1+m)/2,(3+m)/2,c^2*x^2]/ + (f*(m+1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]) + + b*c*(f*x)^(m+2)*HypergeometricPFQ[{1,1+m/2,1+m/2},{3/2+m/2,2+m/2},c^2*x^2]/(Sqrt[-d1*d2]*f^2*(m+1)*(m+2)) /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveQ[d1] && NegativeQ[d2] && Not[IntegerQ[m]] + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(f*x)^m*(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n>0 && Not[PositiveQ[d]] && (IntegerQ[m] || n==1) + + +Int[(f_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + Sqrt[1+c*x]*Sqrt[-1+c*x]/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x])*Int[(f*x)^m*(a+b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n>0 && Not[PositiveQ[d1] && NegativeQ[d2]] && + (IntegerQ[m] || n==1) + + +(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(e*(m+2*p+1)) - + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] - + b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && + (IntegerQ[p] || PositiveQ[d]) && IntegerQ[m] *) + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(e*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d)^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && IntegerQ[p] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(e*(m+2*p+1)) - + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] - + b*f*n*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(c*(m+2*p+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e-c^2*d] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && IntegerQ[m] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(e1*e2*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(c*(m+2*p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && + IntegerQ[m] && IntegerQ[p+1/2] + + +Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + f*(f*x)^(m-1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(e1*e2*(m+2*p+1)) + + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] - + b*f*n*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(c*(m+2*p+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[m,n] && n>0 && m>1 && NeQ[m+2*p+1] && + IntegerQ[m] + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + d^p*(f*x)^m*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] && (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + f*m*(-d)^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + f*m*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(b*c*(n+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] && IntegerQ[p-1/2] + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + f*m*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(b*c*(n+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && EqQ[m+2*p+1] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f*x)^m*(a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + f*m/(b*c*Sqrt[d]*(n+1))*Int[(f*x)^(m-1)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && PositiveQ[d] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_/(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + (f*x)^m*(a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) - + (f*m)/(b*c*Sqrt[-d1*d2]*(n+1))*Int[(f*x)^(m-1)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && PositiveQ[d1] && NegativeQ[d2] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(f*x)^m*(a+b*ArcSinh[c*x])^n/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && Not[PositiveQ[d]] + + +Int[(f_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_/(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + Sqrt[1+c*x]*Sqrt[-1+c*x]/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x])*Int[(f*x)^m*(a+b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && Not[PositiveQ[d1] && NegativeQ[d2]] + + +(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + d^p*(f*x)^m*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] - + c*d^p*(m+2*p+1)/(b*f*(n+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] && + (IntegerQ[p] || PositiveQ[d]) *) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + f*m*(-d)^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] - + c*(-d)^p*(m+2*p+1)/(b*f*(n+1))*Int[(f*x)^(m+1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[p] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - + f*m*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*c*(n+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] - + c*(m+2*p+1)*d^IntPart[p]*(d+e*x^2)^FracPart[p]/(b*f*(n+1)*(1+c^2*x^2)^FracPart[p])* + Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f},x] && EqQ[e-c^2*d] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && PositiveIntegerQ[2*p] + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^(n+1)/(b*c*(n+1)) + + f*m*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(b*c*(n+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] - + c*(m+2*p+1)*(-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(b*f*(n+1)*(1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && RationalQ[n] && n<-1 && IntegerQ[m] && m>-3 && + PositiveIntegerQ[p+1/2] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Sinh[x]^m*Cosh[x]^(2*p+1),x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && (IntegerQ[p] || PositiveQ[d]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^p/c^(m+1)*Subst[Int[(a+b*x)^n*Cosh[x]^m*Sinh[x]^(2*p+1),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[p] && PositiveIntegerQ[m] + + +Int[x_^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^p/c^(m+1)*Subst[Int[(a+b*x)^n*Cosh[x]^m*Sinh[x]^(2*p+1),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[p+1/2] && p>-1 && PositiveIntegerQ[m] && + (PositiveQ[d1] && NegativeQ[d2]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1+c^2*x^2)^FracPart[p]*Int[x^m*(1+c^2*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && Not[(IntegerQ[p] || PositiveQ[d])] + + +Int[x_^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[x^m*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[2*p] && p>-1 && PositiveIntegerQ[m] && + Not[IntegerQ[p] || PositiveQ[d1] && NegativeQ[d2]] + + +Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSinh[c*x])^n/Sqrt[d+e*x^2],(f*x)^m*(d+e*x^2)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[e-c^2*d] && PositiveQ[d] && PositiveIntegerQ[p+1/2] && Not[PositiveIntegerQ[(m+1)/2]] && + IntegerQ[m] && -30 || PositiveIntegerQ[(m-1)/2] && m+p<=0) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e,f,m},x] && NeQ[c^2*d+e] && IntegerQ[p] && (p>0 || PositiveIntegerQ[(m-1)/2] && m+p<=0) + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSinh[c*x])^n,(f*x)^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[e-c^2*d] && PositiveIntegerQ[n] && IntegerQ[p] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCosh[c*x])^n,(f*x)^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,f},x] && NeQ[c^2*d+e] && PositiveIntegerQ[n] && IntegerQ[p] && IntegerQ[m] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(f*x)^m*(d+e*x^2)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && IntegerQ[p] + + +Int[(f_.*x_)^m_.*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(f*x)^m*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,m,n,p},x] + + +Int[(h_.*x_)^m_.*(d_+e_.*x_)^p_*(f_+g_.*x_)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^FracPart[p]*(f+g*x)^FracPart[p]/(d*f+e*g*x^2)^FracPart[p]* + Int[(h*x)^m*(d*f+e*g*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p},x] && EqQ[e*f+d*g] && EqQ[c^2*f^2+g^2] && Not[IntegerQ[p]] + + +Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^IntPart[p]*(d+e*x^2)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f*x)^m*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[c^2*d+e] && Not[IntegerQ[p]] + + + + + +(* ::Subsection::Closed:: *) +(*7.1.5 u (a+b arcsinh(c x))^n*) + + +Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_.+e_.*x_),x_Symbol] := + Subst[Int[(a+b*x)^n*Cosh[x]/(c*d+e*Sinh[x]),x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(d_.+e_.*x_),x_Symbol] := + Subst[Int[(a+b*x)^n*Sinh[x]/(c*d+e*Cosh[x]),x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*ArcSinh[c*x])^n/(e*(m+1)) - + b*c*n/(e*(m+1))*Int[(d+e*x)^(m+1)*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (d+e*x)^(m+1)*(a+b*ArcCosh[c*x])^n/(e*(m+1)) - + b*c*n/(e*(m+1))*Int[(d+e*x)^(m+1)*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[-1+c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c,d,e,m},x] && PositiveIntegerQ[n] && NeQ[m+1] + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*ArcSinh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-1 + + +Int[(d_+e_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^m*(a+b*ArcCosh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[m] && RationalQ[n] && n<-1 + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*Cosh[x]*(c*d+e*Sinh[x])^m,x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && PositiveIntegerQ[m] + + +Int[(d_.+e_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + 1/c^(m+1)*Subst[Int[(a+b*x)^n*(c*d+e*Cosh[x])^m*Sinh[x],x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && PositiveIntegerQ[m] + + +Int[Px_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[ExpandExpression[Px,x],x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] + + +Int[Px_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[ExpandExpression[Px,x],x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] + + +(* Int[Px_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + With[{u=IntHide[Px,x]}, + Dist[(a+b*ArcSinh[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] *) + + +(* Int[Px_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + With[{u=IntHide[Px,x]}, + Dist[(a+b*ArcCosh[c*x])^n,u,x] - + b*c*n*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[SimplifyIntegrand[u*(a+b*ArcCosh[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] *) + + +Int[Px_*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(a+b*ArcSinh[c*x])^n,x],x] /; +FreeQ[{a,b,c,n},x] && PolynomialQ[Px,x] + + +Int[Px_*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + Int[ExpandIntegrand[Px*(a+b*ArcCosh[c*x])^n,x],x] /; +FreeQ[{a,b,c,n},x] && PolynomialQ[Px,x] + + +Int[Px_*(d_.+e_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[Px*(d+e*x)^m,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,m},x] && PolynomialQ[Px,x] + + +Int[Px_*(d_.+e_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[Px*(d+e*x)^m,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[SimplifyIntegrand[u/Sqrt[1-c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,m},x] && PolynomialQ[Px,x] + + +Int[(f_.+g_.*x_)^p_.*(d_+e_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + With[{u=IntHide[(f+g*x)^p*(d+e*x)^m,x]}, + Dist[(a+b*ArcSinh[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[n,p] && NegativeIntegerQ[m] && m+p+1<0 + + +Int[(f_.+g_.*x_)^p_.*(d_+e_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + With[{u=IntHide[(f+g*x)^p*(d+e*x)^m,x]}, + Dist[(a+b*ArcCosh[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PositiveIntegerQ[n,p] && NegativeIntegerQ[m] && m+p+1<0 + + +Int[(f_.+g_.*x_+h_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_/(d_+e_.*x_)^2,x_Symbol] := + With[{u=IntHide[(f+g*x+h*x^2)^p/(d+e*x)^2,x]}, + Dist[(a+b*ArcSinh[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2],x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[n,p] && EqQ[e*g-2*d*h] + + +Int[(f_.+g_.*x_+h_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_/(d_+e_.*x_)^2,x_Symbol] := + With[{u=IntHide[(f+g*x+h*x^2)^p/(d+e*x)^2,x]}, + Dist[(a+b*ArcCosh[c*x])^n,u,x] - b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCosh[c*x])^(n-1)/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[n,p] && EqQ[e*g-2*d*h] + + +Int[Px_*(d_+e_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Px*(d+e*x)^m*(a+b*ArcSinh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[Px_*(d_+e_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Px*(d+e*x)^m*(a+b*ArcCosh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PolynomialQ[Px,x] && PositiveIntegerQ[n] && IntegerQ[m] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f+g*x)^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[Dist[1/Sqrt[1+c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && m>0 && + (m<-2*p-1 || m>3) + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(f+g*x)^m*(d1+e1*x)^p*(d2+e2*x)^p,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[Dist[1/(Sqrt[1+c*x]*Sqrt[-1+c*x]),u,x],x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && + PositiveQ[d1] && NegativeQ[d2] && m>0 && (m<-2*p-1 || m>3) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m>0 && + (n==1 && p>-1 || p>0 || m==1 || m==2 && p<-2) + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && IntegerQ[p+1/2] && + PositiveQ[d1] && NegativeQ[d2] && PositiveIntegerQ[n] && m>0 && (n==1 && p>-1 || p>0 || m==1 || m==2 && p<-2) + + +Int[(f_.+g_.*x_)^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d+e*x^2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + 1/(b*c*Sqrt[d]*(n+1))*Int[(d*g*m+2*e*f*x+e*g*(m+2)*x^2)*(f+g*x)^(m-1)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_*Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d1*d2+e1*e2*x^2)*(a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) - + 1/(b*c*Sqrt[-d1*d2]*(n+1))*Int[(d1*d2*g*m+2*e1*e2*f*x+e1*e2*g*(m+2)*x^2)*(f+g*x)^(m-1)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && PositiveQ[d1] && NegativeQ[d2] && + PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n,(f+g*x)^m*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && PositiveIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n,(f+g*x)^m*(d1+e1*x)^(p-1/2)*(d2+e2*x)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && PositiveIntegerQ[p+1/2] && + PositiveQ[d1] && NegativeQ[d2] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d+e*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + 1/(b*c*Sqrt[d]*(n+1))* + Int[ExpandIntegrand[(f+g*x)^(m-1)*(a+b*ArcSinh[c*x])^(n+1),(d*g*m+e*f*(2*p+1)*x+e*g*(m+2*p+1)*x^2)*(d+e*x^2)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && PositiveIntegerQ[p-1/2] && PositiveQ[d] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (f+g*x)^m*(d1+e1*x)^(p+1/2)*(d2+e2*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) - + 1/(b*c*Sqrt[-d1*d2]*(n+1))* + Int[ExpandIntegrand[(f+g*x)^(m-1)*(a+b*ArcCosh[c*x])^(n+1), + (d1*d2*g*m+e1*e2*f*(2*p+1)*x+e1*e2*g*(m+2*p+1)*x^2)*(d1+e1*x)^(p-1/2)*(d2+e2*x)^(p-1/2),x],x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && PositiveIntegerQ[p-1/2] && + PositiveQ[d1] && NegativeQ[d2] && PositiveIntegerQ[n] && m<0 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_Symbol] := + (f+g*x)^m*(a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + g*m/(b*c*Sqrt[d]*(n+1))*Int[(f+g*x)^(m-1)*(a+b*ArcSinh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && PositiveQ[d] && m>0 && RationalQ[n] && n<-1 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_/(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + (f+g*x)^m*(a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) - + g*m/(b*c*Sqrt[-d1*d2]*(n+1))*Int[(f+g*x)^(m-1)*(a+b*ArcCosh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && PositiveQ[d1] && NegativeQ[d2] && m>0 && + RationalQ[n] && n<-1 + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c^(m+1)*Sqrt[d])*Subst[Int[(a+b*x)^n*(c*f+g*Sinh[x])^m,x],x,ArcSinh[c*x]] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[e-c^2*d] && IntegerQ[m] && PositiveQ[d] && (m>0 || PositiveIntegerQ[n]) + + +Int[(f_+g_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + 1/(c^(m+1)*Sqrt[-d1*d2])*Subst[Int[(a+b*x)^n*(c*f+g*Cosh[x])^m,x],x,ArcCosh[c*x]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && PositiveQ[d1] && NegativeQ[d2] && + (m>0 || PositiveIntegerQ[n]) + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSinh[c*x])^n/Sqrt[d+e*x^2],(f+g*x)^m*(d+e*x^2)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d,e,f,g},x] && EqQ[e-c^2*d] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCosh[c*x])^n/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x]),(f+g*x)^m*(d1+e1*x)^(p+1/2)*(d2+e2*x)^(p+1/2),x],x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && NegativeIntegerQ[p+1/2] && + PositiveQ[d1] && NegativeQ[d2] && PositiveIntegerQ[n] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1+c^2*x^2)^FracPart[p]*Int[(f+g*x)^m*(1+c^2*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[e-c^2*d] && IntegerQ[m] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[(f_+g_.*x_)^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^IntPart[p]*(d+e*x^2)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[(f+g*x)^m*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p-1/2] + + +Int[(f_+g_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/(1-c^2*x^2)^FracPart[p]* + Int[(f+g*x)^m*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[m] && IntegerQ[p-1/2] && + Not[PositiveQ[d1] && NegativeQ[d2]] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Log[h*(f+g*x)^m]*(a+b*ArcSinh[c*x])^(n+1)/(b*c*Sqrt[d]*(n+1)) - + g*m/(b*c*Sqrt[d]*(n+1))*Int[(a+b*ArcSinh[c*x])^(n+1)/(f+g*x),x] /; +FreeQ[{a,b,c,d,e,f,g,h,m},x] && EqQ[e-c^2*d] && PositiveQ[d] && PositiveIntegerQ[n] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.*x_]),x_Symbol] := + Log[h*(f+g*x)^m]*(a+b*ArcCosh[c*x])^(n+1)/(b*c*Sqrt[-d1*d2]*(n+1)) - + g*m/(b*c*Sqrt[-d1*d2]*(n+1))*Int[(a+b*ArcCosh[c*x])^(n+1)/(f+g*x),x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g,h,m},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveQ[d1] && NegativeQ[d2] && PositiveIntegerQ[n] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + d^IntPart[p]*(d+e*x^2)^FracPart[p]/(1+c^2*x^2)^FracPart[p]*Int[Log[h*(f+g*x)^m]*(1+c^2*x^2)^p*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[e-c^2*d] && IntegerQ[p-1/2] && Not[PositiveQ[d]] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d)^IntPart[p]*(d+e*x^2)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[Log[h*(f+g*x)^m]*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n},x] && EqQ[c^2*d+e] && IntegerQ[p-1/2] + + +Int[Log[h_.*(f_.+g_.*x_)^m_.]*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + (-d1*d2)^IntPart[p]*(d1+e1*x)^FracPart[p]*(d2+e2*x)^FracPart[p]/((1+c*x)^FracPart[p]*(-1+c*x)^FracPart[p])* + Int[Log[h*(f+g*x)^m]*(1+c*x)^p*(-1+c*x)^p*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g,h,m,n},x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[p-1/2] && Not[PositiveQ[d1] && NegativeQ[d2]] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x)^m*(f+g*x)^m,x]}, + Dist[a+b*ArcSinh[c*x],u,x] - b*c*Int[Dist[1/Sqrt[1+c^2*x^2],u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NegativeIntegerQ[m+1/2] + + +Int[(d_+e_.*x_)^m_*(f_+g_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x)^m*(f+g*x)^m,x]}, + Dist[a+b*ArcCosh[c*x],u,x] - b*c*Int[Dist[1/(Sqrt[1+c*x]*Sqrt[-1+c*x]),u,x],x]] /; +FreeQ[{a,b,c,d,e,f,g},x] && NegativeIntegerQ[m+1/2] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcSinh[c*x])^n,(d+e*x)^m*(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && IntegerQ[m] + + +Int[(d_+e_.*x_)^m_.*(f_+g_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCosh[c*x])^n,(d+e*x)^m*(f+g*x)^m,x],x] /; +FreeQ[{a,b,c,d,e,f,g,n},x] && IntegerQ[m] + + +Int[u_*(a_.+b_.*ArcSinh[c_.*x_]),x_Symbol] := + With[{v=IntHide[u,x]}, + Dist[a+b*ArcSinh[c*x],v,x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1+c^2*x^2],x],x] /; + InverseFunctionFreeQ[v,x]] /; +FreeQ[{a,b,c},x] + + +Int[u_*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol] := + With[{v=IntHide[u,x]}, + Dist[a+b*ArcCosh[c*x],v,x] - b*c*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[SimplifyIntegrand[v/Sqrt[1-c^2*x^2],x],x] /; + InverseFunctionFreeQ[v,x]] /; +FreeQ[{a,b,c},x] + + +Int[Px_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := + With[{u=ExpandIntegrand[Px*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,n},x] && PolynomialQ[Px,x] && EqQ[e-c^2*d] && IntegerQ[p-1/2] + + +Int[Px_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := + With[{u=ExpandIntegrand[Px*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && PolynomialQ[Px,x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[p-1/2] + + +Int[Px_.*(f_+g_.*(d_+e_.*x_^2)^p_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(f+g*(d+e*x^2)^p)^m*(a+b*ArcSinh[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d,e,f,g},x] && PolynomialQ[Px,x] && EqQ[e-c^2*d] && PositiveIntegerQ[p+1/2] && IntegersQ[m,n] + + +Int[Px_.*(f_+g_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + With[{u=ExpandIntegrand[Px*(f+g*(d1+e1*x)^p*(d2+e2*x)^p)^m*(a+b*ArcCosh[c*x])^n,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{a,b,c,d1,e1,d2,e2,f,g},x] && PolynomialQ[Px,x] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && PositiveIntegerQ[p+1/2] && + IntegersQ[m,n] + + +Int[RFx_*ArcSinh[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[ArcSinh[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[c,x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*ArcCosh[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[ArcCosh[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[c,x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[RFx*(a+b*ArcSinh[c*x])^n,x],x] /; +FreeQ[{a,b,c},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(a_+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[RFx*(a+b*ArcCosh[c*x])^n,x],x] /; +FreeQ[{a,b,c},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] + + +Int[RFx_*(d_+e_.*x_^2)^p_*ArcSinh[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(d+e*x^2)^p*ArcSinh[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[e-c^2*d] && IntegerQ[p-1/2] + + +Int[RFx_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*ArcCosh[c_.*x_]^n_.,x_Symbol] := + With[{u=ExpandIntegrand[(d1+e1*x)^p*(d2+e2*x)^p*ArcCosh[c*x]^n,RFx,x]}, + Int[u,x] /; + SumQ[u]] /; +FreeQ[{c,d1,e1,d2,e2},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && IntegerQ[p-1/2] + + +Int[RFx_*(d_+e_.*x_^2)^p_*(a_+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p,RFx*(a+b*ArcSinh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[e-c^2*d] && IntegerQ[p-1/2] + + +Int[RFx_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d1+e1*x)^p*(d2+e2*x)^p,RFx*(a+b*ArcCosh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d1,e1,d2,e2},x] && RationalFunctionQ[RFx,x] && PositiveIntegerQ[n] && EqQ[e1-c*d1] && EqQ[e2+c*d2] && + IntegerQ[p-1/2] + + +Int[u_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][u*(a+b*ArcSinh[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] + + +Int[u_.*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][u*(a+b*ArcCosh[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] + + + + + +(* ::Subsection::Closed:: *) +(*7.1.6 Miscellaneous inverse hyperbolic sine*) +(**) + + +Int[(a_.+b_.*ArcSinh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(a+b*ArcSinh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,n},x] + + +Int[(a_.+b_.*ArcCosh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(a+b*ArcCosh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,n},x] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*ArcSinh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(a+b*ArcSinh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[(e_.+f_.*x_)^m_.*(a_.+b_.*ArcCosh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(a+b*ArcCosh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +Int[(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(C/d^2+C/d^2*x^2)^p*(a+b*ArcSinh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,A,B,C,n,p},x] && EqQ[B*(1+c^2)-2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[(-C/d^2+C/d^2*x^2)^p*(a+b*ArcCosh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,A,B,C,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(e_.+f_.*x_)^m_.*(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(C/d^2+C/d^2*x^2)^p*(a+b*ArcSinh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && EqQ[B*(1+c^2)-2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[(e_.+f_.*x_)^m_.*(A_.+B_.*x_+C_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_+d_.*x_])^n_.,x_Symbol] := + 1/d*Subst[Int[((d*e-c*f)/d+f*x/d)^m*(-C/d^2+C/d^2*x^2)^p*(a+b*ArcCosh[x])^n,x],x,c+d*x] /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n,p},x] && EqQ[B*(1-c^2)+2*A*c*d] && EqQ[2*c*C-B*d] + + +Int[Sqrt[a_.+b_.*ArcSinh[c_+d_.*x_^2]],x_Symbol] := + x*Sqrt[a+b*ArcSinh[c+d*x^2]] - + Sqrt[Pi]*x*(Cosh[a/(2*b)]-c*Sinh[a/(2*b)])*FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a+b*ArcSinh[c+d*x^2]]]/ + (Sqrt[-(c/b)]*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) + + Sqrt[Pi]*x*(Cosh[a/(2*b)]+c*Sinh[a/(2*b)])*FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a+b*ArcSinh[c+d*x^2]]]/ + (Sqrt[-(c/b)]*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] + + +Int[(a_.+b_.*ArcSinh[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcSinh[c+d*x^2])^n - + 2*b*n*Sqrt[2*c*d*x^2+d^2*x^4]*(a+b*ArcSinh[c+d*x^2])^(n-1)/(d*x) + + 4*b^2*n*(n-1)*Int[(a+b*ArcSinh[c+d*x^2])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] && RationalQ[n] && n>1 + + +Int[1/(a_.+b_.*ArcSinh[c_+d_.*x_^2]),x_Symbol] := + x*(c*Cosh[a/(2*b)]-Sinh[a/(2*b)])*CoshIntegral[(a+b*ArcSinh[c+d*x^2])/(2*b)]/ + (2*b*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[(1/2)*ArcSinh[c+d*x^2]])) + + x*(Cosh[a/(2*b)]-c*Sinh[a/(2*b)])*SinhIntegral[(a+b*ArcSinh[c+d*x^2])/(2*b)]/ + (2*b*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[(1/2)*ArcSinh[c+d*x^2]])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] + + +Int[1/Sqrt[a_.+b_.*ArcSinh[c_+d_.*x_^2]],x_Symbol] := + (c+1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Erfi[Sqrt[a+b*ArcSinh[c+d*x^2]]/Sqrt[2*b]]/ + (2*Sqrt[b]*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) + + (c-1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Erf[Sqrt[a+b*ArcSinh[c+d*x^2]]/Sqrt[2*b]]/ + (2*Sqrt[b]*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] + + +Int[1/(a_.+b_.*ArcSinh[c_+d_.*x_^2])^(3/2),x_Symbol] := + -Sqrt[2*c*d*x^2+d^2*x^4]/(b*d*x*Sqrt[a+b*ArcSinh[c+d*x^2]]) - + (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)]-c*Sinh[a/(2*b)])*FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a+b*ArcSinh[c+d*x^2]]]/ + (Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2]) + + (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)]+c*Sinh[a/(2*b)])*FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a+b*ArcSinh[c+d*x^2]]]/ + (Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2]) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] + + +Int[1/(a_.+b_.*ArcSinh[c_+d_.*x_^2])^2,x_Symbol] := + -Sqrt[2*c*d*x^2+d^2*x^4]/(2*b*d*x*(a+b*ArcSinh[c+d*x^2])) + + x*(Cosh[a/(2*b)]-c*Sinh[a/(2*b)])*CoshIntegral[(a+b*ArcSinh[c+d*x^2])/(2*b)]/ + (4*b^2*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) + + x*(c*Cosh[a/(2*b)]-Sinh[a/(2*b)])*SinhIntegral[(a+b*ArcSinh[c+d*x^2])/(2*b)]/ + (4*b^2*(Cosh[ArcSinh[c+d*x^2]/2]+c*Sinh[ArcSinh[c+d*x^2]/2])) /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] + + +Int[(a_.+b_.*ArcSinh[c_+d_.*x_^2])^n_,x_Symbol] := + -x*(a+b*ArcSinh[c+d*x^2])^(n+2)/(4*b^2*(n+1)*(n+2)) + + Sqrt[2*c*d*x^2+d^2*x^4]*(a+b*ArcSinh[c+d*x^2])^(n+1)/(2*b*d*(n+1)*x) + + 1/(4*b^2*(n+1)*(n+2))*Int[(a+b*ArcSinh[c+d*x^2])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2+1] && RationalQ[n] && n<-1 && n!=-2 + + +Int[Sqrt[a_.+b_.*ArcCosh[1+d_.*x_^2]],x_Symbol] := + 2*Sqrt[a+b*ArcCosh[1+d*x^2]]*Sinh[(1/2)*ArcCosh[1+d*x^2]]^2/(d*x) - + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1+d*x^2]]* + Erfi[(1/Sqrt[2*b])*Sqrt[a+b*ArcCosh[1+d*x^2]]]/(d*x) + + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1+d*x^2]]* + Erf[(1/Sqrt[2*b])*Sqrt[a+b*ArcCosh[1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[Sqrt[a_.+b_.*ArcCosh[-1+d_.*x_^2]],x_Symbol] := + 2*Sqrt[a+b*ArcCosh[-1+d*x^2]]*Cosh[(1/2)*ArcCosh[-1+d*x^2]]^2/(d*x) - + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Cosh[(1/2)*ArcCosh[-1+d*x^2]]* + Erfi[(1/Sqrt[2*b])*Sqrt[a+b*ArcCosh[-1+d*x^2]]]/(d*x) - + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Cosh[(1/2)*ArcCosh[-1+d*x^2]]* + Erf[(1/Sqrt[2*b])*Sqrt[a+b*ArcCosh[-1+d*x^2]]]/(d*x) /; +FreeQ[{a,b,d},x] + + +Int[(a_.+b_.*ArcCosh[c_+d_.*x_^2])^n_,x_Symbol] := + x*(a+b*ArcCosh[c+d*x^2])^n - + 2*b*n*(2*c*d*x^2+d^2*x^4)*(a+b*ArcCosh[c+d*x^2])^(n-1)/(d*x*Sqrt[-1+c+d*x^2]*Sqrt[1+c+d*x^2]) + + 4*b^2*n*(n-1)*Int[(a+b*ArcCosh[c+d*x^2])^(n-2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n>1 + + +Int[1/(a_.+b_.*ArcCosh[1+d_.*x_^2]),x_Symbol] := + x*Cosh[a/(2*b)]*CoshIntegral[(a+b*ArcCosh[1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) - + x*Sinh[a/(2*b)]*SinhIntegral[(a+b*ArcCosh[1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCosh[-1+d_.*x_^2]),x_Symbol] := + -x*Sinh[a/(2*b)]*CoshIntegral[(a+b*ArcCosh[-1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) + + x*Cosh[a/(2*b)]*SinhIntegral[(a+b*ArcCosh[-1+d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/Sqrt[a_.+b_.*ArcCosh[1+d_.*x_^2]],x_Symbol] := + Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Sinh[ArcCosh[1+d*x^2]/2]*Erfi[Sqrt[a+b*ArcCosh[1+d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) + + Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Sinh[ArcCosh[1+d*x^2]/2]*Erf[Sqrt[a+b*ArcCosh[1+d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/Sqrt[a_.+b_.*ArcCosh[-1+d_.*x_^2]],x_Symbol] := + Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Cosh[ArcCosh[-1+d*x^2]/2]*Erfi[Sqrt[a+b*ArcCosh[-1+d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) - + Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Cosh[ArcCosh[-1+d*x^2]/2]*Erf[Sqrt[a+b*ArcCosh[-1+d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCosh[1+d_.*x_^2])^(3/2),x_Symbol] := + -Sqrt[d*x^2]*Sqrt[2+d*x^2]/(b*d*x*Sqrt[a+b*ArcCosh[1+d*x^2]]) + + Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Sinh[ArcCosh[1+d*x^2]/2]*Erfi[Sqrt[a+b*ArcCosh[1+d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)- + Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Sinh[ArcCosh[1+d*x^2]/2]*Erf[Sqrt[a+b*ArcCosh[1+d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCosh[-1+d_.*x_^2])^(3/2),x_Symbol] := + -Sqrt[d*x^2]*Sqrt[-2+d*x^2]/(b*d*x*Sqrt[a+b*ArcCosh[-1+d*x^2]]) + + Sqrt[Pi/2]*(Cosh[a/(2*b)]-Sinh[a/(2*b)])*Cosh[ArcCosh[-1+d*x^2]/2]*Erfi[Sqrt[a+b*ArcCosh[-1+d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) + + Sqrt[Pi/2]*(Cosh[a/(2*b)]+Sinh[a/(2*b)])*Cosh[ArcCosh[-1+d*x^2]/2]*Erf[Sqrt[a+b*ArcCosh[-1+d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCosh[1+d_.*x_^2])^2,x_Symbol] := + -Sqrt[d*x^2]*Sqrt[2+d*x^2]/(2*b*d*x*(a+b*ArcCosh[1+d*x^2])) - + x*Sinh[a/(2*b)]*CoshIntegral[(a+b*ArcCosh[1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) + + x*Cosh[a/(2*b)]*SinhIntegral[(a+b*ArcCosh[1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[1/(a_.+b_.*ArcCosh[-1+d_.*x_^2])^2,x_Symbol] := + -Sqrt[d*x^2]*Sqrt[-2+d*x^2]/(2*b*d*x*(a+b*ArcCosh[-1+d*x^2])) + + x*Cosh[a/(2*b)]*CoshIntegral[(a+b*ArcCosh[-1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) - + x*Sinh[a/(2*b)]*SinhIntegral[(a+b*ArcCosh[-1+d*x^2])/(2*b)]/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) /; +FreeQ[{a,b,d},x] + + +Int[(a_.+b_.*ArcCosh[c_+d_.*x_^2])^n_,x_Symbol] := + -x*(a+b*ArcCosh[c+d*x^2])^(n+2)/(4*b^2*(n+1)*(n+2)) + + (2*c*x^2 +d*x^4)*(a+b*ArcCosh[c+d*x^2])^(n+1)/(2*b*(n+1)*x*Sqrt[-1+c+d*x^2]*Sqrt[1+c+d*x^2]) + + 1/(4*b^2*(n+1)*(n+2))*Int[(a+b*ArcCosh[c+d*x^2])^(n+2),x] /; +FreeQ[{a,b,c,d},x] && EqQ[c^2-1] && RationalQ[n] && n<-1 && n!=-2 + + +Int[ArcSinh[a_.*x_^p_]^n_./x_,x_Symbol] := + 1/p*Subst[Int[x^n*Coth[x],x],x,ArcSinh[a*x^p]] /; +FreeQ[{a,p},x] && PositiveIntegerQ[n] + + +Int[ArcCosh[a_.*x_^p_]^n_./x_,x_Symbol] := + 1/p*Subst[Int[x^n*Tanh[x],x],x,ArcCosh[a*x^p]] /; +FreeQ[{a,p},x] && PositiveIntegerQ[n] + + +Int[u_.*ArcSinh[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCsch[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCosh[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcSech[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[ArcSinh[Sqrt[-1+b_.*x_^2]]^n_./Sqrt[-1+b_.*x_^2],x_Symbol] := + Sqrt[b*x^2]/(b*x)*Subst[Int[ArcSinh[x]^n/Sqrt[1+x^2],x],x,Sqrt[-1+b*x^2]] /; +FreeQ[{b,n},x] + + +Int[ArcCosh[Sqrt[1+b_.*x_^2]]^n_./Sqrt[1+b_.*x_^2],x_Symbol] := + Sqrt[-1+Sqrt[1+b*x^2]]*Sqrt[1+Sqrt[1+b*x^2]]/(b*x)*Subst[Int[ArcCosh[x]^n/(Sqrt[-1+x]*Sqrt[1+x]),x],x,Sqrt[1+b*x^2]] /; +FreeQ[{b,n},x] + + +Int[f_^(c_.*ArcSinh[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[f^(c*x^n)*Cosh[x],x],x,ArcSinh[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[f_^(c_.*ArcCosh[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[f^(c*x^n)*Sinh[x],x],x,ArcCosh[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[n] + + +Int[x_^m_.*f_^(c_.*ArcSinh[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[(-a/b+Sinh[x]/b)^m*f^(c*x^n)*Cosh[x],x],x,ArcSinh[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[m,n] + + +Int[x_^m_.*f_^(c_.*ArcCosh[a_.+b_.*x_]^n_.),x_Symbol] := + 1/b*Subst[Int[(-a/b+Cosh[x]/b)^m*f^(c*x^n)*Sinh[x],x],x,ArcCosh[a+b*x]] /; +FreeQ[{a,b,c,f},x] && PositiveIntegerQ[m,n] + + +Int[ArcSinh[u_],x_Symbol] := + x*ArcSinh[u] - + Int[SimplifyIntegrand[x*D[u,x]/Sqrt[1+u^2],x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[ArcCosh[u_],x_Symbol] := + x*ArcCosh[u] - + Int[SimplifyIntegrand[x*D[u,x]/(Sqrt[-1+u]*Sqrt[1+u]),x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcSinh[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcSinh[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/Sqrt[1+u^2],x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCosh[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCosh[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(Sqrt[-1+u]*Sqrt[1+u]),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[v_*(a_.+b_.*ArcSinh[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcSinh[u]),w,x] - b*Int[SimplifyIntegrand[w*D[u,x]/Sqrt[1+u^2],x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + +Int[v_*(a_.+b_.*ArcCosh[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCosh[u]),w,x] - b*Int[SimplifyIntegrand[w*D[u,x]/(Sqrt[-1+u]*Sqrt[1+u]),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + +Int[E^(n_.*ArcSinh[u_]), x_Symbol] := + Int[(u+Sqrt[1+u^2])^n,x] /; +IntegerQ[n] && PolynomialQ[u,x] + + +Int[x_^m_.*E^(n_.*ArcSinh[u_]), x_Symbol] := + Int[x^m*(u+Sqrt[1+u^2])^n,x] /; +RationalQ[m] && IntegerQ[n] && PolynomialQ[u,x] + + +Int[E^(n_.*ArcCosh[u_]), x_Symbol] := + Int[(u+Sqrt[-1+u]*Sqrt[1+u])^n,x] /; +IntegerQ[n] && PolynomialQ[u,x] + + +Int[x_^m_.*E^(n_.*ArcCosh[u_]), x_Symbol] := + Int[x^m*(u+Sqrt[-1+u]*Sqrt[1+u])^n,x] /; +RationalQ[m] && IntegerQ[n] && PolynomialQ[u,x] + + + + + +(* ::Subsection::Closed:: *) +(*7.2.1 u (a+b arctanh(c x))^n*) + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcTanh[c*x])^n - + b*c*n*Int[x*(a+b*ArcTanh[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + x*(a+b*ArcCoth[c*x])^n - + b*c*n*Int[x*(a+b*ArcCoth[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c},x] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + Defer[Int][(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] && Not[PositiveIntegerQ[n]] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + Defer[Int][(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,n},x] && Not[PositiveIntegerQ[n]] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + -(a+b*ArcTanh[c*x])^n*Log[2*d/(d+e*x)]/e + + b*c*n/e*Int[(a+b*ArcTanh[c*x])^(n-1)*Log[2*d/(d+e*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + -(a+b*ArcCoth[c*x])^n*Log[2*d/(d+e*x)]/e + + b*c*n/e*Int[(a+b*ArcCoth[c*x])^(n-1)*Log[2*d/(d+e*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] + + +Int[ArcTanh[c_.*x_]/(d_+e_.*x_),x_Symbol] := + -ArcTanh[c*d/e]*Log[d+e*x]/e - PolyLog[2,Simp[c*(d+e*x)/(c*d-e),x]]/(2*e) + PolyLog[2,Simp[c*(d+e*x)/(c*d+e),x]]/(2*e) /; +FreeQ[{c,d,e},x] && PositiveQ[c*d/e+1] && NegativeQ[c*d/e-1] + + +(* Int[ArcCoth[c_.*x_]/(d_.+e_.*x_),x_Symbol] := + z /; +FreeQ[{c,d,e},x] *) + + +Int[ArcTanh[c_.*x_]/(d_.+e_.*x_),x_Symbol] := + 1/2*Int[Log[1+c*x]/(d+e*x),x] - 1/2*Int[Log[1-c*x]/(d+e*x),x] /; +FreeQ[{c,d,e},x] + + +Int[ArcCoth[c_.*x_]/(d_.+e_.*x_),x_Symbol] := + 1/2*Int[Log[1+1/(c*x)]/(d+e*x),x] - 1/2*Int[Log[1-1/(c*x)]/(d+e*x),x] /; +FreeQ[{c,d,e},x] + + +Int[(a_+b_.*ArcTanh[c_.*x_])/(d_.+e_.*x_),x_Symbol] := + a/e*Log[RemoveContent[d+e*x,x]] + b*Int[ArcTanh[c*x]/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(a_+b_.*ArcCoth[c_.*x_])/(d_.+e_.*x_),x_Symbol] := + a/e*Log[RemoveContent[d+e*x,x]] + b*Int[ArcCoth[c*x]/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + (d+e*x)^(p+1)*(a+b*ArcTanh[c*x])/(e*(p+1)) - + b*c/(e*(p+1))*Int[(d+e*x)^(p+1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + (d+e*x)^(p+1)*(a+b*ArcCoth[c*x])/(e*(p+1)) - + b*c/(e*(p+1))*Int[(d+e*x)^(p+1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_/x_,x_Symbol] := + 2*(a+b*ArcTanh[c*x])^n*ArcTanh[1-2/(1-c*x)] - + 2*b*c*n*Int[(a+b*ArcTanh[c*x])^(n-1)*ArcTanh[1-2/(1-c*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_/x_,x_Symbol] := + 2*(a+b*ArcCoth[c*x])^n*ArcCoth[1-2/(1-c*x)] - + 2*b*c*n*Int[(a+b*ArcCoth[c*x])^(n-1)*ArcCoth[1-2/(1-c*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && n>1 + + +Int[x_^m_.*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcTanh[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcTanh[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,m},x] && IntegerQ[n] && n>1 && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + x^(m+1)*(a+b*ArcCoth[c*x])^n/(m+1) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCoth[c*x])^(n-1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,m},x] && IntegerQ[n] && n>1 && NeQ[m+1] + + +Int[(d_+e_.*x_)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^p*(a+b*ArcTanh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n,p] + + +Int[(d_+e_.*x_)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x)^p*(a+b*ArcCoth[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && PositiveIntegerQ[n,p] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + Defer[Int][(d+e*x)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && Not[PositiveIntegerQ[n]] + + +Int[(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + Defer[Int][(d+e*x)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] && Not[PositiveIntegerQ[n]] + + +Int[x_^m_.*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/e*Int[x^(m-1)*(a+b*ArcTanh[c*x])^n,x] - + d/e*Int[x^(m-1)*(a+b*ArcTanh[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] && RationalQ[m] && m>0 + + +Int[x_^m_.*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/e*Int[x^(m-1)*(a+b*ArcCoth[c*x])^n,x] - + d/e*Int[x^(m-1)*(a+b*ArcCoth[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] && RationalQ[m] && m>0 + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(x_*(d_+e_.*x_)),x_Symbol] := + (a+b*ArcTanh[c*x])^n*Log[2*e*x/(d+e*x)]/d - + b*c*n/d*Int[(a+b*ArcTanh[c*x])^(n-1)*Log[2*e*x/(d+e*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(x_*(d_+e_.*x_)),x_Symbol] := + (a+b*ArcCoth[c*x])^n*Log[2*e*x/(d+e*x)]/d - + b*c*n/d*Int[(a+b*ArcCoth[c*x])^(n-1)*Log[2*e*x/(d+e*x)]/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTanh[c*x])^n,x] - + e/d*Int[x^(m+1)*(a+b*ArcTanh[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCoth[c*x])^n,x] - + e/d*Int[x^(m+1)*(a+b*ArcCoth[c*x])^n/(d+e*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d^2-e^2] && PositiveIntegerQ[n] && RationalQ[m] && m<-1 + + +Int[x_^m_.*(d_+e_.*x_)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x)^p*(a+b*ArcTanh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || NeQ[a] || IntegerQ[m]) + + +Int[x_^m_.*(d_+e_.*x_)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x)^p*(a+b*ArcCoth[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || NeQ[a] || IntegerQ[m]) + + +Int[x_^m_.*(d_.+e_.*x_)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + b*(d+e*x^2)^p/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcTanh[c*x])/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcTanh[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + b*(d+e*x^2)^p/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcCoth[c*x])/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCoth[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p>0 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + b*n*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n-1)/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcTanh[c*x])^n,x] - + b^2*d*n*(n-1)/(2*p*(2*p+1))*Int[(d+e*x^2)^(p-1)*(a+b*ArcTanh[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p>0 && n>1 + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + b*n*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n-1)/(2*c*p*(2*p+1)) + + x*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n/(2*p+1) + + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCoth[c*x])^n,x] - + b^2*d*n*(n-1)/(2*p*(2*p+1))*Int[(d+e*x^2)^(p-1)*(a+b*ArcCoth[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p>0 && n>1 + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcTanh[c_.*x_])),x_Symbol] := + Log[RemoveContent[a+b*ArcTanh[c*x],x]]/(b*c*d) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcCoth[c_.*x_])),x_Symbol] := + Log[RemoveContent[a+b*ArcCoth[c*x],x]]/(b*c*d) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NeQ[n+1] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NeQ[n+1] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + -2*(a+b*ArcTanh[c*x])*ArcTan[Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) - + I*b*PolyLog[2,-I*Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) + + I*b*PolyLog[2,I*Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])/Sqrt[d_+e_.*x_^2],x_Symbol] := + -2*(a+b*ArcCoth[c*x])*ArcTan[Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) - + I*b*PolyLog[2,-I*Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) + + I*b*PolyLog[2,I*Sqrt[1-c*x]/Sqrt[1+c*x]]/(c*Sqrt[d]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + 1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n*Sech[x],x],x,ArcTanh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -x*Sqrt[1-1/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n*Csch[x],x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcTanh[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcCoth[c*x])^n/Sqrt[1-c^2*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcTanh[c*x])^n/(2*d*(d+e*x^2)) + + (a+b*ArcTanh[c*x])^(n+1)/(2*b*c*d^2*(n+1)) - + b*c*n/2*Int[x*(a+b*ArcTanh[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcCoth[c*x])^n/(2*d*(d+e*x^2)) + + (a+b*ArcCoth[c*x])^(n+1)/(2*b*c*d^2*(n+1)) - + b*c*n/2*Int[x*(a+b*ArcCoth[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcTanh[c_.*x_])/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcTanh[c*x])/(d*Sqrt[d+e*x^2]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcCoth[c*x])/(d*Sqrt[d+e*x^2]) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p<-1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p<-1 && p!=-3/2 + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b*n*(a+b*ArcTanh[c*x])^(n-1)/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcTanh[c*x])^n/(d*Sqrt[d+e*x^2]) + + b^2*n*(n-1)*Int[(a+b*ArcTanh[c*x])^(n-2)/(d+e*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>1 + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_/(d_+e_.*x_^2)^(3/2),x_Symbol] := + -b*n*(a+b*ArcCoth[c*x])^(n-1)/(c*d*Sqrt[d+e*x^2]) + + x*(a+b*ArcCoth[c*x])^n/(d*Sqrt[d+e*x^2]) + + b^2*n*(n-1)*Int[(a+b*ArcCoth[c*x])^(n-2)/(d+e*x^2)^(3/2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + -b*n*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^(n-1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n,x] + + b^2*n*(n-1)/(4*(p+1)^2)*Int[(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p<-1 && n>1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + -b*n*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^(n-1)/(4*c*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n/(2*d*(p+1)) + + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n,x] + + b^2*n*(n-1)/(4*(p+1)^2)*Int[(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n-2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p<-1 && n>1 && p!=-3/2 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) + + 2*c*(p+1)/(b*(n+1))*Int[x*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p<-1 && n<-1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) + + 2*c*(p+1)/(b*(n+1))*Int[x*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n,p] && p<-1 && n<-1 + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + d^p/c*Subst[Int[(a+b*x)^n/Cosh[x]^(2*(p+1)),x],x,ArcTanh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NegativeIntegerQ[2*(p+1)] && (IntegerQ[p] || PositiveQ[d]) + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + d^(p+1/2)*Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(1-c^2*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NegativeIntegerQ[2*(p+1)] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + -(-d)^p/c*Subst[Int[(a+b*x)^n/Sinh[x]^(2*(p+1)),x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NegativeIntegerQ[2*(p+1)] && IntegerQ[p] + + +Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + -(-d)^(p+1/2)*x*Sqrt[(c^2*x^2-1)/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n/Sinh[x]^(2*(p+1)),x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && NegativeIntegerQ[2*(p+1)] && Not[IntegerQ[p]] + + +Int[ArcTanh[c_.*x_]/(d_.+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+c*x]/(d+e*x^2),x] - 1/2*Int[Log[1-c*x]/(d+e*x^2),x] /; +FreeQ[{c,d,e},x] + + +Int[ArcCoth[c_.*x_]/(d_.+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+1/(c*x)]/(d+e*x^2),x] - 1/2*Int[Log[1-1/(c*x)]/(d+e*x^2),x] /; +FreeQ[{c,d,e},x] + + +Int[(a_+b_.*ArcTanh[c_.*x_])/(d_.+e_.*x_^2),x_Symbol] := + a*Int[1/(d+e*x^2),x] + b*Int[ArcTanh[c*x]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(a_+b_.*ArcCoth[c_.*x_])/(d_.+e_.*x_^2),x_Symbol] := + a*Int[1/(d+e*x^2),x] + b*Int[ArcCoth[c*x]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcTanh[c*x],u,x] - b*c*Int[ExpandIntegrand[u/(1-c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (IntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[a+b*ArcCoth[c*x],u,x] - b*c*Int[ExpandIntegrand[u/(1-c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (IntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] + + +Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[p] && PositiveIntegerQ[n] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcTanh[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcCoth[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTanh[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCoth[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcTanh[c*x])^(n+1)/(b*e*(n+1)) + + 1/(c*d)*Int[(a+b*ArcTanh[c*x])^n/(1-c*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcCoth[c*x])^(n+1)/(b*e*(n+1)) + + 1/(c*d)*Int[(a+b*ArcCoth[c*x])^n/(1-c*x),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] + + +Int[x_*(a_.+b_.*ArcTanh[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + x*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) - + 1/(b*c*d*(n+1))*Int[(a+b*ArcTanh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && Not[PositiveIntegerQ[n]] && NeQ[n+1] + + +Int[x_*(a_.+b_.*ArcCoth[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + -x*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) - + 1/(b*c*d*(n+1))*Int[(a+b*ArcCoth[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && Not[PositiveIntegerQ[n]] && NeQ[n+1] + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcTanh[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/e*Int[x^(m-2)*(a+b*ArcCoth[c*x])^n,x] - + d/e*Int[x^(m-2)*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + (a+b*ArcTanh[c*x])^(n+1)/(b*d*(n+1)) + + 1/d*Int[(a+b*ArcTanh[c*x])^n/(x*(1+c*x)),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(x_*(d_+e_.*x_^2)),x_Symbol] := + (a+b*ArcCoth[c*x])^(n+1)/(b*d*(n+1)) + + 1/d*Int[(a+b*ArcCoth[c*x])^n/(x*(1+c*x)),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcTanh[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/d*Int[x^m*(a+b*ArcCoth[c*x])^n,x] - + e/d*Int[x^(m+2)*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + x^m*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*d*(n+1))*Int[x^(m-1)*(a+b*ArcTanh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_/(d_+e_.*x_^2),x_Symbol] := + x^m*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*d*(n+1))*Int[x^(m-1)*(a+b*ArcCoth[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(a_.+b_.*ArcTanh[c_.*x_])/(d_+e_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcTanh[c*x]),x^m/(d+e*x^2),x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[m] && Not[m==1 && NeQ[a]] + + +Int[x_^m_.*(a_.+b_.*ArcCoth[c_.*x_])/(d_+e_.*x_^2),x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCoth[c*x]),x^m/(d+e*x^2),x],x] /; +FreeQ[{a,b,c,d,e},x] && IntegerQ[m] && Not[m==1 && NeQ[a]] + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n/(2*e*(p+1)) + + b*n/(2*c*(p+1))*Int[(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n/(2*e*(p+1)) + + b*n/(2*c*(p+1))*Int[(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && NeQ[p+1] + + +Int[x_*(a_.+b_.*ArcTanh[c_.*x_])^n_/(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)*(d+e*x^2)) + + (1+c^2*x^2)*(a+b*ArcTanh[c*x])^(n+2)/(b^2*e*(n+1)*(n+2)*(d+e*x^2)) + + 4/(b^2*(n+1)*(n+2))*Int[x*(a+b*ArcTanh[c*x])^(n+2)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && n!=-2 + + +Int[x_*(a_.+b_.*ArcCoth[c_.*x_])^n_/(d_+e_.*x_^2)^2,x_Symbol] := + x*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)*(d+e*x^2)) + + (1+c^2*x^2)*(a+b*ArcCoth[c*x])^(n+2)/(b^2*e*(n+1)*(n+2)*(d+e*x^2)) + + 4/(b^2*(n+1)*(n+2))*Int[x*(a+b*ArcCoth[c*x])^(n+2)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n<-1 && n!=-2 + + +Int[x_^2*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c^3*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])/(2*c^2*d*(p+1)) + + 1/(2*c^2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p<-1 && p!=-5/2 + + +Int[x_^2*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + -b*(d+e*x^2)^(p+1)/(4*c^3*d*(p+1)^2) - + x*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])/(2*c^2*d*(p+1)) + + 1/(2*c^2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[p] && p<-1 && p!=-5/2 + + +Int[x_^2*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + -(a+b*ArcTanh[c*x])^(n+1)/(2*b*c^3*d^2*(n+1)) + + x*(a+b*ArcTanh[c*x])^n/(2*c^2*d*(d+e*x^2)) - + b*n/(2*c)*Int[x*(a+b*ArcTanh[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[x_^2*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2)^2,x_Symbol] := + -(a+b*ArcCoth[c*x])^(n+1)/(2*b*c^3*d^2*(n+1)) + + x*(a+b*ArcCoth[c*x])^n/(2*c^2*d*(d+e*x^2)) - + b*n/(2*c)*Int[x*(a+b*ArcCoth[c*x])^(n-1)/(d+e*x^2)^2,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + -b*x^m*(d+e*x^2)^(p+1)/(c*d*m^2) + + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])/(c^2*d*m) - + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[p] && p<-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + -b*x^m*(d+e*x^2)^(p+1)/(c*d*m^2) + + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])/(c^2*d*m) - + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[p] && p<-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + -b*n*x^m*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^(n-1)/(c*d*m^2) + + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n/(c^2*d*m) + + b^2*n*(n-1)/m^2*Int[x^m*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n-2),x] - + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[n,p] && p<-1 && n>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + -b*n*x^m*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^(n-1)/(c*d*m^2) + + x^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n/(c^2*d*m) + + b^2*n*(n-1)/m^2*Int[x^m*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n-2),x] - + (m-1)/(c^2*d*m)*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[n,p] && p<-1 && n>1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[c^2*d+e] && EqQ[m+2*p+2] && RationalQ[n] && n<-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + x^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[c^2*d+e] && EqQ[m+2*p+3] && RationalQ[n] && n>0 && NeQ[m+1] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + x^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n-1),x] /; +FreeQ[{a,b,c,d,e,m,p},x] && EqQ[c^2*d+e] && EqQ[m+2*p+3] && RationalQ[n] && n>0 && NeQ[m+1] + + +Int[x_^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcTanh[c*x])/(m+2) - + b*c*d/(m+2)*Int[x^(m+1)/Sqrt[d+e*x^2],x] + + d/(m+2)*Int[x^m*(a+b*ArcTanh[c*x])/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && NeQ[m+2] + + +Int[x_^m_*Sqrt[d_+e_.*x_^2]*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCoth[c*x])/(m+2) - + b*c*d/(m+2)*Int[x^(m+1)/Sqrt[d+e*x^2],x] + + d/(m+2)*Int[x^m*(a+b*ArcCoth[c*x])/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && NeQ[m+2] + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && IntegerQ[p] && p>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[x^m*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && IntegerQ[p] && p>1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + d*Int[x^m*(d+e*x^2)^(p-1)*(a+b*ArcTanh[c*x])^n,x] - + c^2*d*Int[x^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && RationalQ[p] && p>0 && PositiveIntegerQ[n] && (RationalQ[m] || EqQ[n,1] && IntegerQ[p]) + + +Int[x_^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + d*Int[x^m*(d+e*x^2)^(p-1)*(a+b*ArcCoth[c*x])^n,x] - + c^2*d*Int[x^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[c^2*d+e] && RationalQ[p] && p>0 && PositiveIntegerQ[n] && (RationalQ[m] || EqQ[n,1] && IntegerQ[p]) + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -x^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcTanh[c*x])^n/(c^2*d*m) + + b*n/(c*m)*Int[x^(m-1)*(a+b*ArcTanh[c*x])^(n-1)/Sqrt[d+e*x^2],x] + + (m-1)/(c^2*m)*Int[x^(m-2)*(a+b*ArcTanh[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + -x^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcCoth[c*x])^n/(c^2*d*m) + + b*n/(c*m)*Int[x^(m-1)*(a+b*ArcCoth[c*x])^(n-1)/Sqrt[d+e*x^2],x] + + (m-1)/(c^2*m)*Int[x^(m-2)*(a+b*ArcCoth[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m>1 + + +Int[(a_.+b_.*ArcTanh[c_.*x_])/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -2/Sqrt[d]*(a+b*ArcTanh[c*x])*ArcTanh[Sqrt[1-c*x]/Sqrt[1+c*x]] + + b/Sqrt[d]*PolyLog[2,-Sqrt[1-c*x]/Sqrt[1+c*x]] - + b/Sqrt[d]*PolyLog[2,Sqrt[1-c*x]/Sqrt[1+c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -2/Sqrt[d]*(a+b*ArcCoth[c*x])*ArcTanh[Sqrt[1-c*x]/Sqrt[1+c*x]] + + b/Sqrt[d]*PolyLog[2,-Sqrt[1-c*x]/Sqrt[1+c*x]] - + b/Sqrt[d]*PolyLog[2,Sqrt[1-c*x]/Sqrt[1+c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + 1/Sqrt[d]*Subst[Int[(a+b*x)^n*Csch[x],x],x,ArcTanh[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_/(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -c*x*Sqrt[1-1/(c^2*x^2)]/Sqrt[d+e*x^2]*Subst[Int[(a+b*x)^n*Sech[x],x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && PositiveQ[d] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcTanh[c*x])^n/(x*Sqrt[1-c^2*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(x_*Sqrt[d_+e_.*x_^2]),x_Symbol] := + Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[(a+b*ArcCoth[c*x])^n/(x*Sqrt[1-c^2*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && PositiveIntegerQ[n] && Not[PositiveQ[d]] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_./(x_^2*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -Sqrt[d+e*x^2]*(a+b*ArcTanh[c*x])^n/(d*x) + + b*c*n*Int[(a+b*ArcTanh[c*x])^(n-1)/(x*Sqrt[d+e*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_./(x_^2*Sqrt[d_+e_.*x_^2]),x_Symbol] := + -Sqrt[d+e*x^2]*(a+b*ArcCoth[c*x])^n/(d*x) + + b*c*n*Int[(a+b*ArcCoth[c*x])^(n-1)/(x*Sqrt[d+e*x^2]),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 + + +Int[x_^m_*(a_.+b_.*ArcTanh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcTanh[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcTanh[c*x])^(n-1)/Sqrt[d+e*x^2],x] + + c^2*(m+2)/(m+1)*Int[x^(m+2)*(a+b*ArcTanh[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && m!=-2 + + +Int[x_^m_*(a_.+b_.*ArcCoth[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := + x^(m+1)*Sqrt[d+e*x^2]*(a+b*ArcCoth[c*x])^n/(d*(m+1)) - + b*c*n/(m+1)*Int[x^(m+1)*(a+b*ArcCoth[c*x])^(n-1)/Sqrt[d+e*x^2],x] + + c^2*(m+2)/(m+1)*Int[x^(m+2)*(a+b*ArcCoth[c*x])^n/Sqrt[d+e*x^2],x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n] && n>0 && m<-1 && m!=-2 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + 1/e*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n,x] - + d/e*Int[x^(m-2)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegersQ[m,n,2*p] && p<-1 && m>1 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + 1/e*Int[x^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n,x] - + d/e*Int[x^(m-2)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegersQ[m,n,2*p] && p<-1 && m>1 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + 1/d*Int[x^m*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^n,x] - + e/d*Int[x^(m+2)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegersQ[m,n,2*p] && p<-1 && m<0 && n!=-1 + + +Int[x_^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + 1/d*Int[x^m*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^n,x] - + e/d*Int[x^(m+2)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegersQ[m,n,2*p] && p<-1 && m<0 && n!=-1 + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n+1),x] + + c*(m+2*p+2)/(b*(n+1))*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && p<-1 && n<-1 && NeQ[m+2*p+2] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + x^m*(d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])^(n+1)/(b*c*d*(n+1)) - + m/(b*c*(n+1))*Int[x^(m-1)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n+1),x] + + c*(m+2*p+2)/(b*(n+1))*Int[x^(m+1)*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^(n+1),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[m,n,p] && p<-1 && n<-1 && NeQ[m+2*p+2] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + d^p/c^(m+1)*Subst[Int[(a+b*x)^n*Sinh[x]^m/Cosh[x]^(m+2*(p+1)),x],x,ArcTanh[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && (IntegerQ[p] || PositiveQ[d]) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + d^(p+1/2)*Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]*Int[x^m*(1-c^2*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && Not[IntegerQ[p] || PositiveQ[d]] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + -(-d)^p/c^(m+1)*Subst[Int[(a+b*x)^n*Cosh[x]^m/Sinh[x]^(m+2*(p+1)),x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && IntegerQ[p] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + -(-d)^(p+1/2)*x*Sqrt[(c^2*x^2-1)/(c^2*x^2)]/(c^m*Sqrt[d+e*x^2])*Subst[Int[(a+b*x)^n*Cosh[x]^m/Sinh[x]^(m+2*(p+1)),x],x,ArcCoth[c*x]] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && PositiveIntegerQ[m] && NegativeIntegerQ[m+2*p+1] && Not[IntegerQ[p]] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcTanh[c*x])/(2*e*(p+1)) - + b*c/(2*e*(p+1))*Int[(d+e*x^2)^(p+1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCoth[c*x])/(2*e*(p+1)) - + b*c/(2*e*(p+1))*Int[(d+e*x^2)^(p+1)/(1-c^2*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcTanh[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(1-c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[a+b*ArcCoth[c*x],u,x] - b*c*Int[SimplifyIntegrand[u/(1-c^2*x^2),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcTanh[c*x])^n,x^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || p<-1 && IntegerQ[m] && m!=1) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Int[ExpandIntegrand[(a+b*ArcCoth[c*x])^n,x^m*(d+e*x^2)^p,x],x] /; +FreeQ[{a,b,c,d,e,m},x] && IntegerQ[p] && PositiveIntegerQ[n] && (p>0 || p<-1 && IntegerQ[m] && m!=1) + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_+b_.*ArcTanh[c_.*x_]),x_Symbol] := + a*Int[x^m*(d+e*x^2)^p,x] + b*Int[x^m*(d+e*x^2)^p*ArcTanh[c*x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] + + +Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_+b_.*ArcCoth[c_.*x_]),x_Symbol] := + a*Int[x^m*(d+e*x^2)^p,x] + b*Int[x^m*(d+e*x^2)^p*ArcCoth[c*x],x] /; +FreeQ[{a,b,c,d,e,m,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcTanh[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcTanh[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCoth[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcCoth[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[ArcTanh[u_]*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+u]*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[1-u]*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1+c*x))^2] + + +Int[ArcCoth[u_]*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[SimplifyIntegrand[1+1/u,x]]*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[SimplifyIntegrand[1-1/u,x]]*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1+c*x))^2] + + +Int[ArcTanh[u_]*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[1+u]*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[1-u]*(a+b*ArcTanh[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1-c*x))^2] + + +Int[ArcCoth[u_]*(a_.+b_.*ArcCoth[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + 1/2*Int[Log[SimplifyIntegrand[1+1/u,x]]*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] - + 1/2*Int[Log[SimplifyIntegrand[1-1/u,x]]*(a+b*ArcCoth[c*x])^n/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1-c*x))^2] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcTanh[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) - + b*n/2*Int[(a+b*ArcTanh[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2/(1+c*x))^2] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcCoth[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) - + b*n/2*Int[(a+b*ArcCoth[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2/(1+c*x))^2] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcTanh[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) + + b*n/2*Int[(a+b*ArcTanh[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2/(1-c*x))^2] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_.*Log[u_]/(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcCoth[c*x])^n*PolyLog[2,Together[1-u]]/(2*c*d) + + b*n/2*Int[(a+b*ArcCoth[c*x])^(n-1)*PolyLog[2,Together[1-u]]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[(1-u)^2-(1-2/(1-c*x))^2] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcTanh[c*x])^n*PolyLog[p+1,u]/(2*c*d) + + b*n/2*Int[(a+b*ArcTanh[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1+c*x))^2] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + -(a+b*ArcCoth[c*x])^n*PolyLog[p+1,u]/(2*c*d) + + b*n/2*Int[(a+b*ArcCoth[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1+c*x))^2] + + +Int[(a_.+b_.*ArcTanh[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcTanh[c*x])^n*PolyLog[p+1,u]/(2*c*d) - + b*n/2*Int[(a+b*ArcTanh[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1-c*x))^2] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^n_.*PolyLog[p_,u_]/(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcCoth[c*x])^n*PolyLog[p+1,u]/(2*c*d) - + b*n/2*Int[(a+b*ArcCoth[c*x])^(n-1)*PolyLog[p+1,u]/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e] && RationalQ[n] && n>0 && EqQ[u^2-(1-2/(1-c*x))^2] + + +Int[1/((d_+e_.*x_^2)*(a_.+b_.*ArcCoth[c_.*x_])*(a_.+b_.*ArcTanh[c_.*x_])),x_Symbol] := + (-Log[a+b*ArcCoth[c*x]]+Log[a+b*ArcTanh[c*x]])/(b^2*c*d*(ArcCoth[c*x]-ArcTanh[c*x])) /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] + + +Int[(a_.+b_.*ArcCoth[c_.*x_])^m_.*(a_.+b_.*ArcTanh[c_.*x_])^n_./(d_+e_.*x_^2),x_Symbol] := + (a+b*ArcCoth[c*x])^(m+1)*(a+b*ArcTanh[c*x])^n/(b*c*d*(m+1)) - + n/(m+1)*Int[(a+b*ArcCoth[c*x])^(m+1)*(a+b*ArcTanh[c*x])^(n-1)/(d+e*x^2),x] /; +FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e] && IntegersQ[m,n] && 00 + + +Int[x_^m_.*ArcCoth[a_.+b_.*f_^(c_.+d_.*x_)],x_Symbol] := + 1/2*Int[x^m*Log[1+1/(a+b*f^(c+d*x))],x] - + 1/2*Int[x^m*Log[1-1/(a+b*f^(c+d*x))],x] /; +FreeQ[{a,b,c,d,f},x] && IntegerQ[m] && m>0 + + +Int[u_.*ArcTanh[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCoth[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCoth[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcTanh[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[1/(Sqrt[a_.+b_.*x_^2]*ArcTanh[c_.*x_/Sqrt[a_.+b_.*x_^2]]),x_Symbol] := + 1/c*Log[ArcTanh[c*x/Sqrt[a+b*x^2]]] /; +FreeQ[{a,b,c},x] && EqQ[b-c^2] + + +Int[1/(Sqrt[a_.+b_.*x_^2]*ArcCoth[c_.*x_/Sqrt[a_.+b_.*x_^2]]),x_Symbol] := + -1/c*Log[ArcCoth[c*x/Sqrt[a+b*x^2]]] /; +FreeQ[{a,b,c},x] && EqQ[b-c^2] + + +Int[ArcTanh[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[a_.+b_.*x_^2],x_Symbol] := + ArcTanh[c*x/Sqrt[a+b*x^2]]^(m+1)/(c*(m+1)) /; +FreeQ[{a,b,c,m},x] && EqQ[b-c^2] && NeQ[m+1] + + +Int[ArcCoth[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[a_.+b_.*x_^2],x_Symbol] := + -ArcCoth[c*x/Sqrt[a+b*x^2]]^(m+1)/(c*(m+1)) /; +FreeQ[{a,b,c,m},x] && EqQ[b-c^2] && NeQ[m+1] + + +Int[ArcTanh[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[d_.+e_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2]/Sqrt[d+e*x^2]*Int[ArcTanh[c*x/Sqrt[a+b*x^2]]^m/Sqrt[a+b*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b-c^2] && EqQ[b*d-a*e] + + +Int[ArcCoth[c_.*x_/Sqrt[a_.+b_.*x_^2]]^m_./Sqrt[d_.+e_.*x_^2],x_Symbol] := + Sqrt[a+b*x^2]/Sqrt[d+e*x^2]*Int[ArcCoth[c*x/Sqrt[a+b*x^2]]^m/Sqrt[a+b*x^2],x] /; +FreeQ[{a,b,c,d,e,m},x] && EqQ[b-c^2] && EqQ[b*d-a*e] + + +Int[(c_.+d_.*x_^2)^n_*ArcTanh[a_.*x_],x_Symbol] := + With[{u=IntHide[(c+d*x^2)^n,x]}, + Dist[ArcTanh[a*x],u,x] - + a*Int[Dist[1/(1-a^2*x^2),u,x],x]] /; +FreeQ[{a,c,d},x] && IntegerQ[2*n] && n<=-1 + + +Int[(c_.+d_.*x_^2)^n_*ArcCoth[a_.*x_],x_Symbol] := + With[{u=IntHide[(c+d*x^2)^n,x]}, + Dist[ArcCoth[a*x],u,x] - + a*Int[Dist[1/(1-a^2*x^2),u,x],x]] /; +FreeQ[{a,c,d},x] && IntegerQ[2*n] && n<=-1 + + +If[ShowSteps, + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + ShowStep["","Int[f[x,ArcTanh[a+b*x]]/(1-(a+b*x)^2),x]", + "Subst[Int[f[-a/b+Tanh[x]/b,x],x],x,ArcTanh[a+b*x]]/b",Hold[ + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Sech[x]^(2*(n+1)),x],x], x, tmp]]] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcTanh && EqQ[Discriminant[v,x]*tmp[[1]]^2-D[v,x]^2]] /; +SimplifyFlag && QuadraticQ[v,x] && IntegerQ[n] && n<0 && PosQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]], + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*Sech[x]^(2*(n+1)),x],x], x, tmp] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcTanh && EqQ[Discriminant[v,x]*tmp[[1]]^2-D[v,x]^2]] /; +QuadraticQ[v,x] && IntegerQ[n] && n<0 && PosQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]]] + + +If[ShowSteps, + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + ShowStep["","Int[f[x,ArcCoth[a+b*x]]/(1-(a+b*x)^2),x]", + "Subst[Int[f[-a/b+Coth[x]/b,x],x],x,ArcCoth[a+b*x]]/b",Hold[ + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*(-Csch[x]^2)^(n+1),x],x], x, tmp]]] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcCoth && EqQ[Discriminant[v,x]*tmp[[1]]^2-D[v,x]^2]] /; +SimplifyFlag && QuadraticQ[v,x] && IntegerQ[n] && n<0 && PosQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]], + +Int[u_*v_^n_.,x_Symbol] := + With[{tmp=InverseFunctionOfLinear[u,x]}, + (-Discriminant[v,x]/(4*Coefficient[v,x,2]))^n/Coefficient[tmp[[1]],x,1]* + Subst[Int[SimplifyIntegrand[SubstForInverseFunction[u,tmp,x]*(-Csch[x]^2)^(n+1),x],x], x, tmp] /; + Not[FalseQ[tmp]] && Head[tmp]===ArcCoth && EqQ[Discriminant[v,x]*tmp[[1]]^2-D[v,x]^2]] /; +QuadraticQ[v,x] && IntegerQ[n] && n<0 && PosQ[Discriminant[v,x]] && MatchQ[u,r_.*f_^w_ /; FreeQ[f,x]]] + + +Int[ArcTanh[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Tanh[a+b*x]] + + b*Int[x/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2-1] + + +Int[ArcCoth[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Tanh[a+b*x]] + + b*Int[x/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2-1] + + +Int[ArcTanh[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Coth[a+b*x]] + + b*Int[x/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2-1] + + +Int[ArcCoth[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Coth[a+b*x]] + + b*Int[x/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-d)^2-1] + + +Int[ArcTanh[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Tanh[a+b*x]] + + b*(1-c-d)*Int[x*E^(2*a+2*b*x)/(1-c+d+(1-c-d)*E^(2*a+2*b*x)),x] - + b*(1+c+d)*Int[x*E^(2*a+2*b*x)/(1+c-d+(1+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2-1] + + +Int[ArcCoth[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Tanh[a+b*x]] + + b*(1-c-d)*Int[x*E^(2*a+2*b*x)/(1-c+d+(1-c-d)*E^(2*a+2*b*x)),x] - + b*(1+c+d)*Int[x*E^(2*a+2*b*x)/(1+c-d+(1+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2-1] + + +Int[ArcTanh[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Coth[a+b*x]] + + b*(1+c+d)*Int[x*E^(2*a+2*b*x)/(1+c-d-(1+c+d)*E^(2*a+2*b*x)),x] - + b*(1-c-d)*Int[x*E^(2*a+2*b*x)/(1-c+d-(1-c-d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2-1] + + +Int[ArcCoth[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Coth[a+b*x]] + + b*(1+c+d)*Int[x*E^(2*a+2*b*x)/(1+c-d-(1+c+d)*E^(2*a+2*b*x)),x] - + b*(1-c-d)*Int[x*E^(2*a+2*b*x)/(1-c+d-(1-c-d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Tanh[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Tanh[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d+c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Coth[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Coth[a+b*x]]/(f*(m+1)) + + b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-d-c*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Tanh[a+b*x]]/(f*(m+1)) + + b*(1-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1-c+d+(1-c-d)*E^(2*a+2*b*x)),x] - + b*(1+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1+c-d+(1+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Tanh[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Tanh[a+b*x]]/(f*(m+1)) + + b*(1-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1-c+d+(1-c-d)*E^(2*a+2*b*x)),x] - + b*(1+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1+c-d+(1+c+d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Coth[a+b*x]]/(f*(m+1)) + + b*(1+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1+c-d-(1+c+d)*E^(2*a+2*b*x)),x] - + b*(1-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1-c+d-(1-c-d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Coth[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Coth[a+b*x]]/(f*(m+1)) + + b*(1+c+d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1+c-d-(1+c+d)*E^(2*a+2*b*x)),x] - + b*(1-c-d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*a+2*b*x)/(1-c+d-(1-c-d)*E^(2*a+2*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-d)^2-1] + + +Int[ArcTanh[Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[Tan[a+b*x]] - b*Int[x*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcCoth[Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[Tan[a+b*x]] - b*Int[x*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcTanh[Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[Cot[a+b*x]] - b*Int[x*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[ArcCoth[Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[Cot[a+b*x]] - b*Int[x*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b},x] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[Tan[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[Tan[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[Cot[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[Cot[a+b*x]]/(f*(m+1)) - b/(f*(m+1))*Int[(e+f*x)^(m+1)*Sec[2*a+2*b*x],x] /; +FreeQ[{a,b,e,f},x] && PositiveIntegerQ[m] + + +Int[ArcTanh[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Tan[a+b*x]] + + I*b*Int[x/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c+I*d)^2-1] + + +Int[ArcCoth[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Tan[a+b*x]] + + I*b*Int[x/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c+I*d)^2-1] + + +Int[ArcTanh[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Cot[a+b*x]] + + I*b*Int[x/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-I*d)^2-1] + + +Int[ArcCoth[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Cot[a+b*x]] + + I*b*Int[x/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && EqQ[(c-I*d)^2-1] + + +Int[ArcTanh[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Tan[a+b*x]] + + I*b*(1-c+I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1-c-I*d+(1-c+I*d)*E^(2*I*a+2*I*b*x)),x] - + I*b*(1+c-I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1+c+I*d+(1+c-I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c+I*d)^2-1] + + +Int[ArcCoth[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Tan[a+b*x]] + + I*b*(1-c+I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1-c-I*d+(1-c+I*d)*E^(2*I*a+2*I*b*x)),x] - + I*b*(1+c-I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1+c+I*d+(1+c-I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c+I*d)^2-1] + + +Int[ArcTanh[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcTanh[c+d*Cot[a+b*x]] - + I*b*(1-c-I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1-c+I*d-(1-c-I*d)*E^(2*I*a+2*I*b*x)),x] + + I*b*(1+c+I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1+c-I*d-(1+c+I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-I*d)^2-1] + + +Int[ArcCoth[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + x*ArcCoth[c+d*Cot[a+b*x]] - + I*b*(1-c-I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1-c+I*d-(1-c-I*d)*E^(2*I*a+2*I*b*x)),x] + + I*b*(1+c+I*d)*Int[x*E^(2*I*a+2*I*b*x)/(1+c-I*d-(1+c+I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d},x] && NeQ[(c-I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Tan[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c+I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Tan[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c+I*d+c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c+I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Cot[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Cot[a+b*x]]/(f*(m+1)) + + I*b/(f*(m+1))*Int[(e+f*x)^(m+1)/(c-I*d-c*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && EqQ[(c-I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Tan[a+b*x]]/(f*(m+1)) + + I*b*(1-c+I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-c-I*d+(1-c+I*d)*E^(2*I*a+2*I*b*x)),x] - + I*b*(1+c-I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+c+I*d+(1+c-I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c+I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Tan[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Tan[a+b*x]]/(f*(m+1)) + + I*b*(1-c+I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-c-I*d+(1-c+I*d)*E^(2*I*a+2*I*b*x)),x] - + I*b*(1+c-I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+c+I*d+(1+c-I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c+I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcTanh[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcTanh[c+d*Cot[a+b*x]]/(f*(m+1)) - + I*b*(1-c-I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-c+I*d-(1-c-I*d)*E^(2*I*a+2*I*b*x)),x] + + I*b*(1+c+I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+c-I*d-(1+c+I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-I*d)^2-1] + + +Int[(e_.+f_.*x_)^m_.*ArcCoth[c_.+d_.*Cot[a_.+b_.*x_]],x_Symbol] := + (e+f*x)^(m+1)*ArcCoth[c+d*Cot[a+b*x]]/(f*(m+1)) - + I*b*(1-c-I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1-c+I*d-(1-c-I*d)*E^(2*I*a+2*I*b*x)),x] + + I*b*(1+c+I*d)/(f*(m+1))*Int[(e+f*x)^(m+1)*E^(2*I*a+2*I*b*x)/(1+c-I*d-(1+c+I*d)*E^(2*I*a+2*I*b*x)),x] /; +FreeQ[{a,b,c,d,e,f},x] && PositiveIntegerQ[m] && NeQ[(c-I*d)^2-1] + + +Int[ArcTanh[u_],x_Symbol] := + x*ArcTanh[u] - + Int[SimplifyIntegrand[x*D[u,x]/(1-u^2),x],x] /; +InverseFunctionFreeQ[u,x] + + +Int[ArcCoth[u_],x_Symbol] := + x*ArcCoth[u] - + Int[SimplifyIntegrand[x*D[u,x]/(1-u^2),x],x] /; +InverseFunctionFreeQ[u,x] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcTanh[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcTanh[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(1-u^2),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCoth[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCoth[u])/(d*(m+1)) - + b/(d*(m+1))*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(1-u^2),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1,x]] + + +Int[v_*(a_.+b_.*ArcTanh[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcTanh[u]),w,x] - b*Int[SimplifyIntegrand[w*D[u,x]/(1-u^2),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] && FalseQ[FunctionOfLinear[v*(a+b*ArcTanh[u]),x]] + + +Int[v_*(a_.+b_.*ArcCoth[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCoth[u]),w,x] - b*Int[SimplifyIntegrand[w*D[u,x]/(1-u^2),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] && FalseQ[FunctionOfLinear[v*(a+b*ArcCoth[u]),x]] + + + + + +(* ::Subsection::Closed:: *) +(*7.3.1 Miscellaneous inverse hyperbolic secant*) + + +Int[ArcSech[c_.*x_],x_Symbol] := + x*ArcSech[c*x] + Sqrt[1+c*x]*Sqrt[1/(1+c*x)]*Int[1/(Sqrt[1-c*x]*Sqrt[1+c*x]),x] /; +FreeQ[c,x] + + +Int[ArcCsch[c_.*x_],x_Symbol] := + x*ArcCsch[c*x] + 1/c*Int[1/(x*Sqrt[1+1/(c^2*x^2)]),x] /; +FreeQ[c,x] + + +Int[(a_.+b_.*ArcSech[c_.*x_])^n_,x_Symbol] := + -1/c*Subst[Int[(a+b*x)^n*Sech[x]*Tanh[x],x],x,ArcSech[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcCsch[c_.*x_])^n_,x_Symbol] := + -1/c*Subst[Int[(a+b*x)^n*Csch[x]*Coth[x],x],x,ArcCsch[c*x]] /; +FreeQ[{a,b,c,n},x] + + +Int[(a_.+b_.*ArcSech[c_.*x_])/x_,x_Symbol] := + -Subst[Int[(a+b*ArcCosh[x/c])/x,x],x,1/x] /; +FreeQ[{a,b,c},x] + + +Int[(a_.+b_.*ArcCsch[c_.*x_])/x_,x_Symbol] := + -Subst[Int[(a+b*ArcSinh[x/c])/x,x],x,1/x] /; +FreeQ[{a,b,c},x] + + +Int[x_^m_.*(a_.+b_.*ArcSech[c_.*x_]),x_Symbol] := + x^(m+1)*(a+b*ArcSech[c*x])/(m+1) + + b*Sqrt[1+c*x]/(m+1)*Sqrt[1/(1+c*x)]*Int[x^m/(Sqrt[1-c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c,m},x] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcCsch[c_.*x_]),x_Symbol] := + x^(m+1)*(a+b*ArcCsch[c*x])/(m+1) + + b/(c*(m+1))*Int[x^(m-1)/Sqrt[1+1/(c^2*x^2)],x] /; +FreeQ[{a,b,c,m},x] && NeQ[m+1] + + +Int[x_^m_.*(a_.+b_.*ArcSech[c_.*x_])^n_,x_Symbol] := + -1/c^(m+1)*Subst[Int[(a+b*x)^n*Sech[x]^(m+1)*Tanh[x],x],x,ArcSech[c*x]] /; +FreeQ[{a,b,c,n},x] && IntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcCsch[c_.*x_])^n_,x_Symbol] := + -1/c^(m+1)*Subst[Int[(a+b*x)^n*Csch[x]^(m+1)*Coth[x],x],x,ArcCsch[c*x]] /; +FreeQ[{a,b,c,n},x] && IntegerQ[m] + + +Int[x_^m_.*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(a+b*ArcSech[c*x])^n,x] /; +FreeQ[{a,b,c,m,n},x] + + +Int[x_^m_.*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(a+b*ArcCsch[c*x])^n,x] /; +FreeQ[{a,b,c,m,n},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[(a+b*ArcSech[c*x]),u,x] + b*Sqrt[1+c*x]*Sqrt[1/(1+c*x)]*Int[SimplifyIntegrand[u/(x*Sqrt[1-c*x]*Sqrt[1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_]),x_Symbol] := + With[{u=IntHide[(d+e*x^2)^p,x]}, + Dist[(a+b*ArcCsch[c*x]),u,x] - b*c*x/Sqrt[-c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[-1-c^2*x^2]),x],x]] /; +FreeQ[{a,b,c,d,e},x] && (PositiveIntegerQ[p] || NegativeIntegerQ[p+1/2]) + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegerQ[p] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegerQ[p] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcSech[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + Defer[Int][(d+e*x^2)^p*(a+b*ArcCsch[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,n,p},x] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcSech[c*x])/(2*e*(p+1)) + + b*Sqrt[1+c*x]/(2*e*(p+1))*Sqrt[1/(1+c*x)]*Int[(d+e*x^2)^(p+1)/(x*Sqrt[1-c*x]*Sqrt[1+c*x]),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_]),x_Symbol] := + (d+e*x^2)^(p+1)*(a+b*ArcCsch[c*x])/(2*e*(p+1)) - + b*c*x/(2*e*(p+1)*Sqrt[-c^2*x^2])*Int[(d+e*x^2)^(p+1)/(x*Sqrt[-1-c^2*x^2]),x] /; +FreeQ[{a,b,c,d,e,p},x] && NeQ[p+1] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[(a+b*ArcSech[c*x]),u,x] + b*Sqrt[1+c*x]*Sqrt[1/(1+c*x)]*Int[SimplifyIntegrand[u/(x*Sqrt[1-c*x]*Sqrt[1+c*x]),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_]),x_Symbol] := + With[{u=IntHide[x^m*(d+e*x^2)^p,x]}, + Dist[(a+b*ArcCsch[c*x]),u,x] - b*c*x/Sqrt[-c^2*x^2]*Int[SimplifyIntegrand[u/(x*Sqrt[-1-c^2*x^2]),x],x]] /; +FreeQ[{a,b,c,d,e,m,p},x] && ( + PositiveIntegerQ[p] && Not[NegativeIntegerQ[(m-1)/2] && m+2*p+3>0] || + PositiveIntegerQ[(m+1)/2] && Not[NegativeIntegerQ[p] && m+2*p+3>0] || + NegativeIntegerQ[(m+2*p+1)/2] && Not[NegativeIntegerQ[(m-1)/2]] ) + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegersQ[m,p] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && IntegersQ[m,p] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Sqrt[x^2]/x*Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[m] && IntegerQ[p+1/2] && PositiveQ[e] && Negative[d] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcCosh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e] && IntegerQ[m] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + -Sqrt[d+e*x^2]/(x*Sqrt[e+d/x^2])*Subst[Int[(e+d*x^2)^p*(a+b*ArcSinh[x/c])^n/x^(m+2*(p+1)),x],x,1/x] /; +FreeQ[{a,b,c,d,e,n},x] && EqQ[e-c^2*d] && IntegerQ[m] && IntegerQ[p+1/2] && Not[PositiveQ[e] && Negative[d]] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcSech[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcSech[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[x_^m_.*(d_.+e_.*x_^2)^p_.*(a_.+b_.*ArcCsch[c_.*x_])^n_.,x_Symbol] := + Defer[Int][x^m*(d+e*x^2)^p*(a+b*ArcCsch[c*x])^n,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] + + +Int[ArcSech[a_+b_.*x_],x_Symbol] := + (a+b*x)*ArcSech[a+b*x]/b + + Int[Sqrt[(1-a-b*x)/(1+a+b*x)]/(1-a-b*x),x] /; +FreeQ[{a,b},x] + + +Int[ArcCsch[a_+b_.*x_],x_Symbol] := + (a+b*x)*ArcCsch[a+b*x]/b + + Int[1/((a+b*x)*Sqrt[1+1/(a+b*x)^2]),x] /; +FreeQ[{a,b},x] + + +Int[ArcSech[a_+b_.*x_]^n_,x_Symbol] := + -1/b*Subst[Int[x^n*Sech[x]*Tanh[x],x],x,ArcSech[a+b*x]] /; +FreeQ[{a,b,n},x] + + +Int[ArcCsch[a_+b_.*x_]^n_,x_Symbol] := + -1/b*Subst[Int[x^n*Csch[x]*Coth[x],x],x,ArcCsch[a+b*x]] /; +FreeQ[{a,b,n},x] + + +Int[ArcSech[a_+b_.*x_]/x_,x_Symbol] := + -ArcSech[a+b*x]*Log[1+E^(-2*ArcSech[a+b*x])] + + ArcSech[a+b*x]*Log[1-(1-Sqrt[1-a^2])/(E^ArcSech[a+b*x]*a)] + + ArcSech[a+b*x]*Log[1-(1+Sqrt[1-a^2])/(E^ArcSech[a+b*x]*a)] + + 1/2*PolyLog[2,-E^(-2*ArcSech[a+b*x])] - + PolyLog[2,(1-Sqrt[1-a^2])/(E^ArcSech[a+b*x]*a)] - + PolyLog[2,(1+Sqrt[1-a^2])/(E^ArcSech[a+b*x]*a)] /; +FreeQ[{a,b},x] + + +Int[ArcCsch[a_+b_.*x_]/x_,x_Symbol] := + -ArcCsch[a+b*x]^2 - + ArcCsch[a+b*x]*Log[1-E^(-2*ArcCsch[a+b*x])] + + ArcCsch[a+b*x]*Log[1+(1-Sqrt[1+a^2])*E^ArcCsch[a+b*x]/a] + + ArcCsch[a+b*x]*Log[1+(1+Sqrt[1+a^2])*E^ArcCsch[a+b*x]/a] + + 1/2*PolyLog[2,E^(-2*ArcCsch[a+b*x])] + + PolyLog[2,-(1-Sqrt[1+a^2])*E^ArcCsch[a+b*x]/a] + + PolyLog[2,-(1+Sqrt[1+a^2])*E^ArcCsch[a+b*x]/a] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*ArcSech[a_+b_.*x_],x_Symbol] := + -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcSech[a+b*x]/(b^(m+1)*(m+1)) + + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)*Sqrt[-1+x]*Sqrt[1+x]),x],x,1/(a+b*x)] /; +FreeQ[{a,b,m},x] && IntegerQ[m] && NeQ[m+1] + + +Int[x_^m_.*ArcCsch[a_+b_.*x_],x_Symbol] := + -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcCsch[a+b*x]/(b^(m+1)*(m+1)) + + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)*Sqrt[1+x^2]),x],x,1/(a+b*x)] /; +FreeQ[{a,b,m},x] && IntegerQ[m] && NeQ[m+1] + + +Int[x_^m_.*ArcSech[a_+b_.*x_]^n_,x_Symbol] := + -1/b^(m+1)*Subst[Int[x^n*(-a+Sech[x])^m*Sech[x]*Tanh[x],x],x,ArcSech[a+b*x]] /; +FreeQ[{a,b,n},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*ArcCsch[a_+b_.*x_]^n_,x_Symbol] := + -1/b^(m+1)*Subst[Int[x^n*(-a+Csch[x])^m*Csch[x]*Coth[x],x],x,ArcCsch[a+b*x]] /; +FreeQ[{a,b,n},x] && PositiveIntegerQ[m] + + +Int[u_.*ArcSech[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcCosh[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[u_.*ArcCsch[c_./(a_.+b_.*x_^n_.)]^m_.,x_Symbol] := + Int[u*ArcSinh[a/c+b*x^n/c]^m,x] /; +FreeQ[{a,b,c,n,m},x] + + +Int[E^ArcSech[a_.*x_], x_Symbol] := + x*E^ArcSech[a*x] + Log[x]/a + 1/a*Int[1/(x*(1-a*x))*Sqrt[(1-a*x)/(1+a*x)],x] /; +FreeQ[a,x] + + +Int[E^ArcSech[a_.*x_^p_], x_Symbol] := + x*E^ArcSech[a*x^p] + + p/a*Int[1/x^p,x] + + p*Sqrt[1+a*x^p]/a*Sqrt[1/(1+a*x^p)]*Int[1/(x^p*Sqrt[1+a*x^p]*Sqrt[1-a*x^p]),x] /; +FreeQ[{a,p},x] + + +Int[E^ArcCsch[a_.*x_^p_.], x_Symbol] := + 1/a*Int[1/x^p,x] + Int[Sqrt[1+1/(a^2*x^(2*p))],x] /; +FreeQ[{a,p},x] + + +Int[E^(n_.*ArcSech[u_]), x_Symbol] := + Int[(1/u + Sqrt[(1-u)/(1+u)] + 1/u*Sqrt[(1-u)/(1+u)])^n,x] /; +IntegerQ[n] + + +Int[E^(n_.*ArcCsch[u_]), x_Symbol] := + Int[(1/u + Sqrt[1+1/u^2])^n,x] /; +IntegerQ[n] + + +Int[E^ArcSech[a_.*x_^p_.]/x_, x_Symbol] := + -1/(a*p*x^p) + + Sqrt[1+a*x^p]/a*Sqrt[1/(1+a*x^p)]*Int[Sqrt[1+a*x^p]*Sqrt[1-a*x^p]/x^(p+1),x] /; +FreeQ[{a,p},x] + + +Int[x_^m_.*E^ArcSech[a_.*x_^p_.], x_Symbol] := + x^(m+1)*E^ArcSech[a*x^p]/(m+1) + + p/(a*(m+1))*Int[x^(m-p),x] + + p*Sqrt[1+a*x^p]/(a*(m+1))*Sqrt[1/(1+a*x^p)]*Int[x^(m-p)/(Sqrt[1+a*x^p]*Sqrt[1-a*x^p]),x] /; +FreeQ[{a,m,p},x] && NeQ[m+1] + + +Int[x_^m_.*E^ArcCsch[a_.*x_^p_.], x_Symbol] := + 1/a*Int[x^(m-p),x] + Int[x^m*Sqrt[1+1/(a^2*x^(2*p))],x] /; +FreeQ[{a,m,p},x] + + +Int[x_^m_.*E^(n_.*ArcSech[u_]), x_Symbol] := + Int[x^m*(1/u + Sqrt[(1-u)/(1+u)] + 1/u*Sqrt[(1-u)/(1+u)])^n,x] /; +FreeQ[m,x] && IntegerQ[n] + + +Int[x_^m_.*E^(n_.*ArcCsch[u_]), x_Symbol] := + Int[x^m*(1/u + Sqrt[1+1/u^2])^n,x] /; +FreeQ[m,x] && IntegerQ[n] + + +Int[ArcSech[u_],x_Symbol] := + x*ArcSech[u] + + Sqrt[1-u^2]/(u*Sqrt[-1+1/u]*Sqrt[1+1/u])*Int[SimplifyIntegrand[x*D[u,x]/(u*Sqrt[1-u^2]),x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[ArcCsch[u_],x_Symbol] := + x*ArcCsch[u] - + u/Sqrt[-u^2]*Int[SimplifyIntegrand[x*D[u,x]/(u*Sqrt[-1-u^2]),x],x] /; +InverseFunctionFreeQ[u,x] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcSech[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcSech[u])/(d*(m+1)) + + b*Sqrt[1-u^2]/(d*(m+1)*u*Sqrt[-1+1/u]*Sqrt[1+1/u])*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(u*Sqrt[1-u^2]),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[(c_.+d_.*x_)^m_.*(a_.+b_.*ArcCsch[u_]),x_Symbol] := + (c+d*x)^(m+1)*(a+b*ArcCsch[u])/(d*(m+1)) - + b*u/(d*(m+1)*Sqrt[-u^2])*Int[SimplifyIntegrand[(c+d*x)^(m+1)*D[u,x]/(u*Sqrt[-1-u^2]),x],x] /; +FreeQ[{a,b,c,d,m},x] && NeQ[m+1] && InverseFunctionFreeQ[u,x] && Not[FunctionOfQ[(c+d*x)^(m+1),u,x]] && Not[FunctionOfExponentialQ[u,x]] + + +Int[v_*(a_.+b_.*ArcSech[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcSech[u]),w,x] + b*Sqrt[1-u^2]/(u*Sqrt[-1+1/u]*Sqrt[1+1/u])*Int[SimplifyIntegrand[w*D[u,x]/(u*Sqrt[1-u^2]),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + +Int[v_*(a_.+b_.*ArcCsch[u_]),x_Symbol] := + With[{w=IntHide[v,x]}, + Dist[(a+b*ArcCsch[u]),w,x] - b*u/Sqrt[-u^2]*Int[SimplifyIntegrand[w*D[u,x]/(u*Sqrt[-1-u^2]),x],x] /; + InverseFunctionFreeQ[w,x]] /; +FreeQ[{a,b},x] && InverseFunctionFreeQ[u,x] && Not[MatchQ[v, (c_.+d_.*x)^m_. /; FreeQ[{c,d,m},x]]] + + + +`) + +func resourcesRubi7InverseHyperbolicFunctionsMBytes() ([]byte, error) { + return _resourcesRubi7InverseHyperbolicFunctionsM, nil +} + +func resourcesRubi7InverseHyperbolicFunctionsM() (*asset, error) { + bytes, err := resourcesRubi7InverseHyperbolicFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/7 Inverse hyperbolic functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi8SpecialFunctionsM = []byte(`(* ::Package:: *) + +(* ::Section:: *) +(*Special Function Rules*) + + +(* ::Subsection::Closed:: *) +(*8.1 Error functions*) + + +Int[Erf[a_.+b_.*x_],x_Symbol] := + (a+b*x)*Erf[a+b*x]/b + 1/(b*Sqrt[Pi]*E^(a+b*x)^2) /; +FreeQ[{a,b},x] + + +Int[Erfc[a_.+b_.*x_],x_Symbol] := + (a+b*x)*Erfc[a+b*x]/b - 1/(b*Sqrt[Pi]*E^(a+b*x)^2) /; +FreeQ[{a,b},x] + + +Int[Erfi[a_.+b_.*x_],x_Symbol] := + (a+b*x)*Erfi[a+b*x]/b - E^(a+b*x)^2/(b*Sqrt[Pi]) /; +FreeQ[{a,b},x] + + +Int[Erf[b_.*x_]/x_,x_Symbol] := + 2*b*x/Sqrt[Pi]*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},-b^2*x^2] /; +FreeQ[b,x] + + +Int[Erfc[b_.*x_]/x_,x_Symbol] := + Log[x] - Int[Erf[b*x]/x,x] /; +FreeQ[b,x] + + +Int[Erfi[b_.*x_]/x_,x_Symbol] := + 2*b*x/Sqrt[Pi]*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},b^2*x^2] /; +FreeQ[b,x] + + +Int[x_^m_.*Erf[a_.+b_.*x_],x_Symbol] := + x^(m+1)*Erf[a+b*x]/(m+1) - + 2*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)/E^(a+b*x)^2,x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_^m_.*Erfc[a_.+b_.*x_],x_Symbol] := + x^(m+1)*Erfc[a+b*x]/(m+1) + + 2*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)/E^(a+b*x)^2,x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_^m_.*Erfi[a_.+b_.*x_],x_Symbol] := + x^(m+1)*Erfi[a+b*x]/(m+1) - + 2*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)*E^(a+b*x)^2,x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_*E^(c_.+d_.*x_^2)*Erf[a_.+b_.*x_],x_Symbol] := + E^(c+d*x^2)*Erf[a+b*x]/(2*d) - + b/(d*Sqrt[Pi])*Int[E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_*E^(c_.+d_.*x_^2)*Erfc[a_.+b_.*x_],x_Symbol] := + E^(c+d*x^2)*Erfc[a+b*x]/(2*d) + + b/(d*Sqrt[Pi])*Int[E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_*E^(c_.+d_.*x_^2)*Erfi[a_.+b_.*x_],x_Symbol] := + E^(c+d*x^2)*Erfi[a+b*x]/(2*d) - + b/(d*Sqrt[Pi])*Int[E^(a^2+c+2*a*b*x+(b^2+d)*x^2),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erf[a_.+b_.*x_],x_Symbol] := + x^(m-1)*E^(c+d*x^2)*Erf[a+b*x]/(2*d) - + b/(d*Sqrt[Pi])*Int[x^(m-1)*E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] - + (m-1)/(2*d)*Int[x^(m-2)*E^(c+d*x^2)*Erf[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m>1 + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erfc[a_.+b_.*x_],x_Symbol] := + x^(m-1)*E^(c+d*x^2)*Erfc[a+b*x]/(2*d) + + b/(d*Sqrt[Pi])*Int[x^(m-1)*E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] - + (m-1)/(2*d)*Int[x^(m-2)*E^(c+d*x^2)*Erfc[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m>1 + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erfi[a_.+b_.*x_],x_Symbol] := + x^(m-1)*E^(c+d*x^2)*Erfi[a+b*x]/(2*d) - + b/(d*Sqrt[Pi])*Int[x^(m-1)*E^(a^2+c+2*a*b*x+(b^2+d)*x^2),x] - + (m-1)/(2*d)*Int[x^(m-2)*E^(c+d*x^2)*Erfi[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m>1 + + +Int[E^(c_.+d_.*x_^2)*Erf[b_.*x_]/x_,x_Symbol] := + 2*b*E^c*x/Sqrt[Pi]*HypergeometricPFQ[{1/2,1},{3/2,3/2},d*x^2] /; +FreeQ[b,x] && EqQ[d-b^2] + + +Int[E^(c_.+d_.*x_^2)*Erfc[b_.*x_]/x_,x_Symbol] := + Int[E^(c+d*x^2)/x,x] - Int[E^(c+d*x^2)*Erf[b*x]/x,x] /; +FreeQ[b,x] && EqQ[d-b^2] + + +Int[E^(c_.+d_.*x_^2)*Erfi[b_.*x_]/x_,x_Symbol] := + 2*b*E^c*x/Sqrt[Pi]*HypergeometricPFQ[{1/2,1},{3/2,3/2},d*x^2] /; +FreeQ[b,x] && EqQ[d+b^2] + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erf[a_.+b_.*x_],x_Symbol] := + x^(m+1)*E^(c+d*x^2)*Erf[a+b*x]/(m+1) - + 2*b/((m+1)*Sqrt[Pi])*Int[x^(m+1)*E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] - + 2*d/(m+1)*Int[x^(m+2)*E^(c+d*x^2)*Erf[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erfc[a_.+b_.*x_],x_Symbol] := + x^(m+1)*E^(c+d*x^2)*Erfc[a+b*x]/(m+1) + + 2*b/((m+1)*Sqrt[Pi])*Int[x^(m+1)*E^(-a^2+c-2*a*b*x-(b^2-d)*x^2),x] - + 2*d/(m+1)*Int[x^(m+2)*E^(c+d*x^2)*Erfc[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_*E^(c_.+d_.*x_^2)*Erfi[a_.+b_.*x_],x_Symbol] := + x^(m+1)*E^(c+d*x^2)*Erfi[a+b*x]/(m+1) - + 2*b/((m+1)*Sqrt[Pi])*Int[x^(m+1)*E^(a^2+c+2*a*b*x+(b^2+d)*x^2),x] - + 2*d/(m+1)*Int[x^(m+2)*E^(c+d*x^2)*Erfi[a+b*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[Erf[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*Erf[a+b*x]^2/b - + 4/Sqrt[Pi]*Int[(a+b*x)*Erf[a+b*x]/E^(a+b*x)^2,x] /; +FreeQ[{a,b},x] + + +Int[Erfc[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*Erfc[a+b*x]^2/b + + 4/Sqrt[Pi]*Int[(a+b*x)*Erfc[a+b*x]/E^(a+b*x)^2,x] /; +FreeQ[{a,b},x] + + +Int[Erfi[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*Erfi[a+b*x]^2/b - + 4/Sqrt[Pi]*Int[(a+b*x)*E^(a+b*x)^2*Erfi[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*Erf[b_.*x_]^2,x_Symbol] := + x^(m+1)*Erf[b*x]^2/(m+1) - + 4*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)*E^(-b^2*x^2)*Erf[b*x],x] /; +FreeQ[b,x] && IntegerQ[m] && m!=-1 && (m>0 || OddQ[m]) + + +Int[x_^m_.*Erfc[b_.*x_]^2,x_Symbol] := + x^(m+1)*Erfc[b*x]^2/(m+1) + + 4*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)*E^(-b^2*x^2)*Erfc[b*x],x] /; +FreeQ[b,x] && IntegerQ[m] && m+1!=0 && (m>0 || OddQ[m]) + + +Int[x_^m_.*Erfi[b_.*x_]^2,x_Symbol] := + x^(m+1)*Erfi[b*x]^2/(m+1) - + 4*b/(Sqrt[Pi]*(m+1))*Int[x^(m+1)*E^(b^2*x^2)*Erfi[b*x],x] /; +FreeQ[b,x] && IntegerQ[m] && m+1!=0 && (m>0 || OddQ[m]) + + +Int[x_^m_.*Erf[a_+b_.*x_]^2,x_Symbol] := + 1/b*Subst[Int[(-a/b+x/b)^m*Erf[x]^2,x],x,a+b*x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Erfc[a_+b_.*x_]^2,x_Symbol] := + 1/b*Subst[Int[(-a/b+x/b)^m*Erfc[x]^2,x],x,a+b*x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Erfi[a_+b_.*x_]^2,x_Symbol] := + 1/b*Subst[Int[(-a/b+x/b)^m*Erfi[x]^2,x],x,a+b*x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + + + + +(* ::Subsection::Closed:: *) +(*8.2 Fresnel integral functions*) + + +Int[FresnelS[a_.+b_.*x_],x_Symbol] := + (a+b*x)*FresnelS[a+b*x]/b + Cos[Pi/2*(a+b*x)^2]/(b*Pi) /; +FreeQ[{a,b},x] + + +Int[FresnelC[a_.+b_.*x_],x_Symbol] := + (a+b*x)*FresnelC[a+b*x]/b - Sin[Pi/2*(a+b*x)^2]/(b*Pi) /; +FreeQ[{a,b},x] + + +Int[FresnelS[b_.*x_]/x_,x_Symbol] := + 1/2*I*b*x*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},-1/2*I*b^2*Pi*x^2] - + 1/2*I*b*x*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},1/2*I*b^2*Pi*x^2] /; +FreeQ[b,x] + + +Int[FresnelC[b_.*x_]/x_,x_Symbol] := + 1/2*b*x*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},-1/2*I*b^2*Pi*x^2] + + 1/2*b*x*HypergeometricPFQ[{1/2,1/2},{3/2,3/2},1/2*I*b^2*Pi*x^2] /; +FreeQ[b,x] + + +Int[x_^m_.*FresnelS[a_.+b_.*x_],x_Symbol] := + x^(m+1)*FresnelS[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Sin[Pi/2*(a+b*x)^2],x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_^m_.*FresnelC[a_.+b_.*x_],x_Symbol] := + x^(m+1)*FresnelC[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Cos[Pi/2*(a+b*x)^2],x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[FresnelS[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*FresnelS[a+b*x]^2/b - + 2*Int[(a+b*x)*Sin[Pi/2*(a+b*x)^2]*FresnelS[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[FresnelC[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*FresnelC[a+b*x]^2/b - + 2*Int[(a+b*x)*Cos[Pi/2*(a+b*x)^2]*FresnelC[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[x_^m_*FresnelS[b_.*x_]^2,x_Symbol] := + x^(m+1)*FresnelS[b*x]^2/(m+1) - + 2*b/(m+1)*Int[x^(m+1)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x],x] /; +FreeQ[b,x] && IntegerQ[m] && m+1!=0 && (m>0 && EvenQ[m] || Mod[m,4]==3) + + +Int[x_^m_*FresnelC[b_.*x_]^2,x_Symbol] := + x^(m+1)*FresnelC[b*x]^2/(m+1) - + 2*b/(m+1)*Int[x^(m+1)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x],x] /; +FreeQ[b,x] && IntegerQ[m] && m+1!=0 && (m>0 && EvenQ[m] || Mod[m,4]==3) + + +(* Int[x_^m_.*FresnelS[a_+b_.*x_]^2,x_Symbol] := + 1/b*Subst[Int[(-a/b+x/b)^m*FresnelS[x]^2,x],x,a+b*x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m>0 *) + + +(* Int[x_^m_.*FresnelC[a_+b_.*x_]^2,x_Symbol] := + 1/b*Subst[Int[(-a/b+x/b)^m*FresnelC[x]^2,x],x,a+b*x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m>0 *) + + +Int[x_*Sin[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + -Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/(Pi*b^2) + + 1/(2*Pi*b)*Int[Sin[Pi*b^2*x^2],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] + + +Int[x_*Cos[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/(Pi*b^2) - + 1/(2*Pi*b)*Int[Sin[Pi*b^2*x^2],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] + + +Int[x_^m_*Sin[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + -x^(m-1)*Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/(Pi*b^2) + + 1/(2*Pi*b)*Int[x^(m-1)*Sin[Pi*b^2*x^2],x] + + (m-1)/(Pi*b^2)*Int[x^(m-2)*Cos[Pi/2*b^2*x^2]*FresnelS[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m>1 && Not[Mod[m,4]==2] + + +Int[x_^m_*Cos[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + x^(m-1)*Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/(Pi*b^2) - + 1/(2*Pi*b)*Int[x^(m-1)*Sin[Pi*b^2*x^2],x] - + (m-1)/(Pi*b^2)*Int[x^(m-2)*Sin[Pi/2*b^2*x^2]*FresnelC[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m>1 && Not[Mod[m,4]==2] + + +Int[x_^m_*Sin[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + x^(m+1)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/(m+1) - b*x^(m+2)/(2*(m+1)*(m+2)) + + b/(2*(m+1))*Int[x^(m+1)*Cos[Pi*b^2*x^2],x] - + Pi*b^2/(m+1)*Int[x^(m+2)*Cos[Pi/2*b^2*x^2]*FresnelS[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m<-2 && Mod[m,4]==0 + + +Int[x_^m_*Cos[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + x^(m+1)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/(m+1) - b*x^(m+2)/(2*(m+1)*(m+2)) - + b/(2*(m+1))*Int[x^(m+1)*Cos[Pi*b^2*x^2],x] + + Pi*b^2/(m+1)*Int[x^(m+2)*Sin[Pi/2*b^2*x^2]*FresnelC[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m<-2 && Mod[m,4]==0 + + +Int[x_*Cos[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/(Pi*b^2) - x/(2*Pi*b) + + 1/(2*Pi*b)*Int[Cos[Pi*b^2*x^2],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] + + +Int[x_*Sin[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + -Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/(Pi*b^2) + x/(2*Pi*b) + + 1/(2*Pi*b)*Int[Cos[Pi*b^2*x^2],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] + + +Int[x_^m_*Cos[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + x^(m-1)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/(Pi*b^2) - x^m/(2*b*m*Pi) + + 1/(2*Pi*b)*Int[x^(m-1)*Cos[Pi*b^2*x^2],x] - + (m-1)/(Pi*b^2)*Int[x^(m-2)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m>1 && Not[Mod[m,4]==0] + + +Int[x_^m_*Sin[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + -x^(m-1)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/(Pi*b^2) + x^m/(2*b*m*Pi) + + 1/(2*Pi*b)*Int[x^(m-1)*Cos[Pi*b^2*x^2],x] + + (m-1)/(Pi*b^2)*Int[x^(m-2)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m>1 && Not[Mod[m,4]==0] + + +Int[x_^m_*Cos[c_.*x_^2]*FresnelS[b_.*x_],x_Symbol] := + x^(m+1)*Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/(m+1) - + b/(2*(m+1))*Int[x^(m+1)*Sin[Pi*b^2*x^2],x] + + Pi*b^2/(m+1)*Int[x^(m+2)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m<-1 && Mod[m,4]==2 + + +Int[x_^m_*Sin[c_.*x_^2]*FresnelC[b_.*x_],x_Symbol] := + x^(m+1)*Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/(m+1) - + b/(2*(m+1))*Int[x^(m+1)*Sin[Pi*b^2*x^2],x] - + Pi*b^2/(m+1)*Int[x^(m+2)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x],x] /; +FreeQ[{b,c},x] && EqQ[c-Pi/2*b^2] && IntegerQ[m] && m<-1 && Mod[m,4]==2 + + + + + +(* ::Subsection::Closed:: *) +(*8.3 Exponential integral functions*) + + +Int[ExpIntegralE[n_,a_.+b_.*x_],x_Symbol] := + -ExpIntegralE[n+1,a+b*x]/b /; +FreeQ[{a,b,n},x] + + +Int[x_^m_.*ExpIntegralE[n_,b_.*x_],x_Symbol] := + -x^m*ExpIntegralE[n+1,b*x]/b + + m/b*Int[x^(m-1)*ExpIntegralE[n+1,b*x],x] /; +FreeQ[b,x] && EqQ[m+n] && PositiveIntegerQ[m] + + +Int[ExpIntegralE[1,b_.*x_]/x_,x_Symbol] := + b*x*HypergeometricPFQ[{1,1,1},{2,2,2},-b*x] - EulerGamma*Log[x] - 1/2*Log[b*x]^2 /; +FreeQ[b,x] + + +Int[x_^m_*ExpIntegralE[n_,b_.*x_],x_Symbol] := + x^(m+1)*ExpIntegralE[n,b*x]/(m+1) + + b/(m+1)*Int[x^(m+1)*ExpIntegralE[n-1,b*x],x] /; +FreeQ[b,x] && EqQ[m+n] && IntegerQ[m] && m<-1 + + +Int[x_^m_*ExpIntegralE[n_,b_.*x_],x_Symbol] := + x^m*Gamma[m+1]*Log[x]/(b*(b*x)^m) - x^(m+1)*HypergeometricPFQ[{m+1,m+1},{m+2,m+2},-b*x]/(m+1)^2 /; +FreeQ[{b,m,n},x] && EqQ[m+n] && Not[IntegerQ[m]] + + +Int[x_^m_.*ExpIntegralE[n_,b_.*x_],x_Symbol] := + x^(m+1)*ExpIntegralE[n,b*x]/(m+n) - x^(m+1)*ExpIntegralE[-m,b*x]/(m+n) /; +FreeQ[{b,m,n},x] && NeQ[m+n] + + +Int[x_^m_.*ExpIntegralE[n_,a_+b_.*x_],x_Symbol] := + -x^m*ExpIntegralE[n+1,a+b*x]/b + + m/b*Int[x^(m-1)*ExpIntegralE[n+1,a+b*x],x] /; +FreeQ[{a,b,m,n},x] && (PositiveIntegerQ[m] || NegativeIntegerQ[n] || RationalQ[m,n] && m>0 && n<-1) + + +Int[x_^m_.*ExpIntegralE[n_,a_+b_.*x_],x_Symbol] := + x^(m+1)*ExpIntegralE[n,a+b*x]/(m+1) + + b/(m+1)*Int[x^(m+1)*ExpIntegralE[n-1,a+b*x],x] /; +FreeQ[{a,b,m},x] && (PositiveIntegerQ[n] || RationalQ[m,n] && m<-1 && n>0) && NeQ[m+1] + + +Int[ExpIntegralEi[a_.+b_.*x_],x_Symbol] := + (a+b*x)*ExpIntegralEi[a+b*x]/b - E^(a+b*x)/b /; +FreeQ[{a,b},x] + + +Int[x_^m_.*ExpIntegralEi[a_.+b_.*x_],x_Symbol] := + x^(m+1)*ExpIntegralEi[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*E^(a+b*x)/(a+b*x),x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[ExpIntegralEi[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*ExpIntegralEi[a+b*x]^2/b - + 2*Int[E^(a+b*x)*ExpIntegralEi[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*ExpIntegralEi[b_.*x_]^2,x_Symbol] := + x^(m+1)*ExpIntegralEi[b*x]^2/(m+1) - + 2/(m+1)*Int[x^m*E^(b*x)*ExpIntegralEi[b*x],x] /; +FreeQ[b,x] && PositiveIntegerQ[m] + + +Int[x_^m_.*ExpIntegralEi[a_+b_.*x_]^2,x_Symbol] := + x^(m+1)*ExpIntegralEi[a+b*x]^2/(m+1) + + a*x^m*ExpIntegralEi[a+b*x]^2/(b*(m+1)) - + 2/(m+1)*Int[x^m*E^(a+b*x)*ExpIntegralEi[a+b*x],x] - + a*m/(b*(m+1))*Int[x^(m-1)*ExpIntegralEi[a+b*x]^2,x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +(* Int[x_^m_.*ExpIntegralEi[a_+b_.*x_]^2,x_Symbol] := + b*x^(m+2)*ExpIntegralEi[a+b*x]^2/(a*(m+1)) + + x^(m+1)*ExpIntegralEi[a+b*x]^2/(m+1) - + 2*b/(a*(m+1))*Int[x^(m+1)*E^(a+b*x)*ExpIntegralEi[a+b*x],x] - + b*(m+2)/(a*(m+1))*Int[x^(m+1)*ExpIntegralEi[a+b*x]^2,x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m<-2 *) + + +Int[E^(a_.+b_.*x_)*ExpIntegralEi[c_.+d_.*x_],x_Symbol] := + E^(a+b*x)*ExpIntegralEi[c+d*x]/b - + d/b*Int[E^(a+c+(b+d)*x)/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_.*E^(a_.+b_.*x_)*ExpIntegralEi[c_.+d_.*x_],x_Symbol] := + x^m*E^(a+b*x)*ExpIntegralEi[c+d*x]/b - + d/b*Int[x^m*E^(a+c+(b+d)*x)/(c+d*x),x] - + m/b*Int[x^(m-1)*E^(a+b*x)*ExpIntegralEi[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_*E^(a_.+b_.*x_)*ExpIntegralEi[c_.+d_.*x_],x_Symbol] := + x^(m+1)*E^(a+b*x)*ExpIntegralEi[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*E^(a+c+(b+d)*x)/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*E^(a+b*x)*ExpIntegralEi[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[LogIntegral[a_.+b_.*x_],x_Symbol] := + (a+b*x)*LogIntegral[a+b*x]/b - ExpIntegralEi[2*Log[a+b*x]]/b /; +FreeQ[{a,b},x] + + +Int[LogIntegral[b_.*x_]/x_,x_Symbol] := + -b*x + Log[b*x]*LogIntegral[b*x] /; +FreeQ[b,x] + + +Int[x_^m_.*LogIntegral[a_.+b_.*x_],x_Symbol] := + x^(m+1)*LogIntegral[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)/Log[a+b*x],x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + + + + +(* ::Subsection::Closed:: *) +(*8.4 Trig integral functions*) + + +Int[SinIntegral[a_.+b_.*x_],x_Symbol] := + (a+b*x)*SinIntegral[a+b*x]/b + Cos[a+b*x]/b/; +FreeQ[{a,b},x] + + +Int[CosIntegral[a_.+b_.*x_],x_Symbol] := + (a+b*x)*CosIntegral[a+b*x]/b - Sin[a+b*x]/b /; +FreeQ[{a,b},x] + + +Int[SinIntegral[b_.*x_]/x_,x_Symbol] := + 1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},-I*b*x] + + 1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},I*b*x] /; +FreeQ[b,x] + + +Int[CosIntegral[b_.*x_]/x_,x_Symbol] := + -1/2*I*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},-I*b*x] + + 1/2*I*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},I*b*x] + + EulerGamma*Log[x] + + 1/2*Log[b*x]^2 /; +FreeQ[b,x] + + +Int[x_^m_.*SinIntegral[a_.+b_.*x_],x_Symbol] := + x^(m+1)*SinIntegral[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Sin[a+b*x]/(a+b*x),x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_^m_.*CosIntegral[a_.+b_.*x_],x_Symbol] := + x^(m+1)*CosIntegral[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Cos[a+b*x]/(a+b*x),x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[SinIntegral[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*SinIntegral[a+b*x]^2/b - + 2*Int[Sin[a+b*x]*SinIntegral[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[CosIntegral[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*CosIntegral[a+b*x]^2/b - + 2*Int[Cos[a+b*x]*CosIntegral[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*SinIntegral[b_.*x_]^2,x_Symbol] := + x^(m+1)*SinIntegral[b*x]^2/(m+1) - + 2/(m+1)*Int[x^m*Sin[b*x]*SinIntegral[b*x],x] /; +FreeQ[b,x] && PositiveIntegerQ[m] + + +Int[x_^m_.*CosIntegral[b_.*x_]^2,x_Symbol] := + x^(m+1)*CosIntegral[b*x]^2/(m+1) - + 2/(m+1)*Int[x^m*Cos[b*x]*CosIntegral[b*x],x] /; +FreeQ[b,x] && PositiveIntegerQ[m] + + +Int[x_^m_.*SinIntegral[a_+b_.*x_]^2,x_Symbol] := + x^(m+1)*SinIntegral[a+b*x]^2/(m+1) + + a*x^m*SinIntegral[a+b*x]^2/(b*(m+1)) - + 2/(m+1)*Int[x^m*Sin[a+b*x]*SinIntegral[a+b*x],x] - + a*m/(b*(m+1))*Int[x^(m-1)*SinIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*CosIntegral[a_+b_.*x_]^2,x_Symbol] := + x^(m+1)*CosIntegral[a+b*x]^2/(m+1) + + a*x^m*CosIntegral[a+b*x]^2/(b*(m+1)) - + 2/(m+1)*Int[x^m*Cos[a+b*x]*CosIntegral[a+b*x],x] - + a*m/(b*(m+1))*Int[x^(m-1)*CosIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +(* Int[x_^m_.*SinIntegral[a_+b_.*x_]^2,x_Symbol] := + b*x^(m+2)*SinIntegral[a+b*x]^2/(a*(m+1)) + + x^(m+1)*SinIntegral[a+b*x]^2/(m+1) - + 2*b/(a*(m+1))*Int[x^(m+1)*Sin[a+b*x]*SinIntegral[a+b*x],x] - + b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m<-2 *) + + +(* Int[x_^m_.*CosIntegral[a_+b_.*x_]^2,x_Symbol] := + b*x^(m+2)*CosIntegral[a+b*x]^2/(a*(m+1)) + + x^(m+1)*CosIntegral[a+b*x]^2/(m+1) - + 2*b/(a*(m+1))*Int[x^(m+1)*Cos[a+b*x]*CosIntegral[a+b*x],x] - + b*(m+2)/(a*(m+1))*Int[x^(m+1)*CosIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m<-2 *) + + +Int[Sin[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + -Cos[a+b*x]*SinIntegral[c+d*x]/b + + d/b*Int[Cos[a+b*x]*Sin[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[Cos[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + Sin[a+b*x]*CosIntegral[c+d*x]/b - + d/b*Int[Sin[a+b*x]*Cos[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_.*Sin[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + -x^m*Cos[a+b*x]*SinIntegral[c+d*x]/b + + d/b*Int[x^m*Cos[a+b*x]*Sin[c+d*x]/(c+d*x),x] + + m/b*Int[x^(m-1)*Cos[a+b*x]*SinIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Cos[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Sin[a+b*x]*CosIntegral[c+d*x]/b - + d/b*Int[x^m*Sin[a+b*x]*Cos[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Sin[a+b*x]*CosIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_*Sin[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Sin[a+b*x]*SinIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Sin[a+b*x]*Sin[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Cos[a+b*x]*SinIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_.*Cos[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Cos[a+b*x]*CosIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Cos[a+b*x]*Cos[c+d*x]/(c+d*x),x] + + b/(m+1)*Int[x^(m+1)*Sin[a+b*x]*CosIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[Cos[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + Sin[a+b*x]*SinIntegral[c+d*x]/b - + d/b*Int[Sin[a+b*x]*Sin[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[Sin[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + -Cos[a+b*x]*CosIntegral[c+d*x]/b + + d/b*Int[Cos[a+b*x]*Cos[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_.*Cos[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Sin[a+b*x]*SinIntegral[c+d*x]/b - + d/b*Int[x^m*Sin[a+b*x]*Sin[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Sin[a+b*x]*SinIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Sin[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + -x^m*Cos[a+b*x]*CosIntegral[c+d*x]/b + + d/b*Int[x^m*Cos[a+b*x]*Cos[c+d*x]/(c+d*x),x] + + m/b*Int[x^(m-1)*Cos[a+b*x]*CosIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Cos[a_.+b_.*x_]*SinIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Cos[a+b*x]*SinIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Cos[a+b*x]*Sin[c+d*x]/(c+d*x),x] + + b/(m+1)*Int[x^(m+1)*Sin[a+b*x]*SinIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_*Sin[a_.+b_.*x_]*CosIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Sin[a+b*x]*CosIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Sin[a+b*x]*Cos[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Cos[a+b*x]*CosIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + + + + +(* ::Subsection::Closed:: *) +(*8.5 Hyperbolic integral functions*) + + +Int[SinhIntegral[a_.+b_.*x_],x_Symbol] := + (a+b*x)*SinhIntegral[a+b*x]/b - Cosh[a+b*x]/b/; +FreeQ[{a,b},x] + + +Int[CoshIntegral[a_.+b_.*x_],x_Symbol] := + (a+b*x)*CoshIntegral[a+b*x]/b - Sinh[a+b*x]/b /; +FreeQ[{a,b},x] + + +Int[SinhIntegral[b_.*x_]/x_,x_Symbol] := + 1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},-b*x] + + 1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},b*x] /; +FreeQ[b,x] + + +Int[CoshIntegral[b_.*x_]/x_,x_Symbol] := + -1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},-b*x] + + 1/2*b*x*HypergeometricPFQ[{1,1,1},{2,2,2},b*x] + + EulerGamma*Log[x] + + 1/2*Log[b*x]^2 /; +FreeQ[b,x] + + +Int[x_^m_.*SinhIntegral[a_.+b_.*x_],x_Symbol] := + x^(m+1)*SinhIntegral[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Sinh[a+b*x]/(a+b*x),x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[x_^m_.*CoshIntegral[a_.+b_.*x_],x_Symbol] := + x^(m+1)*CoshIntegral[a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*Cosh[a+b*x]/(a+b*x),x] /; +FreeQ[{a,b,m},x] && NeQ[m+1] + + +Int[SinhIntegral[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*SinhIntegral[a+b*x]^2/b - + 2*Int[Sinh[a+b*x]*SinhIntegral[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[CoshIntegral[a_.+b_.*x_]^2,x_Symbol] := + (a+b*x)*CoshIntegral[a+b*x]^2/b - + 2*Int[Cosh[a+b*x]*CoshIntegral[a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[x_^m_.*SinhIntegral[b_.*x_]^2,x_Symbol] := + x^(m+1)*SinhIntegral[b*x]^2/(m+1) - + 2/(m+1)*Int[x^m*Sinh[b*x]*SinhIntegral[b*x],x] /; +FreeQ[b,x] && PositiveIntegerQ[m] + + +Int[x_^m_.*CoshIntegral[b_.*x_]^2,x_Symbol] := + x^(m+1)*CoshIntegral[b*x]^2/(m+1) - + 2/(m+1)*Int[x^m*Cosh[b*x]*CoshIntegral[b*x],x] /; +FreeQ[b,x] && PositiveIntegerQ[m] + + +Int[x_^m_.*SinhIntegral[a_+b_.*x_]^2,x_Symbol] := + x^(m+1)*SinhIntegral[a+b*x]^2/(m+1) + + a*x^m*SinhIntegral[a+b*x]^2/(b*(m+1)) - + 2/(m+1)*Int[x^m*Sinh[a+b*x]*SinhIntegral[a+b*x],x] - + a*m/(b*(m+1))*Int[x^(m-1)*SinhIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*CoshIntegral[a_+b_.*x_]^2,x_Symbol] := + x^(m+1)*CoshIntegral[a+b*x]^2/(m+1) + + a*x^m*CoshIntegral[a+b*x]^2/(b*(m+1)) - + 2/(m+1)*Int[x^m*Cosh[a+b*x]*CoshIntegral[a+b*x],x] - + a*m/(b*(m+1))*Int[x^(m-1)*CoshIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] + + +(* Int[x_^m_.*SinhIntegral[a_+b_.*x_]^2,x_Symbol] := + b*x^(m+2)*SinhIntegral[a+b*x]^2/(a*(m+1)) + + x^(m+1)*SinhIntegral[a+b*x]^2/(m+1) - + 2*b/(a*(m+1))*Int[x^(m+1)*Sinh[a+b*x]*SinhIntegral[a+b*x],x] - + b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinhIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m<-2 *) + + +(* Int[x_^m_.*CoshIntegral[a_+b_.*x_]^2,x_Symbol] := + b*x^(m+2)*CoshIntegral[a+b*x]^2/(a*(m+1)) + + x^(m+1)*CoshIntegral[a+b*x]^2/(m+1) - + 2*b/(a*(m+1))*Int[x^(m+1)*Cosh[a+b*x]*CoshIntegral[a+b*x],x] - + b*(m+2)/(a*(m+1))*Int[x^(m+1)*CoshIntegral[a+b*x]^2,x] /; +FreeQ[{a,b},x] && IntegerQ[m] && m<-2 *) + + +Int[Sinh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + Cosh[a+b*x]*SinhIntegral[c+d*x]/b - + d/b*Int[Cosh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[Cosh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + Sinh[a+b*x]*CoshIntegral[c+d*x]/b - + d/b*Int[Sinh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_.*Sinh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Cosh[a+b*x]*SinhIntegral[c+d*x]/b - + d/b*Int[x^m*Cosh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Cosh[a+b*x]*SinhIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m>0 + + +Int[x_^m_.*Cosh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Sinh[a+b*x]*CoshIntegral[c+d*x]/b - + d/b*Int[x^m*Sinh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Sinh[a+b*x]*CoshIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m>0 + + +Int[x_^m_*Sinh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Sinh[a+b*x]*SinhIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Sinh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Cosh[a+b*x]*SinhIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_.*Cosh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Cosh[a+b*x]*CoshIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Cosh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Sinh[a+b*x]*CoshIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[Cosh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + Sinh[a+b*x]*SinhIntegral[c+d*x]/b - + d/b*Int[Sinh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[Sinh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + Cosh[a+b*x]*CoshIntegral[c+d*x]/b - + d/b*Int[Cosh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] /; +FreeQ[{a,b,c,d},x] + + +Int[x_^m_.*Cosh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Sinh[a+b*x]*SinhIntegral[c+d*x]/b - + d/b*Int[x^m*Sinh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Sinh[a+b*x]*SinhIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Sinh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + x^m*Cosh[a+b*x]*CoshIntegral[c+d*x]/b - + d/b*Int[x^m*Cosh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] - + m/b*Int[x^(m-1)*Cosh[a+b*x]*CoshIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*Cosh[a_.+b_.*x_]*SinhIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Cosh[a+b*x]*SinhIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Cosh[a+b*x]*Sinh[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Sinh[a+b*x]*SinhIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + +Int[x_^m_*Sinh[a_.+b_.*x_]*CoshIntegral[c_.+d_.*x_],x_Symbol] := + x^(m+1)*Sinh[a+b*x]*CoshIntegral[c+d*x]/(m+1) - + d/(m+1)*Int[x^(m+1)*Sinh[a+b*x]*Cosh[c+d*x]/(c+d*x),x] - + b/(m+1)*Int[x^(m+1)*Cosh[a+b*x]*CoshIntegral[c+d*x],x] /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && m<-1 + + + + + +(* ::Subsection::Closed:: *) +(*8.6 Gamma functions*) + + +Int[Gamma[n_,a_.+b_.*x_],x_Symbol] := + (a+b*x)*Gamma[n,a+b*x]/b - + Gamma[n+1,a+b*x]/b /; +FreeQ[{a,b},x] + + +Int[Gamma[n_,b_*x_]/x_,x_Symbol] := + Gamma[n]*Log[x] - (b*x)^n/n^2*HypergeometricPFQ[{n,n},{1+n,1+n},-b*x] /; +FreeQ[{b,n},x] && Not[IntegerQ[n] && n<=0] + + +Int[x_^m_.*Gamma[n_,b_*x_],x_Symbol] := + x^(m+1)*Gamma[n,b*x]/(m+1) - + x^m*Gamma[m+n+1,b*x]/(b*(m+1)*(b*x)^m) /; +FreeQ[{b,m,n},x] && NeQ[m+1] + + +Int[x_^m_.*Gamma[n_,a_+b_.*x_],x_Symbol] := + Block[{$UseGamma=True}, + x^(m+1)*Gamma[n,a+b*x]/(m+1) + + b/(m+1)*Int[x^(m+1)*(a+b*x)^(n-1)/E^(a+b*x),x]] /; +FreeQ[{a,b,m,n},x] && (PositiveIntegerQ[m] || PositiveIntegerQ[n] || IntegersQ[m,n]) && NeQ[m+1] + + +(* Int[x_^m_.*Gamma[n_,a_+b_.*x_],x_Symbol] := + x^m*(a+b*x)*Gamma[n,a+b*x]/(b*(m+1)) - + x^m*Gamma[n+1,a+b*x]/(b*(m+1)) - + a*m/(b*(m+1))*Int[x^(m-1)*Gamma[n,a+b*x],x] + + m/(b*(m+1))*Int[x^(m-1)*Gamma[n+1,a+b*x],x] /; +FreeQ[{a,b,n},x] && RationalQ[m] && m>0 *) + + +Int[LogGamma[a_.+b_.*x_],x_Symbol] := + PolyGamma[-2,a+b*x]/b /; +FreeQ[{a,b},x] + + +Int[x_^m_.*LogGamma[a_.+b_.*x_],x_Symbol] := + x^m*PolyGamma[-2,a+b*x]/b - + m/b*Int[x^(m-1)*PolyGamma[-2,a+b*x],x] /; +FreeQ[{a,b},x] && RationalQ[m] && m>0 + + +Int[PolyGamma[n_,a_.+b_.*x_],x_Symbol] := + PolyGamma[n-1,a+b*x]/b /; +FreeQ[{a,b,n},x] + + +Int[x_^m_.*PolyGamma[n_,a_.+b_.*x_],x_Symbol] := + x^m*PolyGamma[n-1,a+b*x]/b - + m/b*Int[x^(m-1)*PolyGamma[n-1,a+b*x],x] /; +FreeQ[{a,b,n},x] && RationalQ[m] && m>0 + + +Int[x_^m_.*PolyGamma[n_,a_.+b_.*x_],x_Symbol] := + x^(m+1)*PolyGamma[n,a+b*x]/(m+1) - + b/(m+1)*Int[x^(m+1)*PolyGamma[n+1,a+b*x],x] /; +FreeQ[{a,b,n},x] && RationalQ[m] && m<-1 + + +Int[Gamma[a_.+b_.*x_]^n_.*PolyGamma[0,a_.+b_.*x_],x_Symbol] := + Gamma[a+b*x]^n/(b*n) /; +FreeQ[{a,b,n},x] + + +Int[((a_.+b_.*x_)!)^n_.*PolyGamma[0,c_.+b_.*x_],x_Symbol] := + ((a+b*x)!)^n/(b*n) /; +FreeQ[{a,b,c,n},x] && EqQ[a-c+1] + + + + + +(* ::Subsection::Closed:: *) +(*8.7 Zeta function*) + + +Int[Zeta[2,a_.+b_.*x_],x_Symbol] := + Int[PolyGamma[1,a+b*x],x] /; +FreeQ[{a,b},x] + + +Int[Zeta[s_,a_.+b_.*x_],x_Symbol] := + -Zeta[s-1,a+b*x]/(b*(s-1)) /; +FreeQ[{a,b,s},x] && NeQ[s-1] && NeQ[s-2] + + +Int[x_^m_.*Zeta[2,a_.+b_.*x_],x_Symbol] := + Int[x^m*PolyGamma[1,a+b*x],x] /; +FreeQ[{a,b},x] && RationalQ[m] + + +Int[x_^m_.*Zeta[s_,a_.+b_.*x_],x_Symbol] := + -x^m*Zeta[s-1,a+b*x]/(b*(s-1)) + + m/(b*(s-1))*Int[x^(m-1)*Zeta[s-1,a+b*x],x] /; +FreeQ[{a,b,s},x] && NeQ[s-1] && NeQ[s-2] && RationalQ[m] && m>0 + + +Int[x_^m_.*Zeta[s_,a_.+b_.*x_],x_Symbol] := + x^(m+1)*Zeta[s,a+b*x]/(m+1) + + b*s/(m+1)*Int[x^(m+1)*Zeta[s+1,a+b*x],x] /; +FreeQ[{a,b,s},x] && NeQ[s-1] && NeQ[s-2] && RationalQ[m] && m<-1 + + + + + +(* ::Subsection::Closed:: *) +(*8.8 Polylogarithm function*) + + +Int[PolyLog[n_,a_.*(b_.*x_^p_.)^q_.],x_Symbol] := + x*PolyLog[n,a*(b*x^p)^q] - + p*q*Int[PolyLog[n-1,a*(b*x^p)^q],x] /; +FreeQ[{a,b,p,q},x] && RationalQ[n] && n>0 + + +Int[PolyLog[n_,a_.*(b_.*x_^p_.)^q_.],x_Symbol] := + x*PolyLog[n+1,a*(b*x^p)^q]/(p*q) - + 1/(p*q)*Int[PolyLog[n+1,a*(b*x^p)^q],x] /; +FreeQ[{a,b,p,q},x] && RationalQ[n] && n<-1 + + +Int[PolyLog[n_,c_.*(a_.+b_.*x_)^p_.]/(d_.+e_.*x_),x_Symbol] := + PolyLog[n+1,c*(a+b*x)^p]/(e*p) /; +FreeQ[{a,b,c,d,e,n,p},x] && EqQ[b*d-a*e] + + +Int[PolyLog[n_,a_.*(b_.*x_^p_.)^q_.]/x_,x_Symbol] := + PolyLog[n+1,a*(b*x^p)^q]/(p*q) /; +FreeQ[{a,b,n,p,q},x] + + +Int[x_^m_.*PolyLog[n_,a_.*(b_.*x_^p_.)^q_.],x_Symbol] := + x^(m+1)*PolyLog[n,a*(b*x^p)^q]/(m+1) - + p*q/(m+1)*Int[x^m*PolyLog[n-1,a*(b*x^p)^q],x] /; +FreeQ[{a,b,m,p,q},x] && NeQ[m+1] && RationalQ[n] && n>0 + + +Int[x_^m_.*PolyLog[n_,a_.*(b_.*x_^p_.)^q_.],x_Symbol] := + x^(m+1)*PolyLog[n+1,a*(b*x^p)^q]/(p*q) - + (m+1)/(p*q)*Int[x^m*PolyLog[n+1,a*(b*x^p)^q],x] /; +FreeQ[{a,b,m,p,q},x] && NeQ[m+1] && RationalQ[n] && n<-1 + + +Int[Log[c_.*x_^m_.]^r_.*PolyLog[n_,a_.*(b_.*x_^p_.)^q_.]/x_,x_Symbol] := + Log[c*x^m]^r*PolyLog[n+1,a*(b*x^p)^q]/(p*q) - + m*r/(p*q)*Int[Log[c*x^m]^(r-1)*PolyLog[n+1,a*(b*x^p)^q]/x,x] /; +FreeQ[{a,b,c,m,n,q,r},x] && RationalQ[r] && r>0 + + +Int[PolyLog[n_,c_.*(a_.+b_.*x_)^p_.],x_Symbol] := + x*PolyLog[n,c*(a+b*x)^p] - + p*Int[PolyLog[n-1,c*(a+b*x)^p],x] + + a*p*Int[PolyLog[n-1,c*(a+b*x)^p]/(a+b*x),x] /; +FreeQ[{a,b,c,p},x] && RationalQ[n] && n>0 + + +Int[PolyLog[2,c_.*(a_.+b_.*x_)]/(d_.+e_.*x_),x_Symbol] := + Log[d+e*x]*PolyLog[2,c*(a+b*x)]/e + b/e*Int[Log[d+e*x]*Log[1-a*c-b*c*x]/(a+b*x),x] /; +FreeQ[{a,b,c,d,e},x] + + +Int[(d_.+e_.*x_)^m_.*PolyLog[2,c_.*(a_.+b_.*x_)],x_Symbol] := + (d+e*x)^(m+1)*PolyLog[2,c*(a+b*x)]/(e*(m+1)) + b/(e*(m+1))*Int[(d+e*x)^(m+1)*Log[1-a*c-b*c*x]/(a+b*x),x] /; +FreeQ[{a,b,c,d,e,m},x] && NeQ[m,-1] + + +(* Int[x_^m_.*PolyLog[n_,c_.*(a_.+b_.*x_)^p_.],x_Symbol] := + x^(m+1)*PolyLog[n,c*(a+b*x)^p]/(m+1) - + b*p/(m+1)*Int[x^(m+1)*PolyLog[n-1,c*(a+b*x)^p]/(a+b*x),x] /; +FreeQ[{a,b,c,m,p},x] && RationalQ[n] && n>0 && PositiveIntegerQ[m] *) + + +Int[x_^m_.*PolyLog[n_,c_.*(a_.+b_.*x_)^p_.],x_Symbol] := + -(a^(m+1)-b^(m+1)*x^(m+1))*PolyLog[n,c*(a+b*x)^p]/((m+1)*b^(m+1)) + + p/((m+1)*b^m)*Int[ExpandIntegrand[PolyLog[n-1,c*(a+b*x)^p],(a^(m+1)-b^(m+1)*x^(m+1))/(a+b*x),x],x] /; +FreeQ[{a,b,c,p},x] && GtQ[n,0] && IntegerQ[m] && Not[EqQ[m,-1]] + + +Int[PolyLog[n_,d_.*(F_^(c_.*(a_.+b_.*x_)))^p_.],x_Symbol] := + PolyLog[n+1,d*(F^(c*(a+b*x)))^p]/(b*c*p*Log[F]) /; +FreeQ[{F,a,b,c,d,n,p},x] + + +Int[(e_.+f_.*x_)^m_.*PolyLog[n_,d_.*(F_^(c_.*(a_.+b_.*x_)))^p_.],x_Symbol] := + (e+f*x)^m*PolyLog[n+1,d*(F^(c*(a+b*x)))^p]/(b*c*p*Log[F]) - + f*m/(b*c*p*Log[F])*Int[(e+f*x)^(m-1)*PolyLog[n+1,d*(F^(c*(a+b*x)))^p],x] /; +FreeQ[{F,a,b,c,d,e,f,n,p},x] && RationalQ[m] && m>0 + + +Int[u_*PolyLog[n_,v_],x_Symbol] := + With[{w=DerivativeDivides[v,u*v,x]}, + w*PolyLog[n+1,v] /; + Not[FalseQ[w]]] /; +FreeQ[n,x] + + +Int[u_*Log[w_]*PolyLog[n_,v_],x_Symbol] := + With[{z=DerivativeDivides[v,u*v,x]}, + z*Log[w]*PolyLog[n+1,v] - + Int[SimplifyIntegrand[z*D[w,x]*PolyLog[n+1,v]/w,x],x] /; + Not[FalseQ[z]]] /; +FreeQ[n,x] && InverseFunctionFreeQ[w,x] + + + + + +(* ::Subsection::Closed:: *) +(*8.9 Product logarithm function*) + + +Int[(c_.*ProductLog[a_.+b_.*x_])^p_,x_Symbol] := + (a+b*x)*(c*ProductLog[a+b*x])^p/(b*(p+1)) + + p/(c*(p+1))*Int[(c*ProductLog[a+b*x])^(p+1)/(1+ProductLog[a+b*x]),x] /; +FreeQ[{a,b,c},x] && RationalQ[p] && p<-1 + + +Int[(c_.*ProductLog[a_.+b_.*x_])^p_.,x_Symbol] := + (a+b*x)*(c*ProductLog[a+b*x])^p/b - + p*Int[(c*ProductLog[a+b*x])^p/(1+ProductLog[a+b*x]),x] /; +FreeQ[{a,b,c},x] && Not[RationalQ[p] && p<-1] + + +Int[x_^m_.*(c_.*ProductLog[a_+b_.*x_])^p_.,x_Symbol] := + 1/b*Subst[Int[ExpandIntegrand[(c*ProductLog[x])^p,(-a/b+x/b)^m,x],x],x,a+b*x] /; +FreeQ[{a,b,c,p},x] && PositiveIntegerQ[m] + + +Int[(c_.*ProductLog[a_.*x_^n_])^p_.,x_Symbol] := + x*(c*ProductLog[a*x^n])^p - + n*p*Int[(c*ProductLog[a*x^n])^p/(1+ProductLog[a*x^n]),x] /; +FreeQ[{a,c,n,p},x] && (EqQ[n*(p-1)+1] || IntegerQ[p-1/2] && EqQ[n*(p-1/2)+1]) + + +Int[(c_.*ProductLog[a_.*x_^n_])^p_.,x_Symbol] := + x*(c*ProductLog[a*x^n])^p/(n*p+1) + + n*p/(c*(n*p+1))*Int[(c*ProductLog[a*x^n])^(p+1)/(1+ProductLog[a*x^n]),x] /; +FreeQ[{a,c,n},x] && (IntegerQ[p] && EqQ[n*(p+1)+1] || IntegerQ[p-1/2] && EqQ[n*(p+1/2)+1]) + + +Int[(c_.*ProductLog[a_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^2,x],x,1/x] /; +FreeQ[{a,c,p},x] && NegativeIntegerQ[n] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_.,x_Symbol] := + x^(m+1)*(c*ProductLog[a*x^n])^p/(m+1) - + n*p/(m+1)*Int[x^m*(c*ProductLog[a*x^n])^p/(1+ProductLog[a*x^n]),x] /; +FreeQ[{a,c,m,n,p},x] && NeQ[m+1] && +(IntegerQ[p-1/2] && IntegerQ[2*Simplify[p+(m+1)/n]] && Simplify[p+(m+1)/n]>0 || + Not[IntegerQ[p-1/2]] && IntegerQ[Simplify[p+(m+1)/n]] && Simplify[p+(m+1)/n]>=0) + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_.,x_Symbol] := + x^(m+1)*(c*ProductLog[a*x^n])^p/(m+n*p+1) + + n*p/(c*(m+n*p+1))*Int[x^m*(c*ProductLog[a*x^n])^(p+1)/(1+ProductLog[a*x^n]),x] /; +FreeQ[{a,c,m,n,p},x] && +(EqQ[m+1] || + IntegerQ[p-1/2] && IntegerQ[Simplify[p+(m+1)/n]-1/2] && Simplify[p+(m+1)/n]<0 || + Not[IntegerQ[p-1/2]] && IntegerQ[Simplify[p+(m+1)/n]] && Simplify[p+(m+1)/n]<0) + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_.,x_Symbol] := + Int[x^m*(c*ProductLog[a*x])^p/(1+ProductLog[a*x]),x] + + 1/c*Int[x^m*(c*ProductLog[a*x])^(p+1)/(1+ProductLog[a*x]),x] /; +FreeQ[{a,c,m},x] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_])^p_.,x_Symbol] := + -Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^(m+2),x],x,1/x] /; +FreeQ[{a,c,p},x] && IntegersQ[m,n] && n<0 && NeQ[m+1] + + +Int[1/(d_+d_.*ProductLog[a_.+b_.*x_]),x_Symbol] := + (a+b*x)/(b*d*ProductLog[a+b*x]) /; +FreeQ[{a,b,d},x] + + +Int[ProductLog[a_.+b_.*x_]/(d_+d_.*ProductLog[a_.+b_.*x_]),x_Symbol] := + d*x - + Int[1/(d+d*ProductLog[a+b*x]),x] /; +FreeQ[{a,b,d},x] + + +Int[(c_.*ProductLog[a_.+b_.*x_])^p_/(d_+d_.*ProductLog[a_.+b_.*x_]),x_Symbol] := + c*(a+b*x)*(c*ProductLog[a+b*x])^(p-1)/(b*d) - + c*p*Int[(c*ProductLog[a+b*x])^(p-1)/(d+d*ProductLog[a+b*x]),x] /; +FreeQ[{a,b,c,d},x] && RationalQ[p] && p>0 + + +Int[1/(ProductLog[a_.+b_.*x_]*(d_+d_.*ProductLog[a_.+b_.*x_])),x_Symbol] := + ExpIntegralEi[ProductLog[a+b*x]]/(b*d) /; +FreeQ[{a,b,d},x] + + +Int[1/(Sqrt[c_.*ProductLog[a_.+b_.*x_]]*(d_+d_.*ProductLog[a_.+b_.*x_])),x_Symbol] := + Rt[Pi*c,2]*Erfi[Sqrt[c*ProductLog[a+b*x]]/Rt[c,2]]/(b*c*d) /; +FreeQ[{a,b,c,d},x] && PosQ[c] + + +Int[1/(Sqrt[c_.*ProductLog[a_.+b_.*x_]]*(d_+d_.*ProductLog[a_.+b_.*x_])),x_Symbol] := + Rt[-Pi*c,2]*Erf[Sqrt[c*ProductLog[a+b*x]]/Rt[-c,2]]/(b*c*d) /; +FreeQ[{a,b,c,d},x] && NegQ[c] + + +Int[(c_.*ProductLog[a_.+b_.*x_])^p_/(d_+d_.*ProductLog[a_.+b_.*x_]),x_Symbol] := + (a+b*x)*(c*ProductLog[a+b*x])^p/(b*d*(p+1)) - + 1/(c*(p+1))*Int[(c*ProductLog[a+b*x])^(p+1)/(d+d*ProductLog[a+b*x]),x] /; +FreeQ[{a,b,c,d},x] && RationalQ[p] && p<-1 + + +Int[(c_.*ProductLog[a_.+b_.*x_])^p_./(d_+d_.*ProductLog[a_.+b_.*x_]),x_Symbol] := + Gamma[p+1,-ProductLog[a+b*x]]*(c*ProductLog[a+b*x])^p/(b*d*(-ProductLog[a+b*x])^p) /; +FreeQ[{a,b,c,d,p},x] + + +Int[x_^m_./(d_+d_.*ProductLog[a_+b_.*x_]),x_Symbol] := + 1/b*Subst[Int[ExpandIntegrand[1/(d+d*ProductLog[x]),(-a/b+x/b)^m,x],x],x,a+b*x] /; +FreeQ[{a,b,d},x] && PositiveIntegerQ[m] + + +Int[x_^m_.*(c_.*ProductLog[a_+b_.*x_])^p_./(d_+d_.*ProductLog[a_+b_.*x_]),x_Symbol] := + 1/b*Subst[Int[ExpandIntegrand[(c*ProductLog[x])^p/(d+d*ProductLog[x]),(-a/b+x/b)^m,x],x],x,a+b*x] /; +FreeQ[{a,b,c,d,p},x] && PositiveIntegerQ[m] + + +Int[1/(d_+d_.*ProductLog[a_.*x_^n_]),x_Symbol] := + -Subst[Int[1/(x^2*(d+d*ProductLog[a*x^(-n)])),x],x,1/x] /; +FreeQ[{a,d},x] && NegativeIntegerQ[n] + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + c*x*(c*ProductLog[a*x^n])^(p-1)/d /; +FreeQ[{a,c,d,n,p},x] && EqQ[n*(p-1)+1] + + +Int[ProductLog[a_.*x_^n_.]^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + a^p*ExpIntegralEi[-p*ProductLog[a*x^n]]/(d*n) /; +FreeQ[{a,d},x] && IntegerQ[1/n] && EqQ[p+1/n] + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + Rt[Pi*c*n,2]/(d*n*a^(1/n)*c^(1/n))*Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[c*n,2]] /; +FreeQ[{a,c,d},x] && IntegerQ[1/n] && EqQ[p-1/2+1/n] && PosQ[c*n] + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + Rt[-Pi*c*n,2]/(d*n*a^(1/n)*c^(1/n))*Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[-c*n,2]] /; +FreeQ[{a,c,d},x] && IntegerQ[1/n] && EqQ[p-1/2+1/n] && NegQ[c*n] + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + c*x*(c*ProductLog[a*x^n])^(p-1)/d - + c*(n*(p-1)+1)*Int[(c*ProductLog[a*x^n])^(p-1)/(d+d*ProductLog[a*x^n]),x] /; +FreeQ[{a,c,d},x] && RationalQ[n,p] && n>0 && n*(p-1)+1>0 + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + x*(c*ProductLog[a*x^n])^p/(d*(n*p+1)) - + 1/(c*(n*p+1))*Int[(c*ProductLog[a*x^n])^(p+1)/(d+d*ProductLog[a*x^n]),x] /; +FreeQ[{a,c,d},x] && RationalQ[n,p] && n>0 && n*p+1<0 + + +Int[(c_.*ProductLog[a_.*x_^n_])^p_./(d_+d_.*ProductLog[a_.*x_^n_]),x_Symbol] := + -Subst[Int[(c*ProductLog[a*x^(-n)])^p/(x^2*(d+d*ProductLog[a*x^(-n)])),x],x,1/x] /; +FreeQ[{a,c,d,p},x] && NegativeIntegerQ[n] + + +Int[x_^m_./(d_+d_.*ProductLog[a_.*x_]),x_Symbol] := + x^(m+1)/(d*(m+1)*ProductLog[a*x]) - + m/(m+1)*Int[x^m/(ProductLog[a*x]*(d+d*ProductLog[a*x])),x] /; +FreeQ[{a,d},x] && RationalQ[m] && m>0 + + +Int[1/(x_*(d_+d_.*ProductLog[a_.*x_])),x_Symbol] := + Log[ProductLog[a*x]]/d /; +FreeQ[{a,d},x] + + +Int[x_^m_./(d_+d_.*ProductLog[a_.*x_]),x_Symbol] := + x^(m+1)/(d*(m+1)) - + Int[x^m*ProductLog[a*x]/(d+d*ProductLog[a*x]),x] /; +FreeQ[{a,d},x] && RationalQ[m] && m<-1 + + +Int[x_^m_./(d_+d_.*ProductLog[a_.*x_]),x_Symbol] := + x^m*Gamma[m+1,-(m+1)*ProductLog[a*x]]/ + (a*d*(m+1)*E^(m*ProductLog[a*x])*(-(m+1)*ProductLog[a*x])^m) /; +FreeQ[{a,d,m},x] && Not[IntegerQ[m]] + + +Int[1/(x_*(d_+d_.*ProductLog[a_.*x_^n_.])),x_Symbol] := + Log[ProductLog[a*x^n]]/(d*n) /; +FreeQ[{a,d,n},x] + + +Int[x_^m_./(d_+d_.*ProductLog[a_.*x_^n_]),x_Symbol] := + -Subst[Int[1/(x^(m+2)*(d+d*ProductLog[a*x^(-n)])),x],x,1/x] /; +FreeQ[{a,d},x] && IntegersQ[m,n] && n<0 && NeQ[m+1] + + +Int[(c_.*ProductLog[a_.*x_^n_.])^p_./(x_*(d_+d_.*ProductLog[a_.*x_^n_.])),x_Symbol] := + (c*ProductLog[a*x^n])^p/(d*n*p) /; +FreeQ[{a,c,d,n,p},x] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + c*x^(m+1)*(c*ProductLog[a*x^n])^(p-1)/(d*(m+1)) /; +FreeQ[{a,c,d,m,n,p},x] && NeQ[m+1] && EqQ[m+n*(p-1)+1] + + +Int[x_^m_.*ProductLog[a_.*x_^n_.]^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + a^p*ExpIntegralEi[-p*ProductLog[a*x^n]]/(d*n) /; +FreeQ[{a,d,m,n},x] && IntegerQ[p] && EqQ[m+n*p+1] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_/(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + a^(p-1/2)*c^(p-1/2)*Rt[Pi*c/(p-1/2),2]*Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[c/(p-1/2),2]]/(d*n) /; +FreeQ[{a,c,d,m,n},x] && NeQ[m+1] && IntegerQ[p-1/2] && EqQ[m+n*(p-1/2)+1] && PosQ[c/(p-1/2)] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_/(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + a^(p-1/2)*c^(p-1/2)*Rt[-Pi*c/(p-1/2),2]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[-c/(p-1/2),2]]/(d*n) /; +FreeQ[{a,c,d,m,n},x] && NeQ[m+1] && IntegerQ[p-1/2] && EqQ[m+n*(p-1/2)+1] && NegQ[c/(p-1/2)] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + c*x^(m+1)*(c*ProductLog[a*x^n])^(p-1)/(d*(m+1)) - + c*(m+n*(p-1)+1)/(m+1)*Int[x^m*(c*ProductLog[a*x^n])^(p-1)/(d+d*ProductLog[a*x^n]),x] /; +FreeQ[{a,c,d,m,n,p},x] && NeQ[m+1] && RationalQ[Simplify[p+(m+1)/n]] && Simplify[p+(m+1)/n]>1 + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + x^(m+1)*(c*ProductLog[a*x^n])^p/(d*(m+n*p+1)) - + (m+1)/(c*(m+n*p+1))*Int[x^m*(c*ProductLog[a*x^n])^(p+1)/(d+d*ProductLog[a*x^n]),x] /; +FreeQ[{a,c,d,m,n,p},x] && NeQ[m+1] && RationalQ[Simplify[p+(m+1)/n]] && Simplify[p+(m+1)/n]<0 + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_./(d_+d_.*ProductLog[a_.*x_]),x_Symbol] := + x^m*Gamma[m+p+1,-(m+1)*ProductLog[a*x]]*(c*ProductLog[a*x])^p/ + (a*d*(m+1)*E^(m*ProductLog[a*x])*(-(m+1)*ProductLog[a*x])^(m+p)) /; +FreeQ[{a,c,d,m,p},x] && NeQ[m+1] + + +Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_./(d_+d_.*ProductLog[a_.*x_^n_.]),x_Symbol] := + -Subst[Int[(c*ProductLog[a*x^(-n)])^p/(x^(m+2)*(d+d*ProductLog[a*x^(-n)])),x],x,1/x] /; +FreeQ[{a,c,d,p},x] && NeQ[m+1] && IntegersQ[m,n] && n<0 + + +Int[u_,x_Symbol] := + Subst[Int[SimplifyIntegrand[(x+1)*E^x*SubstFor[ProductLog[x],u,x],x],x],x,ProductLog[x]] /; +FunctionOfQ[ProductLog[x],u,x] + + + +`) + +func resourcesRubi8SpecialFunctionsMBytes() ([]byte, error) { + return _resourcesRubi8SpecialFunctionsM, nil +} + +func resourcesRubi8SpecialFunctionsM() (*asset, error) { + bytes, err := resourcesRubi8SpecialFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/8 Special functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi91IntegrandSimplificationRulesM = []byte(`(* ::Package:: *) + +(************************************************************************) +(* This file was generated automatically by the Mathematica front end. *) +(* It contains Initialization cells from a Notebook file, which *) +(* typically will have the same name as this file except ending in *) +(* ".nb" instead of ".m". *) +(* *) +(* This file is intended to be loaded into the Mathematica kernel using *) +(* the package loading commands Get or Needs. Doing so is equivalent *) +(* to using the Evaluate Initialization Cells menu command in the front *) +(* end. *) +(* *) +(* DO NOT EDIT THIS FILE. This entire file is regenerated *) +(* automatically each time the parent Notebook file is saved in the *) +(* Mathematica front end. Any changes you make to this file will be *) +(* overwritten. *) +(************************************************************************) + + + +(* ::Code:: *) +(* Int[u_.*(v_+w_)^p_.,x_Symbol] := + Int[u*w^p,x] /; +FreeQ[p,x] && EqQ[v] *) + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_^n_.)^p_.,x_Symbol] := + Int[u*(b*x^n)^p,x] /; +FreeQ[{a,b,n,p},x] && EqQ[a] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^n_.)^p_.,x_Symbol] := + Int[u*a^p,x] /; +FreeQ[{a,b,n,p},x] && EqQ[b] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + Int[u*(b*x^n+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[j-2*n] && EqQ[a] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + Int[u*(a+c*x^(2*n))^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[j-2*n] && EqQ[b] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^n_.+c_.*x_^j_.)^p_.,x_Symbol] := + Int[u*(a+b*x^n)^p,x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[j-2*n] && EqQ[c] + + +(* ::Code:: *) +Int[u_.*(a_.*v_+b_.*v_+w_.)^p_.,x_Symbol] := + Int[u*((a+b)*v+w)^p,x] /; +FreeQ[{a,b},x] && Not[FreeQ[v,x]] + + +(* ::Code:: *) +Int[u_.*Pm_^p_,x_Symbol] := + Int[u*Pm^Simplify[p],x] /; +PolyQ[Pm,x] && Not[RationalQ[p]] && FreeQ[p,x] && RationalQ[Simplify[p]] + + +(* ::Code:: *) +Int[a_,x_Symbol] := + a*x /; +FreeQ[a,x] + + +(* ::Code:: *) +Int[a_*(b_+c_.*x_),x_Symbol] := + a*(b+c*x)^2/(2*c) /; +FreeQ[{a,b,c},x] + + +(* ::Code:: *) +Int[-u_,x_Symbol] := + Identity[-1]*Int[u,x] + + +(* ::Code:: *) +Int[Complex[0,a_]*u_,x_Symbol] := + Complex[Identity[0],a]*Int[u,x] /; +FreeQ[a,x] && EqQ[a^2,1] + + +(* ::Code:: *) +Int[a_*u_,x_Symbol] := + a*Int[u,x] /; +FreeQ[a,x] && Not[MatchQ[u, b_*v_ /; FreeQ[b,x]]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + ShowStep["","Int[a*u + b*v + \[CenterEllipsis],x]","a*Integrate[u,x] + b*Integrate[v,x] + \[CenterEllipsis]",Hold[ + IntSum[u,x]]] /; +SimplifyFlag && SumQ[u], + +Int[u_,x_Symbol] := + IntSum[u,x] /; +SumQ[u]] + + +(* ::Code:: *) +Int[(c_.*x_)^m_.*u_,x_Symbol] := + Int[ExpandIntegrand[(c*x)^m*u,x],x] /; +FreeQ[{c,m},x] && SumQ[u] && Not[LinearQ[u,x]] && Not[MatchQ[u,a_+b_.*v_ /; FreeQ[{a,b},x] && InverseFunctionQ[v]]] + + +(* ::Code:: *) +Int[u_.*v_^m_.*(b_*v_)^n_,x_Symbol] := + 1/b^m*Int[u*(b*v)^(m+n),x] /; +FreeQ[{b,n},x] && IntegerQ[m] + + +(* ::Code:: *) +Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + a^(m+1/2)*b^(n-1/2)*Sqrt[b*v]/Sqrt[a*v]*Int[u*v^(m+n),x] /; +FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && PositiveIntegerQ[n+1/2] && IntegerQ[m+n] + + +(* ::Code:: *) +(* Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + b^(n-1/2)*Sqrt[b*v]/(a^(n-1/2)*Sqrt[a*v])*Int[u*(a*v)^(m+n),x] /; +FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && PositiveIntegerQ[n+1/2] && Not[IntegerQ[m+n]] *) + + +(* ::Code:: *) +Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + a^(m-1/2)*b^(n+1/2)*Sqrt[a*v]/Sqrt[b*v]*Int[u*v^(m+n),x] /; +FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && NegativeIntegerQ[n-1/2] && IntegerQ[m+n] + + +(* ::Code:: *) +(* Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + b^(n+1/2)*Sqrt[a*v]/(a^(n+1/2)*Sqrt[b*v])*Int[u*(a*v)^(m+n),x] /; +FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && NegativeIntegerQ[n-1/2] && Not[IntegerQ[m+n]] *) + + +(* ::Code:: *) +Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + a^(m+n)*(b*v)^n/(a*v)^n*Int[u*v^(m+n),x] /; +FreeQ[{a,b,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[m+n] + + +(* ::Code:: *) +Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol] := + b^IntPart[n]*(b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])*Int[u*(a*v)^(m+n),x] /; +FreeQ[{a,b,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[m+n]] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*v_)^m_.*(c_+d_.*v_)^n_.,x_Symbol] := + (b/d)^m*Int[u*(c+d*v)^(m+n),x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[b*c-a*d] && IntegerQ[m] && (Not[IntegerQ[n]] || SimplerQ[c+d*x,a+b*x]) + + +(* ::Code:: *) +Int[u_.*(a_+b_.*v_)^m_*(c_+d_.*v_)^n_,x_Symbol] := + (b/d)^m*Int[u*(c+d*v)^(m+n),x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[b*c-a*d] && PositiveQ[b/d] && Not[IntegerQ[m] || IntegerQ[n]] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*v_)^m_*(c_+d_.*v_)^n_,x_Symbol] := + (a+b*v)^m/(c+d*v)^m*Int[u*(c+d*v)^(m+n),x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[b*c-a*d] && Not[IntegerQ[m] || IntegerQ[n] || PositiveQ[b/d]] + + +(* ::Code:: *) +(* Int[u_.*(a_+b_.*v_)^m_.*(c_+d_.*v_)^m_.,x_Symbol] := + Int[u*(a*c+b*d*v^2)^m,x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[b*c+a*d] && (IntegerQ[m] || PositiveQ[a] && PositiveQ[c]) && + (Not[AlgebraicFunctionQ[u,x]] || Not[MatchQ[v,e_.*x^n_. /; FreeQ[{e,n},x]]]) *) + + +(* ::Code:: *) +(* Int[u_.*(a_+b_.*v_)^m_*(c_+d_.*v_)^m_,x_Symbol] := + (a+b*v)^FracPart[m]*(c+d*v)^FracPart[m]/(a*c+b*d*v^2)^FracPart[m]*Int[u*(a*c+b*d*v^2)^m,x] /; +FreeQ[{a,b,c,d,m},x] && EqQ[b*c+a*d] && Not[IntegerQ[m]] && + (Not[AlgebraicFunctionQ[u,x]] || Not[MatchQ[v,e_.*x^n_. /; FreeQ[{e,n},x]]]) *) + + +(* ::Code:: *) +Int[u_.*(a_.*v_)^m_*(b_.*v_+c_.*v_^2),x_Symbol] := + 1/a*Int[u*(a*v)^(m+1)*(b+c*v),x] /; +FreeQ[{a,b,c},x] && RationalQ[m] && m<=-1 + + +(* ::Code:: *) +Int[u_.*(a_+b_.*v_)^m_*(A_.+B_.*v_+C_.*v_^2),x_Symbol] := + 1/b^2*Int[u*(a+b*v)^(m+1)*Simp[b*B-a*C+b*C*v,x],x] /; +FreeQ[{a,b,A,B,C},x] && EqQ[A*b^2-a*b*B+a^2*C] && RationalQ[m] && m<=-1 + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_^n_.)^m_.*(c_+d_.*x_^q_.)^p_.,x_Symbol] := + (d/a)^p*Int[u*(a+b*x^n)^(m+p)/x^(n*p),x] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[n+q] && IntegerQ[p] && EqQ[a*c-b*d] && Not[IntegerQ[m] && NegQ[n]] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_^n_.)^m_.*(c_+d_.*x_^j_)^p_.,x_Symbol] := + (-b^2/d)^m*Int[u*(a-b*x^n)^(-m),x] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[j-2*n] && EqQ[m+p] && EqQ[b^2*c+a^2*d] && PositiveQ[a] && NegativeQ[d] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_+c_.*x_^2)^p_.,x_Symbol] := + Int[u*Cancel[(b/2+c*x)^(2*p)/c^p],x] /; +FreeQ[{a,b,c},x] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/c^p*Int[u*(b/2+c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] + + +(* ::Code:: *) +Int[Qr_/Pq_,x_Symbol] := + With[{q=Expon[Pq,x],r=Expon[Qr,x]}, + Coeff[Qr,x,r]*Log[RemoveContent[Pq,x]]/(q*Coeff[Pq,x,q]) /; + EqQ[r,q-1] && EqQ[Coeff[Qr,x,r]*D[Pq,x],q*Coeff[Pq,x,q]*Qr]] /; +PolyQ[Pq,x] && PolyQ[Qr,x] + + +(* ::Code:: *) +Int[Qr_*Pq_^p_.,x_Symbol] := + With[{q=Expon[Pq,x],r=Expon[Qr,x]}, + Coeff[Qr,x,r]*Pq^(p+1)/(q*(p+1)*Coeff[Pq,x,q]) /; + EqQ[r,q-1] && EqQ[Coeff[Qr,x,r]*D[Pq,x],q*Coeff[Pq,x,q]*Qr]] /; +FreeQ[p,x] && PolyQ[Pq,x] && PolyQ[Qr,x] && NeQ[p,-1] + + +(* ::Code:: *) +Int[x_^m_.*(a1_+b1_.*x_^n_.)^p_*(a2_+b2_.*x_^n_.)^p_,x_Symbol] := + (a1+b1*x^n)^(p+1)*(a2+b2*x^n)^(p+1)/(2*b1*b2*n*(p+1)) /; +FreeQ[{a1,b1,a2,b2,m,n,p},x] && EqQ[a2*b1+a1*b2,0] && EqQ[m-2*n+1,0] && NeQ[p,-1] + + +(* ::Code:: *) +Int[Qr_*(a_.+b_.*Pq_^n_.)^p_.,x_Symbol] := + With[{q=Expon[Pq,x],r=Expon[Qr,x]}, + Coeff[Qr,x,r]/(q*Coeff[Pq,x,q])*Subst[Int[(a+b*x^n)^p,x],x,Pq] /; + EqQ[r,q-1] && EqQ[Coeff[Qr,x,r]*D[Pq,x],q*Coeff[Pq,x,q]*Qr]] /; +FreeQ[{a,b,n,p},x] && PolyQ[Pq,x] && PolyQ[Qr,x] + + +(* ::Code:: *) +Int[Qr_*(a_.+b_.*Pq_^n_.+c_.*Pq_^n2_.)^p_.,x_Symbol] := + Module[{q=Expon[Pq,x],r=Expon[Qr,x]}, + Coeff[Qr,x,r]/(q*Coeff[Pq,x,q])*Subst[Int[(a+b*x^n+c*x^(2*n))^p,x],x,Pq] /; + EqQ[r,q-1] && EqQ[Coeff[Qr,x,r]*D[Pq,x],q*Coeff[Pq,x,q]*Qr]] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2,2*n] && PolyQ[Pq,x] && PolyQ[Qr,x] + + +(* ::Code:: *) +Int[u_.*(a_.*x_^p_.+b_.*x_^q_.)^m_.,x_Symbol] := + Int[u*x^(m*p)*(a+b*x^(q-p))^m,x] /; +FreeQ[{a,b,p,q},x] && IntegerQ[m] && PosQ[q-p] + + +(* ::Code:: *) +Int[u_.*(a_.*x_^p_.+b_.*x_^q_.+c_.*x_^r_.)^m_.,x_Symbol] := + Int[u*x^(m*p)*(a+b*x^(q-p)+c*x^(r-p))^m,x] /; +FreeQ[{a,b,c,p,q,r},x] && IntegerQ[m] && PosQ[q-p] && PosQ[r-p] + + +(* ::Code:: *) +Int[u_.*Pq_^m_*Qr_^p_,x_Symbol] := + Module[{gcd=PolyGCD[Pq,Qr,x]}, + Int[u*gcd^(m+p)*PolynomialQuotient[Pq,gcd,x]^m*PolynomialQuotient[Qr,gcd,x]^p,x] /; + NeQ[gcd-1]] /; +PositiveIntegerQ[m] && NegativeIntegerQ[p] && PolyQ[Pq,x] && PolyQ[Qr,x] + + +(* ::Code:: *) +Int[u_.*Pq_*Qr_^p_,x_Symbol] := + Module[{gcd=PolyGCD[Pq,Qr,x]}, + Int[u*gcd^(p+1)*PolynomialQuotient[Pq,gcd,x]*PolynomialQuotient[Qr,gcd,x]^p,x] /; + NeQ[gcd-1]] /; +NegativeIntegerQ[p] && PolyQ[Pq,x] && PolyQ[Qr,x] + + + +`) + +func resourcesRubi91IntegrandSimplificationRulesMBytes() ([]byte, error) { + return _resourcesRubi91IntegrandSimplificationRulesM, nil +} + +func resourcesRubi91IntegrandSimplificationRulesM() (*asset, error) { + bytes, err := resourcesRubi91IntegrandSimplificationRulesMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/9.1 Integrand simplification rules.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi92DerivativeIntegrationRulesM = []byte(`(* ::Package:: *) + +(************************************************************************) +(* This file was generated automatically by the Mathematica front end. *) +(* It contains Initialization cells from a Notebook file, which *) +(* typically will have the same name as this file except ending in *) +(* ".nb" instead of ".m". *) +(* *) +(* This file is intended to be loaded into the Mathematica kernel using *) +(* the package loading commands Get or Needs. Doing so is equivalent *) +(* to using the Evaluate Initialization Cells menu command in the front *) +(* end. *) +(* *) +(* DO NOT EDIT THIS FILE. This entire file is regenerated *) +(* automatically each time the parent Notebook file is saved in the *) +(* Mathematica front end. Any changes you make to this file will be *) +(* overwritten. *) +(************************************************************************) + + + +(* ::Code:: *) +Int[Derivative[n_][f_][x_],x_Symbol] := + Derivative[n-1][f][x] /; +FreeQ[{f,n},x] + + +(* ::Code:: *) +Int[(c_.*F_^(a_.+b_.*x_))^p_.*Derivative[n_][f_][x_],x_Symbol] := + (c*F^(a+b*x))^p*Derivative[n-1][f][x] - b*p*Log[F]*Int[(c*F^(a+b*x))^p*Derivative[n-1][f][x],x] /; +FreeQ[{a,b,c,f,F,p},x] && PositiveIntegerQ[n] + + +(* ::Code:: *) +Int[(c_.*F_^(a_.+b_.*x_))^p_.*Derivative[n_][f_][x_],x_Symbol] := + (c*F^(a+b*x))^p*Derivative[n][f][x]/(b*p*Log[F]) - 1/(b*p*Log[F])*Int[(c*F^(a+b*x))^p*Derivative[n+1][f][x],x] /; +FreeQ[{a,b,c,f,F,p},x] && NegativeIntegerQ[n] + + +(* ::Code:: *) +Int[Sin[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] := + Sin[a+b*x]*Derivative[n-1][f][x] - b*Int[Cos[a+b*x]*Derivative[n-1][f][x],x] /; +FreeQ[{a,b,f},x] && PositiveIntegerQ[n] + + +(* ::Code:: *) +Int[Cos[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] := + Cos[a+b*x]*Derivative[n-1][f][x] + b*Int[Sin[a+b*x]*Derivative[n-1][f][x],x] /; +FreeQ[{a,b,f},x] && PositiveIntegerQ[n] + + +(* ::Code:: *) +Int[Sin[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] := + -Cos[a+b*x]*Derivative[n][f][x]/b + 1/b*Int[Cos[a+b*x]*Derivative[n+1][f][x],x] /; +FreeQ[{a,b,f},x] && NegativeIntegerQ[n] + + +(* ::Code:: *) +Int[Cos[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] := + Sin[a+b*x]*Derivative[n][f][x]/b - 1/b*Int[Sin[a+b*x]*Derivative[n+1][f][x],x] /; +FreeQ[{a,b,f},x] && NegativeIntegerQ[n] + + +(* ::Code:: *) +Int[u_*Derivative[n_][f_][x_],x_Symbol] := + Subst[Int[SimplifyIntegrand[SubstFor[Derivative[n-1][f][x],u,x],x],x],x,Derivative[n-1][f][x]] /; +FreeQ[{f,n},x] && FunctionOfQ[Derivative[n-1][f][x],u,x] + + +(* ::Code:: *) +Int[u_*(a_.*Derivative[1][f_][x_]*g_[x_]+a_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] := + a*Subst[Int[SimplifyIntegrand[SubstFor[f[x]*g[x],u,x],x],x],x,f[x]*g[x]] /; +FreeQ[{a,f,g},x] && FunctionOfQ[f[x]*g[x],u,x] + + +(* ::Code:: *) +Int[u_*(a_.*Derivative[m_][f_][x_]*g_[x_]+a_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] := + a*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*g[x],u,x],x],x],x,Derivative[m-1][f][x]*g[x]] /; +FreeQ[{a,f,g,m},x] && EqQ[m1-m+1] && FunctionOfQ[Derivative[m-1][f][x]*g[x],u,x] + + +(* ::Code:: *) +Int[u_*(a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+a_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] := + a*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*Derivative[n-1][g][x],u,x],x],x],x,Derivative[m-1][f][x]*Derivative[n-1][g][x]] /; +FreeQ[{a,f,g,m,n},x] && EqQ[m1-m+1] && EqQ[n1-n+1] && FunctionOfQ[Derivative[m-1][f][x]*Derivative[n-1][g][x],u,x] + + +(* ::Code:: *) +Int[u_*f_[x_]^p_.*(a_.*Derivative[1][f_][x_]*g_[x_]+b_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] := + b*Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p+1)*g[x],u,x],x],x],x,f[x]^(p+1)*g[x]] /; +FreeQ[{a,b,f,g,p},x] && EqQ[a-b*(p+1)] && FunctionOfQ[f[x]^(p+1)*g[x],u,x] + + +(* ::Code:: *) +Int[u_*Derivative[m1_][f_][x_]^p_.* + (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] := + b*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*g[x],u,x],x],x],x, + Derivative[m-1][f][x]^(p+1)*g[x]] /; +FreeQ[{a,b,f,g,m,p},x] && EqQ[m1-m+1] && EqQ[a-b*(p+1)] && FunctionOfQ[Derivative[m-1][f][x]^(p+1)*g[x],u,x] + + +(* ::Code:: *) +Int[u_*g_[x_]^q_.* + (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] := + a*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*g[x]^(q+1),u,x],x],x],x, + Derivative[m-1][f][x]*g[x]^(q+1)] /; +FreeQ[{a,b,f,g,m,q},x] && EqQ[m1-m+1] && EqQ[a*(q+1)-b] && FunctionOfQ[Derivative[m-1][f][x]*g[x]^(q+1),u,x] + + +(* ::Code:: *) +Int[u_*Derivative[m1_][f_][x_]^p_.* + (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] := + b*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x],u,x],x],x],x, + Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]] /; +FreeQ[{a,b,f,g,m,n,p},x] && EqQ[m1-m+1] && EqQ[n1-n+1] && EqQ[a-b*(p+1)] && + FunctionOfQ[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x],u,x] + + +(* ::Code:: *) +Int[u_*f_[x_]^p_.*g_[x_]^q_.*(a_.*Derivative[1][f_][x_]*g_[x_]+b_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] := + a/(p+1)*Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p+1)*g[x]^(q+1),u,x],x],x],x,f[x]^(p+1)*g[x]^(q+1)] /; +FreeQ[{a,b,f,g,p,q},x] && EqQ[a*(q+1)-b*(p+1)] && FunctionOfQ[f[x]^(p+1)*g[x]^(q+1),u,x] + + +(* ::Code:: *) +Int[u_*Derivative[m1_][f_][x_]^p_.*g_[x_]^q_.* + (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] := + a/(p+1)*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*g[x]^(q+1),u,x],x],x],x, + Derivative[m-1][f][x]^(p+1)*g[x]^(q+1)] /; +FreeQ[{a,b,f,g,m,p,q},x] && EqQ[m1-m+1] && EqQ[a*(q+1)-b*(p+1)] && FunctionOfQ[Derivative[m-1][f][x]^(p+1)*g[x]^(q+1),u,x] + + +(* ::Code:: *) +Int[u_*Derivative[m1_][f_][x_]^p_.*Derivative[n1_][g_][x_]^q_.* + (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] := + a/(p+1)*Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1),u,x],x],x],x, + Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1)] /; +FreeQ[{a,b,f,g,m,n,p,q},x] && EqQ[m1-m+1] && EqQ[n1-n+1] && EqQ[a*(q+1)-b*(p+1)] && + FunctionOfQ[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1),u,x] + + +(* ::Code:: *) +Int[f_'[x_]*g_[x_] + f_[x_]*g_'[x_],x_Symbol] := + f[x]*g[x] /; +FreeQ[{f,g},x] + + +(* ::Code:: *) +Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/g_[x_]^2,x_Symbol] := + f[x]/g[x] /; +FreeQ[{f,g},x] + + +(* ::Code:: *) +Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/(f_[x_]*g_[x_]),x_Symbol] := + Log[f[x]/g[x]] /; +FreeQ[{f,g},x] + + + +`) + +func resourcesRubi92DerivativeIntegrationRulesMBytes() ([]byte, error) { + return _resourcesRubi92DerivativeIntegrationRulesM, nil +} + +func resourcesRubi92DerivativeIntegrationRulesM() (*asset, error) { + bytes, err := resourcesRubi92DerivativeIntegrationRulesMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/9.2 Derivative integration rules.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi93PiecewiseLinearFunctionsM = []byte(`(* ::Package:: *) + +(************************************************************************) +(* This file was generated automatically by the Mathematica front end. *) +(* It contains Initialization cells from a Notebook file, which *) +(* typically will have the same name as this file except ending in *) +(* ".nb" instead of ".m". *) +(* *) +(* This file is intended to be loaded into the Mathematica kernel using *) +(* the package loading commands Get or Needs. Doing so is equivalent *) +(* to using the Evaluate Initialization Cells menu command in the front *) +(* end. *) +(* *) +(* DO NOT EDIT THIS FILE. This entire file is regenerated *) +(* automatically each time the parent Notebook file is saved in the *) +(* Mathematica front end. Any changes you make to this file will be *) +(* overwritten. *) +(************************************************************************) + + + +(* ::Code:: *) +Int[u_^m_.,x_Symbol] := + Module[{c=Simplify[D[u,x]]}, + 1/c*Subst[Int[x^m,x],x,u]] /; +FreeQ[m,x] && PiecewiseLinearQ[u,x] + + +(* ::Code:: *) +Int[v_/u_,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + b*x/a - (b*u-a*v)/a*Int[1/u,x] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[v_^n_/u_,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + v^n/(a*n) - (b*u-a*v)/a*Int[v^(n-1)/u,x] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] && RationalQ[n] && n>0 && n!=1 + + +(* ::Code:: *) +Int[1/(u_*v_),x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + b/(b*u-a*v)*Int[1/v,x] - a/(b*u-a*v)*Int[1/u,x] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[1/(u_*Sqrt[v_]),x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + 2*ArcTan[Sqrt[v]/Rt[(b*u-a*v)/a,2]]/(a*Rt[(b*u-a*v)/a,2]) /; + NeQ[b*u-a*v] && PosQ[(b*u-a*v)/a]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[1/(u_*Sqrt[v_]),x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + -2*ArcTanh[Sqrt[v]/Rt[-(b*u-a*v)/a,2]]/(a*Rt[-(b*u-a*v)/a,2]) /; + NeQ[b*u-a*v] && NegQ[(b*u-a*v)/a]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[v_^n_/u_,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + v^(n+1)/((n+1)*(b*u-a*v)) - + a*(n+1)/((n+1)*(b*u-a*v))*Int[v^(n+1)/u,x] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] && RationalQ[n] && n<-1 + + +(* ::Code:: *) +Int[v_^n_/u_,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + v^(n+1)/((n+1)*(b*u-a*v))*Hypergeometric2F1[1,n+1,n+2,-a*v/(b*u-a*v)] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] && Not[IntegerQ[n]] + + +(* ::Code:: *) +Int[1/(Sqrt[u_]*Sqrt[v_]),x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + 2/Rt[a*b,2]*ArcTanh[Rt[a*b,2]*Sqrt[u]/(a*Sqrt[v])] /; + NeQ[b*u-a*v] && PosQ[a*b]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[1/(Sqrt[u_]*Sqrt[v_]),x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + 2/Rt[-a*b,2]*ArcTan[Rt[-a*b,2]*Sqrt[u]/(a*Sqrt[v])] /; + NeQ[b*u-a*v] && NegQ[a*b]] /; +PiecewiseLinearQ[u,v,x] + + +(* ::Code:: *) +Int[u_^m_*v_^n_,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + -u^(m+1)*v^(n+1)/((m+1)*(b*u-a*v)) /; + NeQ[b*u-a*v]] /; +FreeQ[{m,n},x] && PiecewiseLinearQ[u,v,x] && EqQ[m+n+2] && NeQ[m+1] + + +(* ::Code:: *) +Int[u_^m_*v_^n_.,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + u^(m+1)*v^n/(a*(m+1)) - + b*n/(a*(m+1))*Int[u^(m+1)*v^(n-1),x] /; + NeQ[b*u-a*v]] /; +FreeQ[{m,n},x] && PiecewiseLinearQ[u,v,x] (* && NeQ[m+n+2] *) && NeQ[m+1] && ( + RationalQ[m,n] && m<-1 && n>0 && Not[IntegerQ[m+n] && m+n+2<0 && (FractionQ[m] || 2*n+m+1>=0)] || + PositiveIntegerQ[n,m] && n<=m || +(*NegativeIntegerQ[n,m] && n<=m || *) + PositiveIntegerQ[n] && Not[IntegerQ[m]] || + NegativeIntegerQ[m] && Not[IntegerQ[n]]) + + +(* ::Code:: *) +Int[u_^m_*v_^n_.,x_Symbol] := + Module[{a=Simplify[D[u,x]],b=Simplify[D[v,x]]}, + u^(m+1)*v^n/(a*(m+n+1)) - + n*(b*u-a*v)/(a*(m+n+1))*Int[u^m*v^(n-1),x] /; + NeQ[b*u-a*v]] /; +PiecewiseLinearQ[u,v,x] && NeQ[m+n+2] && RationalQ[n] && n>0 && NeQ[m+n+1] && + Not[PositiveIntegerQ[m] && (Not[IntegerQ[n]] || 00 + + +(* ::Code:: *) +(* Int[u_^n_.*Log[a_.+b_.*x_]/(a_.+b_.*x_),x_Symbol] := + Module[{c=Simplify[D[u,x]]}, + u^n*Log[a+b*x]^2/(2*b) - + c*n/(2*b)*Int[u^(n-1)*Log[a+b*x]^2,x]] /; +FreeQ[{a,b},x] && PiecewiseLinearQ[u,x] && RationalQ[n] && n>0 *) + + +(* ::Code:: *) +Int[u_^n_.*(a_.+b_.*x_)^m_.*Log[a_.+b_.*x_],x_Symbol] := + Module[{c=Simplify[D[u,x]]}, + u^n*(a+b*x)^(m+1)*Log[a+b*x]/(b*(m+1)) - + 1/(m+1)*Int[u^n*(a+b*x)^m,x] - + c*n/(b*(m+1))*Int[u^(n-1)*(a+b*x)^(m+1)*Log[a+b*x],x]] /; +FreeQ[{a,b,m},x] && PiecewiseLinearQ[u,x] && Not[LinearQ[u,x]] && RationalQ[n] && n>0 && NeQ[m+1] + + + +`) + +func resourcesRubi93PiecewiseLinearFunctionsMBytes() ([]byte, error) { + return _resourcesRubi93PiecewiseLinearFunctionsM, nil +} + +func resourcesRubi93PiecewiseLinearFunctionsM() (*asset, error) { + bytes, err := resourcesRubi93PiecewiseLinearFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/9.3 Piecewise linear functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi94MiscellaneousIntegrationRulesM = []byte(`(* ::Package:: *) + +(************************************************************************) +(* This file was generated automatically by the Mathematica front end. *) +(* It contains Initialization cells from a Notebook file, which *) +(* typically will have the same name as this file except ending in *) +(* ".nb" instead of ".m". *) +(* *) +(* This file is intended to be loaded into the Mathematica kernel using *) +(* the package loading commands Get or Needs. Doing so is equivalent *) +(* to using the Evaluate Initialization Cells menu command in the front *) +(* end. *) +(* *) +(* DO NOT EDIT THIS FILE. This entire file is regenerated *) +(* automatically each time the parent Notebook file is saved in the *) +(* Mathematica front end. Any changes you make to this file will be *) +(* overwritten. *) +(************************************************************************) + + + +(* ::Code:: *) +(* Int[u_.*(c_.*x_^n_)^p_,x_Symbol] := + c^FracPart[p]*(c*x^n)^FracPart[p]/x^(n*FracPart[p])*Int[u*x^(n*p),x] /; +FreeQ[{c,n,p},x] && Not[IntegerQ[p]] *) + + +(* ::Code:: *) +Int[u_*(c_.*(a_.+b_.* x_)^n_)^p_,x_Symbol] := + c^IntPart[p]*(c*(a+b*x)^n)^FracPart[p]/(a+b*x)^(n*FracPart[p])*Int[u*(a+b*x)^(n*p),x] /; +FreeQ[{a,b,c,n,p},x] && Not[IntegerQ[p]] && Not[MatchQ[u, x^n1_.*v_. /; EqQ[Simplify[n-n1-1]]]] + + +(* ::Code:: *) +Int[u_.*(c_.*(d_*(a_.+b_.* x_))^p_)^q_,x_Symbol] := + (c*(d*(a+b*x))^p)^q/(a+b*x)^(p*q)*Int[u*(a+b*x)^(p*q),x] /; +FreeQ[{a,b,c,d,p,q},x] && Not[IntegerQ[p]] && Not[IntegerQ[q]] + + +(* ::Code:: *) +Int[u_.*(c_.*(d_.*(a_.+b_.* x_)^n_)^p_)^q_,x_Symbol] := + (c*(d*(a+b*x)^n)^p)^q/(a+b*x)^(n*p*q)*Int[u*(a+b*x)^(n*p*q),x] /; +FreeQ[{a,b,c,d,n,p,q},x] && Not[IntegerQ[p]] && Not[IntegerQ[q]] + + +(* ::Code:: *) +Int[(a_.+b_.*F_[c_.*Sqrt[d_.+e_.*x_]/Sqrt[f_.+g_.*x_]])^n_./(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + g/C*Subst[Int[(a+b*F[c*x])^n/x,x],x,Sqrt[d+e*x]/Sqrt[f+g*x]] /; +FreeQ[{a,b,c,d,e,f,g,A,B,C,F},x] && EqQ[e+g] && EqQ[d+f-2] && EqQ[A*e^2+C*d*f] && EqQ[2*C*(d-1)-B*e] + + +(* ::Code:: *) +Int[(a_.+b_.*F_[c_.*Sqrt[1+e_.*x_]/Sqrt[1+g_.*x_]])^n_./(A_.+C_.*x_^2),x_Symbol] := + g/C*Subst[Int[(a+b*F[c*x])^n/x,x],x,Sqrt[1+e*x]/Sqrt[1+g*x]] /; +FreeQ[{a,b,c,e,g,A,C,F},x] && EqQ[e+g] && EqQ[A*e^2+C] + + +(* ::Code:: *) +Int[(a_.+b_.*F_^(c_.*Sqrt[d_.+e_.*x_]/Sqrt[f_.+g_.*x_]))^n_./(A_.+B_.*x_+C_.*x_^2),x_Symbol] := + g/C*Subst[Int[(a+b*F^(c*x))^n/x,x],x,Sqrt[d+e*x]/Sqrt[f+g*x]] /; +FreeQ[{a,b,c,d,e,f,g,A,B,C,F},x] && EqQ[e+g] && EqQ[d+f-2] && EqQ[A*e^2+C*d*f] && EqQ[2*C*(d-1)-B*e] + + +(* ::Code:: *) +Int[(a_.+b_.*F_^(c_.*Sqrt[1+e_.*x_]/Sqrt[1+g_.*x_]))^n_./(A_.+C_.*x_^2),x_Symbol] := + g/C*Subst[Int[(a+b*F^(c*x))^n/x,x],x,Sqrt[1+e*x]/Sqrt[1+g*x]] /; +FreeQ[{a,b,c,e,g,A,C,F},x] && EqQ[e+g] && EqQ[A*e^2+C] + + +(* ::Code:: *) +Int[u_/y_,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Log[RemoveContent[y,x]] /; + Not[FalseQ[q]]] + + +(* ::Code:: *) +Int[u_/(y_*w_),x_Symbol] := + With[{q=DerivativeDivides[y*w,u,x]}, + q*Log[RemoveContent[y*w,x]] /; + Not[FalseQ[q]]] + + +(* ::Code:: *) +Int[u_*y_^m_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*y^(m+1)/(m+1) /; + Not[FalseQ[q]]] /; +FreeQ[m,x] && NeQ[m+1] + + +(* ::Code:: *) +Int[u_*y_^m_.*z_^n_.,x_Symbol] := + With[{q=DerivativeDivides[y*z,u*z^(n-m),x]}, + q*y^(m+1)*z^(m+1)/(m+1) /; + Not[FalseQ[q]]] /; +FreeQ[{m,n},x] && NeQ[m+1] + + +(* ::Code:: *) +Int[u_,x_Symbol] := + With[{v=SimplifyIntegrand[u,x]}, + Int[v,x] /; + SimplerIntegrandQ[v,u,x]] + + +(* ::Code:: *) +Int[u_.*(e_.*Sqrt[a_.+b_.*x_^n_.]+f_.*Sqrt[c_.+d_.*x_^n_.])^m_,x_Symbol] := + (a*e^2-c*f^2)^m*Int[ExpandIntegrand[u*(e*Sqrt[a+b*x^n]-f*Sqrt[c+d*x^n])^(-m),x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NegativeIntegerQ[m] && EqQ[b*e^2-d*f^2] + + +(* ::Code:: *) +Int[u_.*(e_.*Sqrt[a_.+b_.*x_^n_.]+f_.*Sqrt[c_.+d_.*x_^n_.])^m_,x_Symbol] := + (b*e^2-d*f^2)^m*Int[ExpandIntegrand[u*x^(m*n)*(e*Sqrt[a+b*x^n]-f*Sqrt[c+d*x^n])^(-m),x],x] /; +FreeQ[{a,b,c,d,e,f,n},x] && NegativeIntegerQ[m] && EqQ[a*e^2-c*f^2] + + +(* ::Code:: *) +Int[u_^m_.*(a_.*u_^n_+v_)^p_.*w_,x_Symbol] := + Int[u^(m+n*p)*(a+u^(-n)*v)^p*w,x] /; +FreeQ[{a,m,n},x] && IntegerQ[p] && Not[PositiveQ[n]] && Not[FreeQ[v,x]] + + +(* ::Code:: *) +Int[u_*(a_.+b_.*y_)^m_.*(c_.+d_.*v_)^n_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(a+b*x)^m*(c+d*x)^n,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,c,d,m,n},x] && EqQ[y-v] + + +(* ::Code:: *) +Int[u_*(a_.+b_.*y_)^m_.*(c_.+d_.*v_)^n_.*(e_.+f_.*w_)^p_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[y-v] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_*(a_.+b_.*y_)^m_.*(c_.+d_.*v_)^n_.*(e_.+f_.*w_)^p_.*(g_.+h_.*z_)^q_.,x_Symbol] := + With[{r=DerivativeDivides[y,u,x]}, + r*Subst[Int[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p*(g+h*x)^q,x],x,y] /; + Not[FalseQ[r]]] /; +FreeQ[{a,b,c,d,e,f,g,h,m,n,p,q},x] && EqQ[y-v] && EqQ[y-w] && EqQ[y-z] + + +(* ::Code:: *) +Int[u_.*(a_+b_.*y_^n_),x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + a*Int[u,x] + b*q*Subst[Int[x^n,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,n},x] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*y_^n_)^p_,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(a+b*x^n)^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,n,p},x] + + +(* ::Code:: *) +Int[u_.*v_^m_.*(a_.+b_.*y_^n_)^p_.,x_Symbol] := + Module[{q,r}, + q*r*Subst[Int[x^m*(a+b*x^n)^p,x],x,y] /; + Not[FalseQ[r=Divides[y^m,v^m,x]]] && Not[FalseQ[q=DerivativeDivides[y,u,x]]]] /; +FreeQ[{a,b,m,n,p},x] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*y_^n_+c_.*v_^n2_.)^p_,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(a+b*x^n+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && EqQ[y-v] + + +(* ::Code:: *) +Int[u_.*(A_+B_.*y_^n_)(a_.+b_.*v_^n_+c_.*w_^n2_.)^p_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(A+B*x^n)*(a+b*x^n+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,c,A,B,n,p},x] && EqQ[n2-2*n] && EqQ[y-v] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_.*(A_+B_.*y_^n_)(a_.+c_.*w_^n2_.)^p_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(A+B*x^n)*(a+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,c,A,B,n,p},x] && EqQ[n2-2*n] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_.*v_^m_.*(a_.+b_.*y_^n_+c_.*w_^n2_.)^p_.,x_Symbol] := + Module[{q,r}, + q*r*Subst[Int[x^m*(a+b*x^n+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[r=Divides[y^m,v^m,x]]] && Not[FalseQ[q=DerivativeDivides[y,u,x]]]] /; +FreeQ[{a,b,c,m,n,p},x] && EqQ[n2-2*n] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_.*z_^m_.*(A_+B_.*y_^n_)*(a_.+b_.*v_^n_+c_.*w_^n2_.)^p_.,x_Symbol] := + Module[{q,r}, + q*r*Subst[Int[x^m*(A+B*x^n)*(a+b*x^n+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[r=Divides[y^m,z^m,x]]] && Not[FalseQ[q=DerivativeDivides[y,u,x]]]] /; +FreeQ[{a,b,c,A,B,m,n,p},x] && EqQ[n2-2*n] && EqQ[y-v] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_.*z_^m_.*(A_+B_.*y_^n_)*(a_.+c_.*w_^n2_.)^p_.,x_Symbol] := + Module[{q,r}, + q*r*Subst[Int[x^m*(A+B*x^n)*(a+c*x^(2*n))^p,x],x,y] /; + Not[FalseQ[r=Divides[y^m,z^m,x]]] && Not[FalseQ[q=DerivativeDivides[y,u,x]]]] /; +FreeQ[{a,c,A,B,m,n,p},x] && EqQ[n2-2*n] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*y_^n_)^m_.*(c_.+d_.*v_^n_)^p_.,x_Symbol] := + With[{q=DerivativeDivides[y,u,x]}, + q*Subst[Int[(a+b*x^n)^m*(c+d*x^n)^p,x],x,y] /; + Not[FalseQ[q]]] /; +FreeQ[{a,b,c,d,m,n,p},x] && EqQ[y-v] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*y_^n_)^m_.*(c_.+d_.*v_^n_)^p_.*(e_.+f_.*w_^n_)^q_.,x_Symbol] := + With[{r=DerivativeDivides[y,u,x]}, + r*Subst[Int[(a+b*x^n)^m*(c+d*x^n)^p*(e+f*x^n)^q,x],x,y] /; + Not[FalseQ[r]]] /; +FreeQ[{a,b,c,d,e,f,m,n,p,q},x] && EqQ[y-v] && EqQ[y-w] + + +(* ::Code:: *) +Int[u_*F_^v_,x_Symbol] := + With[{q=DerivativeDivides[v,u,x]}, + q*F^v/Log[F] /; + Not[FalseQ[q]]] /; +FreeQ[F,x] + + +(* ::Code:: *) +Int[u_*w_^m_.*F_^v_,x_Symbol] := + With[{q=DerivativeDivides[v,u,x]}, + q*Subst[Int[x^m*F^x,x],x,v] /; + Not[FalseQ[q]]] /; +FreeQ[{F,m},x] && EqQ[w-v] + + +(* ::Code:: *) +Int[u_*(a_+b_.*v_^p_.*w_^p_.)^m_.,x_Symbol] := + With[{c=Simplify[u/(w*D[v,x]+v*D[w,x])]}, + c*Subst[Int[(a+b*x^p)^m,x],x,v*w] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p},x] && IntegerQ[p] + + +(* ::Code:: *) +Int[u_*(a_+b_.*v_^p_.*w_^q_.)^m_.*v_^r_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]+q*v*D[w,x])]}, + c*p/(r+1)*Subst[Int[(a+b*x^(p/(r+1)))^m,x],x,v^(r+1)*w] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q,r},x] && EqQ[p-q*(r+1)] && NeQ[r+1] && IntegerQ[p/(r+1)] + + +(* ::Code:: *) +Int[u_*(a_+b_.*v_^p_.*w_^q_.)^m_.*v_^r_.*w_^s_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]+q*v*D[w,x])]}, + c*p/(r+1)*Subst[Int[(a+b*x^(p/(r+1)))^m,x],x,v^(r+1)*w^(s+1)] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q,r,s},x] && EqQ[p*(s+1)-q*(r+1)] && NeQ[r+1] && IntegerQ[p/(r+1)] + + +(* ::Code:: *) +Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, + c*p*Subst[Int[(b+a*x^p)^m,x],x,v*w^(m*q+1)] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q},x] && EqQ[p+q*(m*p+1)] && IntegerQ[p] && IntegerQ[m] + + +(* ::Code:: *) +(* Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, + -c*q*Subst[Int[(a+b*x^q)^m,x],x,v^(m*p+1)*w] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q},x] && EqQ[p+q*(m*p+1)] && IntegerQ[q] && IntegerQ[m] *) + + +(* ::Code:: *) +Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.*v_^r_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, + -c*q*Subst[Int[(a+b*x^q)^m,x],x,v^(m*p+r+1)*w] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q,r},x] && EqQ[p+q*(m*p+r+1)] && IntegerQ[q] && IntegerQ[m] + + +(* ::Code:: *) +Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.*w_^s_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, + -c*q/(s+1)*Subst[Int[(a+b*x^(q/(s+1)))^m,x],x,v^(m*p+1)*w^(s+1)] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q,s},x] && EqQ[p*(s+1)+q*(m*p+1)] && NeQ[s+1] && IntegerQ[q/(s+1)] && IntegerQ[m] + + +(* ::Code:: *) +Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.*v_^r_.*w_^s_.,x_Symbol] := + With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, + -c*q/(s+1)*Subst[Int[(a+b*x^(q/(s+1)))^m,x],x,v^(m*p+r+1)*w^(s+1)] /; + FreeQ[c,x]] /; +FreeQ[{a,b,m,p,q,r,s},x] && EqQ[p*(s+1)+q*(m*p+r+1)] && NeQ[s+1] && IntegerQ[q/(s+1)] && IntegerQ[m] + + +(* ::Code:: *) +Int[u_*x_^m_.,x_Symbol] := + 1/(m+1)*Subst[Int[SubstFor[x^(m+1),u,x],x],x,x^(m+1)] /; +FreeQ[m,x] && NeQ[m+1] && FunctionOfQ[x^(m+1),u,x] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfLinear[u,x]}, + ShowStep["","Int[F[(a+b*x)^(1/n),x],x]", + "n/b*Subst[Int[x^(n-1)*F[x,-a/b+x^n/b],x],x,(a+b*x)^(1/n)]",Hold[ + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])]]] /; + Not[FalseQ[lst]] && SubstForFractionalPowerQ[u,lst[[3]],x]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfLinear[u,x]}, + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])] /; + Not[FalseQ[lst]] && SubstForFractionalPowerQ[u,lst[[3]],x]]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfQuotientOfLinears[u,x]}, + ShowStep["","Int[F[((a+b*x)/(c+d*x))^(1/n),x],x]", +"n*(b*c-a*d)*Subst[Int[x^(n-1)*F[x,(-a+c*x^n)/(b-d*x^n)]/(b-d*x^n)^2,x],x,((a+b*x)/(c+d*x))^(1/n)]",Hold[ + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])]]] /; + Not[FalseQ[lst]]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfQuotientOfLinears[u,x]}, + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])] /; + Not[FalseQ[lst]]]] + + +(* ::Code:: *) +Int[u_.*(a_.*v_^m_.*w_^n_.*z_^q_.)^p_,x_Symbol] := + a^IntPart[p]*(a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p]))*Int[u*v^(m*p)*w^(n*p)*z^(p*q),x] /; +FreeQ[{a,m,n,p,q},x] && Not[IntegerQ[p]] && Not[FreeQ[v,x]] && Not[FreeQ[w,x]] && Not[FreeQ[z,x]] + + +(* ::Code:: *) +Int[u_.*(a_.*v_^m_.*w_^n_.)^p_,x_Symbol] := + a^IntPart[p]*(a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))*Int[u*v^(m*p)*w^(n*p),x] /; +FreeQ[{a,m,n,p},x] && Not[IntegerQ[p]] && Not[FreeQ[v,x]] && Not[FreeQ[w,x]] + + +(* ::Code:: *) +Int[u_.*(a_.*v_^m_.)^p_,x_Symbol] := + a^IntPart[p]*(a*v^m)^FracPart[p]/v^(m*FracPart[p])*Int[u*v^(m*p),x] /; +FreeQ[{a,m,p},x] && Not[IntegerQ[p]] && Not[FreeQ[v,x]] && Not[EqQ[a,1] && EqQ[m,1]] && Not[EqQ[v,x] && EqQ[m,1]] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^n_)^p_,x_Symbol] := + FullSimplify[Sqrt[a+b*x^n]/(x^(n/2)*Sqrt[b+a/x^n])]*Int[u*x^(n*p)*(b+a*x^(-n))^p,x] /; +FreeQ[{a,b,p},x] && IntegerQ[p+1/2] && NegativeIntegerQ[n] && Not[RationalFunctionQ[u,x]] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*v_^n_)^p_,x_Symbol] := + (a+b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b+a*v^(-n))^FracPart[p])*Int[u*v^(n*p)*(b+a*v^(-n))^p,x] /; +FreeQ[{a,b,p},x] && Not[IntegerQ[p]] && NegativeIntegerQ[n] && BinomialQ[v,x] && Not[LinearQ[v,x]] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^m_.*v_^n_)^p_,x_Symbol] := + (a+b*x^m*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b*x^m+a*v^(-n))^FracPart[p])*Int[u*v^(n*p)*(b*x^m+a*v^(-n))^p,x] /; +FreeQ[{a,b,m,p},x] && Not[IntegerQ[p]] && NegativeIntegerQ[n] && BinomialQ[v,x] + + +(* ::Code:: *) +Int[u_.*(a_.*x_^r_.+b_.*x_^s_.)^m_,x_Symbol] := + With[{v=(a*x^r+b*x^s)^FracPart[m]/(x^(r*FracPart[m])*(a+b*x^(s-r))^FracPart[m])}, + v*Int[u*x^(m*r)*(a+b*x^(s-r))^m,x] /; + Not[EqQ[Simplify[v],1]]] /; +FreeQ[{a,b,m,r,s},x] && Not[IntegerQ[m]] && PosQ[s-r] + + +(* ::Code:: *) +Int[u_/(a_+b_.*x_^n_),x_Symbol] := + With[{v=RationalFunctionExpand[u/(a+b*x^n),x]}, + Int[v,x] /; + SumQ[v]] /; +FreeQ[{a,b},x] && PositiveIntegerQ[n] + + +(* ::Code:: *) +Int[u_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] := + 1/(4^p*c^p)*Int[u*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p] && Not[AlgebraicFunctionQ[u,x]] + + +(* ::Code:: *) +Int[u_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol] := + (a+b*x^n+c*x^(2*n))^p/(b+2*c*x^n)^(2*p)*Int[u*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && Not[IntegerQ[p]] && Not[AlgebraicFunctionQ[u,x]] + + +(* ::Code:: *) +Int[u_/(a_.+b_.*x_^n_.+c_.*x_^n2_.),x_Symbol] := + With[{v=RationalFunctionExpand[u/(a+b*x^n+c*x^(2*n)),x]}, + Int[v,x] /; + SumQ[v]] /; +FreeQ[{a,b,c},x] && EqQ[n2-2*n] && PositiveIntegerQ[n] + + +(* ::Code:: *) +Int[u_./(a_.*x_^m_.+b_.*Sqrt[c_.*x_^n_]),x_Symbol] := + Int[u*(a*x^m-b*Sqrt[c*x^n])/(a^2*x^(2*m)-b^2*c*x^n),x] /; +FreeQ[{a,b,c,m,n},x] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfLinear[u,x]}, + ShowStep["","Int[F[a+b*x],x]","Subst[Int[F[x],x],x,a+b*x]/b",Hold[ + Dist[1/lst[[3]],Subst[Int[lst[[1]],x],x,lst[[2]]+lst[[3]]*x],x]]] /; + Not[FalseQ[lst]]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfLinear[u,x]}, + Dist[1/lst[[3]],Subst[Int[lst[[1]],x],x,lst[[2]]+lst[[3]]*x],x] /; + Not[FalseQ[lst]]]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_/x_,x_Symbol] := + With[{lst=PowerVariableExpn[u,0,x]}, + ShowStep["","Int[F[(c*x)^n]/x,x]","Subst[Int[F[x]/x,x],x,(c*x)^n]/n",Hold[ + 1/lst[[2]]*Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x],x],x],x,(lst[[3]]*x)^lst[[2]]]]] /; + Not[FalseQ[lst]] && NeQ[lst[[2]]]] /; +SimplifyFlag && NonsumQ[u] && Not[RationalFunctionQ[u,x]], + +Int[u_/x_,x_Symbol] := + With[{lst=PowerVariableExpn[u,0,x]}, + 1/lst[[2]]*Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x],x],x],x,(lst[[3]]*x)^lst[[2]]] /; + Not[FalseQ[lst]] && NeQ[lst[[2]]]] /; +NonsumQ[u] && Not[RationalFunctionQ[u,x]]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_*x_^m_.,x_Symbol] := + With[{lst=PowerVariableExpn[u,m+1,x]}, + ShowStep["If g=GCD[m+1,n]>1,","Int[x^m*F[x^n],x]", + "1/g*Subst[Int[x^((m+1)/g-1)*F[x^(n/g)],x],x,x^g]",Hold[ + 1/lst[[2]]*Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x],x],x],x,(lst[[3]]*x)^lst[[2]]]]] /; + Not[FalseQ[lst]] && NeQ[lst[[2]]-m-1]] /; +SimplifyFlag && IntegerQ[m] && m!=-1 && NonsumQ[u] && (m>0 || Not[AlgebraicFunctionQ[u,x]]), + +Int[u_*x_^m_.,x_Symbol] := + With[{lst=PowerVariableExpn[u,m+1,x]}, + 1/lst[[2]]*Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x],x],x],x,(lst[[3]]*x)^lst[[2]]] /; + Not[FalseQ[lst]] && NeQ[lst[[2]]-m-1]] /; +IntegerQ[m] && m!=-1 && NonsumQ[u] && (m>0 || Not[AlgebraicFunctionQ[u,x]])] + + +(* ::Code:: *) +Int[x_^m_*u_,x_Symbol] := + With[{k=Denominator[m]}, + k*Subst[Int[x^(k*(m+1)-1)*ReplaceAll[u,x->x^k],x],x,x^(1/k)]] /; +FractionQ[m] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + ShowStep["","Int[F[Sqrt[a+b*x+c*x^2],x],x]", + "2*Subst[Int[F[(c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2),(-b+2*Sqrt[a]*x)/(c-x^2)]* + (c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2)^2,x],x,(-Sqrt[a]+Sqrt[a+b*x+c*x^2])/x]", + Hold[2*Subst[Int[lst[[1]],x],x,lst[[2]]]]] /; + Not[FalseQ[lst]] && lst[[3]]===1] /; +SimplifyFlag && EulerIntegrandQ[u,x], + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + 2*Subst[Int[lst[[1]],x],x,lst[[2]]] /; + Not[FalseQ[lst]]] /; +EulerIntegrandQ[u,x]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + ShowStep["","Int[F[Sqrt[a+b*x+c*x^2],x],x]", + "2*Subst[Int[F[(a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x),(-a+x^2)/(b+2*Sqrt[c]*x)]* + (a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x)^2,x],x,Sqrt[c]*x+Sqrt[a+b*x+c*x^2]]", + Hold[2*Subst[Int[lst[[1]],x],x,lst[[2]]]]] /; + Not[FalseQ[lst]] && lst[[3]]===2] /; +SimplifyFlag && EulerIntegrandQ[u,x], + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + 2*Subst[Int[lst[[1]],x],x,lst[[2]]] /; + Not[FalseQ[lst]]] /; +EulerIntegrandQ[u,x]] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + ShowStep["","Int[F[Sqrt[a+b*x+c*x^2],x],x]", + "-2*Sqrt[b^2-4*a*c]*Subst[Int[F[-Sqrt[b^2-4*a*c]*x/(c-x^2), + (b*c+c*Sqrt[b^2-4*a*c]+(-b+Sqrt[b^2-4*a*c])*x^2)/(-2*c*(c-x^2))]*x/(c-x^2)^2,x], + x,2*c*Sqrt[a+b*x+c*x^2]/(b-Sqrt[b^2-4*a*c]+2*c*x)]", + Hold[2*Subst[Int[lst[[1]],x],x,lst[[2]]]]] /; + Not[FalseQ[lst]] && lst[[3]]===3] /; +SimplifyFlag && EulerIntegrandQ[u,x], + +Int[u_,x_Symbol] := + With[{lst=FunctionOfSquareRootOfQuadratic[u,x]}, + 2*Subst[Int[lst[[1]],x],x,lst[[2]]] /; + Not[FalseQ[lst]]] /; +EulerIntegrandQ[u,x]] + + +(* ::Code:: *) +Int[1/(a_+b_.*v_^2),x_Symbol] := +(*1/(2*a)*Int[Together[1/(1-Rt[-b/a,2]*v)],x] + 1/(2*a)*Int[Together[1/(1+Rt[-b/a,2]*v)],x] /; *) + 1/(2*a)*Int[Together[1/(1-v/Rt[-a/b,2])],x] + 1/(2*a)*Int[Together[1/(1+v/Rt[-a/b,2])],x] /; +FreeQ[{a,b},x] + + +(* ::Code:: *) +Int[1/(a_+b_.*v_^n_),x_Symbol] := + Dist[2/(a*n),Sum[Int[Together[1/(1-v^2/((-1)^(4*k/n)*Rt[-a/b,n/2]))],x],{k,1,n/2}],x] /; +FreeQ[{a,b},x] && IGtQ[n/2,1] + + +(* ::Code:: *) +Int[1/(a_+b_.*v_^n_),x_Symbol] := + Dist[1/(a*n),Sum[Int[Together[1/(1-v/((-1)^(2*k/n)*Rt[-a/b,n]))],x],{k,1,n}],x] /; +FreeQ[{a,b},x] && IGtQ[(n-1)/2,0] + + +(* ::Code:: *) +Int[v_/(a_+b_.*u_^n_.),x_Symbol] := + Int[ReplaceAll[ExpandIntegrand[PolynomialInSubst[v,u,x]/(a+b*x^n),x],x->u],x] /; +FreeQ[{a,b},x] && PositiveIntegerQ[n] && PolynomialInQ[v,u,x] + + +(* ::Code:: *) +Int[u_,x_Symbol] := + With[{v=NormalizeIntegrand[u,x]}, + Int[v,x] /; + v=!=u] + + +(* ::Code:: *) +Int[u_,x_Symbol] := + With[{v=ExpandIntegrand[u,x]}, + Int[v,x] /; + SumQ[v]] + + +(* ::Code:: *) +Int[u_.*(a_.+b_.*x_^m_.)^p_.*(c_.+d_.*x_^n_.)^q_., x_Symbol] := + (a+b*x^m)^p*(c+d*x^n)^q/x^(m*p)*Int[u*x^(m*p),x] /; +FreeQ[{a,b,c,d,m,n,p,q},x] && EqQ[a+d] && EqQ[b+c] && EqQ[m+n] && EqQ[p+q] + + +(* ::Code:: *) +Int[u_*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_, x_Symbol] := + Sqrt[a+b*x^n+c*x^(2*n)]/((4*c)^(p-1/2)*(b+2*c*x^n))*Int[u*(b+2*c*x^n)^(2*p),x] /; +FreeQ[{a,b,c,n,p},x] && EqQ[n2-2*n] && EqQ[b^2-4*a*c] && IntegerQ[p-1/2] + + +(* ::Code:: *) +If[ShowSteps, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfLinear[u,x]}, + ShowStep["","Int[F[(a+b*x)^(1/n),x],x]", + "n/b*Subst[Int[x^(n-1)*F[x,-a/b+x^n/b],x],x,(a+b*x)^(1/n)]",Hold[ + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])]]] /; + Not[FalseQ[lst]]] /; +SimplifyFlag, + +Int[u_,x_Symbol] := + With[{lst=SubstForFractionalPowerOfLinear[u,x]}, + lst[[2]]*lst[[4]]*Subst[Int[lst[[1]],x],x,lst[[3]]^(1/lst[[2]])] /; + Not[FalseQ[lst]]]] + + +(* ::Code:: *) +Int[u_,x_] := Defer[Int][u,x] + + + +`) + +func resourcesRubi94MiscellaneousIntegrationRulesMBytes() ([]byte, error) { + return _resourcesRubi94MiscellaneousIntegrationRulesM, nil +} + +func resourcesRubi94MiscellaneousIntegrationRulesM() (*asset, error) { + bytes, err := resourcesRubi94MiscellaneousIntegrationRulesMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/9.4 Miscellaneous integration rules.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiIntegrationUtilityFunctionsM = []byte(`(* ::Package:: *) + +(* ::Title:: *) +(*Integration Utility Functions*) + + +(* $TimeLimit is the time constraint in seconds on some potentially expensive routines. *) +If[Not[NumberQ[$TimeLimit]], $TimeLimit=5.0]; + + +(* ::Section::Closed:: *) +(*IntHide[u,x]*) + + +IntHide[u_,x_Symbol] := + Block[{ShowSteps=False,$StepCounter=Null}, Int[u,x]] + + +(* ::Section::Closed:: *) +(*Mapping functions*) + + +(* ::Subsection::Closed:: *) +(*MapAnd[u,x]*) + + +(* MapAnd[f,l] applies f to the elements of list l until False is returned; else returns True *) +MapAnd[f_,lst_] := + Catch[Scan[Function[If[f[#],Null,Throw[False]]],lst];True] + +MapAnd[f_,lst_,x_] := + Catch[Scan[Function[If[f[#,x],Null,Throw[False]]],lst];True] + + +(* ::Subsection::Closed:: *) +(*Map2[func,lst1,lst2]*) + + +Map2[func_,lst1_,lst2_] := + Module[{i}, + ReapList[Do[Sow[func[lst1[[i]],lst2[[i]]]],{i,Length[lst1]}]]] + + +(* ::Subsection::Closed:: *) +(*ReapList[u]*) + + +ReapList[u_] := + With[{lst=Reap[u][[2]]}, + If[lst==={}, lst, lst[[1]]]] + +SetAttributes[ReapList,HoldFirst] + + +(* ::Section::Closed:: *) +(*Numerical type predicates*) + + +(* ::Subsection::Closed:: *) +(*FalseQ[u]*) + + +FalseQ[u_] := u===False + + +(* ::Subsection::Closed:: *) +(*IntegersQ[u]*) + + +(* IntegersQ[m,n,...] returns True if m, n, ... are all explicit integers; else it returns False. *) +IntegersQ[u__] := Catch[Scan[Function[If[IntegerQ[#],Null,Throw[False]]],{u}]; True]; + + +(* ::Subsection::Closed:: *) +(*PositiveIntegerQ[u]*) + + +(* PositiveIntegerQ[m,n,...] returns True if m, n, ... are all explicit positive integers; else it returns False. *) +PositiveIntegerQ[u__] := Catch[Scan[Function[If[IntegerQ[#] && #>0,Null,Throw[False]]],{u}]; True]; + + +(* ::Subsection::Closed:: *) +(*NegativeIntegerQ[u]*) + + +(* NegativeIntegerQ[m,n,...] returns True if m, n, ... are all explicit negative integers; else it returns False. *) +NegativeIntegerQ[u__] := Catch[Scan[Function[If[IntegerQ[#] && #<0,Null,Throw[False]]],{u}]; True]; + + +(* ::Subsection::Closed:: *) +(*FractionQ[u]*) + + +(* FractionQ[m,n,...] returns True if m, n, ... are all explicit fractions; else it returns False. *) +FractionQ[u__] := Catch[Scan[Function[If[Head[#]===Rational,Null,Throw[False]]],{u}]; True] + + +(* ::Subsection::Closed:: *) +(*RationalQ[u]*) + + +(* RationalQ[m,n,...] returns True if m, n, ... are all explicit integers or fractions; else it returns False. *) +RationalQ[u__] := Catch[Scan[Function[If[IntegerQ[#] || Head[#]===Rational,Null,Throw[False]]],{u}]; True] + + +(* ::Subsection::Closed:: *) +(*ComplexNumberQ[u]*) + + +(* ComplexNumberQ[u] returns True if u is an explicit complex number; else it returns False. *) +ComplexNumberQ[u_] := Head[u]===Complex + + +(* ::Subsection::Closed:: *) +(*RealNumericQ[u]*) + + +(* RealNumericQ[u] returns True if u is a real numeric quantity, else returns False. *) +RealNumericQ[u_] := NumericQ[u] && (EqQ[Im[u],0] || EqQ[Im[N[u]],0]) + + +(* ::Subsection::Closed:: *) +(*PositiveQ[u]*) + + +(* PositiveQ[u] returns True if u is a positive numeric quantity, else returns False. *) +PositiveQ[u_] := + With[{v=Simplify[u]}, + RealNumericQ[v] && Re[N[v]]>0] + + +(* ::Subsection::Closed:: *) +(*NegativeQ[u]*) + + +(* NegativeQ[u] returns True if u is a negative numeric quantity, else returns False. *) +NegativeQ[u_] := + With[{v=Simplify[u]}, + RealNumericQ[v] && Re[N[v]]<0] + + +(* ::Subsection::Closed:: *) +(*FractionOrNegativeQ[u]*) + + +(* FractionOrNegativeQ[u] returns True if u is a fraction or negative number; else returns False *) +FractionOrNegativeQ[u__] := Catch[Scan[Function[If[FractionQ[#] || IntegerQ[#] && #<0,Null,Throw[False]]],{u}]; True] + + +(* ::Subsection::Closed:: *) +(*SqrtNumberQ[u]*) + + +(* SqrtNumberQ[u] returns True if u^2 is a rational number; else it returns False. *) +SqrtNumberQ[m_^n_] := + IntegerQ[n] && SqrtNumberQ[m] || IntegerQ[n-1/2] && RationalQ[m] + + +SqrtNumberQ[u_*v_] := + SqrtNumberQ[u] && SqrtNumberQ[v] + + +SqrtNumberQ[u_] := + RationalQ[u] || u===I + + +(* ::Subsection::Closed:: *) +(*SqrtNumberSumQ[u]*) + + +SqrtNumberSumQ[u_] := + SumQ[u] && SqrtNumberQ[First[u]] && SqrtNumberQ[Rest[u]] || + ProductQ[u] && SqrtNumberQ[First[u]] && SqrtNumberSumQ[Rest[u]] + + +(* ::Subsection::Closed:: *) +(*IndependentQ[u,x]*) + + +IndependentQ[u_,x_] := + FreeQ[u,x] + + +(* ::Subsection::Closed:: *) +(*NotIntegrableQ[u,x]*) + + +(* NotIntegrableQ[u,x] returns True if u is definitely not integrable wrt x; else it returns + False if u is, or might be, integrable wrt x. *) +NotIntegrableQ[u_,x_Symbol] := + MatchQ[u,x^m_*Log[a_+b_.*x]^n_ /; FreeQ[{a,b},x] && IntegersQ[m,n] && m<0 && n<0] || + MatchQ[u,f_[x^m_.*Log[a_.+b_.*x]] /; FreeQ[{a,b},x] && IntegerQ[m] && (TrigQ[f] || HyperbolicQ[f])] + + +(* ::Section::Closed:: *) +(*Expression type predicates*) + + +(* ::Subsection::Closed:: *) +(*PowerQ[u]*) + + +(* If u is a power, SumQ[u] returns True; else it returns False. *) +PowerQ[u_] := Head[u]===Power + + +(* ::Subsection::Closed:: *) +(*ProductQ[u]*) + + +(* If u is a product, SumQ[u] returns True; else it returns False. *) +ProductQ[u_] := Head[u]===Times + + +(* ::Subsection::Closed:: *) +(*SumQ[u]*) + + +(* If u is a sum, SumQ[u] returns True; else it returns False. *) +SumQ[u_] := Head[u]===Plus + + +(* ::Subsection::Closed:: *) +(*NonsumQ[u]*) + + +(* If u is not a sum, NonsumQ[u] returns True; else it returns False. *) +NonsumQ[u_] := Head[u]=!=Plus + + +(* ::Subsection::Closed:: *) +(*IntegerPowerQ[u]*) + + +IntegerPowerQ[u_] := + PowerQ[u] && IntegerQ[u[[2]]] + + +(* ::Subsection::Closed:: *) +(*PositiveIntegerPowerQ[u]*) + + +PositiveIntegerPowerQ[u_] := + PowerQ[u] && IntegerQ[u[[2]]] && u[[2]]>0 + + +(* ::Subsection::Closed:: *) +(*FractionalPowerQ[u]*) + + +FractionalPowerQ[u_] := + PowerQ[u] && FractionQ[u[[2]]] + + +(* ::Subsection::Closed:: *) +(*FractionalPowerFreeQ[u]*) + + +FractionalPowerFreeQ[u_] := + If[AtomQ[u], + True, + If[FractionalPowerQ[u] && Not[AtomQ[u[[1]]]], + False, + Catch[Scan[Function[If[FractionalPowerFreeQ[#],Null,Throw[False]]],u];True]]] + + +(* ::Subsection::Closed:: *) +(*ComplexFreeQ[u]*) + + +ComplexFreeQ[u_] := + If[AtomQ[u], + Not[ComplexNumberQ[u]], + Catch[Scan[Function[If[ComplexFreeQ[#],Null,Throw[False]]],u];True]] + + +(* ::Subsection::Closed:: *) +(*ExpQ[u]*) + + +ExpQ[u_] := PowerQ[u] && u[[1]]===E + + +(* ::Subsection::Closed:: *) +(*LogQ[u]*) + + +(* If u is an expression of the form Log[v], LogQ[u] returns True; else it returns False. *) +LogQ[u_] := Head[u]===Log + + +(* ::Subsection::Closed:: *) +(*TrigQ[u]*) + + +(* TrigQ[u] returns True if u or the head of u is a trig function; else returns False *) +TrigQ[u_] := + MemberQ[{Sin,Cos,Tan,Cot,Sec,Csc},If[AtomQ[u],u,Head[u]]] + + +SinQ[u_] := Head[u]===Sin + +CosQ[u_] := Head[u]===Cos + +TanQ[u_] := Head[u]===Tan + +CotQ[u_] := Head[u]===Cot + +SecQ[u_] := Head[u]===Sec + +CscQ[u_] := Head[u]===Csc + + +(* ::Subsection::Closed:: *) +(*HyperbolicQ[u]*) + + +(* HyperbolicQ[u] returns True if u or the head of u is a trig function; else returns False *) +HyperbolicQ[u_] := + MemberQ[{Sinh,Cosh,Tanh,Coth,Sech,Csch},If[AtomQ[u],u,Head[u]]] + + +SinhQ[u_] := Head[u]===Sinh + +CoshQ[u_] := Head[u]===Cosh + +TanhQ[u_] := Head[u]===Tanh + +CothQ[u_] := Head[u]===Coth + +SechQ[u_] := Head[u]===Sech + +CschQ[u_] := Head[u]===Csch + + +(* ::Subsection::Closed:: *) +(*InverseTrigQ[u]*) + + +(* InverseTrigQ[u] returns True if u or the head of u is an inverse trig function; else returns False *) +InverseTrigQ[u_] := + MemberQ[{ArcSin,ArcCos,ArcTan,ArcCot,ArcSec,ArcCsc},If[AtomQ[u],u,Head[u]]] + + +(* ::Subsection::Closed:: *) +(*InverseHyperbolicQ[u]*) + + +(* InverseHyperbolicQ[u] returns True if u or the head of u is an inverse trig function; else returns False *) +InverseHyperbolicQ[u_] := + MemberQ[{ArcSinh,ArcCosh,ArcTanh,ArcCoth,ArcSech,ArcCsch},If[AtomQ[u],u,Head[u]]] + + +(* ::Subsection::Closed:: *) +(*SinCosQ[u]*) + + +SinCosQ[f_] := + MemberQ[{Sin,Cos,Sec,Csc},f] + + +(* ::Subsection::Closed:: *) +(*SinhCoshQ[u]*) + + +SinhCoshQ[f_] := + MemberQ[{Sinh,Cosh,Sech,Csch},f] + + +(* ::Subsection::Closed:: *) +(*CalculusQ[u]*) + + +CalculusFunctions={D,Integrate,Sum,Product,Int,Dif,Subst}; + +(* CalculusQ[u] returns True if the head of u is a calculus function; else returns False *) +CalculusQ[u_] := + MemberQ[CalculusFunctions,Head[u]] + +CalculusFreeQ[u_,x_] := + If[AtomQ[u], + True, + If[CalculusQ[u] && u[[2]]===x || HeldFormQ[u], + False, + Catch[Scan[Function[If[CalculusFreeQ[#,x],Null,Throw[False]]],u];True]]] + + +HeldFormQ[u_] := + If[AtomQ[Head[u]], + MemberQ[{Hold,HoldForm,Defer,Pattern},Head[u]], + HeldFormQ[Head[u]]] + + +(* InverseFunctionQ[u] returns True if u is a call on an inverse function; else returns False. *) +InverseFunctionQ[u_] := + LogQ[u] || InverseTrigQ[u] && Length[u]<=1 || InverseHyperbolicQ[u] || Head[u]===Mods || Head[u]===PolyLog + + +(* If u is free of trig, hyperbolic and calculus functions involving x, + TrigHyperbolicFreeQ[u,x] returns true; else it returns False. *) +TrigHyperbolicFreeQ[u_,x_Symbol] := + If[AtomQ[u], + True, + If[TrigQ[u] || HyperbolicQ[u] || CalculusQ[u], + FreeQ[u,x], + Catch[Scan[Function[If[TrigHyperbolicFreeQ[#,x],Null,Throw[False]]],u];True]]] + + +(* If u is free of inverse, calculus and hypergeometric functions involving x, + InverseFunctionFreeQ[u,x] returns true; else it returns False. *) +InverseFunctionFreeQ[u_,x_Symbol] := + If[AtomQ[u], + True, + If[InverseFunctionQ[u] || CalculusQ[u] || Head[u]===Hypergeometric2F1 || Head[u]===AppellF1, + FreeQ[u,x], + Catch[Scan[Function[If[InverseFunctionFreeQ[#,x],Null,Throw[False]]],u];True]]] + + +(* ElementaryExpressionQ[u] returns True if u is a sum, product, or power and all the operands + are elementary expressions; or if u is a call on a trig, hyperbolic, or inverse function + and all the arguments are elementary expressions; else it returns False. *) +(* ElementaryFunctionQ[u_] := + If[AtomQ[u], + True, + If[SumQ[u] || ProductQ[u] || PowerQ[u] || TrigQ[u] || HyperbolicQ[u] || InverseFunctionQ[u], + Catch[Scan[Function[If[ElementaryFunctionQ[#],Null,Throw[False]]],u];True], + False]] *) + + +(* ::Section::Closed:: *) +(*Equality and inequality predicates*) + + +(* ::Subsection::Closed:: *) +(*Equality predicates*) + + +(* If u-v equals 0, EqQ[u,v] returns True; else it returns False. *) +EqQ[u_] := Quiet[PossibleZeroQ[u]] + +EqQ[u_,v_] := Quiet[PossibleZeroQ[u-v]] + + +(* If u-v equals 0, EqQ[u,v] returns False; else it returns True. *) +NeQ[u_] := Not[Quiet[PossibleZeroQ[u]]] + +NeQ[u_,v_] := Not[Quiet[PossibleZeroQ[u-v]]] + + +(* ::Subsection::Closed:: *) +(*Integer inequality predicates*) + + +(* num is a rational. If u is an integer and u>n, IGtQ[u,n] returns True; else it returns False. *) +IGtQ[u_,n_] := IntegerQ[u] && u>n + + +(* num is a rational. If u is an integer and u=n, IGeQ[u,n] returns True; else it returns False. *) +IGeQ[u_,n_] := IntegerQ[u] && u>=n + + +(* num is a rational. If u is an integer and u<=n, ILeQ[u,n] returns True; else it returns False. *) +ILeQ[u_,n_] := IntegerQ[u] && u<=n + + +(* ::Subsection::Closed:: *) +(*Numeric inequality predicates*) + + +(* num is a rational. If u is a rational and u>n, GtQ[u,n] returns True; else it returns False. *) +GtQ[u_,n_] := RationalQ[u] && u>n + + +(* num is a rational. If u is a rational and u=n, GeQ[u,n] returns True; else it returns False. *) +GeQ[u_,n_] := RationalQ[u] && u>=n + + +(* num is a rational. If u is a rational and u<=n, LeQ[u,n] returns True; else it returns False. *) +LeQ[u_,n_] := RationalQ[u] && u<=n + + +(* ::Section::Closed:: *) +(*Multinomial predicates*) + + +(* ::Subsection::Closed:: *) +(*PolyQ[u,x]*) + + +(* PolyQ[u,x,n] returns True iff u is a polynomial of degree n. *) +PolyQ[u_,x_Symbol,n_Integer] := + If[ListQ[u], + Catch[Scan[Function[If[Not[PolyQ[#,x,n]],Throw[False]]],u]; True], + PolynomialQ[u,x] && Exponent[u,x]==n && Coefficient[u,x,n]=!=0] + + +(* Mathematica's built-in PolynomialQ, Exponent and Coefficient functions can return erroneous results because they do not *) +(* cancel the gcd in pseudo-rational functions that are actually polynomial. *) +(* For some unfully expanded polynomials, the built-in Mathematica function Exponent sometimes returns erronously large degrees. *) +(* For example, Exponent[3*(1+a)*x^4 + 3*x^5 + x^6 - (a+x+x^2)^3, x] incorrectly returns 4 instead of 3. *) +(* Despite what the online help says, PolynomialQ[u,x] returns an error message if x is not a symbol or *) +(* a symbol raised to a symbol power. *) + + +(* If u is a polynomial in x,PolyQ[u,x] returns True; else it returns False. *) +PolyQ[u_,x_Symbol] := + PolynomialQ[u,x] || PolynomialQ[Together[u],x] + +PolyQ[u_,x_Symbol^n_] := + If[PositiveIntegerQ[n], + PolyQ[u,x] && (PolynomialQ[u,x^n] || PolynomialQ[Together[u],x^n]), + If[AtomQ[n], + PolynomialQ[u,x^n] && FreeQ[CoefficientList[u,x^n],x], + False]] /; +FreeQ[n,x] + +PolyQ[u_,u_] := + False + + +(* ::Subsection::Closed:: *) +(*ProperPolyQ[u,x]*) + + +(* If u is a polynomial in x and the constant term is nonzero, ProperPolyQ[u,x] returns True; else it returns False. *) +ProperPolyQ[u_,x_Symbol] := + PolyQ[u,x] && NeQ[Coeff[u,x,0],0] + + +(* ::Subsection::Closed:: *) +(*BinomialQ[u,x]*) + + +(* If u is equivalent to an expression of the form a+b*x^n where n and b are not 0, *) +(* BinomialQ[u,x] returns True; else it returns False. *) +BinomialQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[BinomialQ[#,x]],Throw[False]]],u]; True], + ListQ[BinomialParts[u,x]]] + + +(* If u is equivalent to an expression of the form a+b*x^n, BinomialQ[u,x,n] returns True; else it returns False. *) +BinomialQ[u_,x_Symbol,n_] := + If[ListQ[u], + Catch[Scan[Function[If[Not[BinomialQ[#,x,n]],Throw[False]]],u]; True], + Function[ListQ[#] && #[[3]]===n][BinomialParts[u,x]]] + + +(* ::Subsection::Closed:: *) +(*TrinomialQ[u,x]*) + + +(* If u is equivalent to an expression of the form a+b*x^n+c*x^(2*n) where n, b and c are not 0, *) +(* TrinomialQ[u,x] returns True; else it returns False. *) +TrinomialQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[TrinomialQ[#,x]],Throw[False]]],u]; True], + ListQ[TrinomialParts[u,x]] && Not[QuadraticQ[u,x]] && Not[MatchQ[u,w_^2 /; BinomialQ[w,x]]]] + + +(* ::Subsection::Closed:: *) +(*GeneralizedBinomialQ[u,x]*) + + +(* If u is equivalent to an expression of the form a*x^q+b*x^n where n, q and b are not 0, *) +(* GeneralizedBinomialQ[u,x] returns True; else it returns False. *) +GeneralizedBinomialQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[GeneralizedBinomialQ[#,x]],Throw[False]]],u]; True], + ListQ[GeneralizedBinomialParts[u,x]]] + + +(* ::Subsection::Closed:: *) +(*GeneralizedTrinomialQ[u,x]*) + + +(* If u is equivalent to an expression of the form a*x^q+b*x^n+c*x^(2*n-q) where n, q, b and c are not 0, *) +(* GeneralizedTrinomialQ[u,x] returns True; else it returns False. *) +GeneralizedTrinomialQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[GeneralizedTrinomialQ[#,x]],Throw[False]]],u]; True], + ListQ[GeneralizedTrinomialParts[u,x]]] + + +(* ::Section::Closed:: *) +(*Expression form predicates*) + + +(* ::Subsection::Closed:: *) +(*PosQ[u]*) + + +(* If u is not 0 and has a positive form, PosQ[u] returns True, else it returns False. *) +PosQ[u_] := + PosAux[TogetherSimplify[u]] + +PosAux[u_] := + If[RationalQ[u], + u>0, + If[NumberQ[u], + If[EqQ[Re[u],0], + Im[u]>0, + Re[u]>0], + If[PowerQ[u], + If[IntegerQ[u[[2]]], + EvenQ[u[[2]]] || PosAux[u[[1]]], + True], + If[ProductQ[u], + If[PosAux[First[u]], + PosAux[Rest[u]], + Not[PosAux[Rest[u]]]], + If[NumericQ[u], + With[{v=N[u]}, + If[EqQ[Re[v],0], + Im[v]>0, + Re[v]>0]], + If[SumQ[u], + PosAux[First[u]], + True]]]]]] + + +(* ::Subsection::Closed:: *) +(*NegQ[u]*) + + +NegQ[u_] := + Not[PosQ[u]] && NeQ[u,0] + + +(* ::Subsection::Closed:: *) +(*NiceSqrtQ[u]*) + + +NiceSqrtQ[u_] := + Not[NegativeQ[u]] && NiceSqrtAuxQ[u] + +NiceSqrtAuxQ[u_] := + If[RationalQ[u], + u>0, + If[PowerQ[u], + EvenQ[u[[2]]], + If[ProductQ[u], + NiceSqrtAuxQ[First[u]] && NiceSqrtAuxQ[Rest[u]], + If[SumQ[u], + Function[NonsumQ[#] && NiceSqrtAuxQ[#]] [Simplify[u]], + False]]]] + + +(* ::Subsection::Closed:: *) +(*PerfectSquareQ[u]*) + + +(* If u is a rational number whose squareroot is rational or if u is of the form u1^n1 u2^n2 ... + and n1, n2, ... are even, PerfectSquareQ[u] returns True; else it returns False. *) +PerfectSquareQ[u_] := + If[RationalQ[u], + u>0 && RationalQ[Sqrt[u]], + If[PowerQ[u], + EvenQ[u[[2]]], + If[ProductQ[u], + PerfectSquareQ[First[u]] && PerfectSquareQ[Rest[u]], + If[SumQ[u], + Function[NonsumQ[#] && PerfectSquareQ[#]] [Simplify[u]], + False]]]] + + +(* ::Section::Closed:: *) +(*Simpler functions*) + + +(* ::Subsection::Closed:: *) +(*SimplerQ[u,v]*) + + +(* If u is simpler than v, SimplerQ[u,v] returns True, else it returns False. SimplerQ[u,u] returns False. *) +SimplerQ[u_,v_] := + If[IntegerQ[u], + If[IntegerQ[v], + If[u==v, + False, + If[u==-v, + v<0, + Abs[u]0, + u<-1, + u>=-v]], + SumSimplerAuxQ[Expand[u],Expand[v]]] + + +SumSimplerAuxQ[u_,v_] := + (RationalQ[First[v]] || SumSimplerAuxQ[u,First[v]]) && + (RationalQ[Rest[v]] || SumSimplerAuxQ[u,Rest[v]]) /; +SumQ[v] + +SumSimplerAuxQ[u_,v_] := + SumSimplerAuxQ[First[u],v] || SumSimplerAuxQ[Rest[u],v] /; +SumQ[u] + +SumSimplerAuxQ[u_,v_] := + v=!=0 && + NonnumericFactors[u]===NonnumericFactors[v] && + (NumericFactor[u]/NumericFactor[v]<-1/2 || NumericFactor[u]/NumericFactor[v]==-1/2 && NumericFactor[u]<0) + + +(* ::Subsection::Closed:: *) +(*SimplerIntegrandQ[u,v,x]*) + + +(* SimplerIntegrandQ[u,v,x] returns True iff u is simpler to integrate wrt x than v. *) +SimplerIntegrandQ[u_,v_,x_Symbol] := + Module[{lst=CancelCommonFactors[u,v],u1,v1}, + u1=lst[[1]]; + v1=lst[[2]]; +(*If[Head[u1]===Head[v1] && Length[u1]==Length[v1]==1, + SimplerIntegrandQ[u1[[1]],v1[[1]],x], *) + If[LeafCount[u1]<3/4*LeafCount[v1], + True, + If[RationalFunctionQ[u1,x], + If[RationalFunctionQ[v1,x], + Apply[Plus,RationalFunctionExponents[u1,x]]0, + With[{lst=Exponent[u,x,List]}, + If[Length[lst]==1, + {0, Coefficient[u,x,Exponent[u,x]], Exponent[u,x]}, + If[Length[lst]==2 && lst[[1]]==0, + {Coefficient[u,x,0], Coefficient[u,x,Exponent[u,x]], Exponent[u,x]}, + False]]], + False], + If[PowerQ[u], + If[u[[1]]===x && FreeQ[u[[2]],x], + {0,1,u[[2]]}, + False], + If[ProductQ[u], + If[FreeQ[First[u],x], + With[{lst2=BinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + {First[u]*lst2[[1]],First[u]*lst2[[2]],lst2[[3]]}]], + If[FreeQ[Rest[u],x], + With[{lst1=BinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + {Rest[u]*lst1[[1]],Rest[u]*lst1[[2]],lst1[[3]]}]], + With[{lst1=BinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + With[{lst2=BinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + With[{a=lst1[[1]],b=lst1[[2]],m=lst1[[3]], c=lst2[[1]],d=lst2[[2]],n=lst2[[3]]}, + If[EqQ[a,0], + If[EqQ[c,0], + {0,b*d,m+n}, + If[EqQ[m+n,0], + {b*d,b*c,m}, + False]], + If[EqQ[c,0], + If[EqQ[m+n,0], + {b*d,a*d,n}, + False], + If[EqQ[m,n] && EqQ[a*d+b*c,0], + {a*c,b*d,2*m}, + False]]]]]]]]]], + If[SumQ[u], + If[FreeQ[First[u],x], + With[{lst2=BinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + {First[u]+lst2[[1]],lst2[[2]],lst2[[3]]}]], + If[FreeQ[Rest[u],x], + With[{lst1=BinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + {Rest[u]+lst1[[1]],lst1[[2]],lst1[[3]]}]], + With[{lst1=BinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + With[{lst2=BinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + If[EqQ[lst1[[3]],lst2[[3]]], + {lst1[[1]]+lst2[[1]],lst1[[2]]+lst2[[2]],lst1[[3]]}, + False]]]]]]], + False]]]] + + +(* ::Subsection::Closed:: *) +(*TrinomialParts[u,x]*) + + +(* If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, + TrinomialDegree[u,x] returns n. *) +TrinomialDegree[u_,x_Symbol] := + TrinomialParts[u,x][[4]] + + +(* If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, + TrinomialParts[u,x] returns the list {a,b,c,n}; else it returns False. *) +TrinomialParts[u_,x_Symbol] := + If[PolynomialQ[u,x], + With[{lst=CoefficientList[u,x]}, + If[Length[lst]<3 || EvenQ[Length[lst]] || EqQ[lst[[(Length[lst]+1)/2]],0], + False, + Catch[ + Scan[Function[If[EqQ[#,0],Null,Throw[False]]],Drop[Drop[Drop[lst,{(Length[lst]+1)/2}],1],-1]]; + {First[lst],lst[[(Length[lst]+1)/2]],Last[lst],(Length[lst]-1)/2}]]], + If[PowerQ[u], + If[EqQ[u[[2]],2], + With[{lst=BinomialParts[u[[1]],x]}, + If[AtomQ[lst] || EqQ[lst[[1]],0], + False, + {lst[[1]]^2,2*lst[[1]]*lst[[2]],lst[[2]]^2,lst[[3]]}]], + False], + If[ProductQ[u], + If[FreeQ[First[u],x], + With[{lst2=TrinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + {First[u]*lst2[[1]],First[u]*lst2[[2]],First[u]*lst2[[3]],lst2[[4]]}]], + If[FreeQ[Rest[u],x], + With[{lst1=TrinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + {Rest[u]*lst1[[1]],Rest[u]*lst1[[2]],Rest[u]*lst1[[3]],lst1[[4]]}]], + With[{lst1=BinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + With[{lst2=BinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + With[{a=lst1[[1]],b=lst1[[2]],m=lst1[[3]], c=lst2[[1]],d=lst2[[2]],n=lst2[[3]]}, + If[EqQ[m,n] && NeQ[a*d+b*c,0], + {a*c,a*d+b*c,b*d,m}, + False]]]]]]]], + If[SumQ[u], + If[FreeQ[First[u],x], + With[{lst2=TrinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + False, + {First[u]+lst2[[1]],lst2[[2]],lst2[[3]],lst2[[4]]}]], + If[FreeQ[Rest[u],x], + With[{lst1=TrinomialParts[First[u],x]}, + If[AtomQ[lst1], + False, + {Rest[u]+lst1[[1]],lst1[[2]],lst1[[3]],lst1[[4]]}]], + With[{lst1=TrinomialParts[First[u],x]}, + If[AtomQ[lst1], + With[{lst3=BinomialParts[First[u],x]}, + If[AtomQ[lst3], + False, + With[{lst2=TrinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + With[{lst4=BinomialParts[Rest[u],x]}, + If[AtomQ[lst4], + False, + If[EqQ[lst3[[3]],2*lst4[[3]]], + {lst3[[1]]+lst4[[1]],lst4[[2]],lst3[[2]],lst4[[3]]}, + If[EqQ[lst4[[3]],2*lst3[[3]]], + {lst3[[1]]+lst4[[1]],lst3[[2]],lst4[[2]],lst3[[3]]}, + False]]]], + If[EqQ[lst3[[3]],lst2[[4]]] && NeQ[lst3[[2]]+lst2[[2]],0], + {lst3[[1]]+lst2[[1]],lst3[[2]]+lst2[[2]],lst2[[3]],lst2[[4]]}, + If[EqQ[lst3[[3]],2*lst2[[4]]] && NeQ[lst3[[2]]+lst2[[3]],0], + {lst3[[1]]+lst2[[1]],lst2[[2]],lst3[[2]]+lst2[[3]],lst2[[4]]}, + False]]]]]], + With[{lst2=TrinomialParts[Rest[u],x]}, + If[AtomQ[lst2], + With[{lst4=BinomialParts[Rest[u],x]}, + If[AtomQ[lst4], + False, + If[EqQ[lst4[[3]],lst1[[4]]] && NeQ[lst1[[2]]+lst4[[2]],0], + {lst1[[1]]+lst4[[1]],lst1[[2]]+lst4[[2]],lst1[[3]],lst1[[4]]}, + If[EqQ[lst4[[3]],2*lst1[[4]]] && NeQ[lst1[[3]]+lst4[[2]],0], + {lst1[[1]]+lst4[[1]],lst1[[2]],lst1[[3]]+lst4[[2]],lst1[[4]]}, + False]]]], + If[EqQ[lst1[[4]],lst2[[4]]] && NeQ[lst1[[2]]+lst2[[2]],0] && NeQ[lst1[[3]]+lst2[[3]],0], + {lst1[[1]]+lst2[[1]],lst1[[2]]+lst2[[2]],lst1[[3]]+lst2[[3]],lst1[[4]]}, + False]]]]]]], + False]]]] + + +(* ::Subsection::Closed:: *) +(*GeneralizedBinomialParts[u,x]*) + + +(* If u is equivalent to a generalized binomial of the form a*x^q + b*x^n where a, b, n, and q not equal 0, + GeneralizedBinomialDegree[u,x] returns n-q. *) +GeneralizedBinomialDegree[u_,x_Symbol] := + Function[#[[3]]-#[[4]]][GeneralizedBinomialParts[u,x]] + + +(* If u is equivalent to a generalized binomial of the form a*x^q + b*x^n where a, b, n, and q not equal 0, + GeneralizedBinomialParts[u,x] returns the list {a,b,n,q}; else it returns False. *) +GeneralizedBinomialParts[a_.*x_^q_.+b_.*x_^n_.,x_Symbol] := + {a,b,n,q} /; +FreeQ[{a,b,n,q},x] && PosQ[n-q] + +GeneralizedBinomialParts[a_*u_,x_Symbol] := + With[{lst=GeneralizedBinomialParts[u,x]}, + {a*lst[[1]], a*lst[[2]], lst[[3]], lst[[4]]} /; + ListQ[lst]] /; +FreeQ[a,x] + +GeneralizedBinomialParts[x_^m_.*u_,x_Symbol] := + With[{lst=GeneralizedBinomialParts[u,x]}, + {lst[[1]], lst[[2]], m+lst[[3]], m+lst[[4]]} /; + ListQ[lst] && NeQ[m+lst[[3]],0] && NeQ[m+lst[[4]],0]] /; +FreeQ[m,x] + +GeneralizedBinomialParts[x_^m_.*u_,x_Symbol] := + With[{lst=BinomialParts[u,x]}, + {lst[[1]], lst[[2]], m+lst[[3]], m} /; + ListQ[lst] && NeQ[m+lst[[3]],0]] /; +FreeQ[m,x] + +GeneralizedBinomialParts[u_,x_Symbol] := + False + + +(* ::Subsection::Closed:: *) +(*GeneralizedTrinomialParts[u,x]*) + + +(* If u is equivalent to a generalized trinomial of the form a*x^q + b*x^n + c*x^(2*n-q) where n!=0, q!=0, b!=0 and c!=0, + GeneralizedTrinomialDegree[u,x] returns n-q. *) +GeneralizedTrinomialDegree[u_,x_Symbol] := + Function[#[[4]]-#[[5]]][GeneralizedTrinomialParts[u,x]] + + +(* If u is equivalent to a generalized trinomial of the form a*x^q + b*x^n + c*x^(2*n-q) where n!=0, q!=0, b!=0 and c!=0, + GeneralizedTrinomialParts[u,x] returns the list {a,b,c,n,q}; else it returns False. *) +GeneralizedTrinomialParts[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.,x_Symbol] := + {a,b,c,n,q} /; +FreeQ[{a,b,c,n,q},x] && EqQ[r,2*n-q] + +GeneralizedTrinomialParts[a_*u_,x_Symbol] := + With[{lst=GeneralizedTrinomialParts[u,x]}, + {a*lst[[1]], a*lst[[2]], a*lst[[3]], lst[[4]], lst[[5]]} /; + ListQ[lst]] /; +FreeQ[a,x] + +GeneralizedTrinomialParts[u_,x_Symbol] := + With[{lst=Expon[u,x,List]}, + If[Length[lst]==3 && NeQ[lst[[0]],0] && EqQ[lst[[3]],2*lst[[2]]-lst[[1]]], + {Coeff[u,x,lst[[1]]],Coeff[u,x,lst[[2]]],Coeff[u,x,lst[[3]]],lst[[2]],lst[[1]]}, + False]] /; +PolyQ[u,x] + +GeneralizedTrinomialParts[x_^m_.*u_,x_Symbol] := + With[{lst=GeneralizedTrinomialParts[u,x]}, + {lst[[1]], lst[[2]], lst[[3]], m+lst[[4]], m+lst[[5]]} /; + ListQ[lst] && NeQ[m+lst[[4]],0] && NeQ[m+lst[[5]],0]] /; +FreeQ[m,x] + +GeneralizedTrinomialParts[x_^m_.*u_,x_Symbol] := + With[{lst=TrinomialParts[u,x]}, + {lst[[1]], lst[[2]], lst[[3]], m+lst[[4]], m} /; + ListQ[lst] && NeQ[m+lst[[4]],0]] /; +FreeQ[m,x] + +GeneralizedTrinomialParts[u_,x_Symbol] := + False + + +(* ::Section::Closed:: *) +(*Selection functions*) + + +(* ::Subsection::Closed:: *) +(*IntPart[u]*) + + +(* IntPart[u] returns the sum of the integer terms of u. *) +IntPart[m_*u_,n_:1] := + IntPart[u,m*n] /; +RationalQ[m] + +IntPart[u_,n_:1] := + If[RationalQ[u], + IntegerPart[n*u], + If[SumQ[u], + Map[Function[IntPart[#,n]],u], + 0]] + + +(* ::Subsection::Closed:: *) +(*FracPart[u]*) + + +(* IntPart[u] returns the sum of the non-integer terms of u. *) +FracPart[m_*u_,n_:1] := + FracPart[u,m*n] /; +RationalQ[m] + +FracPart[u_,n_:1] := + If[RationalQ[u], + FractionalPart[n*u], + If[SumQ[u], + Map[Function[FracPart[#,n]],u], + n*u]] + + +(* ::Subsection::Closed:: *) +(*NumericFactor[u] *) + + +(* NumericFactor[u] returns the real numeric factor of u. *) +NumericFactor[u_] := + If[NumberQ[u], + If[EqQ[Im[u],0], + u, + If[EqQ[Re[u],0], + Im[u], + 1]], + If[PowerQ[u], + If[RationalQ[u[[1]]] && FractionQ[u[[2]]], + If[u[[2]]>0, + 1/Denominator[u[[1]]], + 1/Denominator[1/u[[1]]]], + 1], + If[ProductQ[u], + Map[NumericFactor,u], + If[SumQ[u], + If[LeafCount[u]<50, (* Eliminate this kludge! *) + Function[If[SumQ[#], 1, NumericFactor[#]]][ContentFactor[u]], + With[{m=NumericFactor[First[u]],n=NumericFactor[Rest[u]]}, + If[m<0 && n<0, + -GCD[-m,-n], + GCD[m,n]]]], + 1]]]] + + +(* NonnumericFactors[u] returns the product of the factors of u that are not real numbers. *) +NonnumericFactors[u_] := + If[NumberQ[u], + If[EqQ[Im[u],0], + 1, + If[EqQ[Re[u],0], + I, + u]], + If[PowerQ[u], + If[RationalQ[u[[1]]] && FractionQ[u[[2]]], + u/NumericFactor[u], + u], + If[ProductQ[u], + Map[NonnumericFactors,u], + If[SumQ[u], + If[LeafCount[u]<50, (* Eliminate this kludge! *) + Function[If[SumQ[#], u, NonnumericFactors[#]]][ContentFactor[u]], + With[{n=NumericFactor[u]}, + Map[Function[#/n],u]]], + u]]]] + + +(* ::Subsection::Closed:: *) +(*RemoveContent[u,x]*) + + +(* RemoveContent[u,x] returns u with the content free of x removed. *) +RemoveContent[u_,x_Symbol] := + With[{v=NonfreeFactors[u,x]}, + With[{w=Together[v]}, + If[EqQ[FreeFactors[w,x],1], + RemoveContentAux[v,x], + RemoveContentAux[NonfreeFactors[w,x],x]]]] + + +RemoveContentAux[a_^m_*u_.+b_*v_.,x_Symbol] := + If[m>1, + RemoveContentAux[a^(m-1)*u-v,x], + RemoveContentAux[u-a^(1-m)*v,x]] /; +IntegersQ[a,b] && a+b==0 && RationalQ[m] + + +RemoveContentAux[a_^m_.*u_.+a_^n_.*v_.,x_Symbol] := + RemoveContentAux[u+a^(n-m)*v,x] /; +FreeQ[a,x] && RationalQ[m,n] && n-m>=0 + + +RemoveContentAux[a_^m_.*u_.+a_^n_.*v_.+a_^p_.*w_.,x_Symbol] := + RemoveContentAux[u+a^(n-m)*v+a^(p-m)*w,x] /; +FreeQ[a,x] && RationalQ[m,n,p] && n-m>=0 && p-m>=0 + + +RemoveContentAux[u_,x_Symbol] := + If[SumQ[u] && NegQ[First[u]], + -u, + u] + + +(* ::Subsection::Closed:: *) +(*FreeFactors[u,x]*) + + +(* FreeFactors[u,x] returns the product of the factors of u free of x. *) +FreeFactors[u_,x_] := + If[ProductQ[u], + Map[Function[If[FreeQ[#,x],#,1]],u], + If[FreeQ[u,x], + u, + 1]] + + +(* ::Subsection::Closed:: *) +(*NonfreeFactors[u,x]*) + + +(* NonfreeFactors[u,x] returns the product of the factors of u not free of x. *) +NonfreeFactors[u_,x_] := + If[ProductQ[u], + Map[Function[If[FreeQ[#,x],1,#]],u], + If[FreeQ[u,x], + 1, + u]] + + +(* ::Subsection::Closed:: *) +(*FreeTerms[u,x]*) + + +(* FreeTerms[u,x] returns the sum of the terms of u free of x. *) +FreeTerms[u_,x_] := + If[SumQ[u], + Map[Function[If[FreeQ[#,x],#,0]],u], + If[FreeQ[u,x], + u, + 0]] + + +(* ::Subsection::Closed:: *) +(*NonfreeTerms[u,x]*) + + +(* NonfreeTerms[u,x] returns the sum of the terms of u not free of x. *) +NonfreeTerms[u_,x_] := + If[SumQ[u], + Map[Function[If[FreeQ[#,x],0,#]],u], + If[FreeQ[u,x], + 0, + u]] + + +(* ::Subsection::Closed:: *) +(*Polynomial functions*) + + +Expon[expr_,form_] := + Exponent[Together[expr],form] + +Expon[expr_,form_,h_] := + Exponent[Together[expr],form,h] + + +Coeff[expr_,form_] := + Coefficient[Together[expr],form] + +Coeff[expr_,form_,n_] := + With[{coef1=Coefficient[expr,form,n],coef2=Coefficient[Together[expr],form,n]}, + If[Simplify[coef1-coef2]===0, + coef1, + coef2]] + + +(* ::Subsection::Closed:: *) +(*LeadTerm[u]*) + + +LeadTerm[u_] := + If[SumQ[u], + First[u], + u] + + +(* ::Subsection::Closed:: *) +(*RemainingTerms[u]*) + + +RemainingTerms[u_] := + If[SumQ[u], + Rest[u], + 0] + + +(* ::Subsection::Closed:: *) +(*LeadFactor[u]*) + + +(* LeadFactor[u] returns the leading factor of u. *) +LeadFactor[u_] := + If[ProductQ[u], + LeadFactor[First[u]], + If[ComplexNumberQ[u] && Re[u]===0, + If[Im[u]===1, + u, + LeadFactor[Im[u]]], + u]] + + +(* ::Subsection::Closed:: *) +(*RemainingFactors[u]*) + + +(* RemainingFactors[u] returns the remaining factors of u. *) +RemainingFactors[u_] := + If[ProductQ[u], + RemainingFactors[First[u]]*Rest[u], + If[ComplexNumberQ[u] && Re[u]===0, + If[Im[u]===1, + 1, + I*RemainingFactors[Im[u]]], + 1]] + + +(* ::Subsection::Closed:: *) +(*LeadBase[u]*) + + +(* LeadBase[u] returns the base of the leading factor of u. *) +LeadBase[u_] := + With[{v=LeadFactor[u]}, + If[PowerQ[v], + v[[1]], + v]] + + +(* ::Subsection::Closed:: *) +(*LeadDegree[u]*) + + +(* LeadDegree[u] returns the degree of the leading factor of u. *) +LeadDegree[u_] := + With[{v=LeadFactor[u]}, + If[PowerQ[v], + v[[2]], + 1]] + + +(* ::Subsection::Closed:: *) +(*Numer[u]*) + + +(* Numer[u] returns the numerator of u. *) +Numer[m_Integer^n_Rational] := + 1 /; +n<0 + + +Numer[u_*v_] := + Numer[u]*Numer[v] + + +Numer[u_] := Numerator[u] + + +(* ::Subsection::Closed:: *) +(*Denom[u]*) + + +(* Denom[u] returns the denominator of u. *) +Denom[m_Integer^n_Rational] := + m^-n /; +n<0 + + +Denom[u_*v_] := + Denom[u]*Denom[v] + + +Denom[u_] := Denominator[u] + + +(* ::Section::Closed:: *) +(*Multinomial functions*) + + +(* ::Subsection::Closed:: *) +(*Multinomial Recognizer functions*) + + +(* ::Item:: *) +(*LinearQ[u,x] returns True iff u is a polynomial of degree 1*) + + +LinearQ[u_,x_Symbol] := + PolyQ[u,x,1] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*PowerOfLinearQ[u,x] returns True iff u is equivalent to an expression of the form (a+b x)^m.*) + + +PowerOfLinearQ[u_^m_.,x_Symbol] := + FreeQ[m,x] && PolynomialQ[u,x] && If[IntegerQ[m], MatchQ[FactorSquareFree[u], w_^n_. /; FreeQ[n,x] && LinearQ[w,x]], LinearQ[u,x]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*QuadraticQ[u,x] returns True iff u is a polynomial of degree 2 and not a monomial of the form a x^2.*) + + +QuadraticQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[QuadraticQ[#,x]],Throw[False]]],u]; True], + PolyQ[u,x,2] && Not[Coefficient[u,x,0]===0 && Coefficient[u,x,1]===0]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*LinearPairQ[u,v,x] returns True iff u and v are linear not equal x but u/v is a constant wrt x.*) + + +LinearPairQ[u_,v_,x_Symbol] := + LinearQ[u,x] && LinearQ[v,x] && NeQ[u,x] && EqQ[Coefficient[u,x,0]*Coefficient[v,x,1]-Coefficient[u,x,1]*Coefficient[v,x,0],0] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*If u is of the form a*x^n where n!=0 and a!=0, MonomialQ[u,x] returns True; else False.*) + + +MonomialQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[MonomialQ[#,x],Null,Throw[False]]],u]; True], + MatchQ[u, a_.*x^n_. /; FreeQ[{a,n},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*f u[x] is a sum and each term is free of x or an expression of the form a*x^n, MonomialSumQ[u,x] returns True; else it returns False.*) + + +MonomialSumQ[u_,x_Symbol] := + SumQ[u] && Catch[ + Scan[Function[If[FreeQ[#,x] || MonomialQ[#,x], Null, Throw[False]]],u]; + True] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*u is sum whose terms are monomials. MinimumExponent[u,x] returns the exponent of the term having the smallest exponent.*) + + +MinimumMonomialExponent[u_,x_Symbol] := + Module[{n=MonomialExponent[First[u],x]}, + Scan[Function[If[PosQ[n-MonomialExponent[#,x]],n=MonomialExponent[#,x]]],u]; + n] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*u is a monomial. MonomialExponent[u,x] returns the exponent of x in u.*) + + +MonomialExponent[a_.*x_^n_.,x_Symbol] := + n /; +FreeQ[{a,n},x] + + +(* ::Subsection::Closed:: *) +(*Multinomial Match functions*) + + +(* ::Item:: *) +(*LinearMatchQ[u,x] returns True iff u matches patterns of the form a+b x where a and b are free of x.*) + + +LinearMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[LinearMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.+b_.*x /; FreeQ[{a,b},x]]] + + +(* ::Input:: *) +(**) + + +(* ::Item:: *) +(*PowerOfLinearMatchQ[u,x] returns True iff u matches patterns of the form (a+b x)^m where a, b and m are free of x.*) + + +PowerOfLinearMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[PowerOfLinearMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, (a_.+b_.*x)^m_. /; FreeQ[{a,b,m},x]]] + + +(* ::Input:: *) +(**) + + +(* ::Item:: *) +(*QuadraticMatchQ[u,x] returns True iff u matches patterns of the form a+b x+c x^2 or a+c x^2 where a, b and c are free of x.*) + + +QuadraticMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[QuadraticMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.+b_.*x+c_.*x^2 /; FreeQ[{a,b,c},x]] || MatchQ[u, a_.+c_.*x^2 /; FreeQ[{a,c},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*CubicMatchQ[u,x] returns True iff u matches patterns of the form a+b x+c x^2+d x^3, a+b x+d x^3, a+c x^2+d x^3 or a+d x^3 where a, b, c and d are free of x.*) + + +CubicMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[CubicMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.+b_.*x+c_.*x^2+d_.*x^3 /; FreeQ[{a,b,c,d},x]] || + MatchQ[u, a_.+b_.*x+d_.*x^3 /; FreeQ[{a,b,d},x]] || + MatchQ[u, a_.+c_.*x^2+d_.*x^3 /; FreeQ[{a,c,d},x]] || + MatchQ[u, a_.+d_.*x^3 /; FreeQ[{a,d},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*BinomialMatchQ[u,x] returns True iff u matches patterns of the form a+b x^n where a, b and n are free of x.*) + + +BinomialMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[BinomialMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.+b_.*x^n_. /; FreeQ[{a,b,n},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*GeneralizedBinomialMatchQ[u,x] returns True iff u matches patterns of the form a x^q+b x^n where a, b, n and q are free of x.*) + + +GeneralizedBinomialMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[GeneralizedBinomialMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.*x^q_.+b_.*x^n_. /; FreeQ[{a,b,n,q},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*TrinomialMatchQ[u,x] returns True iff u matches patterns of the form a+b x+c x^n or a+c x^(2 n) where a, b, c and n are free of x.*) + + +TrinomialMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[TrinomialMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.+b_.*x^n_.+c_.*x^j_. /; FreeQ[{a,b,c,n},x] && EqQ[j-2*n,0]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*GeneralizedTrinomialMatchQ[u,x] returns True iff u matches patterns of the form a x^q+b x^n+c x^(2 n-q) where a, b, c, n and q are free of x.*) + + +GeneralizedTrinomialMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[GeneralizedTrinomialMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, a_.*x^q_.+b_.*x^n_.+c_.*x^r_. /; FreeQ[{a,b,c,n,q},x] && EqQ[r-(2*n-q),0]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*QuotientOfLinearsMatchQ[u,x] returns True iff u matches patterns of the form e (a+b x)/(c+d x) where a, b, c and d are free of x.*) + + +QuotientOfLinearsMatchQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[QuotientOfLinearsMatchQ[#,x]],Throw[False]]],u]; True], + MatchQ[u, e_.*(a_.+b_.*x)/(c_.+d_.*x) /; FreeQ[{a,b,c,d,e},x]]] + + +(* ::Input:: *) +(* *) + + +(* ::Subsection::Closed:: *) +(*Polynomial Terms functions*) + + +(* If u (x) is an expression of the form a*x^n where n is zero or a positive integer, + PolynomialTermQ[u,x] returns True; else it returns False. *) +PolynomialTermQ[u_,x_Symbol] := + FreeQ[u,x] || MatchQ[u,a_.*x^n_. /; FreeQ[a,x] && IntegerQ[n] && n>0] + + +(* u (x) is a sum. PolynomialTerms[u,x] returns the sum of the polynomial terms of u (x). *) +PolynomialTerms[u_,x_Symbol] := + Map[Function[If[PolynomialTermQ[#,x],#,0]],u] + + +(* u (x) is a sum. NonpolynomialTerms[u,x] returns the sum of the nonpolynomial terms of u (x). *) +NonpolynomialTerms[u_,x_Symbol] := + Map[Function[If[PolynomialTermQ[#,x],0,#]],u] + + +(* ::Subsection::Closed:: *) +(*PseudoBinomial Routines*) + + +(* If u is equivalent to a polynomial of the form a+b*(c+d*x)^n where n\[Element]\[DoubleStruckCapitalZ] and n>2, + PseudoBinomialQ[u,x] returns True; else it returns False. *) +PseudoBinomialQ[u_,x_Symbol] := + ListQ[PseudoBinomialParts[u,x]] + + +(* If u is equivalent to a polynomial of the form a+b*(e+f*x)^n and v to a polynomial of the form c+d*(e+f*x)^n where n\[Element]\[DoubleStruckCapitalZ] and n>2, + PseudoBinomialPairQ[u,v,x] returns True; else it returns False. *) +PseudoBinomialPairQ[u_,v_,x_Symbol] := + With[{lst1=PseudoBinomialParts[u,x]}, + If[AtomQ[lst1], + False, + With[{lst2=PseudoBinomialParts[v,x]}, + If[AtomQ[lst2], + False, + Drop[lst1,2]===Drop[lst2,2]]]]] + + +(* u is pseudo-binomial in x, NormalizePseudoBinomial[u,x] returns u in the form a+b*(c+d*x)^n. *) +NormalizePseudoBinomial[u_,x_Symbol] := + With[{lst=PseudoBinomialParts[u,x]}, + lst[[1]]+lst[[2]]*(lst[[3]]+lst[[4]]*x)^lst[[5]]] + + +(* If u is equivalent to a polynomial of the form a+b*(c+d*x)^n where n\[Element]\[DoubleStruckCapitalZ] and n>2, + PseudoBinomialParts[u,x] returns the list {a,b,c,d,n}; else it returns False. *) +PseudoBinomialParts[u_,x_Symbol] := + If[PolynomialQ[u,x] && Expon[u,x]>2, + Module[{a,c,d,n}, + n=Expon[u,x]; + d=Rt[Coefficient[u,x,n],n]; + c=Coefficient[u,x,n-1]/(n*d^(n-1)); + a=Simplify[u-(c+d*x)^n]; + If[NeQ[a,0] && FreeQ[a,x], + {a,1,c,d,n}, + False]], + False] + + +(* ::Subsection::Closed:: *) +(*Perfect Power Test Function*) + + +(* If u (x) is equivalent to a polynomial raised to an integer power greater than 1, + PerfectPowerTest[u,x] returns u (x) as an expanded polynomial raised to the power; + else it returns False. *) +PerfectPowerTest[u_,x_Symbol] := + If[PolynomialQ[u,x], + Module[{lst=FactorSquareFreeList[u],gcd=0,v=1}, + If[lst[[1]]==={1,1}, + lst=Rest[lst]]; + Scan[Function[gcd=GCD[gcd,#[[2]]]],lst]; + If[gcd>1, + Scan[Function[v=v*#[[1]]^(#[[2]]/gcd)],lst]; + Expand[v]^gcd, + False]], + False] + + +(* ::Subsection::Closed:: *) +(*Square Free Factor Test Function*) + + +(* If u (x) can be square free factored, SquareFreeFactorTest[u,x] returns u (x) in + factored form; else it returns False. *) +(* SquareFreeFactorTest[u_,x_Symbol] := + If[PolynomialQ[u,x], + With[{v=FactorSquareFree[u]}, + If[PowerQ[v] || ProductQ[v], + v, + False]], + False] *) + + +(* ::Section::Closed:: *) +(*Rational function functions*) + + +(* ::Subsection::Closed:: *) +(*RationalFunctionQ*) + + +(* If u is a rational function of x, RationalFunctionQ[u,x] returns True; else it returns False. *) +RationalFunctionQ[u_,x_Symbol] := + If[AtomQ[u] || FreeQ[u,x], + True, + If[IntegerPowerQ[u], + RationalFunctionQ[u[[1]],x], + If[ProductQ[u] || SumQ[u], + Catch[Scan[Function[If[Not[RationalFunctionQ[#,x]],Throw[False]]],u];True], + False]]] + + +(* ::Subsection::Closed:: *) +(*RationalFunctionFactors*) + + +(* RationalFunctionFactors[u,x] returns the product of the factors of u that are rational functions of x. *) +RationalFunctionFactors[u_,x_Symbol] := + If[ProductQ[u], + Map[Function[If[RationalFunctionQ[#,x],#,1]],u], + If[RationalFunctionQ[u,x], + u, + 1]] + + +(* ::Subsection::Closed:: *) +(*NonrationalFunctionFactors*) + + +(* NonrationalFunctionFactors[u,x] returns the product of the factors of u that are not rational functions of x. *) +NonrationalFunctionFactors[u_,x_Symbol] := + If[ProductQ[u], + Map[Function[If[RationalFunctionQ[#,x],1,#]],u], + If[RationalFunctionQ[u,x], + 1, + u]] + + +(* ::Subsection::Closed:: *) +(*RationalFunctionExponents*) + + +(* u is a polynomial or rational function of x. *) +(* RationalFunctionExponents[u,x] returns a list of the exponent of the *) +(* numerator of u and the exponent of the denominator of u. *) +RationalFunctionExponents[u_,x_Symbol] := + If[PolynomialQ[u,x], + {Exponent[u,x],0}, + If[IntegerPowerQ[u], + If[u[[2]]>0, + u[[2]]*RationalFunctionExponents[u[[1]],x], + (-u[[2]])*Reverse[RationalFunctionExponents[u[[1]],x]]], + If[ProductQ[u], + RationalFunctionExponents[First[u],x]+RationalFunctionExponents[Rest[u],x], + If[SumQ[u], + With[{v=Together[u]}, + If[SumQ[v], + Module[{lst1,lst2}, + lst1=RationalFunctionExponents[First[u],x]; + lst2=RationalFunctionExponents[Rest[u],x]; + {Max[lst1[[1]]+lst2[[2]],lst2[[1]]+lst1[[2]]],lst1[[2]]+lst2[[2]]}], + RationalFunctionExponents[v,x]]], + {0,0}]]]] + + +(* ::Subsection::Closed:: *) +(*RationalFunctionExpand*) + + +(* u is a polynomial or rational function of x. *) +(* RationalFunctionExpand[u,x] returns the expansion of the factors of u that are rational functions times the other factors. *) +RationalFunctionExpand[u_*v_^n_,x_Symbol] := + With[{w=RationalFunctionExpand[u,x]}, + If[SumQ[w], + Map[Function[#*v^n],w], + w*v^n]] /; +FractionQ[n] && v=!=x + +RationalFunctionExpand[u_,x_Symbol] := + Module[{v,w}, + v=ExpandIntegrand[u,x]; + If[v=!=u && Not[MatchQ[u, x^m_.*(c_+d_.*x)^p_/(a_+b_.*x^n_) /; FreeQ[{a,b,c,d,p},x] && IntegersQ[m,n] && m==n-1]], + v, + v=ExpandIntegrand[RationalFunctionFactors[u,x],x]; + w=NonrationalFunctionFactors[u,x]; + If[SumQ[v], + Map[Function[#*w],v], + v*w]]] + + +(* ::Subsection::Closed:: *) +(*PolyGCD*) + + +(* u and v are polynomials in x. *) +(* PolyGCD[u,v,x] returns the factors of the gcd of u and v dependent on x. *) +PolyGCD[u_,v_,x_Symbol] := + NonfreeFactors[PolynomialGCD[u,v],x] + + +(* ::Section::Closed:: *) +(*Algebraic function functions*) + + +(* ::Subsection::Closed:: *) +(*AlgebraicFunctionQ*) + + +(* AlgebraicFunctionQ[u,x] returns True iff u is an algebraic of x *) +(* If flag is True, exponents can be nonnumeric. *) +AlgebraicFunctionQ[u_,x_Symbol,flag_:False] := + If[AtomQ[u] || FreeQ[u,x], + True, + If[PowerQ[u] && (RationalQ[u[[2]]] || flag && FreeQ[u[[2]],x]), + AlgebraicFunctionQ[u[[1]],x,flag], + If[ProductQ[u] || SumQ[u], + Catch[Scan[Function[If[Not[AlgebraicFunctionQ[#,x,flag]],Throw[False]]],u];True], + If[ListQ[u], + If[u==={}, + True, + If[AlgebraicFunctionQ[First[u],x,flag], + AlgebraicFunctionQ[Rest[u],x,flag], + False]], + False]]]] + + +(* ::Subsection::Closed:: *) +(*AlgebraicFunctionFactors*) + + +(* AlgebraicFunctionFactors[u,x] returns the product of the factors of u that are algebraic functions of x. *) +(* AlgebraicFunctionFactors[u_,x_Symbol,flag_:False] := + If[ProductQ[u], + Map[Function[If[AlgebraicFunctionQ[#,x,flag],#,1]],u], + If[AlgebraicFunctionQ[u,x,flag],u,1]] *) + + +(* ::Subsection::Closed:: *) +(*NonalgebraicFunctionFactors*) + + +(* NonalgebraicFunctionFactors[u,x] returns the product of the factors of u that are not algebraic functions of x. *) +(* NonalgebraicFunctionFactors[u_,x_Symbol,flag_:False] := + If[ProductQ[u], + Map[Function[If[AlgebraicFunctionQ[#,x,flag],1,#]],u], + If[AlgebraicFunctionQ[u,x,flag],1,u]] *) + + +(* ::Section::Closed:: *) +(*Quotient of linears functions*) + + +(* ::Subsection::Closed:: *) +(*QuotientOfLinearsQ*) + + +(* ::Item:: *) +(*QuotientOfLinearsQ[u,x] returns True iff u is equivalent to an expression of the form (a+b x)/(c+d x) where b!=0 and d!=0.*) + + +QuotientOfLinearsQ[u_,x_Symbol] := + If[ListQ[u], + Catch[Scan[Function[If[Not[QuotientOfLinearsQ[#,x]],Throw[False]]],u]; True], + QuotientOfLinearsP[u,x] && Function[NeQ[#[[2]],0] && NeQ[#[[4]],0]][QuotientOfLinearsParts[u,x]]] + + +QuotientOfLinearsP[a_*u_,x_] := + QuotientOfLinearsP[u,x] /; +FreeQ[a,x] + +QuotientOfLinearsP[a_+u_,x_] := + QuotientOfLinearsP[u,x] /; +FreeQ[a,x] + +QuotientOfLinearsP[1/u_,x_] := + QuotientOfLinearsP[u,x] + +QuotientOfLinearsP[u_,x_] := + True /; +LinearQ[u,x] + +QuotientOfLinearsP[u_/v_,x_] := + True /; +LinearQ[u,x] && LinearQ[v,x] + +QuotientOfLinearsP[u_,x_] := + u===x || FreeQ[u,x] + + +(* ::Subsection::Closed:: *) +(*QuotientOfLinearsParts*) + + +(* If u is equivalent to an expression of the form (a+b*x)/(c+d*x), QuotientOfLinearsParts[u,x] + returns the list {a, b, c, d}. *) +QuotientOfLinearsParts[a_*u_,x_] := + Apply[Function[{a*#1, a*#2, #3, #4}], QuotientOfLinearsParts[u,x]] /; +FreeQ[a,x] + +QuotientOfLinearsParts[a_+u_,x_] := + Apply[Function[{#1+a*#3, #2+a*#4, #3, #4}], QuotientOfLinearsParts[u,x]] /; +FreeQ[a,x] + +QuotientOfLinearsParts[1/u_,x_] := + Apply[Function[{#3, #4, #1, #2}], QuotientOfLinearsParts[u,x]] + +QuotientOfLinearsParts[u_,x_] := + {Coefficient[u,x,0], Coefficient[u,x,1], 1, 0} /; +LinearQ[u,x] + +QuotientOfLinearsParts[u_/v_,x_] := + {Coefficient[u,x,0], Coefficient[u,x,1], Coefficient[v,x,0], Coefficient[v,x,1]} /; +LinearQ[u,x] && LinearQ[v,x] + +QuotientOfLinearsParts[u_,x_] := + If[u===x, + {0, 1, 1, 0}, + If[FreeQ[u,x], + {u, 0, 1, 0}, + Print["QuotientOfLinearsParts error!"]; + {u, 0, 1, 0}]] + + +(* ::Subsection::Closed:: *) +(*SubstForFractionalPowerOfQuotientOfLinears*) + + +(* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n) where m and n>1 are integers, + SubstForFractionalPowerOfQuotientOfLinears[u,x] returns the list {v,n,(a+b*x)/(c+d*x),b*c-a*d} where v is u + with subexpressions of the form ((a+b*x)/(c+d*x))^(m/n) replaced by x^m and x replaced + by (-a+c*x^n)/(b-d*x^n), and all times x^(n-1)/(b-d*x^n)^2; else it returns False. *) +SubstForFractionalPowerOfQuotientOfLinears[u_,x_Symbol] := + Module[{lst=FractionalPowerOfQuotientOfLinears[u,1,False,x]}, + If[AtomQ[lst] || AtomQ[lst[[2]]], + False, + With[{n=lst[[1]],tmp=lst[[2]]}, + lst=QuotientOfLinearsParts[tmp,x]; + With[{a=lst[[1]],b=lst[[2]],c=lst[[3]],d=lst[[4]]}, + If[EqQ[d,0], + False, + lst=Simplify[x^(n-1)*SubstForFractionalPower[u,tmp,n,(-a+c*x^n)/(b-d*x^n),x]/(b-d*x^n)^2]; + {NonfreeFactors[lst,x],n,tmp,FreeFactors[lst,x]*(b*c-a*d)}]]]]] + + +(* If the substitution x=v^(1/n) will not complicate algebraic subexpressions of u, + SubstForFractionalPowerQ[u,v,x] returns True; else it returns False. *) +SubstForFractionalPowerQ[u_,v_,x_Symbol] := + If[AtomQ[u] || FreeQ[u,x], + True, + If[FractionalPowerQ[u], + SubstForFractionalPowerAuxQ[u,v,x], + Catch[Scan[Function[If[Not[SubstForFractionalPowerQ[#,v,x]],Throw[False]]],u];True]]] + +SubstForFractionalPowerAuxQ[u_,v_,x_] := + If[AtomQ[u], + False, + If[FractionalPowerQ[u] && EqQ[u[[1]],v], + True, + Catch[Scan[Function[If[SubstForFractionalPowerAuxQ[#,v,x],Throw[True]]],u];False]]] + + +(* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n), + FractionalPowerOfQuotientOfLinears[u,1,False,x] returns {n,(a+b*x)/(c+d*x)}; else it returns False. *) +FractionalPowerOfQuotientOfLinears[u_,n_,v_,x_] := + If[AtomQ[u] || FreeQ[u,x], + {n,v}, + If[CalculusQ[u], + False, + If[FractionalPowerQ[u] && QuotientOfLinearsQ[u[[1]],x] && Not[LinearQ[u[[1]],x]] && (FalseQ[v] || EqQ[u[[1]],v]), + {LCM[Denominator[u[[2]]],n],u[[1]]}, + Catch[Module[{lst={n,v}}, + Scan[Function[If[AtomQ[lst=FractionalPowerOfQuotientOfLinears[#,lst[[1]],lst[[2]],x]],Throw[False]]],u]; + lst]]]]] + + +(* ::Subsection::Closed:: *) +(*SubstForInverseFunctionOfQuotientOfLinears*) + + +(* If u has a subexpression of the form g[(a+b*x)/(c+d*x)] where g is the inverse of function h + and f[x,g[(a+b*x)/(c+d*x)]] equals u, SubstForInverseFunctionOfQuotientOfLinears[u,x] returns + the list {f[(-a+c*h[x])/(b-d*h[x]),x]*h'[x]/(b-d*h[x])^2, g[(a+b*x)/(c+d*x)], b*c-a*d} *) +SubstForInverseFunctionOfQuotientOfLinears[u_,x_Symbol] := + With[{tmp=InverseFunctionOfQuotientOfLinears[u,x]}, + If[AtomQ[tmp], + False, + With[{h=InverseFunction[Head[tmp]],lst=QuotientOfLinearsParts[tmp[[1]],x]}, + With[{a=lst[[1]],b=lst[[2]],c=lst[[3]],d=lst[[4]]}, + {SubstForInverseFunction[u,tmp,(-a+c*h[x])/(b-d*h[x]),x]*D[h[x],x]/(b-d*h[x])^2, tmp, b*c-a*d}]]]] + + +(* If u has a subexpression of the form g[(a+b*x)/(c+d*x)] where g is an inverse function, + InverseFunctionOfQuotientOfLinears[u,x] returns g[(a+b*x)/(c+d*x)]; else it returns False. *) +InverseFunctionOfQuotientOfLinears[u_,x_Symbol] := + If[AtomQ[u] || CalculusQ[u] || FreeQ[u,x], + False, + If[InverseFunctionQ[u] && QuotientOfLinearsQ[u[[1]],x], + u, + Module[{tmp}, + Catch[ + Scan[Function[If[Not[AtomQ[tmp=InverseFunctionOfQuotientOfLinears[#,x]]],Throw[tmp]]],u]; + False]]]] + + +(* ::Subsubsection::Closed:: *) +(*Substitution for inverse functions*) + + +(* SubstForFractionalPower[u,v,n,w,x] returns u with subexpressions equal to v^(m/n) replaced + by x^m and x replaced by w. *) +SubstForFractionalPower[u_,v_,n_,w_,x_Symbol] := + If[AtomQ[u], + If[u===x, + w, + u], + If[FractionalPowerQ[u] && EqQ[u[[1]],v], + x^(n*u[[2]]), + Map[Function[SubstForFractionalPower[#,v,n,w,x]],u]]] + + +(* SubstForInverseFunction[u,v,w,x] returns u with subexpressions equal to v replaced by x + and x replaced by w. *) +SubstForInverseFunction[u_,v_,x_Symbol] := +(* Module[{a=Coefficient[v[[1]],0],b=Coefficient[v[[1]],1]}, + SubstForInverseFunction[u,v,-a/b+InverseFunction[Head[v]]/b,x]] *) + SubstForInverseFunction[u,v, + (-Coefficient[v[[1]],x,0]+InverseFunction[Head[v]][x])/Coefficient[v[[1]],x,1],x] + +SubstForInverseFunction[u_,v_,w_,x_Symbol] := + If[AtomQ[u], + If[u===x, + w, + u], + If[Head[u]===Head[v] && EqQ[u[[1]],v[[1]]], + x, + Map[Function[SubstForInverseFunction[#,v,w,x]],u]]] + + +(* ::Section::Closed:: *) +(*Absurd number factors*) + + +(* ::Text:: *) +(*Definition: A number is absurd if it is a rational number, a positive rational number raised to a fractional power, or a product of absurd numbers.*) + + +(* AbsurdNumberQ[u] returns True if u is an absurd number, else it returns False. *) +AbsurdNumberQ[u_] := + RationalQ[u] + +AbsurdNumberQ[u_^v_] := + RationalQ[u] && u>0 && FractionQ[v] + +AbsurdNumberQ[u_*v_] := + AbsurdNumberQ[u] && AbsurdNumberQ[v] + + +(* AbsurdNumberFactors[u] returns the product of the factors of u that are absurd numbers. *) +AbsurdNumberFactors[u_] := + If[AbsurdNumberQ[u], + u, + If[ProductQ[u], + Map[AbsurdNumberFactors,u], + NumericFactor[u]]] + + +(* NonabsurdNumberFactors[u] returns the product of the factors of u that are not absurd numbers. *) +NonabsurdNumberFactors[u_] := + If[AbsurdNumberQ[u], + 1, + If[ProductQ[u], + Map[NonabsurdNumberFactors,u], + NonnumericFactors[u]]] + + +(* m must be an absurd number. FactorAbsurdNumber[m] returns the prime factorization of m *) +(* as list of base-degree pairs where the bases are prime numbers and the degrees are rational. *) +FactorAbsurdNumber[m_] := + If[RationalQ[m], + FactorInteger[m], + If[PowerQ[m], + Map[Function[{#[[1]], #[[2]]*m[[2]]}],FactorInteger[m[[1]]]], + CombineExponents[Sort[Flatten[Map[FactorAbsurdNumber,Apply[List,m]],1], Function[#1[[1]]<#2[[1]]]]]]] + + +CombineExponents[lst_] := + If[Length[lst]<2, + lst, + If[lst[[1,1]]==lst[[2,1]], + CombineExponents[Prepend[Drop[lst,2],{lst[[1,1]],lst[[1,2]]+lst[[2,2]]}]], + Prepend[CombineExponents[Rest[lst]],First[lst]]]] + + +(* m, n, ... must be absurd numbers. AbsurdNumberGCD[m,n,...] returns the gcd of m, n, ... *) +AbsurdNumberGCD[seq__] := + With[{lst={seq}}, + If[Length[lst]==1, + First[lst], + AbsurdNumberGCDList[FactorAbsurdNumber[First[lst]],FactorAbsurdNumber[Apply[AbsurdNumberGCD,Rest[lst]]]]]] + + +(* lst1 and lst2 must be absurd number prime factorization lists. *) +(* AbsurdNumberGCDList[lst1,lst2] returns the gcd of the absurd numbers represented by lst1 and lst2. *) +AbsurdNumberGCDList[lst1_,lst2_] := + If[lst1==={}, + Apply[Times,Map[Function[#[[1]]^Min[#[[2]],0]],lst2]], + If[lst2==={}, + Apply[Times,Map[Function[#[[1]]^Min[#[[2]],0]],lst1]], + If[lst1[[1,1]]==lst2[[1,1]], + If[lst1[[1,2]]<=lst2[[1,2]], + lst1[[1,1]]^lst1[[1,2]]*AbsurdNumberGCDList[Rest[lst1],Rest[lst2]], + lst1[[1,1]]^lst2[[1,2]]*AbsurdNumberGCDList[Rest[lst1],Rest[lst2]]], + If[lst1[[1,1]]LeafCount[u]+2, + UnifySum[u,x], + NormalizeIntegrandFactorBase[v,x]]], + Map[Function[NormalizeIntegrandFactor[#,x]],u]]]]]] + + +(* ::Subsection::Closed:: *) +(*NormalizeLeadTermSigns*) + + +NormalizeTogether[u_] := + NormalizeLeadTermSigns[Together[u]] + + +(* NormalizeLeadTermSigns[u] returns an expression equal u but with not more than one sum + factor raised to a integer degree having a lead term with a negative coefficient. *) +NormalizeLeadTermSigns[u_] := + With[{lst=If[ProductQ[u], Map[SignOfFactor,u], SignOfFactor[u]]}, + If[lst[[1]]==1, + lst[[2]], + AbsorbMinusSign[lst[[2]]]]] + + +(* AbsorbMinusSign[u] returns an expression equal to -u. If there is a factor of u of the + form v^m where v is a sum and m is an odd power, the minus sign is distributed over v; + otherwise -u is returned. *) +AbsorbMinusSign[u_.*v_Plus] := + u*(-v) + +AbsorbMinusSign[u_.*v_Plus^m_] := + u*(-v)^m /; +OddQ[m] + +AbsorbMinusSign[u_] := + -u + + +(* ::Subsection::Closed:: *) +(*NormalizeSumFactors*) + + +(* NormalizeSumFactors[u] returns an expression equal u but with the numeric coefficient of + the lead term of sum factors made positive where possible. *) +NormalizeSumFactors[u_] := + If[AtomQ[u] || Head[u]===If || Head[u]===Int || HeldFormQ[u], + u, + If[ProductQ[u], + Function[#[[1]]*#[[2]]][SignOfFactor[Map[NormalizeSumFactors,u]]], + Map[NormalizeSumFactors,u]]] + + +(* SignOfFactor[u] returns the list {n,v} where n*v equals u, n^2 equals 1, and the lead + term of the sum factors of v raised to integer degrees all have positive coefficients. *) +SignOfFactor[u_] := + If[RationalQ[u] && u<0 || SumQ[u] && NumericFactor[First[u]]<0, + {-1, -u}, + If[IntegerPowerQ[u] && SumQ[u[[1]]] && NumericFactor[First[u[[1]]]]<0, + {(-1)^u[[2]], (-u[[1]])^u[[2]]}, + If[ProductQ[u], + Map[SignOfFactor,u], + {1, u}]]] + + +(* ::Subsection::Closed:: *) +(*NormalizePowerOfLinear*) + + +(* u can be square-free factored into an expression of the form (a+b*x)^m. *) +(* NormalizePowerOfLinear[u,x] returns u in the form (a+b*x)^m. *) +NormalizePowerOfLinear[u_,x_Symbol] := + With[{v=FactorSquareFree[u]}, + If[PowerQ[v] && LinearQ[v[[1]],x] && FreeQ[v[[2]],x], + ExpandToSum[v[[1]],x]^v[[2]], + ExpandToSum[v,x]]] + + +(* ::Subsection::Closed:: *) +(*MergeMonomials*) + + +(* ::Subsubsection::Closed:: *) +(*Basis: If m\[Element]\[DoubleStruckCapitalZ] \[And] b c-a d==0, then (a+b z)^m==b^m/d^m (c+d z)^m*) + + +MergeMonomials[u_.*(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.,x_Symbol] := + u*b^m/d^m*(c+d*x)^(m+n) /; +FreeQ[{a,b,c,d},x] && IntegerQ[m] && EqQ[b*c-a*d,0] + + +(* ::Subsubsection::Closed:: *) +(*Basis: If m/n\[Element]\[DoubleStruckCapitalZ], then z^m (c z^n)^p==(c z^n)^(m/n+p)/c^(m/n)*) + + +MergeMonomials[u_.*(a_.+b_.*x_)^m_.*(c_.*(a_.+b_.*x_)^n_.)^p_,x_Symbol] := + u*(c*(a+b*x)^n)^(m/n+p)/c^(m/n) /; +FreeQ[{a,b,c,m,n,p},x] && IntegerQ[m/n] + +MergeMonomials[u_,x_Symbol] := u + + +(* ::Section::Closed:: *) +(*Simplification functions*) + + +(* ::Subsection::Closed:: *) +(*SimplifyIntegrand*) + + +(* SimplifyIntegrand[u,x] simplifies u and returns the result in a standard form recognizable by integration rules. *) +SimplifyIntegrand[u_,x_Symbol] := + Module[{v}, + v=NormalizeLeadTermSigns[NormalizeIntegrandAux[Simplify[u],x]]; + If[LeafCount[v]<4/5*LeafCount[u], + v, + If[v=!=NormalizeLeadTermSigns[u], + v, + u]]] + + +(* ::Subsection::Closed:: *) +(*SimplifyTerm ????*) + + +(* SimplifyTerm[u_,x_Symbol] := + Module[{v=Simplify[u],w}, + w=Together[v]; + NormalizeIntegrand[If[LeafCount[v]0 \[And] v>0, let g=GCD[m,n], then u^m v^n==((u^(m/g) v^(n/g)))^g*) + + +FixSimplify[w_.*u_^m_.*v_^n_] := + With[{z=Simplify[u^(m/GCD[m,n])*v^(n/GCD[m,n])]}, + FixSimplify[w*z^GCD[m,n]]/; + AbsurdNumberQ[z] || SqrtNumberSumQ[z]] /; +RationalQ[m] && FractionQ[n] && SqrtNumberSumQ[u] && SqrtNumberSumQ[v] && PositiveQ[u] && PositiveQ[v] + + +FixSimplify[w_.*u_^m_.*v_^n_] := + With[{z=Simplify[u^(m/GCD[m,-n])*v^(n/GCD[m,-n])]}, + FixSimplify[w*z^GCD[m,-n]]/; + AbsurdNumberQ[z] || SqrtNumberSumQ[z]] /; +RationalQ[m] && FractionQ[n] && SqrtNumberSumQ[u] && SqrtNumberSumQ[1/v] && PositiveQ[u] && PositiveQ[v] + + +(* ::Item:: *) +(*Basis: If m\[Element]\[DoubleStruckCapitalZ] \[And] u<0 \[And] v>0, then u^m v^n==-(((-u))^(m/n) v)^n*) + + +FixSimplify[w_.*u_^m_.*v_^n_] := + With[{z=Simplify[(-u)^(m/GCD[m,n])*v^(n/GCD[m,n])]}, + FixSimplify[-w*z^GCD[m,n]]/; + AbsurdNumberQ[z] || SqrtNumberSumQ[z]] /; +IntegerQ[m] && FractionQ[n] && SqrtNumberSumQ[u] && SqrtNumberSumQ[v] && NegativeQ[u] && PositiveQ[v] + + +FixSimplify[w_.*u_^m_.*v_^n_] := + With[{z=Simplify[(-u)^(m/GCD[m,-n])*v^(n/GCD[m,-n])]}, + FixSimplify[-w*z^GCD[m,-n]]/; + AbsurdNumberQ[z] || SqrtNumberSumQ[z]] /; +IntegerQ[m] && FractionQ[n] && SqrtNumberSumQ[u] && SqrtNumberSumQ[1/v] && NegativeQ[u] && PositiveQ[v] + + +(* ::Item:: *) +(*Basis: If p\[Element]\[DoubleStruckCapitalZ] \[Or] a>0, then a^m (u+v)^p==(a^(m/p) u+a^(m/p) v)^p*) + + +FixSimplify[w_.*a_^m_*(u_+b_^n_*v_.)^p_.] := + With[{c=Simplify[a^(m/p)*b^n]}, + FixSimplify[w*(a^(m/p)*u+c*v)^p] /; + RationalQ[c]] /; +RationalQ[a,b,m,n] && a>0 && b>0 && PositiveIntegerQ[p] + + +(* ::Item:: *) +(*Basis: If p\[Element]\[DoubleStruckCapitalZ], then a^m (a^n u+(-a)^p v)==a^(m+n) (u+(-1)^p a^(p-n) v)*) + + +FixSimplify[w_.*a_^m_.*(a_^n_*u_.+b_^p_.*v_.)] := + FixSimplify[w*a^(m+n)*(u+(-1)^p*a^(p-n)*v)] /; +RationalQ[m] && FractionQ[n] && IntegerQ[p] && p-n>0 && a+b===0 + + +(* ::Item:: *) +(*Basis: If m\[Element]\[DoubleStruckCapitalZ] \[And] b c-a d==0, then (a+b)^m==(b/d)^m (c+d)^m*) + + +FixSimplify[w_.*(a_+b_)^m_.*(c_+d_)^n_] := + With[{q=Simplify[b/d]}, + FixSimplify[w*q^m*(c+d)^(m+n)] /; + FreeQ[q,Plus]] /; +IntegerQ[m] && Not[IntegerQ[n]] && EqQ[b*c-a*d,0] + + +(* ::Item:: *) +(*Basis: If t\[Element]\[DoubleStruckCapitalZ], then (a^m u+a^n v+...)^t==a^(m t) (u+a^(n-m) v+...)^t*) + + +FixSimplify[w_.*(a_^m_.*u_.+a_^n_.*v_.)^t_.] := + FixSimplify[a^(m*t)*w*(u+a^(n-m)*v)^t] /; +Not[RationalQ[a]] && IntegerQ[t] && RationalQ[m,n] && 00, then a^(m/4) Sqrt[b(c+d Sqrt[a])]==Sqrt[b(c a^(m/2)+d a^((m+1)/2))]*) + + +(* FixSimplify[u_.*a_^m_*Sqrt[b_.*(c_+d_.*Sqrt[a_])]] := + Sqrt[Together[b*(c*a^(2*m)+d*a^(2*m+1/2))]]*FixSimplify[u] /; +RationalQ[a,b,c,d,m] && a>0 && Denominator[m]==4 + +FixSimplify[u_.*a_^m_/Sqrt[b_.*(c_+d_.*Sqrt[a_])]] := + FixSimplify[u]/Sqrt[Together[b*(c/a^(2*m)+d/a^(2*m-1/2))]] /; +RationalQ[a,b,c,d,m] && a>0 && Denominator[m]==4 *) + + +(* ::Item:: *) +(*Basis: If w^-n==v, then v^m w^n==v^(m-1)*) + + +FixSimplify[u_.*v_^m_*w_^n_] := + -FixSimplify[u*v^(m-1)] /; +RationalQ[m] && Not[RationalQ[w]] && FractionQ[n] && n<0 && EqQ[v+w^(-n),0] + + +(* ::Item:: *) +(*Basis: If n\[Element]\[DoubleStruckCapitalZ], then v^m (-v)^n==(-1)^n v^(m+n)*) + + +FixSimplify[u_.*v_^m_*w_^n_.] := + (-1)^(n)*FixSimplify[u*v^(m+n)] /; +RationalQ[m] && Not[RationalQ[w]] && IntegerQ[n] && EqQ[v+w,0] + + +(* ::Item:: *) +(*Basis: If n/p\[Element]\[DoubleStruckCapitalZ], then (-v^p)^m v^n==(-1)^(n/p) (-v^p)^(m+n/p)*) + + +FixSimplify[u_.*(-v_^p_.)^m_*w_^n_.] := + (-1)^(n/p)*FixSimplify[u*(-v^p)^(m+n/p)] /; +RationalQ[m] && Not[RationalQ[w]] && IntegerQ[n/p] && EqQ[v-w,0] + + +(* ::Item:: *) +(*Basis: If (n|n/p)\[Element]\[DoubleStruckCapitalZ], then (-v^p)^m (-v)^n==(-1)^(n+n/p) (-v^p)^(m+n/p)*) + + +FixSimplify[u_.*(-v_^p_.)^m_*w_^n_.] := + (-1)^(n+n/p)*FixSimplify[u*(-v^p)^(m+n/p)] /; +RationalQ[m] && Not[RationalQ[w]] && IntegersQ[n,n/p] && EqQ[v+w,0] + + +(* ::Item:: *) +(*Basis: (a-b) (a+b) == a^2-b^2*) + + +FixSimplify[u_.*(a_-b_)^m_.*(a_+b_)^m_.] := + u*(a^2-b^2)^m /; +IntegerQ[m] && AtomQ[a] && AtomQ[b] + + +FixSimplify[u_.*(c_Symbol*d_Symbol^2-e_Symbol*(b_Symbol*d_Symbol-a_Symbol*e_Symbol))^m_.] := + u*(c*d^2-b*d*e+a*e^2)^m /; +RationalQ[m] && OrderedQ[{a,b,c,d,e}] + +FixSimplify[u_.*(c_Symbol*d_Symbol^2+e_Symbol*(-b_Symbol*d_Symbol+a_Symbol*e_Symbol))^m_.] := + u*(c*d^2-b*d*e+a*e^2)^m /; +RationalQ[m] && OrderedQ[{a,b,c,d,e}] + +FixSimplify[u_.*(-c_Symbol*d_Symbol^2+e_Symbol*(b_Symbol*d_Symbol-a_Symbol*e_Symbol))^m_.] := + (-1)^m*u*(c*d^2-b*d*e+a*e^2)^m /; +IntegerQ[m] && OrderedQ[{a,b,c,d,e}] + +FixSimplify[u_.*(-c_Symbol*d_Symbol^2-e_Symbol*(-b_Symbol*d_Symbol+a_Symbol*e_Symbol))^m_.] := + (-1)^m*u*(c*d^2-b*d*e+a*e^2)^m /; +RationalQ[m] && OrderedQ[{a,b,c,d,e}] + + +FixSimplify[u_] := u + + +(* ::Subsection::Closed:: *) +(*SimpFixFactor*) + + +(* SimpFixFactor[(a_.*c_ + b_.*c_)^p_.,x_] := + c^p*SimpFixFactor[(a+b)^p,x] /; +FreeQ[c,x] && IntegerQ[p] && c^p=!=-1 *) + +SimpFixFactor[(a_.*Complex[0,c_] + b_.*Complex[0,d_])^p_.,x_] := + I^p*SimpFixFactor[(a*c+b*d)^p,x] /; +IntegerQ[p] + +SimpFixFactor[(a_.*Complex[0,d_] + b_.*Complex[0,e_]+ c_.*Complex[0,f_])^p_.,x_] := + I^p*SimpFixFactor[(a*d+b*e+c*f)^p,x] /; +IntegerQ[p] + + +SimpFixFactor[(a_.*c_^r_ + b_.*x_^n_.)^p_.,x_] := + c^(r*p)*SimpFixFactor[(a+b/c^r*x^n)^p,x] /; +FreeQ[{a,b,c},x] && IntegersQ[n,p] && AtomQ[c] && RationalQ[r] && r<0 + +SimpFixFactor[(a_. + b_.*c_^r_*x_^n_.)^p_.,x_] := + c^(r*p)*SimpFixFactor[(a/c^r+b*x^n)^p,x] /; +FreeQ[{a,b,c},x] && IntegersQ[n,p] && AtomQ[c] && RationalQ[r] && r<0 + + +SimpFixFactor[(a_.*c_^s_. + b_.*c_^r_.*x_^n_.)^p_.,x_] := + c^(s*p)*SimpFixFactor[(a+b*c^(r-s)*x^n)^p,x] /; +FreeQ[{a,b,c},x] && IntegersQ[n,p] && RationalQ[s,r] && 0=0 || RationalQ[deg] && deg>0), + If[PowerQ[v], + If[bas===v[[1]], + RationalQ[deg+v[[2]]] && (deg+v[[2]]>=0 || RationalQ[deg] && deg>0), + SumQ[v[[1]]] && IntegerQ[v[[2]]] && (Not[IntegerQ[deg]] || IntegerQ[deg/v[[2]]]) && MergeableFactorQ[bas,deg/v[[2]],v[[1]]]], + If[ProductQ[v], + MergeableFactorQ[bas,deg,First[v]] || MergeableFactorQ[bas,deg,Rest[v]], + SumQ[v] && MergeableFactorQ[bas,deg,First[v]] && MergeableFactorQ[bas,deg,Rest[v]]]]] + + +(* ::Subsection::Closed:: *) +(*TrigSimplify*) + + +(* TrigSimplifyQ[u] returns True if TrigSimplify[u] actually simplifies u; else False. *) +TrigSimplifyQ[u_] := + ActivateTrig[u]=!=TrigSimplify[u] + + +(* TrigSimplify[u] returns a bottom-up trig simplification of u. *) +TrigSimplify[u_] := + ActivateTrig[TrigSimplifyRecur[u]] + + +TrigSimplifyRecur[u_] := + If[AtomQ[u], + u, + If[Head[u]===If, + u, + TrigSimplifyAux[Map[TrigSimplifyRecur,u]]]] + +Clear[TrigSimplifyAux] + + +(* Basis: a*z^m+b*z^n == z^m*(a+b*z^(n-m)) *) +TrigSimplifyAux[u_.*(a_.*v_^m_.+b_.*v_^n_.)^p_] := + u*v^(m*p)*TrigSimplifyAux[a+b*v^(n-m)]^p /; +InertTrigQ[v] && IntegerQ[p] && RationalQ[m,n] && m0 && p<0 && m==-n*p + +(* Basis: If n is an integer, Cos[z]^(-n)*(a*Cos[z]^n+b*Sin[z]^n) == a+b*Tan[z]^n *) +TrigSimplifyAux[cos[v_]^m_.*(a_.*cos[v_]^n_.+b_.*sin[v_]^n_.)^p_] := + (a+b*Tan[v]^n)^p /; +IntegersQ[m,n,p] && n>0 && p<0 && m==-n*p *) + + +(* (* Basis: If a^2+b^2=0, 1/(a*Cos[z] + b*Sin[z]) == Cos[z]/a + Sin[z]/b *) +TrigSimplifyAux[(a_.*cos[v_]+b_.*sin[v_])^n_] := + (Cos[v]/a + Sin[v]/b)^(-n) /; +IntegerQ[n] && n<0 && EqQ[a^2+b^2,0] *) + + +TrigSimplifyAux[u_] := u + + +(* ::Section::Closed:: *) +(*Factoring functions*) + + +(* ::Subsection::Closed:: *) +(*ContentFactor*) + + +(* ContentFactor[expn] returns expn with the content of sum factors factored out. *) +(* Basis: a*b+a*c == a*(b+c) *) +ContentFactor[expn_] := + TimeConstrained[ContentFactorAux[expn],$TimeLimit,expn]; + +ContentFactorAux[expn_] := + If[AtomQ[expn], + expn, + If[IntegerPowerQ[expn], + If[SumQ[expn[[1]]] && NumericFactor[expn[[1,1]]]<0, + (-1)^expn[[2]] * ContentFactorAux[-expn[[1]]]^expn[[2]], + ContentFactorAux[expn[[1]]]^expn[[2]]], + If[ProductQ[expn], + Module[{num=1,tmp}, + tmp=Map[Function[If[SumQ[#] && NumericFactor[#[[1]]]<0, num=-num; ContentFactorAux[-#], ContentFactorAux[#]]], expn]; + num*UnifyNegativeBaseFactors[tmp]], + If[SumQ[expn], + With[{lst=CommonFactors[Apply[List,expn]]}, + If[lst[[1]]===1 || lst[[1]]===-1, + expn, + lst[[1]]*Apply[Plus,Rest[lst]]]], + expn]]]] + + +(* UnifyNegativeBaseFactors[u] returns u with factors of the form (-v)^m and v^n where n is an integer replaced with (-1)^n*(-v)^(m+n). *) +(* This should be done automatically by the host CAS! *) +UnifyNegativeBaseFactors[u_.*(-v_)^m_*v_^n_.] := + UnifyNegativeBaseFactors[(-1)^n*u*(-v)^(m+n)] /; +IntegerQ[n] + +UnifyNegativeBaseFactors[u_] := + u + + +(* If lst is a list of n terms, CommonFactors[lst] returns a n+1-element list whose first + element is the product of the factors common to all terms of lst, and whose remaining + elements are quotients of each term divided by the common factor. *) +CommonFactors [lst_] := + Module[{lst1,lst2,lst3,lst4,common,base,num}, + lst1=Map[NonabsurdNumberFactors,lst]; + lst2=Map[AbsurdNumberFactors,lst]; + num=Apply[AbsurdNumberGCD,lst2]; + common=num; + lst2=Map[Function[#/num],lst2]; + While[True, + lst3=Map[LeadFactor,lst1]; + ( If[Apply[SameQ,lst3], + common=common*lst3[[1]]; + lst1=Map[RemainingFactors,lst1], + If[MapAnd[Function[LogQ[#] && IntegerQ[First[#]] && First[#]>0],lst3] && + MapAnd[RationalQ,lst4=Map[Function[FullSimplify[#/First[lst3]]],lst3]], + num=Apply[GCD,lst4]; + common=common*Log[(First[lst3][[1]])^num]; + lst2=Map2[Function[#1*#2/num],lst2,lst4]; + lst1=Map[RemainingFactors,lst1], + lst4=Map[LeadDegree,lst1]; + If[Apply[SameQ,Map[LeadBase,lst1]] && MapAnd[RationalQ,lst4], + num=Smallest[lst4]; + base=LeadBase[lst1[[1]]]; + ( If[num!=0, + common=common*base^num] ); + lst2=Map2[Function[#1*base^(#2-num)],lst2,lst4]; + lst1=Map[RemainingFactors,lst1], + If[Length[lst1]==2 && EqQ[LeadBase[lst1[[1]]]+LeadBase[lst1[[2]]],0] && + NeQ[lst1[[1]],1] && IntegerQ[lst4[[1]]] && FractionQ[lst4[[2]]], + num=Min[lst4]; + base=LeadBase[lst1[[2]]]; + ( If[num!=0, + common=common*base^num] ); + lst2={lst2[[1]]*(-1)^lst4[[1]],lst2[[2]]}; + lst2=Map2[Function[#1*base^(#2-num)],lst2,lst4]; + lst1=Map[RemainingFactors,lst1], + If[Length[lst1]==2 && EqQ[LeadBase[lst1[[1]]]+LeadBase[lst1[[2]]],0] && + NeQ[lst1[[2]],1] && IntegerQ[lst4[[2]]] && FractionQ[lst4[[1]]], + num=Min[lst4]; + base=LeadBase[lst1[[1]]]; + ( If[num!=0, + common=common*base^num] ); + lst2={lst2[[1]],lst2[[2]]*(-1)^lst4[[2]]}; + lst2=Map2[Function[#1*base^(#2-num)],lst2,lst4]; + lst1=Map[RemainingFactors,lst1], + num=MostMainFactorPosition[lst3]; + lst2=ReplacePart[lst2,lst3[[num]]*lst2[[num]],num]; + lst1=ReplacePart[lst1,RemainingFactors[lst1[[num]]],num]]]]]] ); + If[MapAnd[Function[#===1],lst1], + Return[Prepend[lst2,common]]]]] + + +MostMainFactorPosition[lst_List] := + Module[{factor=1,num=1,i}, + Do[If[FactorOrder[lst[[i]],factor]>0,factor=lst[[i]];num=i],{i,Length[lst]}]; + num] + + +FactorOrder[u_,v_] := + If[u===1, + If[v===1, + 0, + -1], + If[v===1, + 1, + Order[u,v]]] + + +Smallest[num1_,num2_] := + If[num1>0, + If[num2>0, + Min[num1,num2], + 0], + If[num2>0, + 0, + Max[num1,num2]]] + +Smallest[lst_List] := + Module[{num=lst[[1]]}, + Scan[Function[num=Smallest[num,#]],Rest[lst]]; + num] + + +(* ::Subsection::Closed:: *) +(*MonomialFactor*) + + +(* MonomialFactor[u,x] returns the list {n,v} where x^n*v==u and n is free of x. *) +MonomialFactor[u_,x_Symbol] := + If[AtomQ[u], + If[u===x, + {1,1}, + {0,u}], + If[PowerQ[u], + If[IntegerQ[u[[2]]], + With[{lst=MonomialFactor[u[[1]],x]}, + {lst[[1]]*u[[2]],lst[[2]]^u[[2]]}], + If[u[[1]]===x && FreeQ[u[[2]],x], + {u[[2]],1}, + {0,u}]], + If[ProductQ[u], + With[{lst1=MonomialFactor[First[u],x],lst2=MonomialFactor[Rest[u],x]}, + {lst1[[1]]+lst2[[1]],lst1[[2]]*lst2[[2]]}], + If[SumQ[u], + Module[{lst,deg}, + lst=Map[Function[MonomialFactor[#,x]],Apply[List,u]]; + deg=lst[[1,1]]; + Scan[Function[deg=MinimumDegree[deg,#[[1]]]],Rest[lst]]; + If[EqQ[deg,0] || RationalQ[deg] && deg<0, + {0,u}, + {deg,Apply[Plus,Map[Function[x^(#[[1]]-deg)*#[[2]]],lst]]}]], + {0,u}]]]] + + +MinimumDegree[deg1_,deg2_] := + If[RationalQ[deg1], + If[RationalQ[deg2], + Min[deg1,deg2], + deg1], + If[RationalQ[deg2], + deg2, + With[{deg=Simplify[deg1-deg2]}, + If[RationalQ[deg], + If[deg>0, + deg2, + deg1], + If[OrderedQ[{deg1,deg2}], + deg1, + deg2]]]]] + + +(* ::Subsection::Closed:: *) +(*ConstantFactor*) + + +(* ConstantFactor[u,x] returns a 2-element list of the factors of u[x] free of x and the + factors not free of u[x]. Common constant factors of the terms of sums are also collected. *) +ConstantFactor[u_,x_Symbol] := + If[FreeQ[u,x], + {u,1}, + If[AtomQ[u], + {1,u}, + If[PowerQ[u] && FreeQ[u[[2]],x], + Module[{lst=ConstantFactor[u[[1]],x],tmp}, + If[IntegerQ[u[[2]]], + {lst[[1]]^u[[2]],lst[[2]]^u[[2]]}, + tmp=PositiveFactors[lst[[1]]]; + If[tmp===1, + {1,u}, + {tmp^u[[2]],(NonpositiveFactors[lst[[1]]]*lst[[2]])^u[[2]]}]]], + If[ProductQ[u], + With[{lst=Map[Function[ConstantFactor[#,x]],Apply[List,u]]}, + {Apply[Times,Map[First,lst]],Apply[Times,Map[Function[#[[2]]],lst]]}], + If[SumQ[u], + With[{lst1=Map[Function[ConstantFactor[#,x]],Apply[List,u]]}, + If[Apply[SameQ,Map[Function[#[[2]]],lst1]], + {Apply[Plus,Map[First,lst1]],lst1[[1,2]]}, + With[{lst2=CommonFactors[Map[First,lst1]]}, + {First[lst2],Apply[Plus,Map2[Times,Rest[lst2],Map[Function[#[[2]]],lst1]]]}]]], + {1,u}]]]]] + + +(* PositiveFactors[u] returns the positive factors of u *) +PositiveFactors[u_] := + If[EqQ[u,0], + 1, + If[RationalQ[u], + Abs[u], + If[PositiveQ[u], + u, + If[ProductQ[u], + Map[PositiveFactors,u], + 1]]]] + + +(* NonpositiveFactors[u] returns the nonpositive factors of u *) +NonpositiveFactors[u_] := + If[EqQ[u,0], + u, + If[RationalQ[u], + Sign[u], + If[PositiveQ[u], + 1, + If[ProductQ[u], + Map[NonpositiveFactors,u], + u]]]] + + +(* ::Section::Closed:: *) +(*Generalized polynomial functions*) + + +(* ::Subsection::Closed:: *) +(*PolynomialInQ*) + + +(* ::Item:: *) +(*If u is a polynomial in v[x], PolynomialInQ[u,v,x] returns True; else it returns False.*) + + +PolynomialInQ[u_,v_,x_Symbol] := + PolynomialInAuxQ[u,NonfreeFactors[NonfreeTerms[v,x],x],x] + + +PolynomialInAuxQ[u_,v_,x_] := + If[u===v, + True, + If[AtomQ[u], + u=!=x, + If[PowerQ[u], + If[PowerQ[v] && u[[1]]===v[[1]], + PositiveIntegerQ[u[[2]]/v[[2]]], + PositiveIntegerQ[u[[2]]] && PolynomialInAuxQ[u[[1]],v,x]], + If[SumQ[u] || ProductQ[u], + Catch[Scan[Function[If[Not[PolynomialInAuxQ[#,v,x]],Throw[False]]],u];True], + False]]]] + + +(* ::Subsection::Closed:: *) +(*ExponentIn*) + + +(* ::Item:: *) +(*If u[v] is a polynomial in w, GeneralizedExponent[u,v,x] returns the degree of the u[x].*) + + +ExponentIn[u_,v_,x_Symbol] := + ExponentInAux[u,NonfreeFactors[NonfreeTerms[v,x],x],x] + + +ExponentInAux[u_,v_,x_] := + If[u===v, + 1, + If[AtomQ[u], + 0, + If[PowerQ[u], + If[PowerQ[v] && u[[1]]===v[[1]], + u[[2]]/v[[2]], + u[[2]]*ExponentInAux[u[[1]],v,x]], + If[ProductQ[u], + Apply[Plus,Map[Function[ExponentInAux[#,v,x]],Apply[List,u]]], + Apply[Max,Map[Function[ExponentInAux[#,v,x]],Apply[List,u]]]]]]] + + +(* ::Subsection::Closed:: *) +(*PolynomialInSubst*) + + +(* ::Item:: *) +(*If u is a polynomial in v[x], PolynomialInSubst[u,v,x] returns the polynomial u in x.*) + + +PolynomialInSubst[u_,v_,x_Symbol] := + With[{w=NonfreeTerms[v,x]}, + ReplaceAll[PolynomialInSubstAux[u,NonfreeFactors[w,x],x],{x->(x-FreeTerms[v,x])/FreeFactors[w,x]}]] + + +PolynomialInSubstAux[u_,v_,x_] := + If[u===v, + x, + If[AtomQ[u], + u, + If[PowerQ[u], + If[PowerQ[v] && u[[1]]===v[[1]], + x^(u[[2]]/v[[2]]), + PolynomialInSubstAux[u[[1]],v,x]^u[[2]]], + Map[Function[PolynomialInSubstAux[#,v,x]],u]]]] + + +(* ::Subsection::Closed:: *) +(*PolynomialDivide*) + + +(* ::Item:: *) +(*u and v are polynomials in x. PolynomialDivide[u,v,x] returns the polynomial quotient of u and v plus the polynomial remainder divided by v.*) + + +PolynomialDivide[u_,v_,x_Symbol] := + Module[{quo=PolynomialQuotient[u,v,x],rem=PolynomialRemainder[u,v,x],free,monomial}, + quo=Apply[Plus,Map[Function[Simp[Together[Coefficient[quo,x,#]*x^#],x]],Exponent[quo,x,List]]]; + rem=Together[rem]; + free=FreeFactors[rem,x]; + rem=NonfreeFactors[rem,x]; + monomial=x^Exponent[rem,x,Min]; + If[NegQ[Coefficient[rem,x,0]], monomial=-monomial]; + rem=Apply[Plus,Map[Function[Simp[Together[Coefficient[rem,x,#]*x^#/monomial],x]],Exponent[rem,x,List]]]; +(*rem=Simplify[rem]; *) + If[BinomialQ[v,x], + quo+free*monomial*rem/ExpandToSum[v,x], + quo+free*monomial*rem/v]] + + +(* ::Item:: *) +(*u[w] and v[w] are polynomials in w. PolynomialDivide[u,v,w,x] returns the polynomial quotient of u[x] and v[x] plus the polynomial remainder divided by v[x] with x replaced by w.*) + + +PolynomialDivide[u_,v_,w_,x_Symbol] := + ReplaceAll[PolynomialDivide[PolynomialInSubst[u,w,x],PolynomialInSubst[v,w,x],x],{x->w}] + + +(* ::Section::Closed:: *) +(*Expansion functions*) + + +(* ::Subsection::Closed:: *) +(*ExpandToSum[u,x]*) + + +(* ::Item:: *) +(*ExpandToSum[u,v,x] returns v expanded into a sum of monomials of x and distributes u over v.*) + + +ExpandToSum[u_,v_,x_Symbol] := + Module[{w=ExpandToSum[v,x],r}, + r=NonfreeTerms[w,x]; + If[SumQ[r], + u*FreeTerms[w,x]+Map[Function[MergeMonomials[u*#,x]],r], + u*FreeTerms[w,x]+MergeMonomials[u*r,x]]] + + +(* ::Item:: *) +(*ExpandToSum[u,x] returns u expanded into a sum of monomials of x.*) + + +ExpandToSum[u_,x_Symbol] := + If[PolyQ[u,x], + Apply[Plus,Map[Function[Coeff[u,x,#]*x^#], Expon[u,x,List]]], + If[BinomialQ[u,x], + Function[#[[1]] + #[[2]]*x^#[[3]]][BinomialParts[u,x]], + If[TrinomialQ[u,x], + Function[#[[1]] + #[[2]]*x^#[[4]] + #[[3]]*x^(2*#[[4]])][TrinomialParts[u,x]], + If[GeneralizedBinomialQ[u,x], + Function[#[[1]]*x^#[[4]] + #[[2]]*x^#[[3]]][GeneralizedBinomialParts[u,x]], + If[GeneralizedTrinomialQ[u,x], + Function[#[[1]]*x^#[[5]] + #[[2]]*x^#[[4]] + #[[3]]*x^(2*#[[4]]-#[[5]])][GeneralizedTrinomialParts[u,x]], + Print["Warning: Unrecognized expression for expansion ",u]; + Expand[u,x]]]]]] + + +(* ::Subsection::Closed:: *) +(*ExpandTrig[u,x]*) + + +ExpandTrig[u_,x_Symbol] := + ActivateTrig[ExpandIntegrand[u,x]] + +ExpandTrig[u_,v_,x_Symbol] := + With[{w=ExpandTrig[v,x],z=ActivateTrig[u]}, + If[SumQ[w], + Map[Function[z*#],w], + z*w]] + + +(* ::Subsection::Closed:: *) +(*ExpandIntegrand[u,x]*) + + +Clear[ExpandIntegrand]; + +ExpandIntegrand[u_,v_,x_Symbol] := + Module[{w=ExpandIntegrand[v,x],r}, + r=NonfreeTerms[w,x]; + If[SumQ[r], + u*FreeTerms[w,x]+Map[Function[MergeMonomials[u*#,x]],r], + u*FreeTerms[w,x]+MergeMonomials[u*r,x]]] + + +(* ExpandIntegrand[u_,x_Symbol] := + With[{nn=FunctionOfPower[u,x]}, + ReplaceAll[ExpandIntegrand[SubstFor[x^nn,u,x],x],x->x^nn] /; + nn!=1] *) + + +(* ::Subsubsection::Closed:: *) +(*Basis: (a+b x)^m/(c+d x)==(b (a+b x)^(m-1))/d+((a d-b c) (a+b x)^(m-1))/(d (c+d x))*) + + +ExpandIntegrand[(a_.+b_.*x_)^m_.*f_^(e_.*(c_.+d_.*x_)^n_.)/(g_.+h_.*x_),x_Symbol] := + With[{tmp=a*h-b*g}, + Module[{k}, + SimplifyTerm[tmp^m/h^m,x]*f^(e*(c+d*x)^n)/(g+h*x) + + Sum[SimplifyTerm[b*tmp^(k-1)/h^k,x]*f^(e*(c+d*x)^n)*(a+b*x)^(m-k),{k,1,m}]]] /; +FreeQ[{a,b,c,d,e,f,g,h},x] && PositiveIntegerQ[m] && EqQ[b*c-a*d,0] + + +ExpandIntegrand[x_^m_.*(e_+f_.*x_)^p_.*F_^(b_.*(c_.+d_.*x_)^n_.),x_Symbol] := + If[PositiveIntegerQ[m,p] && m<=p && (EqQ[n,1] || EqQ[d*e-c*f,0]), + ExpandLinearProduct[(e+f*x)^p*F^(b*(c+d*x)^n),x^m,e,f,x], + If[PositiveIntegerQ[p], + Distribute[x^m*F^(b*(c+d*x)^n)*Expand[(e+f*x)^p,x],Plus,Times], + ExpandIntegrand[F^(b*(c+d*x)^n),x^m*(e+f*x)^p,x]]] /; +FreeQ[{F,b,c,d,e,f,m,n,p},x] + + +ExpandIntegrand[x_^m_.*(e_+f_.*x_)^p_.*F_^(a_.+b_.*(c_.+d_.*x_)^n_.),x_Symbol] := + If[PositiveIntegerQ[m,p] && m<=p && (EqQ[n,1] || EqQ[d*e-c*f,0]), + ExpandLinearProduct[(e+f*x)^p*F^(a+b*(c+d*x)^n),x^m,e,f,x], + If[PositiveIntegerQ[p], + Distribute[x^m*F^(a+b*(c+d*x)^n)*Expand[(e+f*x)^p,x],Plus,Times], + ExpandIntegrand[F^(a+b*(c+d*x)^n),x^m*(e+f*x)^p,x]]] /; +FreeQ[{F,a,b,c,d,e,f,m,n,p},x] + + +ExpandIntegrand[u_.*(a_+b_.*F_^v_)^m_.*(c_+d_.*F_^v_)^n_,x_Symbol] := + With[{w=ReplaceAll[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n,x],x->F^v]}, + Map[Function[u*#],w] /; + SumQ[w]] /; +FreeQ[{F,a,b,c,d},x] && IntegersQ[m,n] && n<0 + + +ExpandIntegrand[u_*(a_.+b_.*x_)^m_.*f_^(e_.*(c_.+d_.*x_)^n_.),x_Symbol] := + With[{v=ExpandIntegrand[u*(a+b*x)^m,x]}, + Distribute[f^(e*(c+d*x)^n)*v,Plus,Times] /; + SumQ[v]] /; +FreeQ[{a,b,c,d,e,f,m,n},x] && PolynomialQ[u,x] + + +ExpandIntegrand[u_*(a_.+b_.*x_)^m_.*Log[c_.*(d_.+e_.*x_^n_.)^p_.],x_Symbol] := + ExpandIntegrand[Log[c*(d+e*x^n)^p],u*(a+b*x)^m,x] /; +FreeQ[{a,b,c,d,e,m,n,p},x] && PolynomialQ[u,x] + + +ExpandIntegrand[u_*f_^(e_.*(c_.+d_.*x_)^n_.),x_Symbol] := + If[EqQ[n,1], + ExpandIntegrand[f^(e*(c+d*x)^n),u,x], + ExpandLinearProduct[f^(e*(c+d*x)^n),u,c,d,x]] /; +FreeQ[{c,d,e,f,n},x] && PolynomialQ[u,x] + + +ExpandIntegrand[F_[u_]^m_.*(a_+b_.*G_[u_])^n_.,x_Symbol] := + ReplaceAll[ExpandIntegrand[(a+b*x)^n/x^m,x],x->G[u]] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && F[u]*G[u]===1 + + +ExpandIntegrand[u_*(a_.+b_.*Log[c_.*(d_.*(e_.+f_.*x_)^p_.)^q_.])^n_,x_Symbol] := + ExpandLinearProduct[(a+b*Log[c*(d*(e+f*x)^p)^q])^n,u,e,f,x] /; +FreeQ[{a,b,c,d,e,f,n,p,q},x] && PolynomialQ[u,x] + + +ExpandIntegrand[u_*(a_.+b_.*F_[c_.+d_.*x_])^n_,x_Symbol] := + ExpandLinearProduct[(a+b*F[c+d*x])^n,u,c,d,x] /; +FreeQ[{a,b,c,d,n},x] && PolynomialQ[u,x] && MemberQ[{ArcSin,ArcCos,ArcSinh,ArcCosh},F] + + +(* ::Subsubsection::Closed:: *) +(*Basis: 1/(a x^n+b Sqrt[c+d x^(2 n)])==(a x^n-b Sqrt[c+d x^(2 n)])/(-b^2 c+(a^2-b^2 d) x^(2 n))*) + + +ExpandIntegrand[u_./(a_.*x_^n_+b_.*Sqrt[c_+d_.*x_^j_]),x_Symbol] := + ExpandIntegrand[u*(a*x^n-b*Sqrt[c+d*x^(2*n)])/(-b^2*c+(a^2-b^2*d)*x^(2*n)),x] /; +FreeQ[{a,b,c,d,n},x] && EqQ[j,2*n,0] + + +(* ::Subsubsection::Closed:: *) +(*Basis: (a+b x)^m/(c+d x)==(b (a+b x)^(m-1))/d+((a d-b c) (a+b x)^(m-1))/(d (c+d x))*) + + +ExpandIntegrand[(a_+b_.*x_)^m_/(c_+d_.*x_),x_Symbol] := + If[RationalQ[a,b,c,d], + ExpandExpression[(a+b*x)^m/(c+d*x),x], + With[{tmp=a*d-b*c}, + Module[{k}, + SimplifyTerm[tmp^m/d^m,x]/(c+d*x) + Sum[SimplifyTerm[b*tmp^(k-1)/d^k,x]*(a+b*x)^(m-k),{k,1,m}]]]] /; +FreeQ[{a,b,c,d},x] && PositiveIntegerQ[m] + + +(* ::Subsubsection::Closed:: *) +(*Basis: ((a+b x)^m (A+B x))/(c+d x)==(B (a+b x)^m)/d+((A d-B c) (a+b x)^m)/(d (c+d x))*) + + +ExpandIntegrand[(a_+b_.*x_)^m_.*(A_+B_.*x_)/(c_+d_.*x_),x_Symbol] := + If[RationalQ[a,b,c,d,A,B], + ExpandExpression[(a+b*x)^m*(A+B*x)/(c+d*x),x], + Module[{tmp1,tmp2}, + tmp1=(A*d-B*c)/d; + tmp2=ExpandIntegrand[(a+b*x)^m/(c+d*x),x]; + tmp2=If[SumQ[tmp2], Map[Function[SimplifyTerm[tmp1*#,x]],tmp2], SimplifyTerm[tmp1*tmp2,x]]; + SimplifyTerm[B/d,x]*(a+b*x)^m + tmp2]] /; +FreeQ[{a,b,c,d,A,B},x] && PositiveIntegerQ[m] + + +(* u is a polynomial in x. ExpandIntegrand[u*(a+b*x)^m,x] expand u*(a+b*x)^m into a sum of terms of the form A*(a+b*x)^n. *) +ExpandIntegrand[u_*(a_+b_.*x_)^m_,x_Symbol] := + Module[{tmp1,tmp2}, + tmp1=ExpandLinearProduct[(a+b*x)^m,u,a,b,x]; + If[Not[IntegerQ[m]] || m>2 && LinearQ[u,x], + tmp1, + tmp2=ExpandExpression[u*(a+b*x)^m,x]; + If[SumQ[tmp2], + If[m>0, + If[Length[tmp2]<=Exponent[u,x]+2, + tmp2, + tmp1], + If[LeafCount[tmp2]<=LeafCount[tmp1]+2, + tmp2, + tmp1]], + tmp1]]] /; +FreeQ[{a,b,m},x] && PolynomialQ[u,x] && + Not[PositiveIntegerQ[m] && MatchQ[u,w_.*(c_+d_.*x)^p_ /; FreeQ[{c,d},x] && IntegerQ[p] && p>m]] + + +(* If u is a polynomial in x, ExpandIntegrand[u*(a+b*x)^m,x] expand u*(a+b*x)^m into a sum of terms of the form A*(a+b*x)^n. *) +ExpandIntegrand[u_*v_^n_*(a_+b_.*x_)^m_,x_Symbol] := + Function[ExpandIntegrand[#[[1]]*(a+b*x)^FractionalPart[m],x] + ExpandIntegrand[#[[2]]*v^n*(a+b*x)^m,x]][ + PolynomialQuotientRemainder[u,v^(-n)*(a+b*x)^(-IntegerPart[m]),x]]/; +FreeQ[{a,b,m},x] && NegativeIntegerQ[n] && Not[IntegerQ[m]] && PolynomialQ[u,x] && PolynomialQ[v,x] && + RationalQ[m] && m<-1 && Exponent[u,x]>=-(n+IntegerPart[m])*Exponent[v,x] + + +(* If u is a polynomial in x, ExpandIntegrand[u*(a+b*x)^m,x] expand u*(a+b*x)^m into a sum of terms of the form A*(a+b*x)^n. *) +ExpandIntegrand[u_*v_^n_*(a_+b_.*x_)^m_,x_Symbol] := + Function[ExpandIntegrand[#[[1]]*(a+b*x)^m,x] + ExpandIntegrand[#[[2]]*v^n*(a+b*x)^m,x]][PolynomialQuotientRemainder[u,v^(-n),x]]/; +FreeQ[{a,b,m},x] && NegativeIntegerQ[n] && Not[IntegerQ[m]] && PolynomialQ[u,x] && PolynomialQ[v,x] && + Exponent[u,x]>=-n*Exponent[v,x] + + +(* ::Subsubsection::Closed:: *) +(*Basis: Let r/s=(-(a/b))^(1/3), then 1/(a+b z^3)==r/(3a(r-s z))+(r(2 r+s z))/(3a(r^2+r s z+s^2 z^2))*) + + +(* ExpandIntegrand[1/(a_+b_.*u_^3),x_Symbol] := + With[{r=Numerator[Rt[-a/b,3]],s=Denominator[Rt[-a/b,3]]}, + r/(3*a*(r-s*u)) + r*(2*r+s*u)/(3*a*(r^2+r*s*u+s^2*u^2))] /; +FreeQ[{a,b},x] *) + + +(* ::Subsubsection::Closed:: *) +(*Basis: Let r/s=Sqrt[-(a/b)], then 1/(a+b z^2)==r/(2a(r-s z))+r/(2a(r+s z))*) + + +ExpandIntegrand[1/(a_+b_.*u_^n_),x_Symbol] := + With[{r=Numerator[Rt[-a/b,2]],s=Denominator[Rt[-a/b,2]]}, + r/(2*a*(r-s*u^(n/2))) + r/(2*a*(r+s*u^(n/2)))] /; +FreeQ[{a,b},x] && PositiveIntegerQ[n/4] + + +(* ::Subsubsection::Closed:: *) +(*Basis: If n\[Element]SuperPlus[\[DoubleStruckCapitalZ]], let r/s=(-(a/b))^(1/n), then 1/(a + b*z^n) == (r*Sum[1/(r - (-1)^(2*(k/n))*s*z), {k, 1, n}])/(a*n)*) + + +ExpandIntegrand[1/(a_+b_.*u_^n_),x_Symbol] := + With[{r=Numerator[Rt[-a/b,n]],s=Denominator[Rt[-a/b,n]]}, + Module[{k}, + Sum[r/(a*n*(r-(-1)^(2*k/n)*s*u)),{k,1,n}]]] /; +FreeQ[{a,b},x] && IntegerQ[n] && n>1 + + +(* ::Subsubsection::Closed:: *) +(*Basis: If (m|n)\[Element]\[DoubleStruckCapitalZ] \[And] 0<=mRt[-a*c,2],x->u^(n/2)}]] /; +FreeQ[{a,c},x] && IntegerQ[n/2] && NegativeIntegerQ[p] + + +ExpandIntegrand[u_^m_.*(a_.+c_.*u_^n_.)^p_,x_Symbol] := + Module[{q}, + ReplaceAll[ExpandIntegrand[1/c^p,x^m*(-q+c*x^(n/2))^p*(q+c*x^(n/2))^p,x],{q->Rt[-a*c,2],x->u}]] /; +FreeQ[{a,c},x] && IntegersQ[m,n/2] && NegativeIntegerQ[p] && 0Rt[b^2-4*a*c,2],x->u^n}]] /; +FreeQ[{a,b,c},x] && IntegerQ[n] && EqQ[j,2*n] && NegativeIntegerQ[p] && NeQ[b^2-4*a*c,0] + + +ExpandIntegrand[u_^m_.*(a_.+b_.*u_^n_.+c_.*u_^j_.)^p_,x_Symbol] := + Module[{q}, + ReplaceAll[ExpandIntegrand[1/(4^p*c^p),x^m*(b-q+2*c*x^n)^p*(b+q+2*c*x^n)^p,x],{q->Rt[b^2-4*a*c,2],x->u}]] /; +FreeQ[{a,b,c},x] && IntegersQ[m,n,j] && EqQ[j,2*n] && NegativeIntegerQ[p] && 0=2 *) + + +ExpandIntegrand[u_/v_,x_Symbol] := + PolynomialDivide[u,v,x] /; +PolynomialQ[u,x] && PolynomialQ[v,x] && Exponent[u,x]>=Exponent[v,x] + + +ExpandIntegrand[u_*(a_.*x_)^p_,x_Symbol] := + ExpandToSum[(a*x)^p,u,x] /; +Not[IntegerQ[p]] && PolynomialQ[u,x] + + +ExpandIntegrand[u_.*v_^p_,x_Symbol] := + ExpandIntegrand[NormalizeIntegrand[v^p,x],u,x] /; +Not[IntegerQ[p]] + + +ExpandIntegrand[u_,x_Symbol] := + With[{v=ExpandExpression[u,x]}, + v /; + SumQ[v]] + + +ExpandIntegrand[u_^m_./(a_+b_.*u_^n_),x_Symbol] := + ExpandBinomial[a,b,m,n,u,x] /; +FreeQ[{a,b},x] && IntegersQ[m,n] && 0(x-a)/b],x]; + lst=Map[Function[SimplifyTerm[#,x]],lst]; + Module[{k}, + Sum[v*lst[[k]]*(a+b*x)^(k-1),{k,1,Length[lst]}]]] /; +FreeQ[{a,b},x] && PolynomialQ[u,x] + + +(* ::Subsection::Closed:: *) +(*ExpandTrigExpand[u,F,v,m,n,x]*) + + +ExpandTrigExpand[u_,F_,v_,m_,n_,x_Symbol] := + With[{w=ReplaceAll[Expand[TrigExpand[F[n*x]]^m,x],x->v]}, + If[SumQ[w], + Map[Function[u*#],w], + u*w]] + + +(* ::Subsection::Closed:: *) +(*ExpandTrigReduce[u,v,x]*) + + +ExpandTrigReduce[u_,v_,x_Symbol] := + With[{w=ExpandTrigReduce[v,x]}, + If[SumQ[w], + Map[Function[u*#],w], + u*w]] + + +(* This is necessary, because TrigReduce expands Sinh[n+v] and Cosh[n+v] to exponential form if n is a number. *) +ExpandTrigReduce[u_.*F_[n_+v_.]^m_.,x_Symbol] := + Module[{nn}, + ExpandTrigReduce[u*F[nn+v]^m,x] /. nn->n]/; +MemberQ[{Sinh,Cosh},F] && IntegerQ[m] && RationalQ[n] + +ExpandTrigReduce[u_,x_Symbol] := + ExpandTrigReduceAux[u,x] + + +ExpandTrigReduceAux[u_,x_Symbol] := + With[{v=Expand[TrigReduce[u]]}, + If[SumQ[v], + Map[Function[NormalizeTrigReduce[#,x]],v], + NormalizeTrigReduce[v,x]]] + + +NormalizeTrigReduce[a_.*F_[u_]^n_.,x_Symbol] := + a*F[ExpandToSum[u,x]]^n /; +FreeQ[{F,a,n},x] && PolynomialQ[u,x] && Exponent[u,x]>0 + +NormalizeTrigReduce[u_,x_Symbol] := u + + +(* ::Subsection::Closed:: *) +(*ExpandTrigToExp[u,v,x]*) + + +Clear[ExpandTrigToExp]; + +ExpandTrigToExp[u_,x_Symbol] := ExpandTrigToExp[1,u,x] + + +ExpandTrigToExp[u_,v_,x_Symbol] := + Module[{w=TrigToExp[v]}, + w=If[SumQ[w], Map[Function[SimplifyIntegrand[u*#,x]],w], SimplifyIntegrand[u*w,x]]; + ExpandIntegrand[FreeFactors[w,x],NonfreeFactors[w,x],x]] + + +(* ::Section::Closed:: *) +(*Distribution functions*) + + +(* ::Subsection::Closed:: *) +(*Distrib*) + + +(* Distrib[u,v] returns the sum of u times each term of v. *) +Distrib[u_,v_] := + If[SumQ[v], + Map[Function[u*#],v], + u*v] + + +(* ::Subsection::Closed:: *) +(*Dist*) + + +(* Dist[u,v] returns the sum of u times each term of v, provided v is free of Int. *) +DownValues[Dist]={}; +UpValues[Dist]={}; + +Dist /: Dist[u_,v_,x_]+Dist[w_,v_,x_] := + If[EqQ[u+w,0], + 0, + Dist[u+w,v,x]] + +Dist /: Dist[u_,v_,x_]-Dist[w_,v_,x_] := + If[EqQ[u-w,0], + 0, + Dist[u-w,v,x]] + +Dist /: w_*Dist[u_,v_,x_] := + Dist[w*u,v,x] /; +w=!=-1 + +Dist[u_,Defer[Dist][v_,w_,x_],x_] := + Dist[u*v,w,x] + +Dist[u_,v_*w_,x_] := + Dist[u*v,w,x] /; +ShowSteps===True && FreeQ[v,Int] && Not[FreeQ[w,Int]] + +Dist[u_,v_,x_] := + If[u===1, + v, + If[u===0, + Print["*** Warning ***: Dist[0,",v,",",x,"]"]; + 0, + If[NumericFactor[u]<0 && NumericFactor[-u]>0, + -Dist[-u,v,x], + If[SumQ[v], + Map[Function[Dist[u,#,x]],v], + If[FreeQ[v,Int], +(* Simp[Simp[u,x]*v,x], *) + Simp[u*v,x], +(*If[ShowSteps=!=True, + Simp[u*v,x], + Module[{w=Simp[u,x]}, + If[w=!=u, + Dist[w,v,x], *) + With[{w=Simp[u*x^2,x]/x^2}, + If[w=!=u && FreeQ[w,x] && w===Simp[w,x] && w===Simp[w*x^2,x]/x^2, + Dist[w,v,x], + If[ShowSteps=!=True, + Simp[u*v,x], + Defer[Dist][u,v,x]]]]]]]]] + + +(* ::Subsection::Closed:: *) +(*DistributeDegree*) + + +(* DistributeDegree[u,m] returns the product of the factors of u each raised to the mth degree. *) +DistributeDegree[u_,m_] := + If[AtomQ[u], + u^m, + If[PowerQ[u], + u[[1]]^(u[[2]]*m), + If[ProductQ[u], + Map[Function[DistributeDegree[#,m]],u], + u^m]]] + + +(* ::Section::Closed:: *) +(*Function of functions*) + + +(* ::Subsection::Closed:: *) +(*FunctionOfPower*) + + +(* FunctionOfPower[u,x] returns the gcd of the integer degrees of x in u. *) +(* FunctionOfPower[u_,x_Symbol] := + FunctionOfPower[u,Null,x] + +FunctionOfPower[u_,n_,x_] := + If[FreeQ[u,x], + n, + If[u===x, + 1, + If[PowerQ[u] && u[[1]]===x && IntegerQ[u[[2]]], + If[n===Null, + u[[2]], + GCD[n,u[[2]]]], + Module[{tmp=n}, + Scan[Function[tmp=FunctionOfPower[#,tmp,x]],u]; + tmp]]]] *) + + +(* ::Subsection::Closed:: *) +(*FunctionOfLinear*) + + +(* If u (x) is equivalent to an expression of the form f (a+b*x) and not the case that a==0 and + b==1, FunctionOfLinear[u,x] returns the list {f (x),a,b}; else it returns False. *) +FunctionOfLinear[u_,x_Symbol] := + With[{lst=FunctionOfLinear[u,False,False,x,False]}, + If[AtomQ[lst] || FalseQ[lst[[1]]] || lst[[1]]===0 && lst[[2]]===1, + False, + {FunctionOfLinearSubst[u,lst[[1]],lst[[2]],x],lst[[1]],lst[[2]]}]] + + +FunctionOfLinear[u_,a_,b_,x_,flag_] := + If[FreeQ[u,x], + {a,b}, + If[CalculusQ[u], + False, + If[LinearQ[u,x], + If[FalseQ[a], + {Coefficient[u,x,0],Coefficient[u,x,1]}, + With[{lst=CommonFactors[{b,Coefficient[u,x,1]}]}, + If[EqQ[Coefficient[u,x,0],0] && Not[flag], + {0,lst[[1]]}, + If[EqQ[b*Coefficient[u,x,0]-a*Coefficient[u,x,1],0], + {a/lst[[2]],lst[[1]]}, + {0,1}]]]], + If[PowerQ[u] && FreeQ[u[[1]],x], + FunctionOfLinear[Log[u[[1]]]*u[[2]],a,b,x,False], + Module[{lst}, + If[ProductQ[u] && NeQ[(lst=MonomialFactor[u,x])[[1]],0], + If[False && IntegerQ[lst[[1]]] && lst[[1]]!=-1 && FreeQ[lst[[2]],x], + If[RationalQ[LeadFactor[lst[[2]]]] && LeadFactor[lst[[2]]]<0, + FunctionOfLinear[DivideDegreesOfFactors[-lst[[2]],lst[[1]]]*x,a,b,x,False], + FunctionOfLinear[DivideDegreesOfFactors[lst[[2]],lst[[1]]]*x,a,b,x,False]], + False], + lst={a,b}; + Catch[ + Scan[Function[lst=FunctionOfLinear[#,lst[[1]],lst[[2]],x,SumQ[u]]; + If[AtomQ[lst],Throw[False]]],u]; + lst]]]]]]] + + +FunctionOfLinearSubst[u_,a_,b_,x_] := + If[FreeQ[u,x], + u, + If[LinearQ[u,x], + Module[{tmp=Coefficient[u,x,1]}, + tmp=If[tmp===b, 1, tmp/b]; + Coefficient[u,x,0]-a*tmp+tmp*x], + If[PowerQ[u] && FreeQ[u[[1]],x], + E^FullSimplify[FunctionOfLinearSubst[Log[u[[1]]]*u[[2]],a,b,x]], + Module[{lst}, + If[ProductQ[u] && NeQ[(lst=MonomialFactor[u,x])[[1]],0], + If[RationalQ[LeadFactor[lst[[2]]]] && LeadFactor[lst[[2]]]<0, + -FunctionOfLinearSubst[DivideDegreesOfFactors[-lst[[2]],lst[[1]]]*x,a,b,x]^lst[[1]], + FunctionOfLinearSubst[DivideDegreesOfFactors[lst[[2]],lst[[1]]]*x,a,b,x]^lst[[1]]], + Map[Function[FunctionOfLinearSubst[#,a,b,x]],u]]]]]] + + +(* DivideDegreesOfFactors[u,n] returns the product of the base of the factors of u raised to + the degree of the factors divided by n. *) +DivideDegreesOfFactors[u_,n_] := + If[ProductQ[u], + Map[Function[LeadBase[#]^(LeadDegree[#]/n)],u], + LeadBase[u]^(LeadDegree[u]/n)] + + +(* ::Subsection::Closed:: *) +(*FunctionOfInverseLinear*) + + +(* If u is a function of an inverse linear binomial of the form 1/(a+b*x), + FunctionOfInverseLinear[u,x] returns the list {a,b}; else it returns False. *) +FunctionOfInverseLinear[u_,x_Symbol] := + FunctionOfInverseLinear[u,Null,x] + +FunctionOfInverseLinear[u_,lst_,x_] := + If[FreeQ[u,x], + lst, + If[u===x, + False, + If[QuotientOfLinearsQ[u,x], + With[{tmp=Drop[QuotientOfLinearsParts[u,x],2]}, + If[tmp[[2]]===0, + False, + If[lst===Null, + tmp, + If[EqQ[lst[[1]]*tmp[[2]]-lst[[2]]*tmp[[1]],0], + lst, + False]]]], + If[CalculusQ[u], + False, + Module[{tmp=lst},Catch[ + Scan[Function[If[AtomQ[tmp=FunctionOfInverseLinear[#,tmp,x]],Throw[False]]],u]; + tmp]]]]]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfExponential*) + + +(* FunctionOfExponentialQ[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x, *) +(* and such an exponential explicitly occurs in u (i.e. not just implicitly in hyperbolic functions). *) +FunctionOfExponentialQ[u_,x_Symbol] := + Block[{$base$=Null,$expon$=Null,$exponFlag$=False}, + FunctionOfExponentialTest[u,x] && $exponFlag$] + + +(* u is a function of F^v where v is linear in x. FunctionOfExponential[u,x] returns F^v. *) +FunctionOfExponential[u_,x_Symbol] := + Block[{$base$=Null,$expon$=Null,$exponFlag$=False}, + FunctionOfExponentialTest[u,x]; + $base$^$expon$] + + +(* u is a function of F^v where v is linear in x. FunctionOfExponentialFunction[u,x] returns u with F^v replaced by x. *) +FunctionOfExponentialFunction[u_,x_Symbol] := + Block[{$base$=Null,$expon$=Null,$exponFlag$=False}, + FunctionOfExponentialTest[u,x]; + SimplifyIntegrand[FunctionOfExponentialFunctionAux[u,x],x]] + + +(* u is a function of F^v where v is linear in x, and the fluid variables $base$=F and $expon$=v. *) +(* FunctionOfExponentialFunctionAux[u,x] returns u with F^v replaced by x. *) +FunctionOfExponentialFunctionAux[u_,x_] := + If[AtomQ[u], + u, + If[PowerQ[u] && FreeQ[u[[1]],x] && LinearQ[u[[2]],x], + If[EqQ[Coefficient[$expon$,x,0],0], + u[[1]]^Coefficient[u[[2]],x,0]*x^FullSimplify[Log[u[[1]]]*Coefficient[u[[2]],x,1]/(Log[$base$]*Coefficient[$expon$,x,1])], + x^FullSimplify[Log[u[[1]]]*Coefficient[u[[2]],x,1]/(Log[$base$]*Coefficient[$expon$,x,1])]], + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + Module[{tmp}, + tmp=x^FullSimplify[Coefficient[u[[1]],x,1]/(Log[$base$]*Coefficient[$expon$,x,1])]; + If[SinhQ[u], + tmp/2-1/(2*tmp), + If[CoshQ[u], + tmp/2+1/(2*tmp), + If[TanhQ[u], + (tmp-1/tmp)/(tmp+1/tmp), + If[CothQ[u], + (tmp+1/tmp)/(tmp-1/tmp), + If[SechQ[u], + 2/(tmp+1/tmp), + 2/(tmp-1/tmp)]]]]]], + If[PowerQ[u] && FreeQ[u[[1]],x] && SumQ[u[[2]]], + FunctionOfExponentialFunctionAux[u[[1]]^First[u[[2]]],x]*FunctionOfExponentialFunctionAux[u[[1]]^Rest[u[[2]]],x], + Map[Function[FunctionOfExponentialFunctionAux[#,x]],u]]]]] + + +(* FunctionOfExponentialTest[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x. *) +(* Before it is called, the fluid variables $base$ and $expon$ should be set to Null and $exponFlag$ to False. *) +(* If u is a function of F^v, $base$ and $expon$ are set to F and v, respectively. *) +(* If an explicit exponential occurs in u, $exponFlag$ is set to True. *) +FunctionOfExponentialTest[u_,x_] := + If[FreeQ[u,x], + True, + If[u===x || CalculusQ[u], + False, + If[PowerQ[u] && FreeQ[u[[1]],x] && LinearQ[u[[2]],x], + $exponFlag$=True; + FunctionOfExponentialTestAux[u[[1]],u[[2]],x], + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + FunctionOfExponentialTestAux[E,u[[1]],x], + If[PowerQ[u] && FreeQ[u[[1]],x] && SumQ[u[[2]]], + FunctionOfExponentialTest[u[[1]]^First[u[[2]]],x] && FunctionOfExponentialTest[u[[1]]^Rest[u[[2]]],x], + Catch[Scan[Function[If[Not[FunctionOfExponentialTest[#,x]],Throw[False]]],u]; True]]]]]] + + +FunctionOfExponentialTestAux[base_,expon_,x_] := + If[$base$===Null, + $base$=base; + $expon$=expon; + True, + Module[{tmp}, + tmp=FullSimplify[Log[base]*Coefficient[expon,x,1]/(Log[$base$]*Coefficient[$expon$,x,1])]; + If[Not[RationalQ[tmp]], + False, + If[EqQ[Coefficient[$expon$,x,0],0] || NeQ[tmp,FullSimplify[Log[base]*Coefficient[expon,x,0]/(Log[$base$]*Coefficient[$expon$,x,0])]], + ( If[PositiveIntegerQ[base,$base$] && base<$base$, + $base$=base; + $expon$=expon; + tmp=1/tmp] ); + $expon$=Coefficient[$expon$,x,1]*x/Denominator[tmp]; + If[tmp<0 && NegQ[Coefficient[$expon$,x,1]], + $expon$=-$expon$; + True, + True], + ( If[PositiveIntegerQ[base,$base$] && base<$base$, + $base$=base; + $expon$=expon; + tmp=1/tmp] ); +(*$expon$=If[SumQ[$expon$], Map[Function[#/Denominator[tmp]],$expon$], $expon$/Denominator[tmp]]; *) + $expon$=$expon$/Denominator[tmp]; + If[tmp<0 && NegQ[Coefficient[$expon$,x,1]], + $expon$=-$expon$; + True, + True]]]]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfTrigOfLinearQ*) + + +(* If u is an algebraic function of trig functions of a linear function of x, + FunctionOfTrigOfLinearQ[u,x] returns True; else it returns False. *) +FunctionOfTrigOfLinearQ[u_,x_Symbol] := + If[MatchQ[u,(c_.+d_.*x)^m_.*(a_.+b_.*trig_[e_.+f_.*x])^n_. /; FreeQ[{a,b,c,d,e,f,m,n},x] && (TrigQ[trig] || HyperbolicQ[trig])], + True, + Not[MemberQ[{Null, False}, FunctionOfTrig[u,Null,x]]] && AlgebraicTrigFunctionQ[u,x]] + + +(* If u is a function of trig functions of v where v is a linear function of x, + FunctionOfTrig[u,x] returns v; else it returns False. *) +FunctionOfTrig[u_,x_Symbol] := + With[{v=FunctionOfTrig[ActivateTrig[u],Null,x]}, + If[v===Null, False, v]] + +FunctionOfTrig[u_,v_,x_] := + If[AtomQ[u], + If[u===x, + False, + v], + If[TrigQ[u] && LinearQ[u[[1]],x], + If[v===Null, + u[[1]], + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=Coefficient[u[[1]],x,0],d=Coefficient[u[[1]],x,1]}, + If[EqQ[a*d-b*c,0] && RationalQ[b/d], + a/Numerator[b/d]+b*x/Numerator[b/d], + False]]], + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + If[v===Null, + I*u[[1]], + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=I*Coefficient[u[[1]],x,0],d=I*Coefficient[u[[1]],x,1]}, + If[EqQ[a*d-b*c,0] && RationalQ[b/d], + a/Numerator[b/d]+b*x/Numerator[b/d], + False]]], + If[CalculusQ[u], + False, + Module[{w=v},Catch[ + Scan[Function[If[FalseQ[w=FunctionOfTrig[#,w,x]],Throw[False]]],u]; + w]]]]]] + + +(* If u is algebraic function of trig functions, AlgebraicTrigFunctionQ[u,x] returns True; else it returns False. *) +AlgebraicTrigFunctionQ[u_,x_Symbol] := + If[AtomQ[u], + True, + If[TrigQ[u] && LinearQ[u[[1]],x], + True, + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + True, + If[PowerQ[u] && FreeQ[u[[2]],x], + AlgebraicTrigFunctionQ[u[[1]],x], + If[ProductQ[u] || SumQ[u], + Catch[Scan[Function[If[Not[AlgebraicTrigFunctionQ[#,x]],Throw[False]]],u]; True], + False]]]]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfHyperbolic*) + + +(* If u is a function of hyperbolic trig functions of v where v is linear in x, + FunctionOfHyperbolic[u,x] returns v; else it returns False. *) +FunctionOfHyperbolic[u_,x_Symbol] := + With[{v=FunctionOfHyperbolic[u,Null,x]}, + If[v===Null, False, v]] + +FunctionOfHyperbolic[u_,v_,x_] := + If[AtomQ[u], + If[u===x, + False, + v], + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + If[v===Null, + u[[1]], + With[{a=Coefficient[v,x,0],b=Coefficient[v,x,1],c=Coefficient[u[[1]],x,0],d=Coefficient[u[[1]],x,1]}, + If[EqQ[a*d-b*c,0] && RationalQ[b/d], + a/Numerator[b/d]+b*x/Numerator[b/d], + False]]], + If[CalculusQ[u], + False, + Module[{w=v},Catch[ + Scan[Function[If[FalseQ[w=FunctionOfHyperbolic[#,w,x]],Throw[False]]],u]; + w]]]]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfQ*) + + +(* v is a function of x. + If u is a function of v, FunctionOfQ[v,u,x] returns True; else it returns False. *) +FunctionOfQ[v_,u_,x_Symbol,PureFlag_:False] := + If[FreeQ[u,x], + False, + If[AtomQ[v], + True, + If[Not[InertTrigFreeQ[u]], + FunctionOfQ[v,ActivateTrig[u],x,PureFlag], + If[ProductQ[v] && NeQ[FreeFactors[v,x],1], + FunctionOfQ[NonfreeFactors[v,x],u,x,PureFlag], + + If[PureFlag, + If[SinQ[v] || CscQ[v], + PureFunctionOfSinQ[u,v[[1]],x], + If[CosQ[v] || SecQ[v], + PureFunctionOfCosQ[u,v[[1]],x], + If[TanQ[v], + PureFunctionOfTanQ[u,v[[1]],x], + If[CotQ[v], + PureFunctionOfCotQ[u,v[[1]],x], + If[SinhQ[v] || CschQ[v], + PureFunctionOfSinhQ[u,v[[1]],x], + If[CoshQ[v] || SechQ[v], + PureFunctionOfCoshQ[u,v[[1]],x], + If[TanhQ[v], + PureFunctionOfTanhQ[u,v[[1]],x], + If[CothQ[v], + PureFunctionOfCothQ[u,v[[1]],x], + FunctionOfExpnQ[u,v,x]=!=False]]]]]]]], + If[SinQ[v] || CscQ[v], + FunctionOfSinQ[u,v[[1]],x], + If[CosQ[v] || SecQ[v], + FunctionOfCosQ[u,v[[1]],x], + If[TanQ[v] || CotQ[v], + FunctionOfTanQ[u,v[[1]],x], + If[SinhQ[v] || CschQ[v], + FunctionOfSinhQ[u,v[[1]],x], + If[CoshQ[v] || SechQ[v], + FunctionOfCoshQ[u,v[[1]],x], + If[TanhQ[v] || CothQ[v], + FunctionOfTanhQ[u,v[[1]],x], + FunctionOfExpnQ[u,v,x]=!=False]]]]]]]]]]] + + +FunctionOfExpnQ[u_,v_,x_] := + If[u===v, + 1, + If[AtomQ[u], + If[u===x, + False, + 0], + If[CalculusQ[u], + False, + If[PowerQ[u] && FreeQ[u[[2]],x], + If[EqQ[u[[1]],v], + If[IntegerQ[u[[2]]], u[[2]], 1], + If[PowerQ[v] && FreeQ[v[[2]],x] && EqQ[u[[1]],v[[1]]], + If[RationalQ[v[[2]]], + If[RationalQ[u[[2]]] && IntegerQ[u[[2]]/v[[2]]] && (v[[2]]>0 || u[[2]]<0), u[[2]]/v[[2]], False], + If[IntegerQ[Simplify[u[[2]]/v[[2]]]], Simplify[u[[2]]/v[[2]]], False]], + FunctionOfExpnQ[u[[1]],v,x]]], + If[ProductQ[u] && NeQ[FreeFactors[u,x],1], + FunctionOfExpnQ[NonfreeFactors[u,x],v,x], + If[ProductQ[u] && ProductQ[v], + Module[{deg1=FunctionOfExpnQ[First[u],First[v],x],deg2}, + If[deg1===False, + False, + deg2=FunctionOfExpnQ[Rest[u],Rest[v],x]; + If[deg1===deg2 && FreeQ[Simplify[u/v^deg1],x], + deg1, + False]]], + With[{lst=Map[Function[FunctionOfExpnQ[#,v,x]],Apply[List,u]]}, + If[MemberQ[lst,False], + False, + Apply[GCD,lst]]]]]]]]] + + +(* ::Subsubsection::Closed:: *) +(*Pure function of trig functions predicates*) + + +(* If u is a pure function of Sin[v] and/or Csc[v], PureFunctionOfSinQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfSinQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && EqQ[u[[1]],v], + SinQ[u] || CscQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfSinQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Cos[v] and/or Sec[v], PureFunctionOfCosQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfCosQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && EqQ[u[[1]],v], + CosQ[u] || SecQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfCosQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Tan[v] and/or Cot[v], PureFunctionOfTanQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfTanQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && EqQ[u[[1]],v], + TanQ[u] || CotQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfTanQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Cot[v], PureFunctionOfCotQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfCotQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && EqQ[u[[1]],v], + CotQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfCotQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* ::Subsubsection::Closed:: *) +(*Function of trig functions predicates*) + + +(* If u is a function of Sin[v], FunctionOfSinQ[u,v,x] returns True; else it returns False. *) +FunctionOfSinQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], + If[OddQuotientQ[u[[1]],v], +(* Basis: If m odd, Sin[m*v]^n is a function of Sin[v]. *) + SinQ[u] || CscQ[u], +(* Basis: If m even, Cos[m*v]^n is a function of Sin[v]. *) + CosQ[u] || SecQ[u]], + If[IntegerPowerQ[u] && TrigQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[EvenQ[u[[2]]], +(* Basis: If m integer and n even, Trig[m*v]^n is a function of Sin[v]. *) + True, + FunctionOfSinQ[u[[1]],v,x]], + If[ProductQ[u], + If[CosQ[u[[1]]] && SinQ[u[[2]]] && EqQ[u[[1,1]],v/2] && EqQ[u[[2,1]],v/2], + FunctionOfSinQ[Drop[u,2],v,x], + Module[{lst}, + lst=FindTrigFactor[Sin,Csc,u,v,False]; + If[ListQ[lst] && EvenQuotientQ[lst[[1]],v], +(* Basis: If m even and n odd, Sin[m*v]^n == Cos[v]*u where u is a function of Sin[v]. *) + FunctionOfSinQ[Cos[v]*lst[[2]],v,x], + lst=FindTrigFactor[Cos,Sec,u,v,False]; + If[ListQ[lst] && OddQuotientQ[lst[[1]],v], +(* Basis: If m odd and n odd, Cos[m*v]^n == Cos[v]*u where u is a function of Sin[v]. *) + FunctionOfSinQ[Cos[v]*lst[[2]],v,x], + lst=FindTrigFactor[Tan,Cot,u,v,True]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Tan[m*v]^n == Cos[v]*u where u is a function of Sin[v]. *) + FunctionOfSinQ[Cos[v]*lst[[2]],v,x], + Catch[Scan[Function[If[Not[FunctionOfSinQ[#,v,x]],Throw[False]]],u];True]]]]]], + Catch[Scan[Function[If[Not[FunctionOfSinQ[#,v,x]],Throw[False]]],u];True]]]]]] + + +(* If u is a function of Cos[v], FunctionOfCosQ[u,v,x] returns True; else it returns False. *) +FunctionOfCosQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], +(* Basis: If m integer, Cos[m*v]^n is a function of Cos[v]. *) + CosQ[u] || SecQ[u], + If[IntegerPowerQ[u] && TrigQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[EvenQ[u[[2]]], +(* Basis: If m integer and n even, Trig[m*v]^n is a function of Cos[v]. *) + True, + FunctionOfCosQ[u[[1]],v,x]], + If[ProductQ[u], + Module[{lst}, + lst=FindTrigFactor[Sin,Csc,u,v,False]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Sin[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *) + FunctionOfCosQ[Sin[v]*lst[[2]],v,x], + lst=FindTrigFactor[Tan,Cot,u,v,True]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Tan[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *) + FunctionOfCosQ[Sin[v]*lst[[2]],v,x], + Catch[Scan[Function[If[Not[FunctionOfCosQ[#,v,x]],Throw[False]]],u];True]]]], + Catch[Scan[Function[If[Not[FunctionOfCosQ[#,v,x]],Throw[False]]],u];True]]]]]] + + +(* If u is a function of the form f[Tan[v],Cot[v]] where f is independent of x, + FunctionOfTanQ[u,v,x] returns True; else it returns False. *) +FunctionOfTanQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], + TanQ[u] || CotQ[u] || EvenQuotientQ[u[[1]],v], + If[PowerQ[u] && EvenQ[u[[2]]] && TrigQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + True, + If[PowerQ[u] && EvenQ[u[[2]]] && SumQ[u[[1]]], + FunctionOfTanQ[Expand[u[[1]]^2],v,x], + If[ProductQ[u], + Module[{lst=ReapList[Scan[Function[If[Not[FunctionOfTanQ[#,v,x]],Sow[#]]],u]]}, + If[lst==={}, + True, + Length[lst]==2 && OddTrigPowerQ[lst[[1]],v,x] && OddTrigPowerQ[lst[[2]],v,x]]], + Catch[Scan[Function[If[Not[FunctionOfTanQ[#,v,x]],Throw[False]]],u];True]]]]]]] + +OddTrigPowerQ[u_,v_,x_] := + If[SinQ[u] || CosQ[u] || SecQ[u] || CscQ[u], + OddQuotientQ[u[[1]],v], + If[PowerQ[u], + OddQ[u[[2]]] && OddTrigPowerQ[u[[1]],v,x], + If[ProductQ[u], + If[NeQ[FreeFactors[u,x],1], + OddTrigPowerQ[NonfreeFactors[u,x],v,x], + Module[{lst=ReapList[Scan[Function[If[Not[FunctionOfTanQ[#,v,x]],Sow[#]]],u]]}, + If[lst==={}, + True, + Length[lst]==1 && OddTrigPowerQ[lst[[1]],v,x]]]], + If[SumQ[u], + Catch[Scan[Function[If[Not[OddTrigPowerQ[#,v,x]],Throw[False]]],u];True], + False]]]] + + +(* u is a function of the form f[Tan[v],Cot[v]] where f is independent of x. +FunctionOfTanWeight[u,v,x] returns a nonnegative number if u is best considered a function +of Tan[v]; else it returns a negative number. *) +FunctionOfTanWeight[u_,v_,x_] := + If[AtomQ[u], + 0, + If[CalculusQ[u], + 0, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], + If[TanQ[u] && EqQ[u[[1]],v], + 1, + If[CotQ[u] && EqQ[u[[1]],v], + -1, + 0]], + If[PowerQ[u] && EvenQ[u[[2]]] && TrigQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[TanQ[u[[1]]] || CosQ[u[[1]]] || SecQ[u[[1]]], + 1, + -1], + If[ProductQ[u], + If[Catch[Scan[Function[If[Not[FunctionOfTanQ[#,v,x]],Throw[False]]],u];True], + Apply[Plus,Map[Function[FunctionOfTanWeight[#,v,x]],Apply[List,u]]], + 0], + Apply[Plus,Map[Function[FunctionOfTanWeight[#,v,x]],Apply[List,u]]]]]]]] + + +(* If u (x) is equivalent to an expression of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]) + where f is independent of x, FunctionOfTrigQ[u,v,x] returns True; else it returns False. *) +FunctionOfTrigQ[u_,v_,x_Symbol] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], + True, + Catch[ + Scan[Function[If[Not[FunctionOfTrigQ[#,v,x]],Throw[False]]],u]; + True]]]] + + +(* ::Subsubsection::Closed:: *) +(*Pure function of hyperbolic functions predicates*) + + +(* If u is a pure function of Sinh[v] and/or Csch[v], PureFunctionOfSinhQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfSinhQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && EqQ[u[[1]],v], + SinhQ[u] || CschQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfSinhQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Cosh[v] and/or Sech[v], PureFunctionOfCoshQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfCoshQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && EqQ[u[[1]],v], + CoshQ[u] || SechQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfCoshQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Tanh[v] and/or Coth[v], PureFunctionOfTanhQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfTanhQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && EqQ[u[[1]],v], + TanhQ[u] || CothQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfTanhQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* If u is a pure function of Coth[v], PureFunctionOfCothQ[u,v,x] returns True; + else it returns False. *) +PureFunctionOfCothQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && EqQ[u[[1]],v], + CothQ[u], + Catch[Scan[Function[If[Not[PureFunctionOfCothQ[#,v,x]],Throw[False]]],u];True]]]] + + +(* ::Subsubsection::Closed:: *) +(*Function of hyperbolic functions predicates*) + + +(* If u is a function of Sinh[v], FunctionOfSinhQ[u,v,x] returns True; else it returns False. *) +FunctionOfSinhQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], + If[OddQuotientQ[u[[1]],v], +(* Basis: If m odd, Sinh[m*v]^n is a function of Sinh[v]. *) + SinhQ[u] || CschQ[u], +(* Basis: If m even, Cos[m*v]^n is a function of Sinh[v]. *) + CoshQ[u] || SechQ[u]], + If[IntegerPowerQ[u] && HyperbolicQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[EvenQ[u[[2]]], +(* Basis: If m integer and n even, Hyper[m*v]^n is a function of Sinh[v]. *) + True, + FunctionOfSinhQ[u[[1]],v,x]], + If[ProductQ[u], + If[CoshQ[u[[1]]] && SinhQ[u[[2]]] && EqQ[u[[1,1]],v/2] && EqQ[u[[2,1]],v/2], + FunctionOfSinhQ[Drop[u,2],v,x], + Module[{lst}, + lst=FindTrigFactor[Sinh,Csch,u,v,False]; + If[ListQ[lst] && EvenQuotientQ[lst[[1]],v], +(* Basis: If m even and n odd, Sinh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *) + FunctionOfSinhQ[Cosh[v]*lst[[2]],v,x], + lst=FindTrigFactor[Cosh,Sech,u,v,False]; + If[ListQ[lst] && OddQuotientQ[lst[[1]],v], +(* Basis: If m odd and n odd, Cosh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *) + FunctionOfSinhQ[Cosh[v]*lst[[2]],v,x], + lst=FindTrigFactor[Tanh,Coth,u,v,True]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Tanh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *) + FunctionOfSinhQ[Cosh[v]*lst[[2]],v,x], + Catch[Scan[Function[If[Not[FunctionOfSinhQ[#,v,x]],Throw[False]]],u];True]]]]]], + Catch[Scan[Function[If[Not[FunctionOfSinhQ[#,v,x]],Throw[False]]],u];True]]]]]] + + +(* If u is a function of Cosh[v], FunctionOfCoshQ[u,v,x] returns True; else it returns False. *) +FunctionOfCoshQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], +(* Basis: If m integer, Cosh[m*v]^n is a function of Cosh[v]. *) + CoshQ[u] || SechQ[u], + If[IntegerPowerQ[u] && HyperbolicQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[EvenQ[u[[2]]], +(* Basis: If m integer and n even, Hyper[m*v]^n is a function of Cosh[v]. *) + True, + FunctionOfCoshQ[u[[1]],v,x]], + If[ProductQ[u], + Module[{lst}, + lst=FindTrigFactor[Sinh,Csch,u,v,False]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Sinh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *) + FunctionOfCoshQ[Sinh[v]*lst[[2]],v,x], + lst=FindTrigFactor[Tanh,Coth,u,v,True]; + If[ListQ[lst], +(* Basis: If m integer and n odd, Tanh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *) + FunctionOfCoshQ[Sinh[v]*lst[[2]],v,x], + Catch[Scan[Function[If[Not[FunctionOfCoshQ[#,v,x]],Throw[False]]],u];True]]]], + Catch[Scan[Function[If[Not[FunctionOfCoshQ[#,v,x]],Throw[False]]],u];True]]]]]] + + +(* If u is a function of the form f[Tanh[v],Coth[v]] where f is independent of x, + FunctionOfTanhQ[u,v,x] returns True; else it returns False. *) +FunctionOfTanhQ[u_,v_,x_] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], + TanhQ[u] || CothQ[u] || EvenQuotientQ[u[[1]],v], + If[PowerQ[u] && EvenQ[u[[2]]] && HyperbolicQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + True, + If[PowerQ[u] && EvenQ[u[[2]]] && SumQ[u[[1]]], + FunctionOfTanhQ[Expand[u[[1]]^2],v,x], + If[ProductQ[u], + With[{lst=ReapList[Scan[Function[If[Not[FunctionOfTanhQ[#,v,x]],Sow[#]]],u]]}, + If[lst==={}, + True, + Length[lst]==2 && OddHyperbolicPowerQ[lst[[1]],v,x] && OddHyperbolicPowerQ[lst[[2]],v,x]]], + Catch[Scan[Function[If[Not[FunctionOfTanhQ[#,v,x]],Throw[False]]],u];True]]]]]]] + +OddHyperbolicPowerQ[u_,v_,x_] := + If[SinhQ[u] || CoshQ[u] || SechQ[u] || CschQ[u], + OddQuotientQ[u[[1]],v], + If[PowerQ[u], + OddQ[u[[2]]] && OddHyperbolicPowerQ[u[[1]],v,x], + If[ProductQ[u], + If[NeQ[FreeFactors[u,x],1], + OddHyperbolicPowerQ[NonfreeFactors[u,x],v,x], + With[{lst=ReapList[Scan[Function[If[Not[FunctionOfTanhQ[#,v,x]],Sow[#]]],u]]}, + If[lst==={}, + True, + Length[lst]==1 && OddHyperbolicPowerQ[lst[[1]],v,x]]]], + If[SumQ[u], + Catch[Scan[Function[If[Not[OddHyperbolicPowerQ[#,v,x]],Throw[False]]],u];True], + False]]]] + + +(* u is a function of the form f[Tanh[v],Coth[v]] where f is independent of x. +FunctionOfTanhWeight[u,v,x] returns a nonnegative number if u is best considered a function +of Tanh[v]; else it returns a negative number. *) +FunctionOfTanhWeight[u_,v_,x_] := + If[AtomQ[u], + 0, + If[CalculusQ[u], + 0, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], + If[TanhQ[u] && EqQ[u[[1]],v], + 1, + If[CothQ[u] && EqQ[u[[1]],v], + -1, + 0]], + If[PowerQ[u] && EvenQ[u[[2]]] && HyperbolicQ[u[[1]]] && IntegerQuotientQ[u[[1,1]],v], + If[TanhQ[u[[1]]] || CoshQ[u[[1]]] || SechQ[u[[1]]], + 1, + -1], + If[ProductQ[u], + If[Catch[Scan[Function[If[Not[FunctionOfTanhQ[#,v,x]],Throw[False]]],u];True], + Apply[Plus,Map[Function[FunctionOfTanhWeight[#,v,x]],Apply[List,u]]], + 0], + Apply[Plus,Map[Function[FunctionOfTanhWeight[#,v,x]],Apply[List,u]]]]]]]] + + +(* If u (x) is equivalent to a function of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v]) + where f is independent of x, FunctionOfHyperbolicQ[u,v,x] returns True; else it returns False. *) +FunctionOfHyperbolicQ[u_,v_,x_Symbol] := + If[AtomQ[u], + u=!=x, + If[CalculusQ[u], + False, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], + True, + Catch[Scan[Function[If[FunctionOfHyperbolicQ[#,v,x],Null,Throw[False]]],u];True]]]] + + +(* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v, + FindTrigFactor[func1,func2,u,v,True] returns the list {w,u/func[w]^n}; else it returns False. *) +(* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v not equal to v, + FindTrigFactor[func1,func2,u,v,False] returns the list {w,u/func[w]^n}; else it returns False. *) +FindTrigFactor[func1_,func2_,u_,v_,flag_] := + If[u===1, + False, + If[(Head[LeadBase[u]]===func1 || Head[LeadBase[u]]===func2) && + OddQ[LeadDegree[u]] && + IntegerQuotientQ[LeadBase[u][[1]],v] && + (flag || NeQ[LeadBase[u][[1]],v]), + {LeadBase[u][[1]], RemainingFactors[u]}, + With[{lst=FindTrigFactor[func1,func2,RemainingFactors[u],v,flag]}, + If[AtomQ[lst], + False, + {lst[[1]], LeadFactor[u]*lst[[2]]}]]]] + + +(* If u/v is an integer, IntegerQuotientQ[u,v] returns True; else it returns False. *) +IntegerQuotientQ[u_,v_] := +(* u===v || EqQ[u,v] || IntegerQ[u/v] *) + IntegerQ[Simplify[u/v]] + +(* If u/v is odd, OddQuotientQ[u,v] returns True; else it returns False. *) +OddQuotientQ[u_,v_] := +(* u===v || EqQ[u,v] || OddQ[u/v] *) + OddQ[Simplify[u/v]] + +(* If u/v is even, EvenQuotientQ[u,v] returns True; else it returns False. *) +EvenQuotientQ[u_,v_] := + EvenQ[Simplify[u/v]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfDensePolynomialsQ*) + + +(* If all occurrences of x in u (x) are in dense polynomials, FunctionOfDensePolynomialsQ[u,x] + returns True; else it returns False. *) +FunctionOfDensePolynomialsQ[u_,x_Symbol] := + If[FreeQ[u,x], + True, + If[PolynomialQ[u,x], + Length[Exponent[u,x,List]]>1, + Catch[ + Scan[Function[If[FunctionOfDensePolynomialsQ[#,x],Null,Throw[False]]],u]; + True]]] + + +(* ::Subsection::Closed:: *) +(*FunctionOfLog*) + + +(* If u (x) is equivalent to an expression of the form f (Log[a*x^n]), FunctionOfLog[u,x] returns + the list {f (x),a*x^n,n}; else it returns False. *) +FunctionOfLog[u_,x_Symbol] := + With[{lst=FunctionOfLog[u,False,False,x]}, + If[AtomQ[lst] || FalseQ[lst[[2]]], + False, + lst]] + + +FunctionOfLog[u_,v_,n_,x_] := + If[AtomQ[u], + If[u===x, + False, + {u,v,n}], + If[CalculusQ[u], + False, + Module[{lst}, + If[LogQ[u] && ListQ[lst=BinomialParts[u[[1]],x]] && EqQ[lst[[1]],0], + If[FalseQ[v] || u[[1]]===v, + {x,u[[1]],lst[[3]]}, + False], + lst={0,v,n}; + Catch[ + {Map[Function[lst=FunctionOfLog[#,lst[[2]],lst[[3]],x]; + If[AtomQ[lst],Throw[False],lst[[1]]]], + u],lst[[2]],lst[[3]]}]]]]] + + +(* ::Subsection::Closed:: *) +(*PowerVariableExpn*) + + +(* If m is an integer, u is an expression of the form f[(c*x)^n] and g=GCD[m,n]>1, + PowerVariableExpn[u,m,x] returns the list {x^(m/g)*f[(c*x)^(n/g)],g,c}; else it returns False. *) +PowerVariableExpn[u_,m_,x_Symbol] := + If[IntegerQ[m], + With[{lst=PowerVariableDegree[u,m,1,x]}, + If[AtomQ[lst], + False, + {x^(m/lst[[1]])*PowerVariableSubst[u,lst[[1]],x], lst[[1]], lst[[2]]}]], + False] + + +PowerVariableDegree[u_,m_,c_,x_Symbol] := + If[FreeQ[u,x], + {m, c}, + If[AtomQ[u] || CalculusQ[u], + False, + If[PowerQ[u] && FreeQ[u[[1]]/x,x], + If[EqQ[m,0] || m===u[[2]] && c===u[[1]]/x, + {u[[2]], u[[1]]/x}, + If[IntegerQ[u[[2]]] && IntegerQ[m] && GCD[m,u[[2]]]>1 && c===u[[1]]/x, + {GCD[m,u[[2]]], c}, + False]], + Catch[Module[{lst={m, c}}, + Scan[Function[lst=PowerVariableDegree[#,lst[[1]],lst[[2]],x];If[AtomQ[lst],Throw[False]]],u]; + lst]]]]] + + +PowerVariableSubst[u_,m_,x_Symbol] := + If[FreeQ[u,x] || AtomQ[u] ||CalculusQ[u], + u, + If[PowerQ[u] && FreeQ[u[[1]]/x,x], + x^(u[[2]]/m), + Map[Function[PowerVariableSubst[#,m,x]],u]]] + + +(* ::Subsection::Closed:: *) +(*EulerIntegrandQ*) + + +(* ::Subsubsection::Closed:: *) +(*Note: If an Euler substitution should be used to integrate u wrt x, EulerIntegrandQ[u,x] returns True.*) + + +EulerIntegrandQ[(a_.*x_+b_.*u_^n_)^p_,x_Symbol] := + True /; +FreeQ[{a,b},x] && IntegerQ[n+1/2] && QuadraticQ[u,x] && (Not[RationalQ[p]] || NegativeIntegerQ[p] && Not[BinomialQ[u,x]]) + + +EulerIntegrandQ[v_^m_.*(a_.*x_+b_.*u_^n_)^p_,x_Symbol] := + True /; +FreeQ[{a,b},x] && EqQ[u,v] && IntegersQ[2*m,n+1/2] && QuadraticQ[u,x] && + (Not[RationalQ[p]] || NegativeIntegerQ[p] && Not[BinomialQ[u,x]]) + + +EulerIntegrandQ[v_^m_.*(a_.*x_+b_.*u_^n_)^p_,x_Symbol] := + True /; +FreeQ[{a,b},x] && EqQ[u,v] && IntegersQ[2*m,n+1/2] && QuadraticQ[u,x] && + (Not[RationalQ[p]] || NegativeIntegerQ[p] && Not[BinomialQ[u,x]]) + + +EulerIntegrandQ[u_^n_*v_^p_,x_Symbol] := + True /; +NegativeIntegerQ[p] && IntegerQ[n+1/2] && QuadraticQ[u,x] && QuadraticQ[v,x] && Not[BinomialQ[v,x]] + +EulerIntegrandQ[u_,x_Symbol] := + False + + +(* ::Subsection::Closed:: *) +(*FunctionOfSquareRootOfQuadratic*) + + +(* +Euler substitution #2: + If u is an expression of the form f (Sqrt[a+b*x+c*x^2],x), f (x,x) is a rational function, and + PosQ[c], FunctionOfSquareRootOfQuadratic[u,x] returns the 3-element list { + f ((a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x),(-a+x^2)/(b+2*Sqrt[c]*x))* + (a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x)^2, + Sqrt[c]*x+Sqrt[a+b*x+c*x^2], 2 }; + +Euler substitution #1: + If u is an expression of the form f (Sqrt[a+b*x+c*x^2],x), f (x,x) is a rational function, and + PosQ[a], FunctionOfSquareRootOfQuadratic[u,x] returns the two element list { + f ((c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2),(-b+2*Sqrt[a]*x)/(c-x^2))* + (c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2)^2, + (-Sqrt[a]+Sqrt[a+b*x+c*x^2])/x, 1 }; + +Euler substitution #3: + If u is an expression of the form f (Sqrt[a+b*x+c*x^2],x), f (x,x) is a rational function, and + NegQ[a] and NegQ[c], FunctionOfSquareRootOfQuadratic[u,x] returns the two element list { + -Sqrt[b^2-4*a*c]* + f (-Sqrt[b^2-4*a*c]*x/(c-x^2),-(b*c+c*Sqrt[b^2-4*a*c]+(-b+Sqrt[b^2-4*a*c])*x^2)/(2*c*(c-x^2)))* + x/(c-x^2)^2, + 2*c*Sqrt[a+b*x+c*x^2]/(b-Sqrt[b^2-4*a*c]+2*c*x), 3 }; + + else it returns False. *) + +FunctionOfSquareRootOfQuadratic[u_,x_Symbol] := + If[MatchQ[u,x^m_.*(a_+b_.*x^n_.)^p_ /; FreeQ[{a,b,m,n,p},x]], + False, + Module[{tmp=FunctionOfSquareRootOfQuadratic[u,False,x]}, + If[AtomQ[tmp] || FalseQ[tmp[[1]]], + False, + tmp=tmp[[1]]; + Module[{a=Coefficient[tmp,x,0],b=Coefficient[tmp,x,1],c=Coefficient[tmp,x,2],sqrt,q,r}, + If[EqQ[a,0] && EqQ[b,0] || EqQ[b^2-4*a*c,0], + False, + If[PosQ[c], + sqrt=Rt[c,2]; + q=a*sqrt+b*x+sqrt*x^2; + r=b+2*sqrt*x; + {Simplify[SquareRootOfQuadraticSubst[u,q/r,(-a+x^2)/r,x]*q/r^2], + Simplify[sqrt*x+Sqrt[tmp]], + 2}, + If[PosQ[a], + sqrt=Rt[a,2]; + q=c*sqrt-b*x+sqrt*x^2; + r=c-x^2; + {Simplify[SquareRootOfQuadraticSubst[u,q/r,(-b+2*sqrt*x)/r,x]*q/r^2], + Simplify[(-sqrt+Sqrt[tmp])/x], + 1}, + sqrt=Rt[b^2-4*a*c,2]; + r=c-x^2; + {Simplify[-sqrt*SquareRootOfQuadraticSubst[u,-sqrt*x/r,-(b*c+c*sqrt+(-b+sqrt)*x^2)/(2*c*r),x]*x/r^2], + FullSimplify[2*c*Sqrt[tmp]/(b-sqrt+2*c*x)], + 3}]]]]]]] + + +FunctionOfSquareRootOfQuadratic[u_,v_,x_Symbol] := + If[AtomQ[u] || FreeQ[u,x], + {v}, + If[PowerQ[u] && FreeQ[u[[2]],x], + If[FractionQ[u[[2]]] && Denominator[u[[2]]]==2 && PolynomialQ[u[[1]],x] && Exponent[u[[1]],x]==2, + If[(FalseQ[v] || u[[1]]===v), + {u[[1]]}, + False], + FunctionOfSquareRootOfQuadratic[u[[1]],v,x]], + If[ProductQ[u] || SumQ[u], + Catch[Module[{lst={v}}, + Scan[Function[lst=FunctionOfSquareRootOfQuadratic[#,lst[[1]],x];If[AtomQ[lst],Throw[False]]],u]; + lst]], + False]]] + + +(* SquareRootOfQuadraticSubst[u,vv,xx,x] returns u with fractional powers replaced by vv raised + to the power and x replaced by xx. *) +SquareRootOfQuadraticSubst[u_,vv_,xx_,x_Symbol] := + If[AtomQ[u] || FreeQ[u,x], + If[u===x, + xx, + u], + If[PowerQ[u] && FreeQ[u[[2]],x], + If[FractionQ[u[[2]]] && Denominator[u[[2]]]==2 && PolynomialQ[u[[1]],x] && Exponent[u[[1]],x]==2, + vv^Numerator[u[[2]]], + SquareRootOfQuadraticSubst[u[[1]],vv,xx,x]^u[[2]]], + Map[Function[SquareRootOfQuadraticSubst[#,vv,xx,x]],u]]] + + +(* ::Section::Closed:: *) +(*Substitution functions*) + + +(* ::Subsection::Closed:: *) +(*Subst*) + + +(* Subst[u,x,w] returns u with x replaced by w and resulting constant terms replaced by 0. *) +Subst[u_,x_Symbol,w_] := + SimplifyAntiderivative[SubstAux[u,x,w],x] + +Subst[u_,Rule[x_Symbol,w_]] := + SimplifyAntiderivative[SubstAux[u,x,w],x] + +Subst[u_,x_,w_] := + SubstAux[u,x,w] + + +SubstAux[a_+b_.*x_,x_Symbol,c_.*F_[z_]^2] := + a*Simplify[1-F[z]^2] /; +FreeQ[{a,b,c},x] && MemberQ[{Sin,Cos,Sec,Csc,Cosh,Tanh,Coth,Sech},F] && EqQ[a+b*c,0] + +SubstAux[a_+b_.*x_,x_Symbol,c_.*F_[z_]^2] := + a*Simplify[1+F[z]^2] /; +FreeQ[{a,b,c},x] && MemberQ[{Tan,Cot,Sinh,Csch},F] && EqQ[a-b*c,0] + + +SubstAux[a_+b_.*x_^2,x_Symbol,c_.*F_[z_]] := + a*Simplify[1-F[z]^2] /; +FreeQ[{a,b,c},x] && MemberQ[{Sin,Cos,Sec,Csc,Cosh,Tanh,Coth,Sech},F] && EqQ[a+b*c^2,0] + +SubstAux[a_+b_.*x_^2,x_Symbol,c_.*F_[z_]] := + a*Simplify[1+F[z]^2] /; +FreeQ[{a,b,c},x] && MemberQ[{Tan,Cot,Sinh,Csch},F] && EqQ[a-b*c^2,0] + + +SubstAux[F_[a_.*x_^m_.],x_Symbol,b_.*x_^n_] := + Extract[{ArcCsc,ArcSec,ArcCot,ArcTan,ArcCos,ArcSin,ArcCsch,ArcSech,ArcCoth,ArcTanh,ArcCosh,ArcSinh}, + Position[{ArcSin,ArcCos,ArcTan,ArcCot,ArcSec,ArcCsc,ArcSinh,ArcCosh,ArcTanh,ArcCoth,ArcSech,ArcCsch},F]][[1]] [x^(-m*n)/(a*b^m)] /; +FreeQ[{a,b},x] && PositiveIntegerQ[m] && NegativeIntegerQ[n] && MemberQ[{ArcSin,ArcCos,ArcTan,ArcCot,ArcSec,ArcCsc,ArcSinh,ArcCosh,ArcTanh,ArcCoth,ArcSech,ArcCsch},F] + + +(* Subst[u,v,w] returns u with all nondummy occurences of v replaced by w *) +SubstAux[u_,v_,w_] := + If[u===v, + w, + If[AtomQ[u], + u, + If[PowerQ[u], + If[PowerQ[v] && u[[1]]===v[[1]] && SumQ[u[[2]]], + SubstAux[u[[1]]^First[u[[2]]],v,w]*SubstAux[u[[1]]^Rest[u[[2]]],v,w], + SubstAux[u[[1]],v,w]^SubstAux[u[[2]],v,w]], + If[Head[u]===Defer[Subst], + If[u[[2]]===v || FreeQ[u[[1]],v], + SubstAux[u[[1]],u[[2]],SubstAux[u[[3]],v,w]], + Defer[Subst][u,v,w]], + If[Head[u]===Defer[Dist], + Defer[Dist][SubstAux[u[[1]],v,w],SubstAux[u[[2]],v,w],u[[3]]], + If[CalculusQ[u] && Not[FreeQ[v,u[[2]]]] || HeldFormQ[u] && Head[u]=!=Defer[AppellF1], + Defer[Subst][u,v,w], + Map[Function[SubstAux[#,v,w]],u]]]]]]] + + +(* ::Subsection::Closed:: *) +(*SimplifyAntiderivative*) + + +(* ::Item::Closed:: *) +(*SimplifyAntiderivative[u,x] returns the simplest, continuous expression whose derivative wrt x equals the derivative of u wrt x.*) + + +(* ::Item:: *) +(*Basis: D[c*F[x], x] == c*D[F[x], x]*) + + +Clear[SimplifyAntiderivative]; + +SimplifyAntiderivative[c_*u_,x_Symbol] := + With[{v=SimplifyAntiderivative[u,x]}, + If[SumQ[v] && NonsumQ[u], + Map[Function[c*#],v], + c*v]] /; +FreeQ[c,x] + + +(* ::Item:: *) +(*Basis: D[Log[c*F[x]], x] == D[Log[F[x]], x]*) + + +SimplifyAntiderivative[Log[c_*u_],x_Symbol] := + SimplifyAntiderivative[Log[u],x] /; +FreeQ[c,x] + + +(* ::Item:: *) +(*Basis: D[Log[F[x]^n], x] == n*D[Log[F[x]], x]*) + + +SimplifyAntiderivative[Log[u_^n_],x_Symbol] := + n*SimplifyAntiderivative[Log[u],x] /; +FreeQ[n,x] + + +(* ::Item:: *) +(*Basis: If F\[Element]{Log,ArcTan,ArcCot}, then D[F[G[x]], x] == -D[F[1/G[x]], x]*) + + +SimplifyAntiderivative[F_[G_[u_]],x_Symbol] := + -SimplifyAntiderivative[F[1/G[u]],x] /; +MemberQ[{Log,ArcTan,ArcCot},F] && MemberQ[{Cot,Sec,Csc,Coth,Sech,Csch},G] + + +(* ::Item:: *) +(*Basis: If F\[Element]{ArcTanh,ArcCoth}, then D[F[G[x]], x] == D[F[1/G[x]], x]*) + + +SimplifyAntiderivative[F_[G_[u_]],x_Symbol] := + SimplifyAntiderivative[F[1/G[u]],x] /; +MemberQ[{ArcTanh,ArcCoth},F] && MemberQ[{Cot,Sec,Csc,Coth,Sech,Csch},G] + + +SimplifyAntiderivative[Log[F_[u_]],x_Symbol] := + -SimplifyAntiderivative[Log[1/F[u]],x] /; +MemberQ[{Cot,Sec,Csc,Coth,Sech,Csch},F] + + +(* ::Item:: *) +(*Basis: D[Log[f^G[x]], x] == Log[f]*D[G[x], x]*) + + +SimplifyAntiderivative[Log[f_^u_],x_Symbol] := + Log[f]*SimplifyAntiderivative[u,x] /; +FreeQ[f,x] + + +(* ::Item:: *) +(*Basis: If a^2+b^2==0, then D[Log[a + b*Tan[F[x]]], x] == (b*D[F[x], x])/a - D[Log[Cos[F[x]]], x]*) + + +SimplifyAntiderivative[Log[a_+b_.*Tan[u_]],x_Symbol] := + b/a*SimplifyAntiderivative[u,x] - SimplifyAntiderivative[Log[Cos[u]],x] /; +FreeQ[{a,b},x] && EqQ[a^2+b^2,0] + + +(* ::Item:: *) +(*Basis: If a^2+b^2==0, then D[Log[a + b*Cot[F[x]]], x] == -((b/a)*D[F[x], x]) - D[Log[Sin[F[x]]], x]*) + + +SimplifyAntiderivative[Log[a_+b_.*Cot[u_]],x_Symbol] := + -b/a*SimplifyAntiderivative[u,x] - SimplifyAntiderivative[Log[Sin[u]],x] /; +FreeQ[{a,b},x] && EqQ[a^2+b^2,0] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*Note: Should also recognize that Tan[z/2]==Sin[z]/(1+Cos[z])==Tan[z]/(1+Sec[z])*) + + +SimplifyAntiderivative[ArcTan[a_.*Tan[u_]],x_Symbol] := + RectifyTangent[u,a,1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcCot[a*Tan[f[x]]], x] == -D[ArcTan[a*Tan[f[x]]], x]*) + + +SimplifyAntiderivative[ArcCot[a_.*Tan[u_]],x_Symbol] := + RectifyTangent[u,a,-1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcTan[a Tanh[z]]==-ArcTan[I a Tan[I z]]*) + + +(* SimplifyAntiderivative[ArcTan[a_.*Tanh[u_]],x_Symbol] := + RectifyTangent[I*u,I*a,-1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] *) + + +(* ::Item:: *) +(*Basis: D[ArcCot[a*Tanh[f[x]]], x] == -D[ArcTan[a*Tanh[f[x]]], x]*) + + +SimplifyAntiderivative[ArcCot[a_.*Tanh[u_]],x_Symbol] := + -SimplifyAntiderivative[ArcTan[a*Tanh[u]],x] /; +FreeQ[a,x] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcTanh[a Tan[z]]==-I ArcTan[I a Tan[z]]*) + + +SimplifyAntiderivative[ArcTanh[a_.*Tan[u_]],x_Symbol] := + RectifyTangent[u,I*a,-I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcCoth[a*Tan[f[x]]], x] == -(I*D[ArcTan[I*a*Tan[f[x]]], x])*) + + +SimplifyAntiderivative[ArcCoth[a_.*Tan[u_]],x_Symbol] := + RectifyTangent[u,I*a,-I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item::Closed:: *) +(*Basis: ArcTanh[a Tanh[z]]==-I ArcTan[a Tan[I z]]*) + + +(* ::Item:: *) +(*Note: Although this makes the antiderivative continuous on the imaginary line, it makes it discontinous on the real line.*) + + +(* SimplifyAntiderivative[ArcTanh[a_.*Tanh[u_]],x_Symbol] := + RectifyTangent[I*u,a,-I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] *) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[Tanh[f[x]]], x] == D[f[x], x]*) + + +SimplifyAntiderivative[ArcTanh[Tanh[u_]],x_Symbol] := + SimplifyAntiderivative[u,x] + + +(* ::Item::Closed:: *) +(*Basis: D[ArcCoth[a*Tanh[f[x]]], x] == -(I*D[ArcTan[a*Tan[I*f[x]]], x])*) + + +(* ::Item:: *) +(*Note: Although this makes the antiderivative continuous on the imaginary line, it makes it discontinous on the real line.*) + + +(* SimplifyAntiderivative[ArcCoth[a_.*Tanh[u_]],x_Symbol] := + RectifyTangent[I*u,a,-I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] *) + + +(* ::Item:: *) +(*Basis: D[ArcCoth[Tanh[f[x]]], x] == D[f[x], x]*) + + +SimplifyAntiderivative[ArcCoth[Tanh[u_]],x_Symbol] := + SimplifyAntiderivative[u,x] + + +(* ::Input:: *) +(* *) + + +SimplifyAntiderivative[ArcCot[a_.*Cot[u_]],x_Symbol] := + RectifyCotangent[u,a,1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcTan[a*Cot[f[x]]], x] == -D[ArcCot[a*Cot[f[x]]], x]*) + + +SimplifyAntiderivative[ArcTan[a_.*Cot[u_]],x_Symbol] := + RectifyCotangent[u,a,-1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcCot[a Coth[z]]==ArcCot[I a Cot[I z]]*) + + +(* SimplifyAntiderivative[ArcCot[a_.*Coth[u_]],x_Symbol] := + RectifyCotangent[I*u,I*a,1,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] *) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a*Coth[f[x]]], x] == -D[ArcTan[Tanh[f[x]]/a], x]*) + + +SimplifyAntiderivative[ArcTan[a_.*Coth[u_]],x_Symbol] := + -SimplifyAntiderivative[ArcTan[Tanh[u]/a],x] /; +FreeQ[a,x] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcCoth[a Cot[z]]==I ArcCot[I a Cot[z]]*) + + +SimplifyAntiderivative[ArcCoth[a_.*Cot[u_]],x_Symbol] := + RectifyCotangent[u,I*a,I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcTanh[a*Cot[f[x]]], x] == I*D[ArcCot[I*a*Cot[f[x]]], x]*) + + +SimplifyAntiderivative[ArcTanh[a_.*Cot[u_]],x_Symbol] := + RectifyCotangent[u,I*a,I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] + + +(* ::Item::Closed:: *) +(*Basis: ArcCoth[a Coth[z]]==-I ArcCot[a Cot[I z]]*) + + +(* ::Item:: *) +(*Note: Although this makes the antiderivative continuous on the imaginary line, it makes it discontinous on the real line.*) + + +(* SimplifyAntiderivative[ArcCoth[a_.*Coth[u_]],x_Symbol] := + RectifyCotangent[I*u,a,-I,x] /; +FreeQ[a,x] && PositiveQ[a^2] && ComplexFreeQ[u] *) + + +(* ::Item:: *) +(*Basis: D[ArcCoth[Coth[f[x]]], x] == D[f[x], x]*) + + +SimplifyAntiderivative[ArcCoth[Coth[u_]],x_Symbol] := + SimplifyAntiderivative[u,x] + + +(* ::Item:: *) +(*Basis: D[ArcTanh[a*Coth[f[x]]], x] == D[ArcTanh[Tanh[f[x]]/a], x]*) + + +SimplifyAntiderivative[ArcTanh[a_.*Coth[u_]],x_Symbol] := + SimplifyAntiderivative[ArcTanh[Tanh[u]/a],x] /; +FreeQ[a,x] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcTanh[Coth[f[x]]], x] == D[f[x], x]*) + + +SimplifyAntiderivative[ArcTanh[Coth[u_]],x_Symbol] := + SimplifyAntiderivative[u,x] + + +(* ::Input:: *) +(* *) + + +SimplifyAntiderivative[ArcTan[c_.*(a_+b_.*Tan[u_])],x_Symbol] := + RectifyTangent[u,a*c,b*c,1,x] /; +FreeQ[{a,b,c},x] && PositiveQ[a^2*c^2] && PositiveQ[b^2*c^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcTanh[a+b Tan[z]]==-I ArcTan[I a+I b Tan[z]]*) + + +SimplifyAntiderivative[ArcTanh[c_.*(a_+b_.*Tan[u_])],x_Symbol] := + RectifyTangent[u,I*a*c,I*b*c,-I,x] /; +FreeQ[{a,b,c},x] && PositiveQ[a^2*c^2] && PositiveQ[b^2*c^2] && ComplexFreeQ[u] + + +(* ::Input:: *) +(* *) + + +SimplifyAntiderivative[ArcTan[c_.*(a_+b_.*Cot[u_])],x_Symbol] := + RectifyCotangent[u,a*c,b*c,1,x] /; +FreeQ[{a,b,c},x] && PositiveQ[a^2*c^2] && PositiveQ[b^2*c^2] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: ArcTanh[a+b Cot[z]]==-I ArcTan[I a+I b Cot[z]]*) + + +SimplifyAntiderivative[ArcTanh[c_.*(a_+b_.*Cot[u_])],x_Symbol] := + RectifyCotangent[u,I*a*c,I*b*c,-I,x] /; +FreeQ[{a,b,c},x] && PositiveQ[a^2*c^2] && PositiveQ[b^2*c^2] && ComplexFreeQ[u] + + +(* ::Input:: *) +(* *) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + b*Tan[F[x]] + c*Tan[F[x]]^2], x] == D[ArcTan[(c + (a - c - 1)*Cos[F[x]]^2 + b*Cos[F[x]]*Sin[F[x]])/(c + (a - c + 1)*Cos[F[x]]^2 + b*Cos[F[x]]*Sin[F[x]])], x] == D[ArcTan[(a + c - 1 + (a - c - 1)*Cos[2*F[x]] + b*Sin[2*F[x]])/(a + c + 1 + (a - c + 1)*Cos[2*F[x]] + b*Sin[2*F[x]])], x]*) + + +SimplifyAntiderivative[ArcTan[a_.+b_.*Tan[u_]+c_.*Tan[u_]^2],x_Symbol] := + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + ArcTan[NormalizeTogether[(a+c-1+(a-c-1)*Cos[2*u]+b*Sin[2*u])/(a+c+1+(a-c+1)*Cos[2*u]+b*Sin[2*u])]], + ArcTan[NormalizeTogether[(c+(a-c-1)*Cos[u]^2+b*Cos[u]*Sin[u])/(c+(a-c+1)*Cos[u]^2+b*Cos[u]*Sin[u])]]] /; +FreeQ[{a,b,c},x] && ComplexFreeQ[u] + +SimplifyAntiderivative[ArcTan[a_.+b_.*(d_.+e_.*Tan[u_])+c_.*(f_.+g_.*Tan[u_])^2],x_Symbol] := + SimplifyAntiderivative[ArcTan[a+b*d+c*f^2+(b*e+2*c*f*g)*Tan[u]+c*g^2*Tan[u]^2],x] /; +FreeQ[{a,b,c},x] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + c*Tan[F[x]]^2], x] == D[ArcTan[(c + (a - c - 1)*Cos[F[x]]^2)/(c + (a - c + 1)*Cos[F[x]]^2)], x] == D[ArcTan[(a + c - 1 + (a - c - 1)*Cos[2*F[x]])/(a + c + 1 + (a - c + 1)*Cos[2*F[x]])], x]*) + + +SimplifyAntiderivative[ArcTan[a_.+c_.*Tan[u_]^2],x_Symbol] := + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + ArcTan[NormalizeTogether[(a+c-1+(a-c-1)*Cos[2*u])/(a+c+1+(a-c+1)*Cos[2*u])]], + ArcTan[NormalizeTogether[(c+(a-c-1)*Cos[u]^2)/(c+(a-c+1)*Cos[u]^2)]]] /; +FreeQ[{a,c},x] && ComplexFreeQ[u] + +SimplifyAntiderivative[ArcTan[a_.+c_.*(f_.+g_.*Tan[u_])^2],x_Symbol] := + SimplifyAntiderivative[ArcTan[a+c*f^2+(2*c*f*g)*Tan[u]+c*g^2*Tan[u]^2],x] /; +FreeQ[{a,c},x] && ComplexFreeQ[u] + + +(* ::Item:: *) +(*Basis: D[c, x] == D[0, x]*) + + +SimplifyAntiderivative[u_,x_Symbol] := + If[FreeQ[u,x], + 0, + If[LogQ[u], + Log[RemoveContent[u[[1]],x]], + If[SumQ[u], + SimplifyAntiderivativeSum[Map[Function[SimplifyAntiderivative[#,x]],u],x], + u]]] + + +(* ::Subsection::Closed:: *) +(*SimplifyAntiderivativeSum*) + + +Clear[SimplifyAntiderivativeSum]; + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Tan[u_]^n_.]+B_.*Log[Cos[u_]],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[a*Cos[u]^n+b*Sin[u]^n,x]] /; +FreeQ[{a,b,A,B},x] && IntegerQ[n] && EqQ[n*A-B,0] + + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Cot[u_]^n_.]+B_.*Log[Sin[u_]],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[a*Sin[u]^n+b*Cos[u]^n,x]] /; +FreeQ[{a,b,A,B},x] && IntegerQ[n] && EqQ[n*A-B,0] + + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Tan[u_]^n_.]+B_.*Log[c_+d_.*Tan[u_]^n_.],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[a*Cos[u]^n+b*Sin[u]^n,x]] + B*Log[RemoveContent[c*Cos[u]^n+d*Sin[u]^n,x]] /; +FreeQ[{a,b,c,d,A,B},x] && IntegerQ[n] && EqQ[A+B,0] + + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Cot[u_]^n_.]+B_.*Log[c_+d_.*Cot[u_]^n_.],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[b*Cos[u]^n+a*Sin[u]^n,x]] + B*Log[RemoveContent[d*Cos[u]^n+c*Sin[u]^n,x]] /; +FreeQ[{a,b,c,d,A,B},x] && IntegerQ[n] && EqQ[A+B,0] + + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Tan[u_]^n_.]+B_.*Log[c_+d_.*Tan[u_]^n_.]+C_.*Log[e_+f_.*Tan[u_]^n_.],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[a*Cos[u]^n+b*Sin[u]^n,x]] + + B*Log[RemoveContent[c*Cos[u]^n+d*Sin[u]^n,x]] + C*Log[RemoveContent[e*Cos[u]^n+f*Sin[u]^n,x]] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && IntegerQ[n] && EqQ[A+B+C,0] + + +SimplifyAntiderivativeSum[v_.+A_.*Log[a_+b_.*Cot[u_]^n_.]+B_.*Log[c_+d_.*Cot[u_]^n_.]+C_.*Log[e_+f_.*Cot[u_]^n_.],x_Symbol] := + SimplifyAntiderivativeSum[v,x] + A*Log[RemoveContent[b*Cos[u]^n+a*Sin[u]^n,x]] + + B*Log[RemoveContent[d*Cos[u]^n+c*Sin[u]^n,x]] + C*Log[RemoveContent[f*Cos[u]^n+e*Sin[u]^n,x]] /; +FreeQ[{a,b,c,d,e,f,A,B,C},x] && IntegerQ[n] && EqQ[A+B+C,0] + + +SimplifyAntiderivativeSum[u_,x_Symbol] := u + + +(* ::Subsection::Closed:: *) +(*RectifyTangent*) + + +(* ::Item::Closed:: *) +(*Basis: ArcTan[I a Tan[f[x]]]==I ArcTanh[a Tan[f[x]]]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[(c/e)*Tan[f[x]]], x] == D[Log[e*Cos[f[x]] + c*Sin[f[x]]]/2 - Log[e*Cos[f[x]] - c*Sin[f[x]]]/2, x]*) + + +(* ::Item:: *) +(*Note: Note unlike the above, the log terms in the following two identities are real-valued when c, e and f[x] are real.*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[(c/e)*Tan[f[x]]], x] == D[Log[c^2 + e^2 - (c^2 - e^2)*Cos[2*f[x]] + 2*c*e*Sin[2*f[x]]]/4 - Log[c^2 + e^2 - (c^2 - e^2)*Cos[2*f[x]] - 2*c*e*Sin[2*f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[(c/e)*Tan[f[x]]], x] == D[Log[e^2 + 2*c*e*Cos[f[x]]*Sin[f[x]] + (c^2 - e^2)*Sin[f[x]]^2]/4 - Log[e^2 - 2*c*e*Cos[f[x]]*Sin[f[x]] + (c^2 - e^2)*Sin[f[x]]^2]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a*Tan[f[x]]], x] == D[f[x] - ArcTan[Sin[2*f[x]]/((1 + a)/(1 - a) + Cos[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a*Tan[f[x]]], x] == D[f[x] + ArcTan[(Cos[f[x]]*Sin[f[x]])/(1/(a - 1) + Sin[f[x]]^2)], x] == D[f[x] - ArcTan[(Cos[f[x]]*Sin[f[x]])/(a/(1 - a) + Cos[f[x]]^2)], x]*) + + +(* ::Item::Bold:: *) +(*Note: If a>0 and f[x] is real, then 1/(1-a)-Sin[f[x]]^2 is nonzero and f[x]-ArcTan[(Cos[f[x]] Sin[f[x]])/(1/(1-a)-Sin[f[x]]^2)] is continuous, unlike ArcTan[a Tan[f[x]]].*) + + +(* RectifyTangent[u,a,b,x] returns an expression whose derivative equals the derivative of b*ArcTan[a*Tan[u]] wrt x. *) +RectifyTangent[u_,a_,b_,x_Symbol] := + If[MatchQ[Together[a],d_.*Complex[0,c_]], + Module[{c=a/I,e}, + If[NegativeQ[c], + RectifyTangent[u,-a,-b,x], + If[EqQ[c,1], + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + I*b*ArcTanh[Sin[2*u]]/2, + I*b*ArcTanh[2*Cos[u]*Sin[u]]/2], + e=SmartDenominator[c]; + c=c*e; +(* If[EvenQ[Denominator[NumericFactor[Together[u]]]], + I*b*Log[RemoveContent[c^2+e^2-(c^2-e^2)*Cos[2*u]+2*c*e*Sin[2*u],x]]/4 - + I*b*Log[RemoveContent[c^2+e^2-(c^2-e^2)*Cos[2*u]-2*c*e*Sin[2*u],x]]/4, + I*b*Log[RemoveContent[e^2+2*c*e*Cos[u]*Sin[u]+(c^2-e^2)*Sin[u]^2,x]]/4 - + I*b*Log[RemoveContent[e^2-2*c*e*Cos[u]*Sin[u]+(c^2-e^2)*Sin[u]^2,x]]/4]]]], *) + I*b*Log[RemoveContent[e*Cos[u]+c*Sin[u],x]]/2 - + I*b*Log[RemoveContent[e*Cos[u]-c*Sin[u],x]]/2]]], + If[NegativeQ[a], + RectifyTangent[u,-a,-b,x], + If[EqQ[a,1], + b*SimplifyAntiderivative[u,x], + Module[{c,numr,denr}, + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + c=Simplify[(1+a)/(1-a)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] - b*ArcTan[NormalizeLeadTermSigns[denr*Sin[2*u]/(numr+denr*Cos[2*u])]], + If[PositiveQ[a-1], + c=Simplify[1/(a-1)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] + b*ArcTan[NormalizeLeadTermSigns[denr*Cos[u]*Sin[u]/(numr+denr*Sin[u]^2)]], + c=Simplify[a/(1-a)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] - b*ArcTan[NormalizeLeadTermSigns[denr*Cos[u]*Sin[u]/(numr+denr*Cos[u]^2)]]]]]]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item::Closed:: *) +(*Basis: ArcTan[I a+I b Tan[f[x]]]==I ArcTanh[a+b Tan[f[x]]]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[c/e + (d/e)*Tan[f[x]]], x] == D[Log[(c + e)^2 + d^2 + ((c + e)^2 - d^2)*Cos[2*f[x]] + 2*(c + e)*d*Sin[2*f[x]]]/4 - Log[(c - e)^2 + d^2 + ((c - e)^2 - d^2)*Cos[2*f[x]] + 2*(c - e)*d*Sin[2*f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[c/e + (d/e)*Tan[f[x]]], x] == D[Log[(c + e)^2 + 2*(c + e)*d*Cos[f[x]]*Sin[f[x]] - ((c + e)^2 - d^2)*Sin[f[x]]^2]/4 - Log[(c - e)^2 + 2*(c - e)*d*Cos[f[x]]*Sin[f[x]] - ((c - e)^2 - d^2)*Sin[f[x]]^2]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + b*Tan[f[x]]], x] == D[f[x] + ArcTan[(2*a*b*Cos[2*f[x]] - (1 + a^2 - b^2)*Sin[2*f[x]])/(a^2 + (1 + b)^2 + (1 + a^2 - b^2)*Cos[2*f[x]] + 2*a*b*Sin[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + b*Tan[f[x]]], x] == D[f[x] - ArcTan[(a*b - 2*a*b*Cos[f[x]]^2 + (1 + a^2 - b^2)*Cos[f[x]]*Sin[f[x]])/(b*(1 + b) + (1 + a^2 - b^2)*Cos[f[x]]^2 + 2*a*b*Cos[f[x]]*Sin[f[x]])], x]*) + + +(* RectifyTangent[u,a,b,r,x] returns an expression whose derivative equals the derivative of r*ArcTan[a+b*Tan[u]] wrt x. *) +RectifyTangent[u_,a_,b_,r_,x_Symbol] := + If[MatchQ[Together[a],d_.*Complex[0,c_]] && MatchQ[Together[b],d_.*Complex[0,c_]], + Module[{c=a/I,d=b/I,e}, + If[NegativeQ[d], + RectifyTangent[u,-a,-b,-r,x], + e=SmartDenominator[Together[c+d*x]]; + c=c*e; + d=d*e; + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + I*r*Log[RemoveContent[Simplify[(c+e)^2+d^2]+Simplify[(c+e)^2-d^2]*Cos[2*u]+Simplify[2*(c+e)*d]*Sin[2*u],x]]/4 - + I*r*Log[RemoveContent[Simplify[(c-e)^2+d^2]+Simplify[(c-e)^2-d^2]*Cos[2*u]+Simplify[2*(c-e)*d]*Sin[2*u],x]]/4, + I*r*Log[RemoveContent[Simplify[(c+e)^2]+Simplify[2*(c+e)*d]*Cos[u]*Sin[u]-Simplify[(c+e)^2-d^2]*Sin[u]^2,x]]/4 - + I*r*Log[RemoveContent[Simplify[(c-e)^2]+Simplify[2*(c-e)*d]*Cos[u]*Sin[u]-Simplify[(c-e)^2-d^2]*Sin[u]^2,x]]/4]]], + If[NegativeQ[b], + RectifyTangent[u,-a,-b,-r,x], + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + r*SimplifyAntiderivative[u,x] + + r*ArcTan[Simplify[(2*a*b*Cos[2*u]-(1+a^2-b^2)*Sin[2*u])/(a^2+(1+b)^2+(1+a^2-b^2)*Cos[2*u]+2*a*b*Sin[2*u])]], + r*SimplifyAntiderivative[u,x] - + r*ArcTan[ActivateTrig[Simplify[(a*b-2*a*b*cos[u]^2+(1+a^2-b^2)*cos[u]*sin[u])/(b*(1+b)+(1+a^2-b^2)*cos[u]^2+2*a*b*cos[u]*sin[u])]]]]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item::Closed:: *) +(*Basis: D[ArcTanh[a + b*Tanh[f[x]]], x] == D[f[x] - ArcTanh[(2*a*b*Cosh[2*f[x]] - (1 - a^2 - b^2)*Sinh[2*f[x]])/(a^2 - (1 + b)^2 - (1 - a^2 - b^2)*Cosh[2*f[x]] + 2*a*b*Sinh[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[a + b*Tanh[f[x]]], x] == D[f[x] - ArcTanh[(a*b - 2*a*b*Cosh[f[x]]^2 + (1 - a^2 - b^2)*Cosh[f[x]]*Sinh[f[x]])/(b*(1 + b) + (1 - a^2 - b^2)*Cosh[f[x]]^2 - 2*a*b*Cosh[f[x]]*Sinh[f[x]])], x]*) + + +(* SimplifyAntiderivative[ArcTanh[c_.*(a_+b_.*Tanh[u_])],x_Symbol] := + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + SimplifyAntiderivative[u,x] - + ArcTanh[NormalizeTogether[(2*a*b*c^2*Cosh[2*u]-(1-a^2*c^2-b^2*c^2)*Sinh[2*u])/(a^2*c^2-(1+b*c)^2-(1-a^2*c^2-b^2*c^2)*Cosh[2*u]+2*a*b*c^2*Sinh[2*u])]], + SimplifyAntiderivative[u,x] - + ArcTanh[NormalizeTogether[(a*b*c^2-2*a*b*c^2*Cosh[u]^2+(1-a^2*c^2-b^2*c^2)*Cosh[u]*Sinh[u])/(b*c*(1+b*c)+(1-a^2*c^2-b^2*c^2)*Cosh[u]^2-2*a*b*c^2*Cosh[u]*Sinh[u])]]] /; +FreeQ[{a,b,c},x] *) + + +(* ::Subsection::Closed:: *) +(*RectifyCotangent*) + + +(* ::Item::Closed:: *) +(*Basis: ArcCot[I a Cot[f[x]]]==-I ArcCoth[a Cot[f[x]]]*) + + +(* ::Item:: *) +(*Basis: D[ArcCoth[(c/e)*Cot[f[x]]], x] == D[Log[c*Cos[f[x]] + e*Sin[f[x]]]/2 - Log[c*Cos[f[x]] - e*Sin[f[x]]]/2, x]*) + + +(* ::Item:: *) +(*Note: Note unlike the above, the log terms in the following two identities are real-valued when c, e and f[x] are real.*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[(c/e)*Cot[f[x]]], x] == D[Log[c^2 + e^2 + (c^2 - e^2)*Cos[2*f[x]] + 2*c*e*Sin[2*f[x]]]/4 - Log[c^2 + e^2 + (c^2 - e^2)*Cos[2*f[x]] - 2*c*e*Sin[2*f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[(c/e)*Cot[f[x]]], x] == D[Log[e^2 + (c^2 - e^2)*Cos[f[x]]^2 + 2*c*e*Cos[f[x]]*Sin[f[x]]]/4 - Log[e^2 + (c^2 - e^2)*Cos[f[x]]^2 - 2*c*e*Cos[f[x]]*Sin[f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcCot[a*Cot[f[x]]], x] == D[f[x] + ArcTan[Sin[2*f[x]]/((1 + a)/(1 - a) - Cos[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcCot[a*Cot[f[x]]], x] == D[f[x] - ArcTan[(Cos[f[x]]*Sin[f[x]])/(1/(a - 1) + Cos[f[x]]^2)], x] == D[f[x] + ArcTan[(Cos[f[x]]*Sin[f[x]])/(a/(1 - a) + Sin[f[x]]^2)], x]*) + + +(* RectifyCotangent[u,a,b,x] returns an expression whose derivative equals the derivative of b*ArcCot[a*Cot[u]] wrt x. *) +RectifyCotangent[u_,a_,b_,x_Symbol] := + If[MatchQ[Together[a],d_.*Complex[0,c_]], + Module[{c=a/I,e}, + If[NegativeQ[c], + RectifyCotangent[u,-a,-b,x], + If[EqQ[c,1], + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + -I*b*ArcTanh[Sin[2*u]]/2, + -I*b*ArcTanh[2*Cos[u]*Sin[u]]/2], + e=SmartDenominator[c]; + c=c*e; +(* If[EvenQ[Denominator[NumericFactor[Together[u]]]], + -I*b*Log[RemoveContent[c^2+e^2+(c^2-e^2)*Cos[2*u]+2*c*e*Sin[2*u],x]]/4 + + I*b*Log[RemoveContent[c^2+e^2+(c^2-e^2)*Cos[2*u]-2*c*e*Sin[2*u],x]]/4, + -I*b*Log[RemoveContent[e^2+(c^2-e^2)*Cos[u]^2+2*c*e*Cos[u]*Sin[u],x]]/4 + + I*b*Log[RemoveContent[e^2+(c^2-e^2)*Cos[u]^2-2*c*e*Cos[u]*Sin[u],x]]/4]]]], *) + -I*b*Log[RemoveContent[c*Cos[u]+e*Sin[u],x]]/2 + + I*b*Log[RemoveContent[c*Cos[u]-e*Sin[u],x]]/2]]], + If[NegativeQ[a], + RectifyCotangent[u,-a,-b,x], + If[EqQ[a,1], + b*SimplifyAntiderivative[u,x], + Module[{c,numr,denr}, + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + c=Simplify[(1+a)/(1-a)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] + b*ArcTan[NormalizeLeadTermSigns[denr*Sin[2*u]/(numr-denr*Cos[2*u])]], + If[PositiveQ[a-1], + c=Simplify[1/(a-1)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] - b*ArcTan[NormalizeLeadTermSigns[denr*Cos[u]*Sin[u]/(numr+denr*Cos[u]^2)]], + c=Simplify[a/(1-a)]; + numr=SmartNumerator[c]; + denr=SmartDenominator[c]; + b*SimplifyAntiderivative[u,x] + b*ArcTan[NormalizeLeadTermSigns[denr*Cos[u]*Sin[u]/(numr+denr*Sin[u]^2)]]]]]]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item::Closed:: *) +(*Basis: ArcTan[I a+I b Cot[f[x]]]==I ArcTanh[a+b Cot[f[x]]]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[c/e + (d/e)*Cot[f[x]]], x] == D[Log[(c + e)^2 + d^2 - ((c + e)^2 - d^2)*Cos[2*f[x]] + 2*(c + e)*d*Sin[2*f[x]]]/4 - Log[(c - e)^2 + d^2 - ((c - e)^2 - d^2)*Cos[2*f[x]] + 2*(c - e)*d*Sin[2*f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[c/e + (d/e)*Cot[f[x]]], x] == D[Log[(c + e)^2 - ((c + e)^2 - d^2)*Cos[f[x]]^2 + 2*(c + e)*d*Cos[f[x]]*Sin[f[x]]]/4 - Log[(c - e)^2 - ((c - e)^2 - d^2)*Cos[f[x]]^2 + 2*(c - e)*d*Cos[f[x]]*Sin[f[x]]]/4, x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + b*Cot[f[x]]], x] == D[-f[x] - ArcTan[(2*a*b*Cos[2*f[x]] + (1 + a^2 - b^2)*Sin[2*f[x]])/(a^2 + (1 + b)^2 - (1 + a^2 - b^2)*Cos[2*f[x]] + 2*a*b*Sin[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTan[a + b*Cot[f[x]]], x] == D[-f[x] - ArcTan[(a*b - 2*a*b*Sin[f[x]]^2 + (1 + a^2 - b^2)*Cos[f[x]]*Sin[f[x]])/(b*(1 + b) + (1 + a^2 - b^2)*Sin[f[x]]^2 + 2*a*b*Cos[f[x]]*Sin[f[x]])], x]*) + + +(* RectifyCotangent[u,a,b,r,x] returns an expression whose derivative equals the derivative of r*ArcTan[a+b*Cot[u]] wrt x. *) +RectifyCotangent[u_,a_,b_,r_,x_Symbol] := + If[MatchQ[Together[a],d_.*Complex[0,c_]] && MatchQ[Together[b],d_.*Complex[0,c_]], + Module[{c=a/I,d=b/I,e}, + If[NegativeQ[d], + RectifyTangent[u,-a,-b,-r,x], + e=SmartDenominator[Together[c+d*x]]; + c=c*e; + d=d*e; + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + I*r*Log[RemoveContent[Simplify[(c+e)^2+d^2]-Simplify[(c+e)^2-d^2]*Cos[2*u]+Simplify[2*(c+e)*d]*Sin[2*u],x]]/4 - + I*r*Log[RemoveContent[Simplify[(c-e)^2+d^2]-Simplify[(c-e)^2-d^2]*Cos[2*u]+Simplify[2*(c-e)*d]*Sin[2*u],x]]/4, + I*r*Log[RemoveContent[Simplify[(c+e)^2]-Simplify[(c+e)^2-d^2]*Cos[u]^2+Simplify[2*(c+e)*d]*Cos[u]*Sin[u],x]]/4 - + I*r*Log[RemoveContent[Simplify[(c-e)^2]-Simplify[(c-e)^2-d^2]*Cos[u]^2+Simplify[2*(c-e)*d]*Cos[u]*Sin[u],x]]/4]]], + If[NegativeQ[b], + RectifyCotangent[u,-a,-b,-r,x], + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + -r*SimplifyAntiderivative[u,x] - + r*ArcTan[Simplify[(2*a*b*Cos[2*u]+(1+a^2-b^2)*Sin[2*u])/(a^2+(1+b)^2-(1+a^2-b^2)*Cos[2*u]+2*a*b*Sin[2*u])]], + -r*SimplifyAntiderivative[u,x] - + r*ArcTan[ActivateTrig[Simplify[(a*b-2*a*b*sin[u]^2+(1+a^2-b^2)*cos[u]*sin[u])/(b*(1+b)+(1+a^2-b^2)*sin[u]^2+2*a*b*cos[u]*sin[u])]]]]]] + + +(* ::Input:: *) +(* *) + + +(* ::Item::Closed:: *) +(*Basis: D[ArcTanh[a + b*Coth[f[x]]], x] == D[f[x] - ArcTanh[(2*a*b*Cosh[2*f[x]] - (1 - a^2 - b^2)*Sinh[2*f[x]])/(-a^2 + (1 + b)^2 - (1 - a^2 - b^2)*Cosh[2*f[x]] + 2*a*b*Sinh[2*f[x]])], x]*) + + +(* ::Item:: *) +(*Basis: D[ArcTanh[a + b*Coth[f[x]]], x] == D[f[x] - ArcTanh[(a*b + 2*a*b*Sinh[f[x]]^2 - (1 - a^2 - b^2)*Cosh[f[x]]*Sinh[f[x]])/(b*(1 + b) - (1 - a^2 - b^2)*Sinh[f[x]]^2 + 2*a*b*Cosh[f[x]]*Sinh[f[x]])], x]*) + + +(* SimplifyAntiderivative[ArcTanh[c_.*(a_+b_.*Coth[u_])],x_Symbol] := + If[EvenQ[Denominator[NumericFactor[Together[u]]]], + SimplifyAntiderivative[u,x] - + ArcTanh[NormalizeTogether[(2*a*b*c^2*Cosh[2*u]-(1-a^2*c^2-b^2*c^2)*Sinh[2*u])/(-a^2*c^2+(1+b*c)^2-(1-a^2*c^2-b^2*c^2)*Cosh[2*u]+2*a*b*c^2*Sinh[2*u])]], + SimplifyAntiderivative[u,x] - + ArcTanh[NormalizeTogether[(a*b*c^2+2*a*b*c^2*Sinh[u]^2-(1-a^2*c^2-b^2*c^2)*Cosh[u]*Sinh[u])/(b*c*(1+b*c)-(1-a^2*c^2-b^2*c^2)*Sinh[u]^2+2*a*b*c^2*Cosh[u]*Sinh[u])]]] /; +FreeQ[{a,b,c},x] *) + + +SmartNumerator[u_^n_] := SmartDenominator[u^(-n)] /; RationalQ[n] && n<0 + +SmartNumerator[u_*v_] := SmartNumerator[u]*SmartNumerator[v] + +SmartNumerator[u_] := Numerator[u] + + +SmartDenominator[u_^n_] := SmartNumerator[u^(-n)] /; RationalQ[n] && n<0 + +SmartDenominator[u_*v_] := SmartDenominator[u]*SmartDenominator[v] + +SmartDenominator[u_] := Denominator[u] + + +(* ::Subsection::Closed:: *) +(*SubstFor*) + + +(* u is a function of v. SubstFor[w,v,u,x] returns w times u with v replaced by x. *) +SubstFor[w_,v_,u_,x_] := + SimplifyIntegrand[w*SubstFor[v,u,x],x] + + +(* u is a function of v. SubstFor[v,u,x] returns u with v replaced by x. *) +SubstFor[v_,u_,x_] := + If[AtomQ[v], + Subst[u,v,x], + If[Not[InertTrigFreeQ[u]], + SubstFor[v,ActivateTrig[u],x], + If[NeQ[FreeFactors[v,x],1], + SubstFor[NonfreeFactors[v,x],u,x/FreeFactors[v,x]], + + If[SinQ[v], + SubstForTrig[u,x,Sqrt[1-x^2],v[[1]],x], + If[CosQ[v], + SubstForTrig[u,Sqrt[1-x^2],x,v[[1]],x], + If[TanQ[v], + SubstForTrig[u,x/Sqrt[1+x^2],1/Sqrt[1+x^2],v[[1]],x], + If[CotQ[v], + SubstForTrig[u,1/Sqrt[1+x^2],x/Sqrt[1+x^2],v[[1]],x], + If[SecQ[v], + SubstForTrig[u,1/Sqrt[1-x^2],1/x,v[[1]],x], + If[CscQ[v], + SubstForTrig[u,1/x,1/Sqrt[1-x^2],v[[1]],x], + + If[SinhQ[v], + SubstForHyperbolic[u,x,Sqrt[1+x^2],v[[1]],x], + If[CoshQ[v], + SubstForHyperbolic[u,Sqrt[-1+x^2],x,v[[1]],x], + If[TanhQ[v], + SubstForHyperbolic[u,x/Sqrt[1-x^2],1/Sqrt[1-x^2],v[[1]],x], + If[CothQ[v], + SubstForHyperbolic[u,1/Sqrt[-1+x^2],x/Sqrt[-1+x^2],v[[1]],x], + If[SechQ[v], + SubstForHyperbolic[u,1/Sqrt[-1+x^2],1/x,v[[1]],x], + If[CschQ[v], + SubstForHyperbolic[u,1/x,1/Sqrt[1+x^2],v[[1]],x], + + SubstForAux[u,v,x]]]]]]]]]]]]]]]] + + +(* u is a function of v. SubstForAux[u,v,x] returns u with v replaced by x. *) +SubstForAux[u_,v_,x_] := + If[u===v, + x, + If[AtomQ[u], + If[PowerQ[v] && FreeQ[v[[2]],x] && EqQ[u,v[[1]]], + x^Simplify[1/v[[2]]], + u], + If[PowerQ[u] && FreeQ[u[[2]],x], + If[EqQ[u[[1]],v], + x^u[[2]], + If[PowerQ[v] && FreeQ[v[[2]],x] && EqQ[u[[1]],v[[1]]], + x^Simplify[u[[2]]/v[[2]]], + SubstForAux[u[[1]],v,x]^u[[2]]]], + If[ProductQ[u] && NeQ[FreeFactors[u,x],1], + FreeFactors[u,x]*SubstForAux[NonfreeFactors[u,x],v,x], + If[ProductQ[u] && ProductQ[v], + SubstForAux[First[u],First[v],x], + Map[Function[SubstForAux[#,v,x]],u]]]]]] + + +(* u (v) is an expression of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]). *) +(* SubstForTrig[u,sin,cos,v,x] returns the expression f (sin,cos,sin/cos,cos/sin,1/cos,1/sin). *) +SubstForTrig[u_,sin_,cos_,v_,x_] := + If[AtomQ[u], + u, + If[TrigQ[u] && IntegerQuotientQ[u[[1]],v], + If[u[[1]]===v || EqQ[u[[1]],v], + If[SinQ[u], + sin, + If[CosQ[u], + cos, + If[TanQ[u], + sin/cos, + If[CotQ[u], + cos/sin, + If[SecQ[u], + 1/cos, + 1/sin]]]]], + Map[Function[SubstForTrig[#,sin,cos,v,x]], + ReplaceAll[TrigExpand[Head[u][Simplify[u[[1]]/v]*x]],x->v]]], + If[ProductQ[u] && CosQ[u[[1]]] && SinQ[u[[2]]] && EqQ[u[[1,1]],v/2] && EqQ[u[[2,1]],v/2], + sin/2*SubstForTrig[Drop[u,2],sin,cos,v,x], + Map[Function[SubstForTrig[#,sin,cos,v,x]],u]]]] + + +(* u (v) is an expression of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v]). *) +(* SubstForHyperbolic[u,sinh,cosh,v,x] returns the expression + f (sinh,cosh,sinh/cosh,cosh/sinh,1/cosh,1/sinh). *) +SubstForHyperbolic[u_,sinh_,cosh_,v_,x_] := + If[AtomQ[u], + u, + If[HyperbolicQ[u] && IntegerQuotientQ[u[[1]],v], + If[u[[1]]===v || EqQ[u[[1]],v], + If[SinhQ[u], + sinh, + If[CoshQ[u], + cosh, + If[TanhQ[u], + sinh/cosh, + If[CothQ[u], + cosh/sinh, + If[SechQ[u], + 1/cosh, + 1/sinh]]]]], + Map[Function[SubstForHyperbolic[#,sinh,cosh,v,x]], + ReplaceAll[TrigExpand[Head[u][Simplify[u[[1]]/v]*x]],x->v]]], + If[ProductQ[u] && CoshQ[u[[1]]] && SinhQ[u[[2]]] && EqQ[u[[1,1]],v/2] && EqQ[u[[2,1]],v/2], + sinh/2*SubstForHyperbolic[Drop[u,2],sinh,cosh,v,x], + Map[Function[SubstForHyperbolic[#,sinh,cosh,v,x]],u]]]] + + +(* ::Subsection::Closed:: *) +(*SubstForFractionalPowerOfLinear*) + + +(* If u has a subexpression of the form (a+b*x)^(m/n) where m and n>1 are integers, + SubstForFractionalPowerOfLinear[u,x] returns the list {v,n,a+b*x,1/b} where v is u + with subexpressions of the form (a+b*x)^(m/n) replaced by x^m and x replaced + by -a/b+x^n/b, and all times x^(n-1); else it returns False. *) +SubstForFractionalPowerOfLinear[u_,x_Symbol] := + With[{lst=FractionalPowerOfLinear[u,1,False,x]}, + If[AtomQ[lst] || FalseQ[lst[[2]]], + False, + With[{n=lst[[1]], a=Coefficient[lst[[2]],x,0], b=Coefficient[lst[[2]],x,1]}, + With[{tmp=Simplify[x^(n-1)*SubstForFractionalPower[u,lst[[2]],n,-a/b+x^n/b,x]]}, + {NonfreeFactors[tmp,x],n,lst[[2]],FreeFactors[tmp,x]/b}]]]] + + +(* If u has a subexpression of the form (a+b*x)^(m/n), + FractionalPowerOfLinear[u,1,False,x] returns {n,a+b*x}; else it returns False. *) +FractionalPowerOfLinear[u_,n_,v_,x_] := + If[AtomQ[u] || FreeQ[u,x], + {n,v}, + If[CalculusQ[u], + False, + If[FractionalPowerQ[u] && LinearQ[u[[1]],x] && (FalseQ[v] || EqQ[u[[1]],v]), + {LCM[Denominator[u[[2]]],n],u[[1]]}, + Catch[Module[{lst={n,v}}, + Scan[Function[If[AtomQ[lst=FractionalPowerOfLinear[#,lst[[1]],lst[[2]],x]],Throw[False]]],u]; + lst]]]]] + + +(* ::Subsection::Closed:: *) +(*InverseFunctionOfLinear*) + + +(* If u has a subexpression of the form g[a+b*x] where g is an inverse function, + InverseFunctionOfLinear[u,x] returns g[a+b*x]; else it returns False. *) +InverseFunctionOfLinear[u_,x_Symbol] := + If[AtomQ[u] || CalculusQ[u] || FreeQ[u,x], + False, + If[InverseFunctionQ[u] && LinearQ[u[[1]],x], + u, + Module[{tmp}, + Catch[ + Scan[Function[If[Not[AtomQ[tmp=InverseFunctionOfLinear[#,x]]],Throw[tmp]]],u]; + False]]]] + + +(* ::Subsection::Closed:: *) +(*TryPureTanSubst*) + + +TryPureTanSubst[u_,x_Symbol] := +(* Not[MatchQ[u,Log[v_]]] && + Not[MatchQ[u,f_[v_]^2 /; LinearQ[v,x]]] && + Not[MatchQ[u,ArcTan[a_.*Tan[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcTan[a_.*Cot[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcCot[a_.*Tan[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcCot[a_.*Cot[v_]] /; FreeQ[a,x]]] && + u===ExpandIntegrand[u,x] *) + Not[MatchQ[u,F_[c_.*(a_.+b_.*G_[v_])] /; FreeQ[{a,b,c},x] && MemberQ[{ArcTan,ArcCot,ArcTanh,ArcCoth},F] && MemberQ[{Tan,Cot,Tanh,Coth},G] && LinearQ[v,x]]] + + +(* ::Subsection::Closed:: *) +(*TryPureTanhSubst*) + + +TryTanhSubst[u_,x_Symbol] := + FalseQ[FunctionOfLinear[u,x]] && + Not[MatchQ[u,r_.*(s_+t_)^n_. /; IntegerQ[n] && n>0]] && +(*Not[MatchQ[u,Log[f_[x]^2] /; SinhCoshQ[f]]] && *) + Not[MatchQ[u,Log[v_]]] && + Not[MatchQ[u,1/(a_+b_.*f_[x]^n_) /; SinhCoshQ[f] && IntegerQ[n] && n>2]] && + Not[MatchQ[u,f_[m_.*x]*g_[n_.*x] /; IntegersQ[m,n] && SinhCoshQ[f] && SinhCoshQ[g]]] && + Not[MatchQ[u,r_.*(a_.*s_^m_)^p_ /; FreeQ[{a,m,p},x] && Not[m===2 && (s===Sech[x] || s===Csch[x])]]] && + u===ExpandIntegrand[u,x] + + +TryPureTanhSubst[u_,x_Symbol] := + Not[MatchQ[u,Log[v_]]] && + Not[MatchQ[u,ArcTanh[a_.*Tanh[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcTanh[a_.*Coth[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcCoth[a_.*Tanh[v_]] /; FreeQ[a,x]]] && + Not[MatchQ[u,ArcCoth[a_.*Coth[v_]] /; FreeQ[a,x]]] && + u===ExpandIntegrand[u,x] + + +(* ::Section::Closed:: *) +(*Inert trig functions*) + + +(* ::Subsection::Closed:: *) +(*InertTrigQ*) + + +InertTrigQ[f_] := MemberQ[{sin,cos,tan,cot,sec,csc},f] + +InertTrigQ[f_,g_] := + If[f===g, + InertTrigQ[f], + InertReciprocalQ[f,g] || InertReciprocalQ[g,f]] + +InertTrigQ[f_,g_,h_] := InertTrigQ[f,g] && InertTrigQ[g,h] + + +InertReciprocalQ[f_,g_] := f===sin && g===csc || f===cos && g===sec || f===tan && g===cot + + +InertTrigFreeQ[u_] := FreeQ[u,sin] && FreeQ[u,cos] && FreeQ[u,tan] && FreeQ[u,cot] && FreeQ[u,sec] && FreeQ[u,csc] + + +(* ::Subsection::Closed:: *) +(*ActivateTrig[u]*) + + +ActivateTrig[u_] := + ReplaceAll[u,{sin->Sin,cos->Cos,tan->Tan,cot->Cot,sec->Sec,csc->Csc}] + + +(* ::Subsection::Closed:: *) +(*DeactivateTrig[u,x]*) + + +(* u is a function of trig functions of a linear function of x. *) +(* DeactivateTrig[u,x] returns u with the trig functions replaced with inert trig functions. *) + + +DeactivateTrig[(c_.+d_.*x_)^m_.*(a_.+b_.*trig_[e_.+f_.*x_])^n_.,x_] := + (c+d*x)^m*(a+b*DeactivateTrig[trig[e+f*x],x])^n /; +FreeQ[{a,b,c,d,e,f,m,n},x] && (TrigQ[trig] || HyperbolicQ[trig]) + + +DeactivateTrig[u_,x_] := + UnifyInertTrigFunction[FixInertTrigFunction[DeactivateTrigAux[u,x],x],x] + + +DeactivateTrigAux[u_,x_] := + If[AtomQ[u], + u, + If[TrigQ[u] && LinearQ[u[[1]],x], + With[{v=ExpandToSum[u[[1]],x]}, + Switch[Head[u], + Sin, ReduceInertTrig[sin,v], + Cos, ReduceInertTrig[cos,v], + Tan, ReduceInertTrig[tan,v], + Cot, ReduceInertTrig[cot,v], + Sec, ReduceInertTrig[sec,v], + Csc, ReduceInertTrig[csc,v]]], + If[HyperbolicQ[u] && LinearQ[u[[1]],x], + With[{v=ExpandToSum[I*u[[1]],x]}, + Switch[Head[u], + Sinh, -I*ReduceInertTrig[sin,v], + Cosh, ReduceInertTrig[cos,v], + Tanh, -I*ReduceInertTrig[tan,v], + Coth, I*ReduceInertTrig[cot,v], + Sech, ReduceInertTrig[sec,v], + Csch, I*ReduceInertTrig[csc,v]]], + Map[Function[DeactivateTrigAux[#,x]],u]]]] + + +(* ::Subsection::Closed:: *) +(*FixInertTrigFunction[u,x]*) + + +Clear[FixInertTrigFunction] + +FixInertTrigFunction[a_*u_,x_] := + a*FixInertTrigFunction[u,x] /; +FreeQ[a,x] + +FixInertTrigFunction[u_.*(a_*(b_+v_))^n_,x_] := + FixInertTrigFunction[u*(a*b+a*v)^n,x] /; +FreeQ[{a,b,n},x] && Not[FreeQ[v,x]] + + +FixInertTrigFunction[csc[v_]^m_.*(c_.*sin[w_])^n_.,x_] := + sin[v]^(-m)*(c*sin[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[sec[v_]^m_.*(c_.*cos[w_])^n_.,x_] := + cos[v]^(-m)*(c*cos[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[cot[v_]^m_.*(c_.*tan[w_])^n_.,x_] := + tan[v]^(-m)*(c*tan[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[tan[v_]^m_.*(c_.*cot[w_])^n_.,x_] := + cot[v]^(-m)*(c*cot[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[cos[v_]^m_.*(c_.*sec[w_])^n_.,x_] := + sec[v]^(-m)*(c*sec[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[sin[v_]^m_.*(c_.*csc[w_])^n_.,x_] := + csc[v]^(-m)*(c*csc[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + + +FixInertTrigFunction[sec[v_]^m_.*(c_.*sin[w_])^n_.,x_] := + cos[v]^(-m)*(c*sin[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[csc[v_]^m_.*(c_.*cos[w_])^n_.,x_] := + sin[v]^(-m)*(c*cos[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[cos[v_]^m_.*(c_.*tan[w_])^n_.,x_] := + sec[v]^(-m)*(c*tan[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[sin[v_]^m_.*(c_.*cot[w_])^n_.,x_] := + csc[v]^(-m)*(c*cot[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[sin[v_]^m_.*(c_.*sec[w_])^n_.,x_] := + csc[v]^(-m)*(c*sec[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[cos[v_]^m_.*(c_.*csc[w_])^n_.,x_] := + sec[v]^(-m)*(c*csc[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + + +FixInertTrigFunction[cot[v_]^m_.*(c_.*sin[w_])^n_.,x_] := + tan[v]^(-m)*(c*sin[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[tan[v_]^m_.*(c_.*cos[w_])^n_.,x_] := + cot[v]^(-m)*(c*cos[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[csc[v_]^m_.*(c_.*tan[w_])^n_.,x_] := + sin[v]^(-m)*(c*tan[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[sec[v_]^m_.*(c_.*cot[w_])^n_.,x_] := + cos[v]^(-m)*(c*cot[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[cot[v_]^m_.*(c_.*sec[w_])^n_.,x_] := + tan[v]^(-m)*(c*sec[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + +FixInertTrigFunction[tan[v_]^m_.*(c_.*csc[w_])^n_.,x_] := + cot[v]^(-m)*(c*csc[w])^n /; +FreeQ[{c,n},x] && IntegerQ[m] + + +FixInertTrigFunction[sec[v_]^m_.*sec[w_]^n_.,x_] := + cos[v]^(-m)*cos[w]^(-n) /; +IntegersQ[m,n] + +FixInertTrigFunction[csc[v_]^m_.*csc[w_]^n_.,x_] := + sin[v]^(-m)*sin[w]^(-n) /; +IntegersQ[m,n] + + +FixInertTrigFunction[u_*tan[v_]^m_.*(a_+b_.*sin[w_])^n_.,x_] := + sin[v]^m/cos[v]^m*FixInertTrigFunction[u*(a+b*sin[w])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_*cot[v_]^m_.*(a_+b_.*sin[w_])^n_.,x_] := + cos[v]^m/sin[v]^m*FixInertTrigFunction[u*(a+b*sin[w])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_*tan[v_]^m_.*(a_+b_.*cos[w_])^n_.,x_] := + sin[v]^m/cos[v]^m*FixInertTrigFunction[u*(a+b*cos[w])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_*cot[v_]^m_.*(a_+b_.*cos[w_])^n_.,x_] := + cos[v]^m/sin[v]^m*FixInertTrigFunction[u*(a+b*cos[w])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + + +FixInertTrigFunction[cot[v_]^m_.*(a_.+b_.*(c_.*sin[w_])^p_.)^n_.,x_] := + tan[v]^(-m)*(a+b*(c*sin[w])^p)^n /; +FreeQ[{a,b,c,n,p},x] && IntegerQ[m] + +FixInertTrigFunction[tan[v_]^m_.*(a_.+b_.*(c_.*cos[w_])^p_.)^n_.,x_] := + cot[v]^(-m)*(a+b*(c*cos[w])^p)^n /; +FreeQ[{a,b,c,n,p},x] && IntegerQ[m] + + +FixInertTrigFunction[u_.*(c_.*sin[v_]^n_.)^p_.*w_,x_] := + (c*sin[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,sin,x] + +FixInertTrigFunction[u_.*(c_.*cos[v_]^n_.)^p_.*w_,x_] := + (c*cos[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,cos,x] + +FixInertTrigFunction[u_.*(c_.*tan[v_]^n_.)^p_.*w_,x_] := + (c*tan[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,tan,x] + +FixInertTrigFunction[u_.*(c_.*cot[v_]^n_.)^p_.*w_,x_] := + (c*cot[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,cot,x] + +FixInertTrigFunction[u_.*(c_.*sec[v_]^n_.)^p_.*w_,x_] := + (c*sec[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,sec,x] + +FixInertTrigFunction[u_.*(c_.*csc[v_]^n_.)^p_.*w_,x_] := + (c*csc[v]^n)^p*FixInertTrigFunction[u*w,x] /; +FreeQ[{c,p},x] && PowerOfInertTrigSumQ[w,csc,x] + + +FixInertTrigFunction[u_.*sec[v_]^n_.*w_,x_] := + cos[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cos,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*csc[v_]^n_.*w_,x_] := + sin[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,sin,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*sec[v_]^n_.*w_,x_] := + cos[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,sin,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*csc[v_]^n_.*w_,x_] := + sin[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cos,x] && IntegerQ[n] + + +FixInertTrigFunction[u_.*cot[v_]^n_.*w_,x_] := + tan[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,tan,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*cos[v_]^n_.*w_,x_] := + sec[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,tan,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*cos[v_]^n_*w_,x_] := + sec[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,tan,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*csc[v_]^n_.*w_,x_] := + sin[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,tan,x] && IntegerQ[n] + + +FixInertTrigFunction[u_.*tan[v_]^n_.*w_,x_] := + cot[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cot,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*sin[v_]^n_.*w_,x_] := + csc[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cot,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*sin[v_]^n_.*w_,x_] := + csc[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cot,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*sec[v_]^n_.*w_,x_] := + cos[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,cot,x] && IntegerQ[n] + + +FixInertTrigFunction[u_.*cos[v_]^n_.*w_,x_] := + sec[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,sec,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*cot[v_]^n_.*w_,x_] := + tan[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,sec,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*csc[v_]^n_.*w_,x_] := + sin[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,sec,x] && IntegerQ[n] + + +FixInertTrigFunction[u_.*sin[v_]^n_.*w_,x_] := + csc[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,csc,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*tan[v_]^n_.*w_,x_] := + cot[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,csc,x] && IntegerQ[n] + +FixInertTrigFunction[u_.*sec[v_]^n_.*w_,x_] := + cos[v]^(-n)*FixInertTrigFunction[u*w,x] /; +PowerOfInertTrigSumQ[w,csc,x] && IntegerQ[n] + + +FixInertTrigFunction[u_.*tan[v_]^m_.*(a_.*sin[v_]+b_.*cos[v_])^n_.,x_] := + sin[v]^m*cos[v]^(-m)*FixInertTrigFunction[u*(a*sin[v]+b*cos[v])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_.*cot[v_]^m_.*(a_.*sin[v_]+b_.*cos[v_])^n_.,x_] := + cos[v]^m*sin[v]^(-m)*FixInertTrigFunction[u*(a*sin[v]+b*cos[v])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_.*sec[v_]^m_.*(a_.*sin[v_]+b_.*cos[v_])^n_.,x_] := + cos[v]^(-m)*FixInertTrigFunction[u*(a*sin[v]+b*cos[v])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + +FixInertTrigFunction[u_.*csc[v_]^m_.*(a_.*sin[v_]+b_.*cos[v_])^n_.,x_] := + sin[v]^(-m)*FixInertTrigFunction[u*(a*sin[v]+b*cos[v])^n,x] /; +FreeQ[{a,b,n},x] && IntegerQ[m] + + +FixInertTrigFunction[f_[v_]^m_.*(A_.+B_.*g_[v_]+C_.*g_[v_]^2),x_] := + g[v]^(-m)*(A+B*g[v]+C*g[v]^2) /; +FreeQ[{A,B,C},x] && IntegerQ[m] && (InertReciprocalQ[f,g] || InertReciprocalQ[g,f]) + +FixInertTrigFunction[f_[v_]^m_.*(A_.+C_.*g_[v_]^2),x_] := + g[v]^(-m)*(A+C*g[v]^2) /; +FreeQ[{A,C},x] && IntegerQ[m] && (InertReciprocalQ[f,g] || InertReciprocalQ[g,f]) + + +FixInertTrigFunction[f_[v_]^m_.*(A_.+B_.*g_[v_]+C_.*g_[v_]^2)*(a_.+b_.*g_[v_])^n_.,x_] := + g[v]^(-m)*(A+B*g[v]+C*g[v]^2)*(a+b*g[v])^n /; +FreeQ[{a,b,A,B,C,n},x] && IntegerQ[m] && (InertReciprocalQ[f,g] || InertReciprocalQ[g,f]) + +FixInertTrigFunction[f_[v_]^m_.*(A_.+C_.*g_[v_]^2)*(a_.+b_.*g_[v_])^n_.,x_] := + g[v]^(-m)*(A+C*g[v]^2)*(a+b*g[v])^n /; +FreeQ[{a,b,A,C,n},x] && IntegerQ[m] && (InertReciprocalQ[f,g] || InertReciprocalQ[g,f]) + + +FixInertTrigFunction[u_,x_] := u + + +PowerOfInertTrigSumQ[u_,func_,x_] := + MatchQ[u, (a_.+b_.*(c_.*func[w_])^n_.)^p_. /; FreeQ[{a,b,c,n,p},x] && Not[EqQ[a,0] && (IntegerQ[p] || EqQ[n,1])]] || + MatchQ[u, (a_.+b_.*(d_.*func[w_])^p_.+c_.*(d_.*func[w_])^q_.)^n_. /; FreeQ[{a,b,c,d,n,p,q},x]] + + +(* ::Subsection::Closed:: *) +(*ReduceInertTrig[func,a,b,x]*) + + +ReduceInertTrig[func_,m_.*(n_.*Pi+u_.)+v_.] := + ReduceInertTrig[func,m*n,m*u+v] /; +RationalQ[m,n] + +ReduceInertTrig[func_,m_.*Complex[0,mz_]*(n_.*Complex[0,nz_]*Pi+u_.)+v_.] := + ReduceInertTrig[func,-m*mz*n*nz,m*mz*I*u+v] /; +RationalQ[m,mz,n,nz] + +ReduceInertTrig[func_,u_] := + func[u] + + +(* func is an inert function and m is rational *) +(* ReduceInertTrig[func,m_,u_] returns func[m*Pi+u] with m reduced 0<=m<1/2. *) +ReduceInertTrig[func_,m_,u_] := + If[m<0, + If[m>=-1/4, + func[m*Pi+u], + Switch[func, + sin, -ReduceInertTrig[sin,-m,-u], + cos, ReduceInertTrig[cos,-m,-u], + tan, -ReduceInertTrig[tan,-m,-u], + cot, -ReduceInertTrig[cot,-m,-u], + sec, ReduceInertTrig[sec,-m,-u], + csc, -ReduceInertTrig[csc,-m,-u]]], + If[m>=2, + ReduceInertTrig[func,Mod[m,2],u], + If[m>=1, + Switch[func, + sin, -ReduceInertTrig[sin,m-1,u], + cos, -ReduceInertTrig[cos,m-1,u], + tan, ReduceInertTrig[tan,m-1,u], + cot, ReduceInertTrig[cot,m-1,u], + sec, -ReduceInertTrig[sec,m-1,u], + csc, -ReduceInertTrig[csc,m-1,u]], + If[m>=1/2, + Switch[func, + sin, ReduceInertTrig[cos,m-1/2,u], + cos, -ReduceInertTrig[sin,m-1/2,u], + tan, -ReduceInertTrig[cot,m-1/2,u], + cot, -ReduceInertTrig[tan,m-1/2,u], + sec, -ReduceInertTrig[csc,m-1/2,u], + csc, ReduceInertTrig[sec,m-1/2,u]], + func[m*Pi+u]]]]] /; +RationalQ[m] + + +(* ::Subsection::Closed:: *) +(*UnifyInertTrigFunction[u,x]*) + + +Clear[UnifyInertTrigFunction] + + +UnifyInertTrigFunction[a_*u_,x_] := + a*UnifyInertTrigFunction[u,x] /; +FreeQ[a,x] + + +(* ::Subsubsection:: *) +(*Cosine to sine*) + + +(* ::Subsubsection::Closed:: *) +(*1.0 (a sin)^m (b trg)^n*) + + +(* ::Text:: *) +(*(a Cos[e+f x])^m (b Csc[e+f x])^n == (a Sin[e+Pi/2+f x])^m (-b Sec[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*cos[e_.+f_.*x_])^m_.*(b_.*csc[e_.+f_.*x_])^n_.,x_] := + (a*sin[e+Pi/2+f*x])^m*(-b*sec[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Text:: *) +(*(a Cos[e+f x])^m (b Sec[e+f x])^n == (a Sin[e+Pi/2+f x])^m (b Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*cos[e_.+f_.*x_])^m_.*(b_.*sec[e_.+f_.*x_])^n_.,x_] := + (a*sin[e+Pi/2+f*x])^m*(b*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*1.1.1 (a+b sin)^n*) + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^n == (a+b Sin[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^n_.,x_] := + (a+b*sin[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,n},x] + + +(* ::Subsubsection::Closed:: *) +(*1.1.2 (g cos)^p (a+b sin)^m*) + + +(* ::Text:: *) +(*(g Sin[e+f x])^p (a+b Cos[e+f x])^m == (g Cos[e-Pi/2+f x])^p (a-b Sin[e-Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*sin[e_.+f_.*x_])^p_.*(a_+b_.*cos[e_.+f_.*x_])^m_.,x_] := + (g*cos[e-Pi/2+f*x])^p*(a-b*sin[e-Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Text:: *) +(*(g Csc[e+f x])^p (a+b Cos[e+f x])^m == (g Sec[e-Pi/2+f x])^p (a-b Sin[e-Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*cos[e_.+f_.*x_])^m_.,x_] := + (g*sec[e-Pi/2+f*x])^p*(a-b*sin[e-Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Subsubsection::Closed:: *) +(*1.1.3 (g tan)^p (a+b sin)^m*) + + +(* ::Text:: *) +(*(g Cot[e+f x])^p (a+b Cos[e+f x])^m == (-g Tan[e-Pi/2+f x])^p (a-b Sin[e-Pi/2+f x])^m*) + + +(* ::Text:: *) +(*(g Cot[e+f x])^p (a+b Cos[e+f x])^m == (-g Tan[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*cot[e_.+f_.*x_])^p_.*(a_+b_.*cos[e_.+f_.*x_])^m_.,x_] := + If[True, + (-g*tan[e-Pi/2+f*x])^p*(a-b*sin[e-Pi/2+f*x])^m, + (-g*tan[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m] /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Text:: *) +(*(g Tan[e+f x])^p (a+b Cos[e+f x])^m == (-g Cot[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*cos[e_.+f_.*x_])^m_.,x_] := + (-g*cot[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Subsubsection::Closed:: *) +(*1.2.1 (a+b sin)^m (c+d sin)^n*) + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_] := + (a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (c+d Sec[e+f x])^n == (a+b Sin[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (a+b*sin[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n*) + + +(* ::Text:: *) +(*(g Sin[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (g Cos[e-Pi/2+f x])^p (a-b Sin[e-Pi/2+f x])^m (c-d Sin[e-Pi/2+f x])^n*) + + +(* ::Text:: *) +(*(g Sin[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (-g Cos[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*sin[e_.+f_. x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_] := + If[IntegerQ[2*p] && p<0 && IntegerQ[2*n], + (g*cos[e-Pi/2+f*x])^p*(a-b*sin[e-Pi/2+f*x])^m*(c-d*sin[e-Pi/2+f*x])^n, + (-g*cos[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n] /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Csc[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (g Sec[e-Pi/2+f x])^p (a-b Sin[e-Pi/2+f x])^m (c-d Sin[e-Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*csc[e_.+f_.*x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_] := + (g*sec[e-Pi/2+f*x])^p*(a-b*sin[e-Pi/2+f*x])^m*(c-d*sin[e-Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Subsubsection::Closed:: *) +(*1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n*) + + +(* ::Text:: *) +(*(g Cos[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (g Sin[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*cos[e_.+f_.*x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_] := + (g*sin[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Cos[e+f x])^p (a+b Cos[e+f x])^m (c+d Sec[e+f x])^n == (g Sin[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*cos[e_.+f_.*x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (g*sin[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Sec[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n == (g Csc[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*sec[e_.+f_.*x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.,x_] := + (g*csc[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Sec[e+f x])^p (a+b Cos[e+f x])^m (c+d Sec[e+f x])^n == (g Csc[e+Pi/2+f x])^p (a+b Sin[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*sec[e_.+f_.*x_])^p_.*(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (g*csc[e+Pi/2+f*x])^p*(a+b*sin[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Subsubsection::Closed:: *) +(*1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin)*) + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (c+d Cos[e+f x])^n (A+B Cos[e+f x]) == (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n (A+B Sin[e+Pi/2+f x])*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.*(A_.+B_.*cos[e_.+f_.*x_]),x_] := + (a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n*(A+B*sin[e+Pi/2+f*x]) /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*1.4.1 (a+b sin)^m (A+B sin+C sin^2)*) + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2) == (a+b Sin[e+Pi/2+f x])^m (A+B Sin[e+Pi/2+f x]+C Sin[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(A_.+B_.*cos[e_.+f_.*x_]+C_.*cos[e_.+f_.*x_]^2),x_] := + (a+b*sin[e+Pi/2+f*x])^m*(A+B*sin[e+Pi/2+f*x]+C*sin[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,e,f,A,B,C,m},x] + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (A+C Cos[e+f x]^2) == (a+b Sin[e+Pi/2+f x])^m (A+C Sin[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(A_.+C_.*cos[e_.+f_.*x_]^2),x_] := + (a+b*sin[e+Pi/2+f*x])^m*(A+C*sin[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,e,f,A,C,m},x] + + +(* ::Subsubsection::Closed:: *) +(*1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2)*) + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (c+d Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) == (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n (A+B Sin[e+Pi/2+f x]+C Sin[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.*(A_.+B_.*cos[e_.+f_.*x_]+C_.*cos[e_.+f_.*x_]^2),x_] := + (a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n*(A+B*sin[e+Pi/2+f*x]+C*sin[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n},x] + + +(* ::Text:: *) +(*(a+b Cos[e+f x])^m (c+d Cos[e+f x])^n (A+C Cos[e+f x]^2) == (a+b Sin[e+Pi/2+f x])^m (c+d Sin[e+Pi/2+f x])^n (A+C Sin[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(c_.+d_.*cos[e_.+f_.*x_])^n_.*(A_.+C_.*cos[e_.+f_.*x_]^2),x_] := + (a+b*sin[e+Pi/2+f*x])^m*(c+d*sin[e+Pi/2+f*x])^n*(A+C*sin[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,d,e,f,A,C,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*1.7 (d trig)^m (a+b (c sin)^n)^p*) + + +(* ::Text:: *) +(*(a+b (c Cos[e+f x])^n)^p == (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_,x_] := + (a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,e,f,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cos[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (d Sin[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cos[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (d*sin[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sin[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (-d Cos[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sin[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cos[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cot[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (-d Tan[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cot[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*tan[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Tan[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (-d Cot[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*tan[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cot[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Csc[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (-d Sec[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*csc[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*sec[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sec[e+f x])^m (a+b (c Cos[e+f x])^n)^p == (d Csc[e+Pi/2+f x])^m (a+b (c Sin[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sec[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cos[e_.+f_.*x_])^n_)^p_.,x_] := + (d*csc[e+Pi/2+f*x])^m*(a+b*(c*sin[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(a+b Cos[e+f x]^n)^m (A+B Cos[e+f x]^n) == (a+b Sin[e+Pi/2+f x]^n)^m (A+B Sin[e+Pi/2+f x]^n)*) + + +UnifyInertTrigFunction[(a_.+b_.*cos[e_.+f_.*x_]^n_)^m_.*(A_.+B_.*cos[e_.+f_.*x_]^n_),x_] := + (a+b*sin[e+Pi/2+f*x]^n)^m*(A+B*sin[e+Pi/2+f*x]^n) /; +FreeQ[{a,b,e,f,A,B,m,n},x] && Not[EqQ[a,0] && IntegerQ[m]] + + +(* ::Subsubsection:: *) +(*Cotangent to tangent*) + + +(* ::Subsubsection::Closed:: *) +(*2.0 (a trg)^m (b tan)^n*) + + +(* ::Text:: *) +(*(a Cos[e+f x])^m (b Cot[e+f x])^n == (a Sin[e+Pi/2+f x])^m (-b Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*cos[e_.+f_.*x_])^m_.*(b_.*cot[e_.+f_.*x_])^n_.,x_] := + (a*sin[e+Pi/2+f*x])^m*(-b*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Text:: *) +(*(a Sin[e+f x])^m (b Cot[e+f x])^n == (a Cos[e-Pi/2+f x])^m (-b Tan[e-Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*sin[e_.+f_.*x_])^m_.*(b_.*cot[e_.+f_.*x_])^n_.,x_] := + (a*cos[e-Pi/2+f*x])^m*(-b*tan[e-Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Text:: *) +(*(a Csc[e+f x])^m (b Cot[e+f x])^n == (a Sec[e-Pi/2+f x])^m (-b Tan[e-Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*csc[e_.+f_.*x_])^m_.*(b_.*cot[e_.+f_.*x_])^n_.,x_] := + (a*sec[e-Pi/2+f*x])^m*(-b*tan[e-Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Text:: *) +(*(a Sec[e+f x])^m (b Cot[e+f x])^n == (a Csc[e+Pi/2+f x])^m (-b Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*cot[e_.+f_.*x_])^n_.,x_] := + (a*csc[e+Pi/2+f*x])^m*(-b*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.1.1 (a+b tan)^n*) + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^n == (a-b Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^n_.,x_] := + (a-b*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.1.2 (d sec)^m (a+b tan)^n*) + + +(* ::Text:: *) +(*(d Csc[e+f x])^m (a+b Cot[e+f x])^n == (d Sec[e-Pi/2+f x])^m (a-b Tan[e-Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(d_.*csc[e_.+f_.*x_])^m_.*(a_+b_.*cot[e_.+f_.*x_])^n_.,x_] := + (d*sec[e-Pi/2+f*x])^m*(a-b*tan[e-Pi/2+f*x])^n /; +FreeQ[{a,b,d,e,f,m,n},x] + + +(* ::Text:: *) +(*(d Sin[e+f x])^m (a+b Cot[e+f x])^n == (d Cos[e-Pi/2+f x])^m (a-b Tan[e-Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(d_.*sin[e_.+f_.*x_])^m_.*(a_+b_.*cot[e_.+f_.*x_])^n_.,x_] := + (d*cos[e-Pi/2+f*x])^m*(a-b*tan[e-Pi/2+f*x])^n /; +FreeQ[{a,b,d,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.1.3 (d sin)^m (a+b tan)^n*) + + +(* ::Text:: *) +(*(d Cos[e+f x])^m (a+b Cot[e+f x])^n == (d Sin[e+Pi/2+f x])^m (a-b Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(d_.*cos[e_.+f_.*x_])^m_.*(a_+b_.*cot[e_.+f_.*x_])^n_.,x_] := + (d*sin[e+Pi/2+f*x])^m*(a-b*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,d,e,f,m,n},x] + + +(* ::Text:: *) +(*(d Sec[e+f x])^m (a+b Cot[e+f x])^n == (d Csc[e+Pi/2+f x])^m (a-b Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*cot[e_.+f_.*x_])^n_.,x_] := + (d*csc[e+Pi/2+f*x])^m*(a-b*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,d,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.2.1 (a+b tan)^m (c+d tan)^n*) + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (c+d Cot[e+f x])^n == (a-b Tan[e+Pi/2+f x])^m (c-d Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*cot[e_.+f_.*x_])^n_.,x_] := + (a-b*tan[e+Pi/2+f*x])^m*(c-d*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n*) + + +(* ::Text:: *) +(*(g Cot[e+f x])^p (a+b Cot[e+f x])^m (c+d Cot[e+f x])^n == (-g Tan[e+Pi/2+f x])^p (a-b Tan[e+Pi/2+f x])^m (c-d Tan[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*cot[e_.+f_.*x_])^p_.*(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*cot[e_.+f_.*x_])^n_.,x_] := + (-g*tan[e+Pi/2+f*x])^p*(a-b*tan[e+Pi/2+f*x])^m*(c-d*tan[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Cot[e+f x])^p (a+b Cot[e+f x])^m (c+d Tan[e+f x])^n == (-g Tan[e+Pi/2+f x])^p (a-b Tan[e+Pi/2+f x])^m (c-d Cot[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*cot[e_.+f_.*x_])^p_.*(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*tan[e_.+f_.*x_])^n_.,x_] := + (-g*tan[e+Pi/2+f*x])^p*(a-b*tan[e+Pi/2+f*x])^m*(c-d*cot[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Subsubsection::Closed:: *) +(*2.3.1 (a+b tan)^m (c+d tan)^n (A+B tan)*) + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (c+d Cot[e+f x])^n (A+B Cot[e+f x]) == (a-b Tan[e+Pi/2+f x])^m (c-d Tan[e+Pi/2+f x])^n (A-B Tan[e+Pi/2+f x])*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*cot[e_.+f_.*x_])^n_.*(A_.+B_.*cot[e_.+f_.*x_]),x_] := + (a-b*tan[e+Pi/2+f*x])^m*(c-d*tan[e+Pi/2+f*x])^n*(A-B*tan[e+Pi/2+f*x]) /; +FreeQ[{a,b,c,d,e,f,A,B,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.4.1 (a+b tan)^m (A+B tan+C tan^2)*) + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (A+B Cot[e+f x]+C Cot[e+f x]^2) == (a-b Tan[e+Pi/2+f x])^m (A-B Tan[e+Pi/2+f x]+C Tan[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(A_.+B_.*cot[e_.+f_.*x_]+C_.*cot[e_.+f_.*x_]^2),x_] := + (a-b*tan[e+Pi/2+f*x])^m*(A-B*tan[e+Pi/2+f*x]+C*tan[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,e,f,A,B,C,m},x] + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (A+C Cot[e+f x]^2) == (a-b Tan[e+Pi/2+f x])^m (A+C Tan[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(A_.+C_.*cot[e_.+f_.*x_]^2),x_] := + (a-b*tan[e+Pi/2+f*x])^m*(A+C*tan[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,e,f,A,C,m},x] + + +(* ::Subsubsection::Closed:: *) +(*2.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2)*) + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (c+d Cot[e+f x])^n (A+B Cot[e+f x]+C Cot[e+f x]^2) == *) +(* (a-b Tan[e+Pi/2+f x])^m (c-d Tan[e+Pi/2+f x])^n (A-B Tan[e+Pi/2+f x]+C Tan[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*cot[e_.+f_.*x_])^n_.*(A_.+B_.*cot[e_.+f_.*x_]+C_.*cot[e_.+f_.*x_]^2),x_] := + (a-b*tan[e+Pi/2+f*x])^m*(c-d*tan[e+Pi/2+f*x])^n*(A-B*tan[e+Pi/2+f*x]+C*tan[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,d,e,f,A,B,C,m,n},x] + + +(* ::Text:: *) +(*(a+b Cot[e+f x])^m (c+d Cot[e+f x])^n (A+C Cot[e+f x]^2) == (a-b Tan[e+Pi/2+f x])^m (c-d Tan[e+Pi/2+f x])^n (A+C Tan[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*cot[e_.+f_.*x_])^m_.*(c_.+d_.*cot[e_.+f_.*x_])^n_.*(A_.+C_.*cot[e_.+f_.*x_]^2),x_] := + (a-b*tan[e+Pi/2+f*x])^m*(c-d*tan[e+Pi/2+f*x])^n*(A+C*tan[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,c,d,e,f,A,C,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*2.7 trig^m (a+b (c tan)^n)^p*) + + +(* ::Text:: *) +(*(a+b (c Cot[e+f x])^n)^p == (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_,x_] := + (a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,e,f,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cos[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (d Sin[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cos[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (d*sin[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sin[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (-d Cos[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sin[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cos[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cot[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (-d Tan[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cot[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*tan[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Tan[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (-d Cot[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*tan[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cot[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Csc[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (-d Sec[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*csc[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*sec[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sec[e+f x])^m (a+b (c Cot[e+f x])^n)^p == (d Csc[e+Pi/2+f x])^m (a+b (-c Tan[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sec[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*cot[e_.+f_.*x_])^n_)^p_.,x_] := + (d*csc[e+Pi/2+f*x])^m*(a+b*(-c*tan[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Subsubsection:: *) +(*Cosecant to secant*) + + +(* ::Subsubsection::Closed:: *) +(*3.1.1 (a+b sec)^n*) + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^n == (a+b Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^n_.,x_] := + (a+b*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,e,f,n},x] + + +(* ::Subsubsection::Closed:: *) +(*3.1.2 (d sec)^n (a+b sec)^m*) + + +(* ::Text:: *) +(*(g Sec[e+f x])^p (a+b Sec[e+f x])^m == (g Csc[e+Pi/2+f x])^p (a+b Csc[e+Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*sec[e_.+f_.*x_])^p_.*(a_+b_.*sec[e_.+f_.*x_])^m_.,x_] := + (g*csc[e+Pi/2+f*x])^p*(a+b*csc[e+Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Subsubsection::Closed:: *) +(*3.1.3 (d sin)^n (a+b sec)^m*) + + +(* ::Text:: *) +(*(g Sin[e+f x])^p (a+b Sec[e+f x])^m == (g Cos[e-Pi/2+f x])^p (a-b Csc[e-Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*sin[e_.+f_.*x_])^p_.*(a_+b_.*sec[e_.+f_.*x_])^m_.,x_] := + (g*cos[e-Pi/2+f*x])^p*(a-b*csc[e-Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Text:: *) +(*(g Csc[e+f x])^p (a+b Sec[e+f x])^m == (g Sec[e-Pi/2+f x])^p (a-b Csc[e-Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*csc[e_.+f_.*x_])^p_.*(a_+b_.*sec[e_.+f_.*x_])^m_.,x_] := + (g*sec[e-Pi/2+f*x])^p*(a-b*csc[e-Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Subsubsection::Closed:: *) +(*3.1.4 (d tan)^n (a+b sec)^m*) + + +(* ::Text:: *) +(*(g Tan[e+f x])^p (a+b Sec[e+f x])^m == (-g Cot[e+Pi/2+f x])^p (a+b Csc[e+Pi/2+f x])^m*) + + +UnifyInertTrigFunction[(g_.*tan[e_.+f_.*x_])^p_.*(a_+b_.*sec[e_.+f_.*x_])^m_.,x_] := + (-g*cot[e+Pi/2+f*x])^p*(a+b*csc[e+Pi/2+f*x])^m /; +FreeQ[{a,b,e,f,g,m,p},x] + + +(* ::Subsubsection::Closed:: *) +(*3.2.1 (a+b sec)^m (c+d sec)^n*) + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (c+d Sec[e+f x])^n == (a+b Csc[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (a+b*csc[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,m,n},x] + + +(* ::Subsubsection::Closed:: *) +(*3.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n*) + + +(* ::Text:: *) +(*(g Sec[e+f x])^p (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n == (g Csc[e+Pi/2+f x])^p (a+b Csc[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*sec[e_.+f_.*x_])^p_.*(a_.+b_.*sec[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (g*csc[e+Pi/2+f*x])^p*(a+b*csc[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Text:: *) +(*(g Cos[e+f x])^p (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n == (g Sin[e+Pi/2+f x])^p (a+b Csc[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n*) + + +UnifyInertTrigFunction[(g_.*cos[e_.+f_.*x_])^p_.*(a_.+b_.*sec[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.,x_] := + (g*sin[e+Pi/2+f*x])^p*(a+b*csc[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n /; +FreeQ[{a,b,c,d,e,f,g,m,n,p},x] + + +(* ::Subsubsection::Closed:: *) +(*3.3.1 (a+b sec)^m (d sec)^n (A+B sec)*) + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x]) == (a+b Csc[e+Pi/2+f x])^m (d Csc[e+Pi/2+f x])^n (A+B Csc[e+Pi/2+f x])*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(d_.*sec[e_.+f_.*x_])^n_.*(A_.+B_.*sec[e_.+f_.*x_]),x_] := + (a+b*csc[e+Pi/2+f*x])^m*(d*csc[e+Pi/2+f*x])^n*(A+B*csc[e+Pi/2+f*x]) /; +FreeQ[{a,b,d,e,f,A,B,m,n},x] + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (c+d Sec[e+f x])^n (A+B Sec[e+f x])^p == (a+b Csc[e+Pi/2+f x])^m (c+d Csc[e+Pi/2+f x])^n (A+B Csc[e+Pi/2+f x])^p*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(c_.+d_.*sec[e_.+f_.*x_])^n_.*(A_.+B_.*sec[e_.+f_.*x_])^p_.,x_] := + (a+b*csc[e+Pi/2+f*x])^m*(c+d*csc[e+Pi/2+f*x])^n*(A+B*csc[e+Pi/2+f*x])^p /; +FreeQ[{a,b,c,d,e,f,A,B,m,n,p},x] + + +(* ::Subsubsection::Closed:: *) +(*3.4.1 (a+b sec)^m (A+B sec+C sec^2)*) + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2) == (a+b Csc[e+Pi/2+f x])^m (A+B Csc[e+Pi/2+f x]+C Csc[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(A_.+B_.*sec[e_.+f_.*x_]+C_.*sec[e_.+f_.*x_]^2),x_] := + (a+b*csc[e+Pi/2+f*x])^m*(A+B*csc[e+Pi/2+f*x]+C*csc[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,e,f,A,B,C,m},x] + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2) == (a+b Csc[e+Pi/2+f x])^m (A+C Csc[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(A_.+C_.*sec[e_.+f_.*x_]^2),x_] := + (a+b*csc[e+Pi/2+f*x])^m*(A+C*csc[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,e,f,A,C,m},x] + + +(* ::Subsubsection::Closed:: *) +(*3.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2)*) + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2) == *) +(* (a+b Csc[e+Pi/2+f x])^m (d Csc[e+Pi/2+f x])^n (A+B Csc[e+Pi/2+f x]+C Csc[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(d_.*sec[e_.+f_.*x_])^n_.*(A_.+B_.*sec[e_.+f_.*x_]+C_.*sec[e_.+f_.*x_]^2),x_] := + (a+b*csc[e+Pi/2+f*x])^m*(d*csc[e+Pi/2+f*x])^n*(A+B*csc[e+Pi/2+f*x]+C*csc[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,d,e,f,A,B,C,m,n},x] + + +(* ::Text:: *) +(*(a+b Sec[e+f x])^m (d Sec[e+f x])^n (A+C Sec[e+f x]^2) == (a+b Csc[e+Pi/2+f x])^m (d Csc[e+Pi/2+f x])^n (A+C Csc[e+Pi/2+f x]^2)*) + + +UnifyInertTrigFunction[(a_.+b_.*sec[e_.+f_.*x_])^m_.*(d_.*sec[e_.+f_.*x_])^n_.*(A_.+C_.*sec[e_.+f_.*x_]^2),x_] := + (a+b*csc[e+Pi/2+f*x])^m*(d*csc[e+Pi/2+f*x])^n*(A+C*csc[e+Pi/2+f*x]^2) /; +FreeQ[{a,b,d,e,f,A,C,m,n},x] + + +UnifyInertTrigFunction[u_,x_] := u + + +(* ::Subsubsection::Closed:: *) +(*3.7 (d trig)^m (a+b (c sec)^n)^p*) + + +(* ::Text:: *) +(*(a+b (c Csc[e+f x])^n)^p == (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_,x_] := + (a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,e,f,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cos[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (d Sin[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cos[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (d*sin[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sin[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (-d Cos[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sin[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cos[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Cot[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (-d Tan[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*cot[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*tan[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Tan[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (-d Cot[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*tan[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*cot[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Csc[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (-d Sec[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*csc[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (-d*sec[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[n,2] && EqQ[p,1]] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Text:: *) +(*(d Sec[e+f x])^m (a+b (c Csc[e+f x])^n)^p == (d Csc[e+Pi/2+f x])^m (a+b (-c Sec[e+Pi/2+f x])^n)^p*) + + +UnifyInertTrigFunction[(d_.*sec[e_.+f_.*x_])^m_.*(a_.+b_.*(c_.*csc[e_.+f_.*x_])^n_)^p_.,x_] := + (d*csc[e+Pi/2+f*x])^m*(a+b*(-c*sec[e+Pi/2+f*x])^n)^p /; +FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[EqQ[a,0] && IntegerQ[p]] + + +(* ::Subsection::Closed:: *) +(*KnownTrigIntegrandQ[u,x]*) + + +KnownSineIntegrandQ[u_,x_Symbol] := + KnownTrigIntegrandQ[{sin,cos},u,x] + + +KnownTangentIntegrandQ[u_,x_Symbol] := + KnownTrigIntegrandQ[{tan},u,x] + + +KnownCotangentIntegrandQ[u_,x_Symbol] := + KnownTrigIntegrandQ[{cot},u,x] + + +KnownSecantIntegrandQ[u_,x_Symbol] := + KnownTrigIntegrandQ[{sec,csc},u,x] + + +KnownTrigIntegrandQ[list_,u_,x_Symbol] := + u===1 || + MatchQ[u,(a_.+b_.*func_[e_.+f_.*x])^m_. /; MemberQ[list,func] && FreeQ[{a,b,e,f,m},x]] || + MatchQ[u,(a_.+b_.*func_[e_.+f_.*x])^m_.*(A_.+B_.*func_[e_.+f_.*x]) /; MemberQ[list,func] && FreeQ[{a,b,e,f,A,B,m},x]] || + MatchQ[u,(A_.+C_.*func_[e_.+f_.*x]^2) /; MemberQ[list,func] && FreeQ[{e,f,A,C},x]] || + MatchQ[u,(A_.+B_.*func_[e_.+f_.*x]+C_.*func_[e_.+f_.*x]^2) /; MemberQ[list,func] && FreeQ[{e,f,A,B,C},x]] || + MatchQ[u,(a_.+b_.*func_[e_.+f_.*x])^m_.*(A_.+C_.*func_[e_.+f_.*x]^2) /; MemberQ[list,func] && FreeQ[{a,b,e,f,A,C,m},x]] || + MatchQ[u,(a_.+b_.*func_[e_.+f_.*x])^m_.*(A_.+B_.*func_[e_.+f_.*x]+C_.*func_[e_.+f_.*x]^2) /; MemberQ[list,func] && FreeQ[{a,b,e,f,A,B,C,m},x]] + + +(* ::Section::Closed:: *) +(*Derivative divides function*) + + +(* If the derivative of u wrt x is a constant wrt x, PiecewiseLinearQ[u,x] returns True; + else it returns False. *) +PiecewiseLinearQ[u_,v_,x_Symbol] := + PiecewiseLinearQ[u,x] && PiecewiseLinearQ[v,x] + +PiecewiseLinearQ[u_,x_Symbol] := + LinearQ[u,x] (* && Not[MonomialQ[u,x]] *) || + MatchQ[u,Log[c_.*F_^(v_)] /; FreeQ[{F,c},x] && LinearQ[v,x]] || + MatchQ[u,F_[G_[v_]] /; LinearQ[v,x] && MemberQ[{ + {ArcTanh,Tanh},{ArcTanh,Coth},{ArcCoth,Coth},{ArcCoth,Tanh}, + {ArcTan,Tan},{ArcTan,Cot},{ArcCot,Cot},{ArcCot,Tan} + },{F,G}]] + + +(* If u divided by y is free of x, Divides[y,u,x] returns the quotient; else it returns False. *) +Divides[y_,u_,x_Symbol] := + With[{v=Simplify[u/y]}, + If[FreeQ[v,x], + v, + False]] + + +(* If y not equal to x, y is easy to differentiate wrt x, and u divided by the derivative of y + is free of x, DerivativeDivides[y,u,x] returns the quotient; else it returns False. *) +DerivativeDivides[y_,u_,x_Symbol] := + If[MatchQ[y,a_.*x /; FreeQ[a,x]], + False, + If[If[PolynomialQ[y,x], PolynomialQ[u,x] && Exponent[u,x]==Exponent[y,x]-1, EasyDQ[y,x]], + Module[{v=Block[{ShowSteps=False}, D[y,x]]}, + If[EqQ[v,0], + False, + v=Simplify[u/v]; + If[FreeQ[v,x], + v, + False]]], + False]] + + +(* If u is easy to differentiate wrt x, EasyDQ[u,x] returns True; else it returns False. *) +EasyDQ[u_.*x_^m_.,x_Symbol] := + EasyDQ[u,x] /; +FreeQ[m,x] + +EasyDQ[u_,x_Symbol] := + If[AtomQ[u] || FreeQ[u,x] || Length[u]==0, + True, + If[CalculusQ[u], + False, + If[Length[u]==1, + EasyDQ[u[[1]],x], + If[BinomialQ[u,x] || ProductOfLinearPowersQ[u,x], + True, + If[RationalFunctionQ[u,x] && RationalFunctionExponents[u,x]==={1,1}, + True, + If[ProductQ[u], + If[FreeQ[First[u],x], + EasyDQ[Rest[u],x], + If[FreeQ[Rest[u],x], + EasyDQ[First[u],x], + False]], + If[SumQ[u], + EasyDQ[First[u],x] && EasyDQ[Rest[u],x], + If[Length[u]==2, + If[FreeQ[u[[1]],x], + EasyDQ[u[[2]],x], + If[FreeQ[u[[2]],x], + EasyDQ[u[[1]],x], + False]], + False]]]]]]]] + + +(* ProductOfLinearPowersQ[u,x] returns True iff u is a product of factors of the form v^n where v is linear in x. *) +ProductOfLinearPowersQ[u_,x_Symbol] := + FreeQ[u,x] || + MatchQ[u, v_^n_. /; LinearQ[v,x] && FreeQ[n,x]] || + ProductQ[u] && ProductOfLinearPowersQ[First[u],x] && ProductOfLinearPowersQ[Rest[u],x] + + +(* ::Section::Closed:: *) +(*Simplest nth root function*) + + +(* ::Subsection::Closed:: *) +(*Rt[u,n]*) + + +(* Rt[u,n] returns the simplest nth root of u. *) +Rt[u_,n_Integer] := + RtAux[TogetherSimplify[u],n] + + +(* ::Subsection::Closed:: *) +(*NthRoot[u,n]*) + + +NthRoot[u_,n_] := u^(1/n) + + +(* ::Subsection::Closed:: *) +(*TrigSquare[u]*) + + +(* If u is an expression of the form a-a*Sin[z]^2 or a-a*Cos[z]^2, TrigSquare[u] returns Cos[z]^2 or Sin[z]^2 respectively, else it returns False. *) +TrigSquare[u_] := + If[SumQ[u], + With[{lst=SplitSum[Function[SplitProduct[TrigSquareQ,#]],u]}, + If[Not[AtomQ[lst]] && EqQ[lst[[1,2]]+lst[[2]],0], + If[Head[lst[[1,1]][[1]]]===Sin, + lst[[2]]*Cos[lst[[1,1]][[1,1]]]^2, + lst[[2]]*Sin[lst[[1,1]][[1,1]]]^2], + False]], + False] + + +(* ::Section::Closed:: *) +(*Simplest nth root helper functions*) + + +(* ::Subsection::Closed:: *) +(*RtAux[u,n]*) + + +RtAux[u_,n_] := + If[PowerQ[u], + u[[1]]^(u[[2]]/n), + If[ProductQ[u], + Module[{lst}, + lst=SplitProduct[PositiveQ,u]; + If[ListQ[lst], + RtAux[lst[[1]],n]*RtAux[lst[[2]],n], + lst=SplitProduct[NegativeQ,u]; + If[ListQ[lst], + If[EqQ[lst[[1]],-1], + With[{v=lst[[2]]}, + If[PowerQ[v] && NegativeQ[v[[2]]], + 1/RtAux[-v[[1]]^(-v[[2]]),n], + If[ProductQ[v], + If[ListQ[SplitProduct[SumBaseQ,v]], + lst=SplitProduct[AllNegTermQ,v]; + If[ListQ[lst], + RtAux[-lst[[1]],n]*RtAux[lst[[2]],n], + lst=SplitProduct[NegSumBaseQ,v]; + If[ListQ[lst], + RtAux[-lst[[1]],n]*RtAux[lst[[2]],n], + lst=SplitProduct[SomeNegTermQ,v]; + If[ListQ[lst], + RtAux[-lst[[1]],n]*RtAux[lst[[2]],n], + lst=SplitProduct[SumBaseQ,v]; + RtAux[-lst[[1]],n]*RtAux[lst[[2]],n]]]], + lst=SplitProduct[AtomBaseQ,v]; + If[ListQ[lst], + RtAux[-lst[[1]],n]*RtAux[lst[[2]],n], + RtAux[-First[v],n]*RtAux[Rest[v],n]]], + If[OddQ[n], + -RtAux[v,n], + NthRoot[u,n]]]]], + RtAux[-lst[[1]],n]*RtAux[-lst[[2]],n]], + lst=SplitProduct[AllNegTermQ,u]; + If[ListQ[lst] && ListQ[SplitProduct[SumBaseQ,lst[[2]]]], + RtAux[-lst[[1]],n]*RtAux[-lst[[2]],n], + lst=SplitProduct[NegSumBaseQ,u]; + If[ListQ[lst] && ListQ[SplitProduct[NegSumBaseQ,lst[[2]]]], + RtAux[-lst[[1]],n]*RtAux[-lst[[2]],n], + Map[Function[RtAux[#,n]],u]]]]]], + With[{v=TrigSquare[u]}, + If[Not[AtomQ[v]], + RtAux[v,n], + If[OddQ[n] && NegativeQ[u], + -RtAux[-u,n], + If[ComplexNumberQ[u], + With[{a=Re[u],b=Im[u]}, + If[Not[IntegerQ[a] && IntegerQ[b]] && IntegerQ[a/(a^2+b^2)] && IntegerQ[b/(a^2+b^2)], +(* Basis: a+b*I==1/(a/(a^2+b^2)-b/(a^2+b^2)*I) *) + 1/RtAux[a/(a^2+b^2)-b/(a^2+b^2)*I,n], + NthRoot[u,n]]], + If[OddQ[n] && NegQ[u] && PosQ[-u], + -RtAux[-u,n], + NthRoot[u,n]]]]]]]] + + +(* ::Subsection::Closed:: *) +(*Factor base predicates*) + + +(* If u is an atom or an atom raised to an odd degree, AtomBaseQ[u] returns True; else it returns False. *) +AtomBaseQ[u_] := + AtomQ[u] || PowerQ[u] && OddQ[u[[2]]] && AtomBaseQ[u[[1]]] + + +(* If u is an sum or a sum raised to an odd degree, SumBaseQ[u] returns True; else it returns False. *) +SumBaseQ[u_] := + SumQ[u] || PowerQ[u] && OddQ[u[[2]]] && SumBaseQ[u[[1]]] + + +(* If u is a sum or a sum raised to an odd degree whose lead term has a negative form, NegSumBaseQ[u] returns True; else it returns False. *) +NegSumBaseQ[u_] := + SumQ[u] && NegQ[First[u]] || PowerQ[u] && OddQ[u[[2]]] && NegSumBaseQ[u[[1]]] + + +(* If all terms of u have a negative form, AllNegTermQ[u] returns True; else it returns False. *) +AllNegTermQ[u_] := + If[PowerQ[u] && OddQ[u[[2]]], + AllNegTermQ[u[[1]]], + If[SumQ[u], + NegQ[First[u]] && AllNegTermQ[Rest[u]], + NegQ[u]]] + + +(* If some term of u has a negative form, SomeNegTermQ[u] returns True; else it returns False. *) +SomeNegTermQ[u_] := + If[PowerQ[u] && OddQ[u[[2]]], + SomeNegTermQ[u[[1]]], + If[SumQ[u], + NegQ[First[u]] || SomeNegTermQ[Rest[u]], + NegQ[u]]] + + +(* ::Subsection::Closed:: *) +(*TrigSquareQ[u]*) + + +(* If u is an expression of the form Sin[z]^2 or Cos[z]^2, TrigSquareQ[u] returns True, else it returns False. *) +TrigSquareQ[u_] := + PowerQ[u] && EqQ[u[[2]],2] && MemberQ[{Sin,Cos},Head[u[[1]]]] + + +(* ::Subsection::Closed:: *) +(*SplitProduct[func,u]*) + + +(* If func[v] is True for a factor v of u, SplitProduct[func,u] returns {v, u/v} where v is the first such factor; else it returns False. *) +SplitProduct[func_,u_] := + If[ProductQ[u], + If[func[First[u]], + {First[u], Rest[u]}, + With[{lst=SplitProduct[func,Rest[u]]}, + If[AtomQ[lst], + False, + {lst[[1]],First[u]*lst[[2]]}]]], + If[func[u], + {u, 1}, + False]] + + +(* ::Subsection::Closed:: *) +(*SplitSum[func,u]*) + + +(* If func[v] is nonatomic for a term v of u, SplitSum[func,u] returns {func[v], u-v} where v is the first such term; else it returns False. *) +SplitSum[func_,u_] := + If[SumQ[u], + If[Not[AtomQ[func[First[u]]]], + {func[First[u]], Rest[u]}, + With[{lst=SplitSum[func,Rest[u]]}, + If[AtomQ[lst], + False, + {lst[[1]],First[u]+lst[[2]]}]]], + If[Not[AtomQ[func[u]]], + {func[u], 0}, + False]] + + +(* ::Section::Closed:: *) +(*IntSum*) + + +(* If u is free of x or of the form c*(a+b*x)^m, IntSum[u,x] returns the antiderivative of u wrt x; + else it returns d*Int[v,x] where d*v=u and d is free of x. *) +IntSum[u_,x_Symbol] := + Simp[FreeTerms[u,x]*x,x] + IntTerm[NonfreeTerms[u,x],x] + + +(* If u is of the form c*(a+b*x)^m, IntTerm[u,x] returns the antiderivative of u wrt x; + else it returns d*Int[v,x] where d*v=u and d is free of x. *) +IntTerm[c_./v_,x_Symbol] := + Simp[c*Log[RemoveContent[v,x]]/Coefficient[v,x,1],x] /; +FreeQ[c,x] && LinearQ[v,x] + +IntTerm[c_.*v_^m_.,x_Symbol] := + Simp[c*v^(m+1)/(Coefficient[v,x,1]*(m+1)),x] /; +FreeQ[{c,m},x] && NeQ[m,-1] && LinearQ[v,x] + +IntTerm[u_,x_Symbol] := + Map[Function[IntTerm[#,x]],u] /; +SumQ[u] + +IntTerm[u_,x_Symbol] := + Dist[FreeFactors[u,x], Int[NonfreeFactors[u,x],x], x] + + +(* ::Section::Closed:: *) +(*Fix integration rules functions*) + + +(* ::Subsection::Closed:: *) +(*RuleName[name]*) + + +$RuleNameList={}; + + +RuleName[name_] := + (AppendTo[$RuleNameList,name]; Null) + + +(* ::Subsection::Closed:: *) +(*FixIntRules[rulelist]*) + + +ClearAll[FixIntRules,FixIntRule,FixRhsIntRule] + + +FixIntRules[] := + (DownValues[Int]=FixIntRules[DownValues[Int]]; Null) + + +FixIntRules[rulelist_] := + Module[{IntDownValues=DownValues[Int],SubstDownValues=DownValues[Subst], + SimpDownValues=DownValues[Simp],DistDownValues=DownValues[Dist],lst,object}, + object=PrintTemporary[Row[{"Modifying ",Length[rulelist]," integration rules to distribute coefficients over sums..."}]]; + Clear[Int,Subst,Simp,Dist]; + SetAttributes[{Simp,Dist,Int,Subst},HoldAll]; + lst=Map[Function[FixIntRule[#,#[[1,1,2,1]]]],rulelist]; + DownValues[Int]=IntDownValues; + DownValues[Subst]=SubstDownValues; + DownValues[Simp]=SimpDownValues; + DownValues[Dist]=DistDownValues; + ClearAttributes[{Simp,Dist,Int,Subst},HoldAll]; + NotebookDelete[object]; + lst] + + +(* ::Subsection::Closed:: *) +(*FixIntRule[rule]*) + + +FixIntRule[RuleDelayed[lhs_,F_[G_[list_,F_[u_+v_,test2_]],test1_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list,F[u+v,test2]],test1]],{{2,1,2,1,1}->FixRhsIntRule[u,x],{2,1,2,1,2}->FixRhsIntRule[v,x]}] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,G_[list_,F_[u_+v_,test2_]]],x_] := + ReplacePart[RuleDelayed[lhs,G[list,F[u+v,test2]]],{{2,2,1,1}->FixRhsIntRule[u,x],{2,2,1,2}->FixRhsIntRule[v,x]}] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,F_[G_[list_,u_+v_],test_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list,u+v],test]],{{2,1,2,1}->FixRhsIntRule[u,x],{2,1,2,2}->FixRhsIntRule[v,x]}] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,G_[list_,u_+v_]],x_] := + ReplacePart[RuleDelayed[lhs,G[list,u+v]],{{2,2,1}->FixRhsIntRule[u,x],{2,2,2}->FixRhsIntRule[v,x]}] /; +(G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,F_[u_+v_,test_]],x_] := + ReplacePart[RuleDelayed[lhs,F[u+v,test]],{{2,1,1}->FixRhsIntRule[u,x],{2,1,2}->FixRhsIntRule[v,x]}] /; +F===Condition + +FixIntRule[RuleDelayed[lhs_,u_+v_],x_] := + ReplacePart[RuleDelayed[lhs,u+v],{{2,1}->FixRhsIntRule[u,x],{2,2}->FixRhsIntRule[v,x]}] + + +FixIntRule[RuleDelayed[lhs_,F_[G_[list1_,F_[H_[list2_,u_],test2_]],test1_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list1,F[H[list2,u],test2]],test1]],{2,1,2,1,2}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) && (H===With || H===Module || H===Block) + +FixIntRule[RuleDelayed[lhs_,F_[G_[list1_,H_[list2_,u_]],test1_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list1,H[list2,u]],test1]],{2,1,2,2}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) && (H===With || H===Module || H===Block) + +FixIntRule[RuleDelayed[lhs_,F_[G_[list_,F_[H_[str1_,str2_,str3_,J_[u_]],test2_]],test1_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list,F[H[str1,str2,str3,J[u]],test2]],test1]],{2,1,2,1,4,1}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) && H===ShowStep && J===Hold + +FixIntRule[RuleDelayed[lhs_,F_[G_[list_,F_[u_,test2_]],test1_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list,F[u,test2]],test1]],{2,1,2,1}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,G_[list_,F_[u_,test2_]]],x_] := + ReplacePart[RuleDelayed[lhs,G[list,F[u,test2]]],{2,2,1}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,F_[G_[list_,u_],test_]],x_] := + ReplacePart[RuleDelayed[lhs,F[G[list,u],test]],{2,1,2}->FixRhsIntRule[u,x]] /; +F===Condition && (G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,G_[list_,u_]],x_] := + ReplacePart[RuleDelayed[lhs,G[list,u]],{2,2}->FixRhsIntRule[u,x]] /; +(G===With || G===Module || G===Block) + +FixIntRule[RuleDelayed[lhs_,F_[u_,test_]],x_] := + ReplacePart[RuleDelayed[lhs,F[u,test]],{2,1}->FixRhsIntRule[u,x]] /; +F===Condition + +FixIntRule[RuleDelayed[lhs_,u_],x_] := + ReplacePart[RuleDelayed[lhs,u],{2}->FixRhsIntRule[u,x]] + + +(* ::Subsection::Closed:: *) +(*FixRhsIntRule[rhs,x]*) + + +SetAttributes[FixRhsIntRule,HoldAll]; + +FixRhsIntRule[u_+v_,x_] := + FixRhsIntRule[u,x]+FixRhsIntRule[v,x] + +FixRhsIntRule[u_-v_,x_] := + FixRhsIntRule[u,x]-FixRhsIntRule[v,x] + +FixRhsIntRule[-u_,x_] := + -FixRhsIntRule[u,x] + +FixRhsIntRule[a_*u_,x_] := + Dist[a,u,x] /; +MemberQ[{Int,Subst},Head[Unevaluated[u]]] + +FixRhsIntRule[u_,x_] := + If[Head[Unevaluated[u]]===Dist && Length[Unevaluated[u]]==2, + Insert[Unevaluated[u],x,3], + If[MemberQ[{Int,Subst,Defer[Int],Simp,Dist},Head[Unevaluated[u]]], + u, + Simp[u,x]]] +`) + +func resourcesRubiIntegrationUtilityFunctionsMBytes() ([]byte, error) { + return _resourcesRubiIntegrationUtilityFunctionsM, nil +} + +func resourcesRubiIntegrationUtilityFunctionsM() (*asset, error) { + bytes, err := resourcesRubiIntegrationUtilityFunctionsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/Integration Utility Functions.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiMakerubimxfileM = []byte(`(* ::Package:: *) + +(* ::Title:: *) +(*Make Rubi MX file*) + + +(* ::Subsubtitle:: *) +(*Run this package to generate a fast-loading Rubi4.8.mx file that can be read using a Get ["Rubi4.8.mx"] command.*) + + +(* ::Subsubtitle:: *) +(*Set the control variable ShowSteps to True before running this package to generate a ShowRubi4.8.mx file that enables Rubi to display the steps required to integrate an expression.*) + + +(* ::Subsubtitle:: *) +(*Note that MX files cannot be exchanged between different operating systems or versions of Mathematica.*) + + +ShowSteps=False; + + +Get[NotebookDirectory[]<>"ShowStep Routines.m"]; +Get[NotebookDirectory[]<>"Integration Utility Functions.m"]; + +Clear[Int]; +Int::usage="Int[expn,var] returns the indefinite integral of with respect to ."; + +LoadPackage["8.1 Integrand Simplification Rules"]; + +LoadPackage["1.1 Linear Product Rules"]; +LoadPackage["1.2 Quadratic Product Rules"]; +LoadPackage["1.3 Binomial Product Rules"]; +LoadPackage["1.4 Trinomial Product Rules"]; +LoadPackage["1.5 Algebraic Function Rules"]; + +LoadPackage["8.3 Piecewise Linear Function Rules"]; +LoadPackage["2 Exponential Function Rules"]; + +LoadPackage["3.1 Sine Function Rules"]; +LoadPackage["3.2 Tangent Function Rules"]; +LoadPackage["3.3 Secant Function Rules"]; +LoadPackage["3.4 Trig Function Rules"]; + +LoadPackage["4 Hyperbolic Function Rules"]; + +LoadPackage["5 Inverse Trig Function Rules"]; + +LoadPackage["6 Inverse Hyperbolic Function Rules"]; + +LoadPackage["7 Special Function Rules"]; + +LoadPackage["8.2 Derivative Integration Rules"]; +LoadPackage["8.4 Miscellaneous Integration Rules"]; +FixIntRules[]; + + +If[ShowSteps, StepFunction[Int]]; + + +SetDirectory[NotebookDirectory[]]; +DumpSave[If[ShowSteps===True, "StepRubi4.8.mx", "Rubi4.8.mx"], "Global`+"`"+`"]; +ResetDirectory[]; +`) + +func resourcesRubiMakerubimxfileMBytes() ([]byte, error) { + return _resourcesRubiMakerubimxfileM, nil +} + +func resourcesRubiMakerubimxfileM() (*asset, error) { + bytes, err := resourcesRubiMakerubimxfileMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/MakeRubiMxFile.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiReadme = []byte(`A snapshot of the Rubi rules by Albert D. Rich from http://www.apmaths.uwo.ca/~arich/. We sincerely thank Albert for his work on building this impressive collection of integration rules and making them freely available for use. + +This README is the only file in this directory that is not part of the Rubi distribution. The other files are not modified in any way. + +The snapshot currently contains Rubi version: 4.12.1 +`) + +func resourcesRubiReadmeBytes() ([]byte, error) { + return _resourcesRubiReadme, nil +} + +func resourcesRubiReadme() (*asset, error) { + bytes, err := resourcesRubiReadmeBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/README", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiRubiM = []byte(`(* ::Package:: *) + +(* ::Title:: *) +(*Rubi (Rule-Based Integrator) Package*) + + +BeginPackage["Rubi`+"`"+`"]; + + +Int::usage = +"Int[expn,var] returns the antiderivative (indefinite integral) of with respect to . +Int[list,var] returns a list of the antiderivatives of the elements of with respect to ."; + + +IntShowSteps::usage = "IntShowSteps[expn,var] shows all the rules and intermediate steps required to integrate with respect to , and returns Null."; + + +Dist::usage = "Dist[expn1,expn2,var] distributes over ."; +Subst::usage = "Subst[expn1,var,expn2] substitutes for in ."; + + +ShowSteps::usage = "If ShowSteps is True and the ShowSteps package has been loaded, integration steps are displayed."; +$StepCounter::usage = "If the ShowSteps package has been loaded and $StepCounter is an integer, it is incremented each time an integration rule is applied."; + + +$RuleColor::usage = "$RuleColor is the color used to display rules when showing integration steps. The default rule color is red." +$ConditionColor::usage = "$ConditionColor is the color used to display application conditions when showing integration steps. The default condition color is blue." + + +sin::usage = "Inert sine function"; +cos::usage = "Inert cosine function"; +tan::usage = "Inert tangent function"; +cot::usage = "Inert cotangent function"; +sec::usage = "Inert secant function"; +csc::usage = "Inert cosecant function"; + + +Begin["`+"`"+`Private`+"`"+`"]; + + +LoadRules[filename_String] := + Module[{object}, + object=PrintTemporary["Loading "<>filename<>".m..."]; + Get[NotebookDirectory[]<>filename<>".m"]; + NotebookDelete[object]; + Null] + + +Unprotect[Int]; Clear[Int]; + + +SetAttributes [Int, {Listable}]; + + +ShowSteps = Global`+"`"+`$LoadShowSteps===True; + + +LoadRules["ShowStep routines"]; +LoadRules["Integration utility functions"]; +LoadRules["9.1 Integrand simplification rules"]; + + +LoadRules["1.1.1 Linear binomial products"]; +LoadRules["1.1.3 General binomial products"]; + +LoadRules["1.2.1 Quadratic trinomial products"]; +LoadRules["1.2.2 Quartic trinomial products"]; +LoadRules["1.2.3 General trinomial products"]; +LoadRules["1.2.4 Improper trinomial products"]; + +LoadRules["1.1.4 Improper binomial products"]; +LoadRules["1.3 Miscellaneous algebraic functions"]; + + +LoadRules["9.3 Piecewise linear functions"]; +LoadRules["2 Exponentials"]; +LoadRules["3 Logarithms"]; +LoadRules["4.1 Sine"]; +LoadRules["4.2 Tangent"]; +LoadRules["4.3 Secant"]; +LoadRules["4.4 Miscellaneous trig functions"]; +LoadRules["5 Inverse trig functions"]; +LoadRules["6 Hyperbolic functions"]; +LoadRules["7 Inverse hyperbolic functions"]; +LoadRules["8 Special functions"]; +LoadRules["9.2 Derivative integration rules"]; +LoadRules["9.4 Miscellaneous integration rules"]; + + +FixIntRules[]; + + +If[Global`+"`"+`$LoadShowSteps===True, StepFunction[Int]]; + + +Protect[Int]; + + +End []; +EndPackage []; +`) + +func resourcesRubiRubiMBytes() ([]byte, error) { + return _resourcesRubiRubiM, nil +} + +func resourcesRubiRubiM() (*asset, error) { + bytes, err := resourcesRubiRubiMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/Rubi.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiRubi412Nb = []byte(`(* Content-type: application/mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 7.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 145, 7] +NotebookDataLength[ 18593, 419] +NotebookOptionsPosition[ 17655, 387] +NotebookOutlinePosition[ 18064, 404] +CellTagsIndexPosition[ 18021, 401] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ + +Cell[CellGroupData[{ +Cell[TextData[{ + StyleBox["Rubi", + FontSlant->"Italic"], + " 4.12.1" +}], "Title", + CellChangeTimes->{{3.4897813784127054`+"`"+`*^9, 3.4897813794267073`+"`"+`*^9}, + 3.518320509030015*^9, {3.5769613958072157`+"`"+`*^9, 3.576961399157221*^9}, { + 3.5818113211358624`+"`"+`*^9, 3.581811322505864*^9}, {3.588014284106391*^9, + 3.5880142845587916`+"`"+`*^9}, {3.5933144297049866`+"`"+`*^9, + 3.5933144300349865`+"`"+`*^9}, {3.5953446914325457`+"`"+`*^9, 3.595344691902547*^9}, { + 3.612238596642627*^9, 3.612238596892227*^9}, {3.6235574469855204`+"`"+`*^9, + 3.623557447305521*^9}, {3.6336310694366074`+"`"+`*^9, 3.633631070093645*^9}, { + 3.6414292510530057`+"`"+`*^9, 3.6414292518174067`+"`"+`*^9}, {3.6669851230264883`+"`"+`*^9, + 3.66698512435649*^9}, {3.6809646450469093`+"`"+`*^9, 3.6809646561585455`+"`"+`*^9}, { + 3.682799968357607*^9, 3.682799968746629*^9}, {3.683231370531602*^9, + 3.6832313714826565`+"`"+`*^9}, {3.685735947623728*^9, 3.6857359497737303`+"`"+`*^9}, { + 3.686789919766761*^9, 3.6867899200487766`+"`"+`*^9}, {3.687230099297368*^9, + 3.6872300995693836`+"`"+`*^9}, {3.6892138635040665`+"`"+`*^9, 3.689213863730079*^9}, { + 3.690146376179472*^9, 3.690146376547493*^9}, {3.6919485296883698`+"`"+`*^9, + 3.6919485300683703`+"`"+`*^9}, {3.6932827954675274`+"`"+`*^9, + 3.6932827957483277`+"`"+`*^9}, {3.695091542008456*^9, 3.6950915425794888`+"`"+`*^9}, { + 3.6966170132343836`+"`"+`*^9, 3.696617013554384*^9}, {3.6976798751471663`+"`"+`*^9, + 3.6976798774002953`+"`"+`*^9}, {3.702673552626842*^9, 3.7026735531568427`+"`"+`*^9}, { + 3.705885647832942*^9, 3.705885649283745*^9}, {3.706152621368204*^9, + 3.7061526217682047`+"`"+`*^9}, {3.7063765173533025`+"`"+`*^9, 3.706376517609317*^9}, { + 3.706393141129644*^9, 3.7063931434617777`+"`"+`*^9}, {3.7090906465542336`+"`"+`*^9, + 3.7090906473842807`+"`"+`*^9}}, + TextAlignment->Center], + +Cell["Rule-Based Integrator", "Subtitle", + CellChangeTimes->{{3.4897814556796412`+"`"+`*^9, 3.4897814580976458`+"`"+`*^9}, { + 3.5769621785283117`+"`"+`*^9, 3.576962186368323*^9}}, + TextAlignment->Center, + FontWeight->"Bold"], + +Cell["Crafted by Albert D. 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time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiShowstepRoutinesM = []byte(`(* ::Package:: *) + +(* ShowSteps controls the use of special definitions when defining rules AND display of steps when simplify expressions. *) +If[ShowSteps=!=False, ShowSteps=True]; +SimplifyFlag=True; + + +$StepCounter = 0; +$RuleColor = Red; +$ConditionColor = Blue; + + +Unprotect[IntShowSteps]; Clear[IntShowSteps]; + +IntShowSteps[u_,x_Symbol] := + Block[{ShowSteps=True}, + FixedPoint[ + Function[CellPrint[ExpressionCell[#,"Input"]]; + ReplaceAll[#,{Defer[Int]->Int,Defer[Dist]->Dist,Defer[Subst]->Subst}]],Int[u,x]]; + Null] + +Protect[IntShowSteps]; + + +(* If func is a function defined using properly defined transformation rules, + StepFunction[func] modifies the rules to display steps when the control + variable ShowSteps is True. *) +StepFunction[func_] := + Module[{lst=DownValues[func]}, + Print["Modifying ",Length[lst]," integration rules to display steps..."]; + Block[{ShowStep,SimplifyFlag}, + Monitor[Do[ + lst[[i]]=ModifyRule[i,lst[[i]],SimplifyFlag], + {i,1,Length[lst]}], + ProgressIndicator[i,{1,Length[lst]+1},ImageSize->{500,20}]]]; + ClearDownValues[func]; + SetDownValues[func,lst]; + Unprotect[func]] + +StepFunction[func1_,func2_] := + Block[{lst,num=0,ShowStep,SimplifyFlag}, + lst=Map[Function[ModifyRule[num++,#, SimplifyFlag]],DownValues[func1]]; + ClearDownValues[func1]; + SetDownValues[func2,ReplaceAll[lst,{func1->func2}]]] + + +(* rule is a rule as an expression of the form RuleDelayed[lhs,rhs]. + flag has the value SimplifyFlag. ModifyRule[rule,flag] + formats the rule's left hand side as a string (lhsStrg) in InputForm, + formats the rule's conditions as a string (condStrg) in StandardForm, + formats the rule's let statements as a string (letStrg) in StandardForm, + splices the conditions string and let strings together as a string (condStrg) in StandardForm, + formats the rule's right hand side as a string (rhsStrg) in InputForm, + then it returns rule with the body replaced with the expression + ShowStep[num,condStrg,lhsStrg,rhsStrg,rhs] /; flag. *) +ModifyRule[num_,rule_RuleDelayed, flag_] := + Module[{lhsStrg,rhsStrg,condStrg,letStrg}, + If[Not[FreeQ[Hold[rule],ShowStep]] || + Not[FreeQ[Hold[rule],Identity]] || + Not[FreeQ[Hold[rule],DeactivateTrig]] || + Not[FreeQ[Hold[rule],Defer[Int]]] || + Not[FreeQ[Hold[rule],Preprocess]], + rule, + lhsStrg=FormatLhs[rule]; + If[rule[[2,0]]===Condition, + condStrg=FormatConditions[Extract[rule,{2,2},Defer]]; + If[rule[[2,1,0]]===With || rule[[2,1,0]]===Module || rule[[2,1,0]]===Block, + letStrg=FormatLets[Extract[rule,{2,1,1},Defer]]; + If[rule[[2,1,2,0]]===Condition, + condStrg=SpliceConditionString[condStrg,letStrg,FormatConditions[Extract[rule,{2,1,2,2},Defer]]]; + rhsStrg=FormatRhs[Extract[rule,{2,1,2,1},Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,condStrg,lhsStrg,rhsStrg,Extract[rule,{2,1,2,1},Hold]], {2,1,2,1}], flag], + condStrg=SpliceConditionString[condStrg,letStrg,""]; + rhsStrg=FormatRhs[Extract[rule,{2,1,2},Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,condStrg,lhsStrg,rhsStrg,Extract[rule,{2,1,2},Hold]], {2,1,2}], flag]], + condStrg=SpliceConditionString[condStrg,"",""]; + rhsStrg=FormatRhs[Extract[rule,{2,1},Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,condStrg,lhsStrg,rhsStrg,Extract[rule,{2,1},Hold]], {2,1}], flag]], + If[rule[[2,0]]===With || rule[[2,0]]===Module || rule[[2,0]]===Block, + letStrg=FormatLets[Extract[rule,{2,1},Defer]]; + If[rule[[2,2,0]]===Condition, + condStrg=FormatConditions[Extract[rule,{2,2,2},Defer]]; + condStrg=SpliceConditionString["",letStrg,condStrg]; + rhsStrg=FormatRhs[Extract[rule,{2,2,1},Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,condStrg,lhsStrg,rhsStrg,Extract[rule,{2,2,1},Hold]], {2,2,1}], flag], + condStrg=SpliceConditionString["",letStrg,""]; + rhsStrg=FormatRhs[Extract[rule,{2,2},Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,condStrg,lhsStrg,rhsStrg,Extract[rule,{2,2},Hold]], {2,2}], flag]], + rhsStrg=FormatRhs[Extract[rule,2,Defer]]; + WrapCondition[ReplacePart[rule, ShowStep[num,"",lhsStrg,rhsStrg,Extract[rule,2,Hold]], 2], flag]]]]] + + +WrapCondition[rule_RuleDelayed,condition_] := + ReplacePart[ReplacePart[rule,Append[Extract[rule,{2},Hold],condition],{2}],Condition,{2,0}] + + +(* rule is a rule as an expression of the form RuleDelayed[lhs,rhs]. + FormatLhs[rule] returns returns a string for the lhs of the rule in InputForm with the + pattern tags "_." and "_" removed from the dummy variable names. *) +FormatLhs[rule_] := + Module[{lhs=Extract[rule,{1,1},Defer],conditions,func,var}, + ( If[rule[[2,0]]===Condition, + conditions=Extract[rule,{2,2},Defer]; + If[conditions[[1,0]]===FunctionOfQ, + func=conditions[[1,1]]; + var=conditions[[1,2]]; + lhs=ReplaceVariable[lhs,var,func], + If[conditions[[1,0]]===And && MemberQ[conditions,FunctionOfQ,{3},Heads->True], + func=conditions[[1,Position[conditions,FunctionOfQ,{3},1][[1,2]],1]]; + var=conditions[[1,Position[conditions,FunctionOfQ,{3},1][[1,2]],2]]; + lhs=ReplaceVariable[lhs,var,func]]]] ); + DropDefer[StringReplace[ToString[lhs, InputForm],{"_Symbol"->"", "_."->"", "_"->""}]]] + +(* ReplaceVariable[lhs,var,func] returns lhs with the var replaced by F[func] *) +ReplaceVariable[lhs_,var_,func_] := + Block[{F}, + If[PatternEqualQ[lhs[[1,1]],var], + ReplacePart[lhs,F[func],{1,1}], + If[PatternEqualQ[lhs[[1,1,1]],var], + ReplacePart[lhs,F[func],{1,1,1}], + If[PatternEqualQ[lhs[[1,1,2]],var], + ReplacePart[lhs,F[func],{1,1,2}], + If[PatternEqualQ[lhs[[1,1,3]],var], + ReplacePart[lhs,F[func],{1,1,3}], + Print["Function of expression variable not found: ",lhs," ",var," ",func]; + Abort[]]]]]] + +PatternEqualQ[pattern_,var_] := + Head[pattern]===Pattern && pattern[[1]]===var + + +(* rhs is an expression of the form Defer[...] where ... is the right hand side of a rule. + FormatRhs[rhs] returns a string for the rhs of the rule in InputForm. *) +FormatRhs[rhs_] := + Block[{SubstFor,Rt,F}, + DropDefer[ToString[ReplaceAll[rhs,{ + SubstFor[v_,u_,x_] -> F[x], + Rt[u_,2] -> Sqrt[u], + Rt[u_,n_] -> u^(1/n) + }], InputForm]]] + + +(* strg is a string of the form "Defer[...]". DropDefer[strg] returns the string "...", with + all occurrences of the string "Int[" replaced with "Integrate" and "Dif" with "D" so they will + display using integral and differential signs when the string is converted to StandardForm. *) +DropDefer[strg_] := + StringDrop[StringDrop[StringReplace[strg,{ + "Int["->"Integrate[", + "Dif["->"D[" +(*, "sin["->"Sin[", "cos["->"Cos[", "tan["->"Tan[", "cot["->"Cot[", "sec["->"Sec[", "csc["->"csc[", + "sinh["->"Sinh[", "cosh["->"Cosh[", "tanh["->"Tanh[", "coth["->"Coth[", "sech["->"Sech[", "csch["->"csch[" *) + }],6],-1] + + +(* conditions is an expression comprising the conditions on a rule. FormatConditions[conditions] + replaces expressions of the form Not[FalseQ[u]] with u in conditions, + deletes expressions of the form FreeQ[u,x], FunctionOfQ[u,v,x] and EasyDQ[u,x] in conditions, + replaces expressions of the form Rt[u,2] with Sqrt[u] and Rt[u,n] with u^(1/n) in conditions, + and then returns the conditions as a string formatted in Mathematica's StandardForm. *) +FormatConditions[conditions_] := + If[conditions[[1,0]]===Not && conditions[[1,1,0]]===FalseQ, + FormatConditions[Extract[conditions,{1,1,1},Defer]], + If[conditions[[1,0]]===FreeQ, + "", + If[conditions[[1,0]]===And && MemberQ[conditions,FreeQ,{3},Heads->True], + FormatConditions[DeleteCondition[FreeQ,conditions]], + If[conditions[[1,0]]===FunctionOfQ, + "", + If[conditions[[1,0]]===And && MemberQ[conditions,FunctionOfQ,{3},Heads->True], + FormatConditions[DeleteCondition[FunctionOfQ,conditions]], + If[conditions[[1,0]]===EasyDQ, + "", + If[conditions[[1,0]]===And && MemberQ[conditions,EasyDQ,{3},Heads->True], + FormatConditions[DeleteCondition[EasyDQ,conditions]], + ToConditionString[conditions] <> ","]]]]]]] + +DeleteCondition[func_,conditions_] := + If[Quiet[Head[Extract[conditions,{1,3},Defer]]===Extract], + If[Position[conditions,func,{3},1][[1,2]]==1, + Extract[conditions,{1,2},Defer], + Extract[conditions,{1,1},Defer]], + Delete[conditions,{1,Position[conditions,func,{3},1][[1,2]]}]] + + +(* let is a body of a With, Module or Block statement. FormatLets[let] returns the assignment as a + string in the form "var=expression, then" formatted in Mathematica's StandardForm. *) +FormatLets[let_] := + If[MatchQ[let,Defer[{u_}]], + If[let[[1,1,0]]===Set && + let[[1,1,2,0]]===Block && + Extract[let,{1,1,2,1},Defer]===Defer[{ShowSteps=False}], + If[let[[1,1,2,2,0]]===Simplify, + ToConditionString[Extract[let,{1,1,1},Defer]] <> "=" <> + ToConditionString[Extract[let,{1,1,2,2,1},Defer]] <> ", then", + ToConditionString[Extract[let,{1,1,1},Defer]] <> "=" <> + ToConditionString[Extract[let,{1,1,2,2},Defer]] <> ", then"], + ToConditionString[Extract[let,{1,1},Defer]] <> ", then"], + ToConditionString[let] <> ", then"] + + +(* conditions is an expression comprising the conditions on a rule. ToConditionString[conditions] + replace calls in the conditions on the function Rt[u,2] with Sqrt[u] and Rt[u,n] with u^(1/n), + and then returns the conditions as a string formatted in Mathematica's StandardForm. *) +ToConditionString[conditions_] := + ToString[ReplaceAll[conditions,{ + Rt[u_,2]->Sqrt[u], + Rt[u_,n_]->u^(1/n) + }],StandardForm] + + +(* cond1, lets and cond2 are strings. SpliceConditionString[cond1,lets,cond2] concatenates and + returns the condition and let strings along with the "if" and "let" to make a human readable + condition string. *) +SpliceConditionString[cond1_,lets_,cond2_] := + If[cond2==="", + If[lets==="", + If[cond1==="", + "", + "If " <> cond1], + If[cond1==="", + "Let " <> lets, + "If " <> cond1 <> " let " <> lets]], + If[lets==="", + If[cond1==="", + "If " <> cond2, + "If " <> cond1 <> " if " <> cond2], + If[cond1==="", + "Let " <> lets <> " if " <> cond2, + "If " <> cond1 <> " let " <> lets <> " if " <> cond2]]] + + +(* condStrg, lhsStrg and rhsStrg are the strings required to display the rule being applied. + rhs is the expression on the right side of the rule (i.e. the consequent of the rule). + If ShowSteps is True, ShowStep[num,condStrg,lhsStrg,rhsStrg,rhs] displays the rule being applied, + sets SimplifyFlag to False to turn off further simplification, and release the hold on the rhs + of the rule. *) +ShowStep[condStrg_,lhsStrg_,rhsStrg_,rhs_] := ( + If[IntegerQ[$StepCounter], $StepCounter++]; + If[ShowSteps, + Print["Rule: ",Style[condStrg,$ConditionColor]]; + Print[" ",Style[ToExpression["Defer["<>lhsStrg<>"]"],$RuleColor],Style[" \[LongRightArrow] ",Bold],Style[ToExpression["Defer["<>rhsStrg<>"]"],$RuleColor]]; + Block[{SimplifyFlag=False}, + ReleaseHold[rhs]], + ReleaseHold[rhs]] ) + +ShowStep[num_,condStrg_,lhsStrg_,rhsStrg_,rhs_] := ( + If[IntegerQ[$StepCounter], $StepCounter++]; + If[ShowSteps, + Print["Rule ",num+1,": ",Style[condStrg,$ConditionColor]]; + Print[" ",Style[ToExpression["Defer["<>lhsStrg<>"]"],$RuleColor],Style[" \[LongRightArrow] ",Bold],Style[ToExpression["Defer["<>rhsStrg<>"]"],$RuleColor]]; + Block[{SimplifyFlag=False}, + ReleaseHold[rhs]], + ReleaseHold[rhs]] ) + + +(* Note: Clear[func] also eliminates 2-D display of functions like Integrate. *) +ClearDownValues[func_Symbol] := ( + Unprotect[func]; + DownValues[func]={}; + Protect[func]) + + +(* Note: A bug in earlier versions of Mathematica prevents setting more than 529 DownValues using a simple assignment! *) +SetDownValues[func_Symbol,lst_List] := ( + Unprotect[func]; + ( If[$VersionNumber>=8, + DownValues[func]=lst, + DownValues[func]=Take[lst,Min[529,Length[lst]]]; + Scan[Function[ReplacePart[ReplacePart[#,#[[1,1]],1],SetDelayed,0]],Drop[lst,Min[529,Length[lst]]]]] ); + Protect[func]) +`) + +func resourcesRubiShowstepRoutinesMBytes() ([]byte, error) { + return _resourcesRubiShowstepRoutinesM, nil +} + +func resourcesRubiShowstepRoutinesM() (*asset, error) { + bytes, err := resourcesRubiShowstepRoutinesMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi/ShowStep Routines.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubiM = []byte(`LoadRubi::usage = "`+"`"+`LoadRubi[]`+"`"+` will load the Rubi integration package by Albert Rich."; +LoadRubi[] := Get["__res__/rubi_loader.m"]; +Attributes[LoadRubi] = {Protected}; +Tests`+"`"+`LoadRubi = { + ESimpleExamples[ + (*Just test a few cases for a sanity check. Full test suite can be run + outside of test suite.*) + ESameTest[Null, LoadRubiBundledSnapshot[]], + ESameTest[x^3/3 + 2*x, Rubi`+"`"+`Int[x^2+2, x]], + ESameTest[(-3*ArcTanh[Cosh[a+b*x]])/(8*b)+(3*Coth[a+b*x]*Csch[a+b*x])/(8*b)-(Coth[a+b*x]*Csch[a+b*x]^3)/(4*b), Rubi`+"`"+`Int[Csch[a + b*x]^5,x]], + ESameTest[Log[x], Rubi`+"`"+`Int[1/x, x]], + ESameTest[-(a/(4 x^4))-b/(3 x^3), Rubi`+"`"+`Int[(a+b x)/x^5,x]], + ], ETests[ + ESameTest[x^3/3, Rubi`+"`"+`Int[x^2, x]], + ], EKnownFailures[ + (*Equal, but SameQ cannot tell:*) + ESameTest[-((3/2+2 x)^2/(3 x^2)), Rubi`+"`"+`Int[(3/2+2 x)/x^3,x]], + (*Will not work until we add support for FixIntRules. Coefficients + will not distribute.*) + ESameTest[-(3/8) ArcTanh[Cos[a+x]]-3/8 Cot[a+x] Csc[a+x]-1/4 Cot[a+x] Csc[a+x]^3, Rubi`+"`"+`Int[csc[a+x]^5,x]], + ] +}; + +LoadRubiSnapshot[snapshotLoc_] := ( + Get[snapshotLoc]; + $ContextPath = Prepend[$ContextPath, "Rubi`+"`"+`"]; +); +LoadRubiBundledSnapshot[] := LoadRubiSnapshot["__rubi_snapshot__/rubi_snapshot/rubi_snapshot.expred"]; +SaveRubiSnapshot[snapshotLoc_] := Save[snapshotLoc, "Rubi`+"`"+`*"]; +`) + +func resourcesRubiMBytes() ([]byte, error) { + return _resourcesRubiM, nil +} + +func resourcesRubiM() (*asset, error) { + bytes, err := resourcesRubiMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesRubi_loaderM = []byte(`(* ::Package:: *) + +(* ::Title:: *) +(*Rubi (Rule-Based Integrator) Package*) +(* Clone https://github.com/corywalker/rubi into the root dir of Expreduce. *) + + + +BeginPackage["Rubi`+"`"+`"] + +Int::usage = +"Int[expn,var] returns the antiderivative (indefinite integral) of with respect to . +Int[list,var] returns a list of the antiderivatives of the elements of with respect to ."; + + +IntShowSteps::usage = "IntShowSteps[expn,var] shows all the rules and intermediate steps required to integrate with respect to , and returns Null."; + + +Dist::usage = "Dist[expn1,expn2,var] distributes over ."; +Subst::usage = "Subst[expn1,var,expn2] substitutes for in ."; + + +ShowSteps::usage = "If ShowSteps is True and the ShowSteps package has been loaded, integration steps are displayed."; +$StepCounter::usage = "If the ShowSteps package has been loaded and $StepCounter is an integer, it is incremented each time an integration rule is applied."; + + +$RuleColor::usage = "$RuleColor is the color used to display rules when showing integration steps. The default rule color is red." +$ConditionColor::usage = "$ConditionColor is the color used to display application conditions when showing integration steps. The default condition color is blue." + + +sin::usage = "Inert sine function"; +cos::usage = "Inert cosine function"; +tan::usage = "Inert tangent function"; +cot::usage = "Inert cotangent function"; +sec::usage = "Inert secant function"; +csc::usage = "Inert cosecant function"; + + +Begin["`+"`"+`Private`+"`"+`"] + +(*LoadRules[filename_String] := + Module[{object}, + object=PrintTemporary["Loading "<>filename<>".m..."]; + Get[NotebookDirectory[]<>filename<>".m"]; + NotebookDelete[object]; + Null]*) + +LoadRules[filename_String] := ( + Print["Loading "<>filename<>".m..."]; + loadRes = Get["__res__/rubi/"<>filename<>".m"]; + If[loadRes === $Failed, Print["Loading failed!"]]; +) + + +Unprotect[Int]; Clear[Int]; + + +SetAttributes [Int, {Listable}]; + + +ShowSteps = Global`+"`"+`$LoadShowSteps===True; + + +Print["Loading Rubi integration suite by Albert Rich."]; + +(*LoadRules["ShowStep routines"];*) +LoadRules["Integration Utility Functions"]; +LoadRules["9.1 Integrand simplification rules"]; + + +LoadRules["1.1.1 Linear binomial products"]; +LoadRules["1.1.3 General binomial products"]; + +LoadRules["1.2.1 Quadratic trinomial products"]; +LoadRules["1.2.2 Quartic trinomial products"]; +LoadRules["1.2.3 General trinomial products"]; +LoadRules["1.2.4 Improper trinomial products"]; + +LoadRules["1.1.4 Improper binomial products"]; +LoadRules["1.3 Miscellaneous algebraic functions"]; + + +LoadRules["9.3 Piecewise linear functions"]; +LoadRules["2 Exponentials"]; +LoadRules["3 Logarithms"]; +LoadRules["4.1 Sine"]; +LoadRules["4.2 Tangent"]; +LoadRules["4.3 Secant"]; +LoadRules["4.4 Miscellaneous trig functions"]; +LoadRules["5 Inverse trig functions"]; +LoadRules["6 Hyperbolic functions"]; +LoadRules["7 Inverse hyperbolic functions"]; +LoadRules["8 Special functions"]; +LoadRules["9.2 Derivative integration rules"]; +LoadRules["9.4 Miscellaneous integration rules"]; + + +FixIntRules[]; + + +(*If[Global`+"`"+`$LoadShowSteps===True, StepFunction[Int]];*) + + +Protect[Int]; + + +End []; +EndPackage []; + +(*Helper debug functions in global context.*) + +printDownValues[args_, sym_] := + Print /@ Map[{#, MatchQ[args, #]} &, + Map[List @@ (#[[1]][[1]]) &, DownValues[sym]]]; +isPartialMatch[args_, dv_] := MatchQ[args, List @@ (dv[[1]][[1]])]; +isFullMatch[args_, dv_, sym_] := + Replace[args, dv /. sym -> List] =!= args; +printPartitalMatchingDownValues[args_, sym_, n_] := + Select[DownValues[sym], isPartialMatch[args, #] &, n]; +printMatchingDownValues[args_, sym_, n_] := + Select[DownValues[sym], isFullMatch[args, #, sym] &, n]; + +`) + +func resourcesRubi_loaderMBytes() ([]byte, error) { + return _resourcesRubi_loaderM, nil +} + +func resourcesRubi_loaderM() (*asset, error) { + bytes, err := resourcesRubi_loaderMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/rubi_loader.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesSimplifyM = []byte(`Simplify::usage = "`+"`"+`Simplify[expr]`+"`"+` attempts to perform simplification operations on `+"`"+`expr`+"`"+`."; +booleanSimplify[exp_] := exp //. { + (*a_ && a_ :> a,*) + (*a_ || a_ :> a,*) + + !x_ || !y_ :> !(x && y), + !x_ || !y_ || !z :> !(x && y && z), + (*This is a generalization of the above rule, but causes issues*) + (*This problem actually happens outside of Expreduce as well.*) + (*The issue is that the Or with the pattern inside evaluates*) + (*immediately to just the pattern, due to how Or works. This*) + (*happens before pattern matching, causing all kinds of*) + (*expressions, even outside Or expressions, to match.*) + (*Or[match__?(AllTrue[#, (Head[#] == Not &)] &)] :> Not[(#[[1]] &) /@ match],*) + + (*a_ || (a_ && b_) :> a,*) + (*a_ || !a_ && b_ :> a || b,*) + + (a_ || b_) && (a_ || c_) :> a || (b && c), + (b_ || a_) && (c_ || a_) :> a || (b && c), + Or[___, a_, ___, Not[And[___, a_, ___] | a_], ___] :> True, + Or[___, Not[And[___, a_, ___] | a_], ___, a_, ___] :> True, + Or[x1___, a_, x2___, And[x3___, a_, x4___], x5___] :> Or[a, x1, x2, x5], + Or[x1___, And[x2___, a_, x3___], x4___, a_, x5___] :> Or[a, x1, x4, x5], + Or[x1___, a_, x2___, a_, x3___] :> Or[a, x1, x2, x3], + Or[x1___, a_, x2___, And[x3___, !a_, x4___], x5___] :> Or[a, x1, x2, And[x3, x4], x5], + Or[x1___, And[x2___, !a_, x3___], x4___, a_, x5___] :> Or[a, x1, And[x2, x3], x4, x5], + + (*Dual of these rules.*) + And[___, a_, ___, Not[Or[___, a_, ___] | a_], ___] :> False, + And[___, Not[Or[___, a_, ___] | a_], ___, a_, ___] :> False, + And[x1___, a_, x2___, Or[x3___, a_, x4___], x5___] :> And[a, x1, x2, x5], + And[x1___, Or[x2___, a_, x3___], x4___, a_, x5___] :> And[a, x1, x4, x5], + And[x1___, a_, x2___, a_, x3___] :> And[a, x1, x2, x3], + And[x1___, a_, x2___, Or[x3___, !a_, x4___], x5___] :> And[a, x1, x2, Or[x3, x4], x5], + And[x1___, Or[x2___, !a_, x3___], x4___, a_, x5___] :> And[a, x1, Or[x2, x3], x4, x5] +}; +trigSimplify[exp_] := exp //. { + (Rational[1,2]*c_.*(a_ + b_*Cos[inner_]) /; a == -b) :> c*a*Sin[inner/2]^2 +}; +tryFactoringOutTerm[e_] := e; +tryFactoringOutTerm[p_Plus] := Module[{cterms, simplest, i, tryVal}, + simplest = p; + cterms = ((ExpreduceConstantTerm /@ (List @@ p)) // Transpose)[[1]]; + For[i = 1, i <= Length[cterms], i++, + tryVal = cterms[[i]] (p/cterms[[i]] // Distribute); + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[simplest], + simplest = tryVal]; + ]; + simplest + ]; +simplifyInner[exp_] := Module[{e = exp, tryVal}, + e = booleanSimplify[e]; + e = trigSimplify[e]; + inferredPower = 1; + If[MatchQ[e, s_Plus^n_Integer], + inferredPower := Replace[e, s_Plus^n_Integer -> n]]; + If[inferredPower < 5, + (*Similar to Expand, but not using ReplaceAll*) + tryVal = FixedPoint[Replace[#, expandRules]&, e]; + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[e], e = tryVal]; + ]; + (*Avoid expressions containing expensive expressions to expand.*) + If[FreeQ[e, a_Plus^(b_Integer?((# >= 5) &))], + tryVal = ComplexExpand[e]; + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[e], e = tryVal]; + ]; + tryVal = Together[e]; + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[e], e = tryVal]; + tryVal = Private`+"`"+`myFactorCommonTerms[e]; + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[e], e = tryVal]; + tryVal = Private`+"`"+`tryFactoringOutTerm[e]; + If[ExpreduceLeafCountSimplify[tryVal] < ExpreduceLeafCountSimplify[e], e = tryVal]; + e = trigSimplify[e]; + (* also need to try complexexpand to simplify cases like (-1)^(1/3) (1 + I Sqrt[3]) *) + e = Replace[e, Sqrt[inner_] :> Sqrt[FactorTerms[inner]]]; + e +]; +Simplify[exp_] := Map[simplifyInner, exp, {0, Infinity}]; +Attributes[Simplify] = {Protected}; +Tests`+"`"+`Simplify = { + ESimpleExamples[ + EComment["`+"`"+`Simplify`+"`"+` can simplify some boolean expressions."], + ESameTest[b, b && b // Simplify], + ESameTest[False, a && b && !b // Simplify], + ESameTest[a || (b && c), (a || b) && (a || c) // Simplify], + ESameTest[a || b, a || ! a && b // Simplify], + ESameTest[True, a || b || ! a || ! b // Simplify] + ], ETests[ + (*Seems wrong, but this is the right behavior*) + ESameTest[a, a || (a && Infinity) // Simplify], + ESameTest[a || (b && c), (b || a) && (c || a) // Simplify], + + ESameTest[True, (a || b || c || Not[(a && b && c)]) // Simplify], + ESameTest[True, (a || b || c || Not[(a && b && c && d)]) // Simplify], + ESameTest[True, (a || b || c || d || Not[(a && b && c)]) // Simplify], + ESameTest[True, (a || b || c || d || Not[a]) // Simplify], + ESameTest[True, (az || b || cz || Not[(ay && b && cy && dy)]) // Simplify], + + ESameTest[a, a || (a && b && c && d) // Simplify], + ESameTest[d, d || (a && b && c && d) // Simplify], + ESameTest[d || e, d || e || (a && b && c && d) // Simplify], + ESameTest[b || d, d || b || (a && b && c && d) // Simplify // Sort], + ESameTest[True, d || b || (a && b && c && d) || ! b // Simplify], + ESameTest[foo[True], foo[d || b || (a && b && c && d) || ! b] // Simplify], + ESameTest[d || e || (a && b && c), d || e || (a && b && c) // Simplify], + ESameTest[z, z || z // Simplify], + ESameTest[a || z, z || a || z // Simplify // Sort], + ESameTest[a, a || a && b // Simplify], + ESameTest[a || b, a || !a && b // Simplify], + ESameTest[a || b || c, a || c || ! a && b // Simplify // Sort], + ESameTest[a || b || c, a || c || ! a && ! c && b // Simplify // Sort], + ESameTest[a || c || !b, a || c || ! a && ! c && ! b // Simplify // Sort], + ESameTest[a || c || (! b && d), a || c || ! a && ! c && ! b && d // Simplify // Sort], + ESameTest[a || c || Not[b], c || a || Not[b] // Simplify // Sort], + ESameTest[False, And[x1, a, x2, Not[Or[x3, a, x4]], x5] // Simplify], + ESameTest[a && x1 && x2 && x5, And[x1, a, x2, Or[x3, a, x4], x5] // Simplify], + ESameTest[a && b, a&&b&&a//Simplify], + ESameTest[(-32+32 x-16 x^2)^9, (16 + 8 (-6 + 4 x - 2 x^2))^9 // Simplify], + ESameTest[1, (1/3+(1/3)*(-2)^(-1/3)*2^(-2/3)*(1+(0+1*I)*3^(1/2))+(1/6)*(-1)^(1/3)*(1+(0+-1*I)*3^(1/2)))^2//Simplify], + ESameTest[0, (1/3+(1/3)*(-2)^(-1/3)*2^(-2/3)*(1+(0+-1*I)*3^(1/2))+(1/6)*(-1)^(1/3)*(1+(0+1*I)*3^(1/2)))^2//Simplify], + ] +}; + +FullSimplify::usage = "`+"`"+`FullSimplify[expr]`+"`"+` attempts to perform full simplification operations on `+"`"+`expr`+"`"+`."; +FullSimplify[e_] := Simplify[e] //. { + E^(-x_Symbol)+E^x_Symbol:>2 Cosh[x] +}; +Attributes[FullSimplify] = {Protected}; +Tests`+"`"+`FullSimplify = { + ESimpleExamples[ + ESameTest[2 Cosh[x], E^-x+E^x//FullSimplify], + ] +}; +`) + +func resourcesSimplifyMBytes() ([]byte, error) { + return _resourcesSimplifyM, nil +} + +func resourcesSimplifyM() (*asset, error) { + bytes, err := resourcesSimplifyMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/simplify.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesSolveM = []byte(`Solve::usage = "`+"`"+`Solve[eqn, var]`+"`"+` solves `+"`"+`eqn`+"`"+` for `+"`"+`var`+"`"+`."; + +countVar[expr_, var_Symbol] := + If[expr === var, 1, Count[expr, var, -1]]; + +containsOneOccurrence[eqn_Equal, var_Symbol] := + Count[eqn, var, -1] == 1; +containsZeroOccurrences[eqn_Equal, var_Symbol] := + Count[eqn, var, -1] == 0; + +checkedConditionalExpression[e_, c_] := + Module[{nextC = 1, res, reps, newC = -1}, + While[True, If[Count[e, C[nextC], -1] > 0, nextC++, Break[]]]; + res = ConditionalExpression[e, c]; + reps = {}; + While[True, + If[Count[res, C[newC], -1] > 0, + AppendTo[reps, C[newC] -> C[-newC]]; newC--, + Break[]]]; + res = res /. C[n_Integer?Positive] :> C[n + Length[reps]]; + res /. reps]; + +applyInverse[lhs_Plus -> rhs_, var_Symbol] := Module[{nonVarParts}, + nonVarParts = Select[lhs, (countVar[#, var] === 0) &]; + varParts = Select[lhs, (countVar[#, var] =!= 0) &]; + {varParts -> rhs - nonVarParts}]; +applyInverse[lhs_Times -> rhs_, var_Symbol] := Module[{nonVarParts}, + nonVarParts = Select[lhs, (countVar[#, var] === 0) &]; + varParts = Select[lhs, (countVar[#, var] =!= 0) &]; + {varParts -> rhs / nonVarParts}]; +applyInverse[base_^pow_ -> rhs_, var_Symbol] := If[countVar[base, var] =!= 0, + (* var in base *) + Switch[pow, + -1, {base -> 1/rhs}, + 2, {base -> -Sqrt[rhs]//PowerExpand, base -> Sqrt[rhs]//PowerExpand}, + 3, {base->rhs^(1/3),base->-(-1)^(1/3) rhs^(1/3),base->(-1)^(2/3) rhs^(1/3)}, + 4, {base->-rhs^(1/4),base->-I rhs^(1/4),base->I rhs^(1/4),base->rhs^(1/4)}, + 5, {base->rhs^(1/5),base->-(-1)^(1/5) rhs^(1/5),base->(-1)^(2/5) rhs^(1/5),base->-(-1)^(3/5) rhs^(1/5),base->(-1)^(4/5) rhs^(1/5)}, + (* Not implemented yet *) + _Integer, SolveError, + (* Similar to the general case but without the message.*) + Rational[1, n_], {base -> rhs ^ (1/pow)}, + _, Message[Solve::ifun, Solve];{base -> rhs ^ (1/pow)} + ], + (* else, var in power *) + If[rhs === 0, + {}, + Switch[base, + _Integer, + {pow-> + ConditionalExpression[ + If[base > 0, + (2 I Pi C[1])/Log[base]+Log[rhs]/Log[base], + ((2 I Pi C[1])+Log[rhs])/Log[base], + ], + C[1]\[Element]Integers + ] + }, + E, If[MatchQ[rhs, _Real], + {pow -> Log[rhs]}, + {pow -> ConditionalExpression[2 I Pi C[1] + Log[rhs], + C[1] \[Element] Integers]}, + ], + _, Message[Solve::ifun, Solve];{pow -> Log[rhs]/Log[base]} + ]] +]; +applyInverse[Log[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[E^rhs, -Pi < Im[rhs] <= Pi]}; +applyInverse[Abs[lhs_] -> rhs_, var_Symbol] := + (Message[Solve::ifun, Solve];{lhs -> -rhs, lhs -> rhs}); +(* Trig inverses *) +applyInverse[Sin[lhs_] -> rhs_, var_Symbol] := + {lhs->checkedConditionalExpression[Pi-ArcSin[rhs]+2 Pi C[-1],C[-1]\[Element]Integers], + lhs->checkedConditionalExpression[ArcSin[rhs]+2 Pi C[-1],C[-1]\[Element]Integers]}; +applyInverse[Cos[lhs_] -> rhs_, var_Symbol] := + {lhs->checkedConditionalExpression[-ArcCos[rhs]+2 Pi C[-1],C[-1]\[Element]Integers], + lhs->checkedConditionalExpression[ArcCos[rhs]+2 Pi C[-1],C[-1]\[Element]Integers]}; +applyInverse[Tan[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[ArcTan[rhs] + Pi C[1], + C[1] \[Element] Integers]}; +applyInverse[Cot[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[ArcCot[rhs] + Pi C[1], + C[1] \[Element] Integers]}; +applyInverse[Sec[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[-ArcCos[1/rhs] + 2 Pi C[1], + C[1] \[Element] Integers], lhs -> + ConditionalExpression[ArcCos[1/rhs] + 2 Pi C[1], + C[1] \[Element] Integers]}; +applyInverse[Csc[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[Pi - ArcSin[1/rhs] + 2 Pi C[1], + C[1] \[Element] Integers], lhs -> + ConditionalExpression[ArcSin[1/rhs] + 2 Pi C[1], + C[1] \[Element] Integers]}; +(* Inverses for inverse trig functions *) +applyInverse[ArcSin[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[ + Sin[rhs], (Re[rhs] == -(Pi/2) && Im[rhs] >= 0) || -(Pi/2) < + Re[rhs] < Pi/2 || (Re[rhs] == Pi/2 && Im[rhs] <= 0)]}; +applyInverse[ArcCos[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[ + Cos[rhs], (Re[rhs] == 0 && Im[rhs] >= 0) || + 0 < Re[rhs] < Pi || (Re[rhs] == Pi && Im[rhs] <= 0)]}; +applyInverse[ArcTan[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[ + Tan[rhs], (Re[rhs] == -(Pi/2) && Im[rhs] < 0) || -(Pi/2) < + Re[rhs] < Pi/2 || (Re[rhs] == Pi/2 && Im[rhs] > 0)]}; +applyInverse[Cosh[lhs_] -> rhs_, var_Symbol] := + {lhs -> ConditionalExpression[-ArcCosh[rhs] + 2 I \[Pi] C[1], + C[1] \[Element] Integers], lhs -> + ConditionalExpression[ArcCosh[rhs] + 2 I \[Pi] C[1], + C[1] \[Element] Integers]} + +(* Base case: *) + +isolate[var_Symbol -> rest_, var_Symbol] := {var -> rest}; +isolate[lhs_ -> rhs_, var_Symbol] := Module[{inverseApplied}, + (* Switch sides if needed to get var on LHS: *) + + If[(countVar[rhs, var] === 1) && (countVar[lhs, var] === 0), + Return[isolate[rhs -> lhs, var]]]; + + (* Assert var occurs only once in the LHS: *) + + If[!((countVar[lhs, var] === 1) && (countVar[rhs, var] === 0)), + Return[$Failed]]; + + inverseApplied = applyInverse[lhs -> rhs, var]; + If[Head[inverseApplied] =!= List, + Print["Solve error: Finding inverse failed for ", lhs -> rhs, + ", var: ", var]; Return[SolveFailed]]; + + allIsolated = isolate[#, var]& /@ inverseApplied; + Join[Sequence @@ allIsolated] + ]; +checkSolution[eqn_Equal, sol_] := Module[{checkRes}, + checkRes = eqn /. sol // Simplify; + checkRes =!= False +]; +isolateInEqn[eqn_Equal, var_Symbol] := Module[{isolated}, + isolated = {#}& /@ isolate[Rule @@ eqn, var]; + (*Check that the isolate procedure returned sane results.*) + If[!AllTrue[isolated, (Head[#[[1]]] == Rule)&], Return[SolveFailed]]; + (*Canonicalize solutions.*) + isolated = isolated//Simplify//Sort; + isolated = Select[isolated, checkSolution[eqn, #]&]; + isolated +]; + +collect[eqn_Equal, var_Symbol] := Module[{toTry, collected, continue, foundSimpler, toTryFns}, + collected = eqn; + continue = True; + (*These rewrite rules cannot change the meaning of the solution. That is, + applying Exp to both sides would not be a suitable technique here.*) + While[continue, + foundSimpler = False; + toTryFns = { + (#[[1]]-#[[2]]==0)&, + (Expand[#])&, + (ExpandAll[#])& + }; + Do[ + toTry = toTryFn[collected]; + If[Not[foundSimpler] && countVar[toTry, var] < countVar[collected, var], + collected = toTry; + foundSimpler = True; + ]; + , {toTryFn, toTryFns}]; + continue = foundSimpler; + ]; + collected +]; + +(* Expanded equations are defined in equationdata.m. *) +solveQuadratic[a_.*x_^2 + b_.*x_ + c_., x_] := + solveQuadratic[a,b,c,x]/;FreeQ[{a,b,c},x]; +solveCubic[d_.*x_^3 + c_.*x_^2 + b_.*x_ + a_., x_] := + solveCubic[d,c,b,a,x]/;FreeQ[{a,b,c,d},x]; +solveCubic[d_.*x_^3 + c_.*x_^2 + a_., x_] := + solveCubic[d,c,0,a,x]/;FreeQ[{a,c,d},x]; +solveCubic[d_.*x_^3 + b_.*x_ + a_., x_] := + solveCubic[d,0,b,a,x]/;FreeQ[{a,b,d},x]; +solveQuartic[e_.*x_^4 + d_.*x_^3 + c_.*x_^2 + b_.*x_ + a_., x_] := + solveQuartic[e,d,c,b,a,x]/;FreeQ[{a,b,c,d,e},x]; +solveQuartic[e_.*x_^4 + d_.*x_^3 + a_, x_] := + solveQuartic[e,d,0,0,a,x]/;FreeQ[{a,d,e},x]; +solveQuartic[e_.*x_^4 + d_.*x_^3 , x_] := + {{x->0},{x->0},{x->0},{x->-(d/e)}}/;FreeQ[{d,e},x]; + +(* Solve using u-substitution for polynomial-like forms.*) +uSubstitute::usage = + "uSubstitute[eqn, var] takes a non-polynomial `+"`"+`eqn`+"`"+` and attempts a \ +u-substitution such that `+"`"+`eqn`+"`"+` takes a polynomial form. If the \ +substitution succeeds, the function returns `+"`"+`{transformed, uValue}`+"`"+`. \ +The `+"`"+`transformed`+"`"+` equation will include the `+"`"+`uPlaceholder`+"`"+` symbol as \ +the u symbol. The function returns an error symbol if the \ +substitution fails to produce a polynomial form."; +uSubstitute[theEqn_, theVar_Symbol] := + Module[{eqn = theEqn, var = theVar, expGcd, uValue, transformed, + exponents}, + (* Attempt to extract exponents of polynomial-like equations. + We wish to ignore any zero-valued exponents. *) + + exponents = DeleteCases[Exponent[eqn, var, List], 0]; + If[Length[exponents] === 0, Return[$Failed]]; + (* Find the signed GCD of the exponents. + This seems to work for many of the problem cases, + but may not yield a useful polynomial form for all equations. *) + + expGcd = GCD @@ exponents*Sign[exponents[[1]]]; + uValue = var^expGcd; + (* Rewrite the variable with our u-value. *) + + transformed = + eqn /. var -> uPlaceholder^(1/expGcd) // PowerExpand; + (* If our transformed equation is polynomial, we have succeeded. *) + + If[PolynomialQ[transformed, var], {transformed, uValue}, $Failed] + ]; + +uSubstitutionSolve::usage = + "uSubstitutionSolve[eqn, var] takes a non-polynomial `+"`"+`eqn`+"`"+` and \ +attempts a u-substitution such that `+"`"+`eqn`+"`"+` takes a polynomial form in \ +order to solve the equation. The function returns the solve result or \ +`+"`"+`$Failed`+"`"+` in case of an issue."; +uSubstitutionSolve[theEqn_, theVar_Symbol] := + Module[{eqn = theEqn, var = theVar, transformed, uSolved, solved}, + transformed = uSubstitute[eqn, var]; + (* If the u- + substitution fails to produce a polynomial form and instead \ +returns an error symbol, fail this solution attempt. *) + + If[Head[transformed] =!= List, Return[$Failed]]; + (* Find the roots of the equation that was transformed into a \ +polynomial form. *) + + uSolved = Solve[transformed[[1]] == 0, uPlaceholder]; + (* For each of the roots, solve for the original variable. *) + If[Head[uSolved] =!= List, Return[$Failed]]; + + solved = + Map[Solve[transformed[[2]] == #[[1, 2]], var] &, uSolved]; + (* There may be some roots that have no solution for the original \ +variable. Throw these out. *) + + solved = Select[solved, (Length[#] >= 1) &]; + (* No way to handle multiple solutions yet. *) + + If[AnyTrue[solved, (Length[#] != 1) &], Return[$Failed]]; + (* Collect the unique solutions of eqn and return. *) + + Map[First, solved] // DeleteDuplicates // Sort + ]; + +(* Following method described in: *) +(*Sterling, L, Bundy, A, Byrd, L, O'Keefe, R & Silver, B 1982, Solving Symbolic Equations with PRESS. in*) +(*Computer Algebra - Lecture Notes in Computer Science. vol. 7. DOI: 10.1007/3-540-11607-9_13*) +(* Available at: http://www.research.ed.ac.uk/portal/files/413486/Solving_Symbolic_Equations_%20with_PRESS.pdf *) +Solve[eqn_Equal, var_Symbol] := Module[{degree, collected, fullSimplified, poly}, + If[containsZeroOccurrences[eqn, var], Return[{}]]; + (* Attempt isolation *) + If[containsOneOccurrence[eqn, var], Return[isolateInEqn[eqn, var]]]; + + (* Attempt polynomial solve *) + poly = eqn[[1]]-eqn[[2]]; + If[PolynomialQ[poly, var], + degree = Exponent[poly, var]; + If[degree === 2, Return[solveQuadratic[poly//Expand, var]//Simplify//Sort]]; + If[degree === 3, Return[solveCubic[poly//Expand, var]//Simplify//Sort]]; + If[degree === 4, Return[solveQuartic[poly//Expand, var]//Simplify//Sort]]; + Print["Encountered unsolvable polynomial in Solve: ", poly];, + + (* else *) + + uSubstituted = uSubstitutionSolve[eqn[[1]]-eqn[[2]], var]; + If[Head[uSubstituted] === List, Return[uSubstituted]]; + ]; + + collected = collect[eqn, var]; + If[containsOneOccurrence[collected, var], Return[isolateInEqn[collected, var]]]; + + (*Try FullSimplify then solve.*) + fullSimplified = eqn // FullSimplify; + If[fullSimplified =!= eqn, Return[Solve[fullSimplified, var]]]; + + Print["Solve found no solutions for ", eqn, " for ", var]; + SolveFailed + ]; +solveMultOrdered[eqns_List, vars_List] := + Module[{firstSol, secondSol, toSolve}, + If[Length[eqns] =!= 2 || Length[vars] =!= 2, Return[SolveFailed]]; + firstSol = Solve[eqns[[1]], vars[[1]]]; + Print[firstSol]; + If[Length[firstSol] =!= 1, Return[TooManySols1]]; + toSolve := eqns[[2]] /. firstSol[[1, 1]]; + secondSol = Solve[toSolve, vars[[2]]]; + Print[secondSol]; + If[Length[secondSol] =!= 1, Return[TooManySols2]]; + firstSol = firstSol /. secondSol[[1]]; + {{firstSol[[1, 1]], secondSol[[1, 1]]}} + ]; +Solve[eqn_Equal, vars_List] := Solve[eqn, vars[[Length[vars]]]]; +Solve[eqns_List, vars_List] := Module[{res, currVarOrder}, + res = Do[ + res = solveMultOrdered[eqns, currVarOrder]; + If[Head[res] === List, Return[res]]; + , {currVarOrder, Permutations[vars]}]; + If[res === Null, NoneFound, res] + ]; + +(* Special cases for Solve: *) +Solve[False, _] := {}; +Solve[True, _] := {{}}; +Solve[{}, _] := {{}}; +Solve[x_Symbol+a_.*E^x_Symbol==0, x_Symbol] := {{x->-ProductLog[a]}}; +(* Currently needed for Apart: *) +(*Orderless matching would be nice here*) +Solve[{a_.*x_Symbol+b_.*y_Symbol==c_,d_.*x_Symbol+e_.*y_Symbol==f_},{x_Symbol,y_Symbol}] := {{x->-((c e-b f)/(b d-a e)),y->-((-c d+a f)/(b d-a e))}} /;FreeQ[{a,b,c,d,e,f},x]&&FreeQ[{a,b,c,d,e,f},y] +Solve[{a_.*x_Symbol+b_.*y_Symbol==c_,d_.*x_Symbol==f_},{x_Symbol,y_Symbol}] := {{x->f/d,y->-((-c d+a f)/(b d))}}/;FreeQ[{a,b,c,d,f},x]&&FreeQ[{a,b,c,d,f},y] + +Attributes[Solve] = {Protected}; +normSol[s_List] := + Sort[(# /. + ConditionalExpression[e_, a_And] :> + ConditionalExpression[e, a // Sort]) & /@ s]; +Tests`+"`"+`Solve = { + ESimpleExamples[ + ESameTest[{{x->Log[y]/Log[a+b]}}, Solve[(a+b)^x==y,x]], + ESameTest[{{x -> -Sqrt[-3 + y]}, {x -> Sqrt[-3 + y]}}, Solve[y == x^2 + 3, x]], + ESameTest[{{y->1-Sqrt[-2+2 x-x^2]},{y->1+Sqrt[-2+2 x-x^2]}}, Solve[2==x^2+y^2+(x-2)^2+(y-2)^2,y]], + (*ESameTest[{{x -> (-b - Sqrt[b^2 - 4 a c])/(2 a)}, {x -> (-b + Sqrt[b^2 - 4 a c])/(2 a)}}, Solve[a*x^2 + b*x + c == 0, x]],*) + (*ESameTest[{{x -> (-b - Sqrt[b^2 - 4 a c + 4 a d])/(2 a)}, {x -> (-b + Sqrt[b^2 - 4 a c + 4 a d])/(2 a)}}, Print[a,b,c,d,x];Solve[a*x^2 + b*x + c == d, x]]*) + ], ETests[ + (* BASIC ISOLATION *) + + (* Sanity checks *) + ESameTest[{{x -> 0}}, Solve[x == 0, x]], + ESameTest[{{x -> b}}, Solve[x == b, x]], + (* Inverses of addition *) + ESameTest[{{a -> 1.5}}, Solve[a + 1.5 == 3, a]], + (* Inverses of multiplication/division *) + ESameTest[{{x -> b/a}}, Solve[x*a == b, x]], + ESameTest[{{x -> a b}}, Solve[x/a == b, x]], + ESameTest[{{x -> -(b/m)}}, Solve[m*x + b == 0, x]], + ESameTest[{{x -> (-b + c)/m}}, Solve[m*x + b == c, x]], + (* Inverse of exponentiation, var in base *) + ESameTest[{{x -> -Sqrt[a]}, {x -> Sqrt[a]}}, Solve[x^2 == a, x]], + ESameTest[{{x->b^(1/a)}}, Solve[x^a==b,x]], + ESameTest[{{x -> -Sqrt[-3 + y]}, {x -> Sqrt[-3 + y]}}, Solve[y == x^2 + 3, x]], + ESameTest[{{x->2^(1/(5 y+Sin[y]))}}, Solve[x^(5y+Sin[y])==2,x]], + ESameTest[{{a->-b},{a->b}}, Solve[a^2==b^2,a]], + ESameTest[{}, Solve[x^(1/2)==-1,x]], + (* Inverse of exponentiation, var in base: To a fractional power *) + ESameTest[{{x->y^2}}, Solve[Sqrt[x]==y,x]], + ESameTest[{{x->y^9}}, Solve[x^(1/9)==y,x]], + ESameTest[{}, Solve[x^(1/6)==-I,x]], + (* Inverse of exponentiation, var in power *) + ESameTest[{{x->Log[y]/Log[a+b]}}, Solve[(a+b)^x==y,x]], + ESameTest[{{x->ConditionalExpression[1/2 (2 I Pi C[1]+Log[5/4]),C[1]\[Element]Integers]}}, Solve[4E^(2x)==5,x]], + ESameTest[{{x->ConditionalExpression[2 I Pi C[1]+Log[5],C[1]\[Element]Integers]}}, Solve[E^x==5,x]], + ESameTest[{}, Solve[E^x==0,x]], + ESameTest[{{x->ConditionalExpression[2 I Pi C[1],C[1]\[Element]Integers]}}, Solve[E^x==1,x]], + (*ESameTest[{{x->2.4663 Log[y]}}, Solve[1.5^x==y,x]],*) + ESameTest[{{x->ConditionalExpression[(2 I Pi C[1])/Log[3]+Log[y]/Log[3],C[1]\[Element]Integers]}}//Simplify, Solve[3^x == y, x]], + ESameTest[{{x->ConditionalExpression[(2 I Pi C[1])/Log[2]+Log[y]/Log[2],C[1]\[Element]Integers]}}//Simplify, Solve[2^x == y, x]], + ESameTest[{{x->ConditionalExpression[(2 I Pi C[1]+Log[y])/(I Pi+Log[2]),C[1]\[Element]Integers]}}, Solve[(-2)^x == y, x]], + ESameTest[{{x->ConditionalExpression[(2 I Pi C[1]+Log[y])/(I Pi+Log[3]),C[1]\[Element]Integers]}}, Solve[(-3)^x == y, x]], + (* Inverse of log *) + ESameTest[{{x->ConditionalExpression[E^(b+y-z)-y,-PiConditionalExpression[Pi-ArcSin[E^y]+2 Pi C[1],C[1]\[Element]Integers&&-PiConditionalExpression[ArcSin[E^y]+2 Pi C[1],C[1]\[Element]Integers&&-Pi-2-b},{a->2-b}}, Solve[Abs[a+b]==2,a]], + (* Inverse of Sin *) + ESameTest[{{x->ConditionalExpression[-b-ArcSin[c-d]+2 Pi C[1],C[1]\[Element]Integers]},{x->ConditionalExpression[-b+Pi+ArcSin[c-d]+2 Pi C[1],C[1]\[Element]Integers]}}, Solve[Sin[x+b]+c==d,x]], + (* Inverse of Cos *) + ESameTest[{{a->ConditionalExpression[-b-ArcCos[2]+2 Pi C[1],C[1]\[Element]Integers]},{a->ConditionalExpression[-b+ArcCos[2]+2 Pi C[1],C[1]\[Element]Integers]}}, Solve[Cos[a+b]==2,a]], + (* Solve nested trig functions *) + ESameTest[{{x->ConditionalExpression[-ArcSin[ArcCos[y]-2 Pi C[2]]+2 Pi C[1],C[2]\[Element]Integers&&C[1]\[Element]Integers]},{x->ConditionalExpression[Pi+ArcSin[ArcCos[y]-2 Pi C[2]]+2 Pi C[1],C[2]\[Element]Integers&&C[1]\[Element]Integers]},{x->ConditionalExpression[Pi-ArcSin[ArcCos[y]+2 Pi C[2]]+2 Pi C[1],C[2]\[Element]Integers&&C[1]\[Element]Integers]},{x->ConditionalExpression[ArcSin[ArcCos[y]+2 Pi C[2]]+2 Pi C[1],C[2]\[Element]Integers&&C[1]\[Element]Integers]}}//normSol, Solve[Cos[Sin[x]]==y,x]//normSol], + (* Solve combination of Sin and ArcSin *) + ESameTest[{{x->-b+y}}, Solve[Sin[ArcSin[x+b]]==y,x]], + ESameTest[{{x->ConditionalExpression[Pi-ArcSin[Sin[y]]+2 Pi C[1],((Re[y]==-(Pi/2)&&Im[y]>=0)||-(Pi/2)ConditionalExpression[ArcSin[Sin[y]]+2 Pi C[1],((Re[y]==-(Pi/2)&&Im[y]>=0)||-(Pi/2) -1.66352}, {x -> 1.66352}}", "Solve[PDF[NormalDistribution[0, 1], x] == .1, x]"], + + (* POLYNOMIALS *) + ESameTest[{{x->(-b-Sqrt[b^2-4 a c])/(2 a)},{x->(-b+Sqrt[b^2-4 a c])/(2 a)}}, Solve[a*x^2==-b*x-c,x]], + ESameTest[Solve[a x^2==-b x-foo[x],x], Solve[a*x^2==-b*x-foo[x],x]], + ESameTest[{{x->-((I Sqrt[c])/Sqrt[a])},{x->(I Sqrt[c])/Sqrt[a]}}, Solve[a*x^2==-c,x]], + ESameTest[{{x->-I Sqrt[c]},{x->I Sqrt[c]}}, Solve[x^2==-c,x]], + + (* COLLECTION *) + ESameTest[{{x->4}}, Solve[3x+1==2x+5,x]], + ESameTest[{{x->17/15}}, Solve[5(4x-3)+4 (6 x+1)==7 (2 x+3)+2,x]], + + ESameTest[{}, Solve[x^(1/2) == -2, x]], + (* Probably needs solution checking for this to work. *) + ESameTest[{}, Solve[x^(1/4) == -2, x]], + + (* From Sympy. *) + ESameTest[{}, Solve[False, x]], + ESameTest[{}, Solve[False, x]], + ESameTest[{}, Solve[False, x]], + ESameTest[{{}}, Solve[{}, x]], + ESameTest[{{x -> 0}}, Solve[x == 0, x]], + ESameTest[{{x -> 0}}, Solve[x == 0, x]], + ESameTest[{}, Solve[1 + x^(1/2) == 0, x]], + ESameTest[{{x -> 0}}, Solve[2*x == 0, x]], + ESameTest[{}, Solve[x^2 + -Pi == 0, pi]], + ESameTest[{{y -> 3}}, Solve[-3 + y == 0, y]], + ESameTest[{{x -> -a}}, Solve[a + x == 0, x]], + ESameTest[{{x -> -1}}, Solve[y + x*y == 0, x]], + ESameTest[{{x -> 16}}, Solve[-2 + x^(1/4) == 0, x]], + ESameTest[{{x -> 27}}, Solve[-3 + x^(1/3) == 0, x]], + ESameTest[{{x -> 4}}, Solve[-2 + x^(1/2) == 0, x]], + ESameTest[{{x -> 1}}, Solve[-1 + x^(1/2) == 0, x]], + ESameTest[{{x -> 2/3}}, Solve[-2 + 3*x == 0, x]], + ESameTest[{{x -> 2/3}}, Solve[-2 + 3*x == 0, x]], + ESameTest[{}, Solve[1 + x^(1/3) + x^(1/2) == 0, x]], + ESameTest[{{x -> -2}, {x -> 2}}, Solve[-4 + x^2 == 0, x]], + ESameTest[{{x -> -2}, {x -> 2}}, Solve[-4 + x^2 == 0, x]], + ESameTest[{{x -> -1}, {x -> 1}}, Solve[-1 + x^2 == 0, x]], + ESameTest[{{x -> -1}, {x -> 1}}, Solve[-1 + x^2 == 0, x]], + ESameTest[{{x -> -1}, {x -> 1}}, Solve[-1 + x^2 == 0, x]], + ESameTest[{{x -> y^3}}, Solve[x + -y^3 == 0, x]], + ESameTest[{{x -> -ProductLog[-1]}}, Solve[x + -E^x == 0, x]], + ESameTest[{{x -> 0}, {x -> 27}}, Solve[x*(-3 + x^(1/3)) == 0, x]], + ESameTest[{{x -> -2^(1/4)}}, Solve[2^(1/4) + x == 0, x]], + ESameTest[{{x -> -2^(1/2)}}, Solve[2^(1/2) + x == 0, x]], + ESameTest[{{x -> (0 + -1*I)}, {x -> (0 + 1*I)}}, Solve[1 + x^2 == 0, x]], + ESameTest[{{x -> -y*E^((-1)*y)}}, Solve[y + x*E^y == 0, x]], + ESameTest[{{x -> 0}, {x -> 1}}, Solve[4*x*(1 + -x^(1/2)) == 0, x]], + ESameTest[{{x -> (a^(-1)*y)^(b^(-1))}}, Solve[a*x^b + -y == 0, x]], + ESameTest[{{x -> 0}, {x -> a^(-2)}}, Solve[4*x*(1 + -a*x^(1/2)) == 0, x]], + ESameTest[{{x -> ConditionalExpression[(0 + 1*I)*Pi + (0 + 2*I)*Pi*C[1], Element[C[1], Integers]]}}//Simplify, Solve[1 + E^x == 0, x]], + ESameTest[{{x -> y^3}}, Solve[y^(-2)*(1 + -y^2)^(-1/2)*(x + -y^3) == 0, x]], + ESameTest[{{x -> y^3}}, Solve[y^(-2)*(1 + -y^2)^(-1/2)*(x + -y^3) == 0, x]], + ESameTest[{{x -> ConditionalExpression[(0 + 2*I)*Pi*C[1]*Log[3]^(-1) + Log[3]^(-1)*Log[10], Element[C[1], Integers]]}}//Simplify, Solve[-10 + 3^x == 0, x]], + ESameTest[{{x -> ConditionalExpression[(0 + 2*I)*Pi*C[1]*Log[3]^(-1) + Log[3]^(-1)*Log[10], Element[C[1], Integers]]}}//Simplify, Solve[10 + -3^x == 0, x]], + ESameTest[{{y -> x^(1/3)}, {y -> -(-1)^(1/3)*x^(1/3)}, {y -> (-1)^(2/3)*x^(1/3)}}, Solve[x + -y^3 == 0, y]], + ESameTest[{{x -> ConditionalExpression[-ArcCos[(1/2)*y] + 2*Pi*C[1], Element[C[1], Integers]]}, {x -> ConditionalExpression[ArcCos[(1/2)*y] + 2*Pi*C[1], Element[C[1], Integers]]}}, Solve[-y + 2*Cos[x] == 0, x]], + ESameTest[{{x -> -(1 + -a^(1/2))^(1/2)}, {x -> (1 + -a^(1/2))^(1/2)}, {x -> -(1 + a^(1/2))^(1/2)}, {x -> (1 + a^(1/2))^(1/2)}}, Solve[-a + (-1 + x^2)^2 == 0, x]], + ], EKnownFailures[ + ESameTest[{{x->-2 I},{x->-2 I-2 y}}//normSol, Solve[Abs[x+2I+y]==y,x]//normSol], + ], +}; + +ExpreduceTestSolve[fn_] := Module[{testproblems, testi, runSolveTest, testp, res, nCorrect, isCorrect}, + testproblems = ReadList[fn]; + testproblems = Select[testproblems, (MatchQ[#[[2]],_Symbol])&]; + Print[Length[testproblems]]; + + testi = 1; + nCorrect = 0; + + runSolveTest[thei_Integer] := ( + testp = testproblems[[thei]]; + res = Solve[testp[[1]], testp[[2]]] // Timing; + isCorrect = res[[2]] === testp[[3]]; + Print[{testp[[1]], testp[[2]], testp[[3]], res[[2]], res[[1]], isCorrect}]; + If[isCorrect, Print[ + blaESameTest[testp[[3]], blaSolve[testp[[1]], testp[[2]]]] + ]]; + If[isCorrect, nCorrect++]; + ); + + While[testi <= Length[testproblems], + Print[{testi, testproblems[[testi]]}]; + runSolveTest[testi]; + testi = testi+1; + ]; + Print[nCorrect]; +]; + +ConditionalExpression::usage = "`+"`"+`ConditionalExpression[expr, conditions]`+"`"+` represents `+"`"+`expr`+"`"+` which is conditional on `+"`"+`conditions`+"`"+`."; +Attributes[ConditionalExpression] = {Protected}; +ConditionalExpression[e_, True] := e; +ConditionalExpression[e_, False] := Undefined; +ConditionalExpression[ConditionalExpression[a_, b_], c_] := + ConditionalExpression[a, c && b]; +`) + +func resourcesSolveMBytes() ([]byte, error) { + return _resourcesSolveM, nil +} + +func resourcesSolveM() (*asset, error) { + bytes, err := resourcesSolveMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/solve.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesSortM = []byte(`Sort::usage = "`+"`"+`Sort[list]`+"`"+` sorts the elements in list according to `+"`"+`Order`+"`"+`."; +Attributes[Sort] = {Protected}; +Tests`+"`"+`Sort = { + ESimpleExamples[ + EComment["Sort a list of numbers:"], + ESameTest[{-5.1, 2, 3.2, 6}, Sort[{6, 2, 3.2, -5.1}]], + EComment["Sort a list of strings:"], + ESameTest[{"a", "aa", "hello", "zzz"}, Sort[{"hello", "a", "aa", "zzz"}]], + EComment["Sort a list of symbols:"], + ESameTest[{a, b, c, d}, Sort[{d, a, b, c}]], + EComment["Sort a list of heterogenous expressions:"], + ESameTest[{5, h, bar[a^x], foo[y, 2]}, Sort[{5, h, foo[y, 2], bar[a^x]}]] + ], EFurtherExamples[ + EComment["The object to sort need not be a list:"], + ESameTest[foo[a, b, c, d], Sort[foo[d, a, b, c]]] + ], ETests[ + ESameTest[{x, 2*x, 2*x^2, y, 2*y, 2*y^2}, Sort[{x, 2*x, y, 2*y, 2*y^2, 2*x^2}]] + ], EKnownFailures[ + (*Appears to be broken when we switch to the Private`+"`"+` context.*) + (*Minimal example: Sort[{Private`+"`"+`foo[Private`+"`"+`x]^2,(Private`+"`"+`a+Private`+"`"+`b)^2}]*) + ESameTest[{1/a,a,a^2,1/b^2,1/b,b,b^2,1/(a+b)^2,a+b,a+b,(a+b)^2,1/foo[x]^2,foo[x],foo[x]^2}, Sort[{a^-1,a,a^2,b^-2,b^-1,b,b^2,(a+b)^-2,(a+b),(a+b)^2,a+b,foo[x],foo[x]^-2,foo[x]^2}]] + ] +}; + +Order::usage = "`+"`"+`Order[e1, e2]`+"`"+` returns 1 if `+"`"+`e1`+"`"+` should come before `+"`"+`e2`+"`"+` in canonical ordering, -1 if it should come after, and 0 if the two expressions are equal."; +Attributes[Order] = {Protected}; +Tests`+"`"+`Order = { + ESimpleExamples[ + EComment["Find the relative order of symbols:"], + ESameTest[1, Order[a, b]], + ESameTest[-1, Order[b, a]], + ESameTest[1, Order[a, aa]], + EComment["Find the relative order of numbers:"], + ESameTest[-1, Order[2, 1.]], + ESameTest[1, Order[1, 2]], + ESameTest[0, Order[1, 1]], + EComment["Find the relative order of strings:"], + ESameTest[1, Order["a", "b"]], + ESameTest[-1, Order["b", "a"]], + EComment["Find the relative order of heterogenous types:"], + ESameTest[-1, Order[ab, 1]], + ESameTest[1, Order[1, ab]], + ESameTest[-1, Order[y[a], x]], + EComment["Find the relative order of rationals:"], + ESameTest[1, Order[Rational[-5, 3], Rational[-4, 6]]], + ESameTest[-1, Order[Rational[4, 6], .6]], + ESameTest[1, Order[.6, Rational[4, 6]]], + ESameTest[1, Order[Rational[4, 6], .7]], + EComment["Find the relative order of expressions:"], + ESameTest[0, Order[bar[x, y], bar[x, y]]], + ESameTest[1, Order[fizz[bar[x, y]], fizz[bar[x, y, a]]]] + ], ETests[ + (*Symbol ordering*) + ESameTest[0, Order[a, a]], + ESameTest[1, Order[a, b]], + ESameTest[-1, Order[b, a]], + ESameTest[1, Order[a, aa]], + ESameTest[1, Order[aa, aab]], + ESameTest[-1, Order[aab, aa]], + ESameTest[-1, Order[aa, a]], + ESameTest[-1, Order[ab, aa]], + (*Number ordering*) + ESameTest[-1, Order[2, 1.]], + ESameTest[1, Order[1, 2]], + ESameTest[0, Order[1, 1]], + ESameTest[0, Order[1., 1.]], + ESameTest[1, Order[1, 1.]], + ESameTest[-1, Order[1., 1]], + (*Symbols vs numbers*) + ESameTest[-1, Order[ab, 1]], + ESameTest[1, Order[1, ab]], + (*Sort expressions*) + ESameTest[-1, Order[foo[x, y], bar[x, y]]], + ESameTest[1, Order[bar[x, y], foo[x, y]]], + ESameTest[0, Order[bar[x, y], bar[x, y]]], + ESameTest[1, Order[bar[x, y], bar[x, y, z]]], + ESameTest[1, Order[bar[x, y], bar[x, y, a]]], + ESameTest[1, Order[bar[x, y], bar[y, z]]], + ESameTest[-1, Order[bar[x, y], bar[w, x]]], + ESameTest[-1, Order[fizz[foo[x, y]], fizz[bar[x, y]]]], + ESameTest[1, Order[fizz[bar[x, y]], fizz[foo[x, y]]]], + ESameTest[0, Order[fizz[bar[x, y]], fizz[bar[x, y]]]], + ESameTest[1, Order[fizz[bar[x, y]], fizz[bar[x, y, z]]]], + ESameTest[1, Order[fizz[bar[x, y]], fizz[bar[x, y, a]]]], + ESameTest[1, Order[fizz[bar[x, y]], fizz[bar[y, z]]]], + ESameTest[-1, Order[fizz[bar[x, y]], fizz[bar[w, x]]]], + ESameTest[-1, Order[fizz[foo[x, y]], fizz[bar[a, y]]]], + ESameTest[-1, Order[fizz[foo[x, y]], fizz[bar[z, y]]]], + ESameTest[1, Order[1, a[b]]], + ESameTest[1, Order[1., a[b]]], + ESameTest[-1, Order[a[b], 1]], + ESameTest[-1, Order[a[b], 1.]], + ESameTest[1, Order[x, y[a]]], + ESameTest[1, Order[x, w[a]]], + ESameTest[-1, Order[w[a], x]], + ESameTest[-1, Order[y[a], x]], + ESameTest[-1, Order[y[], x]], + ESameTest[-1, Order[y, x]], + ESameTest[-1, Order[w[], x]], + ESameTest[1, Order[w, x]], + (*Test Rational ordering*) + ESameTest[0, Order[Rational[4, 6], Rational[2, 3]]], + ESameTest[1, Order[Rational[4, 6], Rational[5, 3]]], + ESameTest[-1, Order[Rational[5, 3], Rational[4, 6]]], + ESameTest[1, Order[Rational[-5, 3], Rational[-4, 6]]], + ESameTest[-1, Order[Rational[4, 6], .6]], + ESameTest[1, Order[.6, Rational[4, 6]]], + ESameTest[1, Order[Rational[4, 6], .7]], + ESameTest[-1, Order[.7, Rational[4, 6]]], + ESameTest[-1, Order[Rational[4, 6], 0]], + ESameTest[1, Order[Rational[4, 6], 1]], + ESameTest[1, Order[0, Rational[4, 6]]], + ESameTest[-1, Order[1, Rational[4, 6]]], + ESameTest[1, Order[Rational[4, 6], a]], + ESameTest[-1, Order[a, Rational[4, 6]]], + (*Test String ordering*) + ESameTest[-1, Order["a", 5]], + ESameTest[-1, Order["a", 5.5]], + ESameTest[-1, Order["a", 5/2]], + ESameTest[1, Order["a", x]], + ESameTest[1, Order["a", x^2]], + ESameTest[1, Order["a", x^2]], + ESameTest[-1, Order["b", "a"]], + ESameTest[1, Order["a", "b"]], + ESameTest[1, Order["a", "aa"]], + (*Test polynomial ordering*) + ESameTest[-1, Order[x^3,4x^2]], + ESameTest[-1, Order[x^3,4x^2]], + ESameTest[1, Order[1,4x^2]], + ESameTest[1, Order[1,4*Sin[x]^2]], + ESameTest[-1, Order[5x^2,4x^2]], + ESameTest[-1, Order[4x^3,4x^2]], + ESameTest[1, Order[x^2,4x^2]], + ESameTest[1, Order[x^2,foo[x]]], + ESameTest[1, Order[x^2,x*y]], + ESameTest[-1, Order[3x^3,4x^2]], + ESameTest[-1, Order[d*g,d*f]], + ESameTest[0, Order[d*g,d*g]], + ESameTest[1, Order[d*g,d*h]], + ESameTest[1, Order[d*g,e*g]], + ESameTest[1, Order[d*g,e*h]], + ESameTest[-1, Order[d g, e f]], + ESameTest[1, Order[x^2*y,x*y^2]], + ESameTest[1, Order[x^4*y^2,x^2*y^4]], + ESameTest[1, Order[x^2*y,2*x*y^2]], + ESameTest[1, Order[c, 5 * b * c]], + ESameTest[{-1,-1.,-0.1,0,0.1,0.11,2,2,2.,0.5^x,2^x,x,2 x,x^2,x^x,x^(2 x),xxx,2 y}, Sort[{-1,-1.,0.1,0.11,2.,-.1,2,0,2,2*x,2*y,x,xxx,2^x,x^2,x^x,x^(2*x),.5^x}]], + ESameTest[-1, Order[foo[x],-x]], + ESameTest[-1, Order[a[b,piz,foo[]],a[b,foo[]]]], + ESameTest[{-I,1/A,A,1/k,k}, Sort[{A^(-1), k^(-1), (-I), A, k}]], + ], EKnownFailures[ + ESameTest[{a,A,b,B}, Sort[{a,A,b,B}]], + ESameTest[1, Order[a + b - c, a + c]] + ] +}; +`) + +func resourcesSortMBytes() ([]byte, error) { + return _resourcesSortM, nil +} + +func resourcesSortM() (*asset, error) { + bytes, err := resourcesSortMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/sort.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesSpecialsymsM = []byte(`Infinity::usage = "`+"`"+`Infinity`+"`"+` represents the mathematical concept of infinity."; +Plus[Infinity, _, rest___] := Infinity + rest; +Plus[-Infinity, _, rest___] := -Infinity + rest; +Plus[Infinity, -Infinity, rest___] := Indeterminate + rest; +Attributes[Infinity] = {ReadProtected, Protected}; +Tests`+"`"+`Infinity = { + ESimpleExamples[ + ESameTest[Infinity, Infinity - 1], + ESameTest[Infinity, Infinity - 990999999], + ESameTest[Infinity, Infinity - 990999999.], + ESameTest[Indeterminate, Infinity - Infinity], + ESameTest[-Infinity, Infinity*-1], + ESameTest[-Infinity, -Infinity + 1], + ESameTest[-Infinity, -Infinity + 999], + ESameTest[Infinity, -Infinity*-1], + ESameTest[0, 1/Infinity] + ], EKnownFailures[ + (*I can't simplify this type of infinity until I have ;/ rules*) + ESameTest[Infinity, Infinity*2] + ] +}; + +ComplexInfinity::usage = "`+"`"+`ComplexInfinity`+"`"+` represents an an infinite quantity that extends in an unknown direction in the complex plane."; +Attributes[ComplexInfinity] = {Protected}; +Tests`+"`"+`ComplexInfinity = { + ESimpleExamples[ + ESameTest[ComplexInfinity, 0^(-1)], + ESameTest[ComplexInfinity, a/0], + ESameTest[ComplexInfinity, ComplexInfinity * foo[x]], + ESameTest[ComplexInfinity, Factorial[-1]] + ] +}; + +Indeterminate::usage = "`+"`"+`Indeterminate`+"`"+` represents an indeterminate form."; +Attributes[Indeterminate] = {Protected}; +Tests`+"`"+`Indeterminate = { + ESimpleExamples[ + ESameTest[Indeterminate, 0/0], + ESameTest[Indeterminate, Infinity - Infinity], + ESameTest[Indeterminate, 0 * Infinity], + ESameTest[Indeterminate, 0 * ComplexInfinity], + ESameTest[Indeterminate, 0^0] + ] +}; + +Pi::usage = "`+"`"+`Pi`+"`"+` is the constant of pi."; +N[Pi] := 3.141592653589793238462643383279502884197; +Times[Pi, a_Real, rest___] := N[Pi] * a * rest; +Attributes[Pi] = {ReadProtected, Constant, Protected}; + +E::usage = "`+"`"+`E`+"`"+` is the constant for the base of the natural logarithm."; +N[E] := 2.71828182845904523536028747135266249775724709370; +Times[E, a_Real, rest___] := N[E] * a * rest; +Attributes[E] = {ReadProtected, Constant, Protected}; +`) + +func resourcesSpecialsymsMBytes() ([]byte, error) { + return _resourcesSpecialsymsM, nil +} + +func resourcesSpecialsymsM() (*asset, error) { + bytes, err := resourcesSpecialsymsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/specialsyms.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesStatsM = []byte(`NormalDistribution::usage = "`+"`"+`NormalDistribution[mu, sigma]`+"`"+` is a normal distribution with a mean `+"`"+`mu`+"`"+` and standard deviation of `+"`"+`sigma`+"`"+`."; +Attributes[NormalDistribution] = {ReadProtected, Protected}; +Tests`+"`"+`NormalDistribution = { + ESimpleExamples[ + ESameTest[E^(-((x-\[Mu])^2/(2 \[Sigma]^2)))/(Sqrt[2 \[Pi]] \[Sigma]), PDF[NormalDistribution[\[Mu],\[Sigma]],x]] + ] +}; + +LogNormalDistribution::usage = "`+"`"+`LogNormalDistribution[mu, sigma]`+"`"+` is a lognormal distribution with a mean `+"`"+`mu`+"`"+` and standard deviation of `+"`"+`sigma`+"`"+`."; +Attributes[LogNormalDistribution] = {ReadProtected, Protected}; +Tests`+"`"+`LogNormalDistribution = { + ESimpleExamples[ + ESameTest[Piecewise[{{1/(E^((-\[Mu] + Log[x])^2/(2*\[Sigma]^2))*Sqrt[2*Pi]*x*\[Sigma]), x > 0}}, 0], PDF[LogNormalDistribution[\[Mu],\[Sigma]],x]] + ] +}; + +PDF::usage = "`+"`"+`PDF[dist, var]`+"`"+` calculates the PDF of `+"`"+`dist`+"`"+`."; +PDF[NormalDistribution[mu_, sigma_], x_] := E^(-((-mu+x)^2/(2 sigma^2)))/(Sqrt[2 \[Pi]] sigma); +PDF[LogNormalDistribution[mu_, sigma_], x_] := Piecewise[{{1/(E^((-mu + Log[x])^2/(2*sigma^2))*Sqrt[2*Pi]*sigma*x), x > 0}}, 0] +Attributes[PDF] = {ReadProtected, Protected}; +Tests`+"`"+`PDF = { + ESimpleExamples[ + ESameTest[E^(-((x-\[Mu])^2/(2 \[Sigma]^2)))/(Sqrt[2 \[Pi]] \[Sigma]), PDF[NormalDistribution[\[Mu],\[Sigma]],x]], + ESameTest[Piecewise[{{1/(E^((-\[Mu] + Log[x])^2/(2*\[Sigma]^2))*Sqrt[2*Pi]*x*\[Sigma]), x > 0}}, 0], PDF[LogNormalDistribution[\[Mu],\[Sigma]],x]] + ] +}; +`) + +func resourcesStatsMBytes() ([]byte, error) { + return _resourcesStatsM, nil +} + +func resourcesStatsM() (*asset, error) { + bytes, err := resourcesStatsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/stats.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesStringM = []byte(`ToString::usage = "`+"`"+`ToString[expr, form]`+"`"+` converts `+"`"+`expr`+"`"+` into a string using printing method `+"`"+`form`+"`"+`."; +ToString[a_] := ToString[a, OutputForm]; +Attributes[ToString] = {Protected}; +Tests`+"`"+`ToString = { + ESimpleExamples[ + ESameTest["a^2", ToString[Global`+"`"+`a^2, InputForm]], + ESameTest["\sin \\left(1\\right)", ToString[Sin[1], TeXForm]], + ESameTest["Hello World", "Hello World" // ToString] + ], ETests[ + ESameTest["\sin \\left(\\right)", ToString[Sin[], TeXForm]], + ESameTest["\sin \\left(1,2\\right)", ToString[Sin[1, 2], TeXForm]], + ] +}; + +StringJoin::usage = "`+"`"+`s1 <> s2 <> ...`+"`"+` can join a list of strings into a single string."; +(*For some reason this is fast for StringJoin[Table["x", {k,2000}]/.List->Sequence]*) +(*but slow for StringJoin[Table["x", {k,2000}]]*) +(*StringJoin[{args___}]": "StringJoin[args]",*) +(*This rule runs much faster, probably because it avoids*) +(*OrderlessIsMatchQ*) +StringJoin[list_List] := StringJoin[list /. List->Sequence]; +Attributes[StringJoin] = {Flat, OneIdentity, Protected}; +Tests`+"`"+`StringJoin = { + ESimpleExamples[ + ESameTest["Hello World", "Hello" <> " " <> "World"], + ESameTest["If a=2, then a^2=4", "If a=2, then " <> ToString[Global`+"`"+`a^2, InputForm] <> "=" <> ToString[a^2 /. a -> 2, InputForm]] + ], EFurtherExamples[ + EComment["The `+"`"+`StringJoin`+"`"+` of nothing is the empty string:"], + ESameTest["", StringJoin[]], + EComment["If `+"`"+`StringJoin`+"`"+` receives any non-string arguments, the expression does not evaluate:"], + ESameTest["Hello" <> 5, StringJoin["Hello", 5]], + EComment["This function takes `+"`"+`List`+"`"+` arguments as well:"], + ESameTest["abc", StringJoin[{"a", "b", "c"}]] + ] +}; + +Infix::usage = "`+"`"+`Infix[expr, sep]`+"`"+` represents `+"`"+`expr`+"`"+` in infix form with separator `+"`"+`sep`+"`"+` when converted to a string."; +Attributes[Infix] = {Protected}; +Tests`+"`"+`Infix = { + ESimpleExamples[ + ESameTest["bar|fuzz|zip", Infix[foo[Global`+"`"+`bar, Global`+"`"+`fuzz, Global`+"`"+`zip], "|"] // ToString] + ] +}; + +StringLength::usage = "`+"`"+`StringLength[s]`+"`"+` returns the length of s."; +Attributes[StringLength] = {Listable, Protected}; +Tests`+"`"+`StringLength = { + ESimpleExamples[ + ESameTest[5, StringLength["Hello"]] + ] +}; + +StringTake::usage = "`+"`"+`StringTake[s, {start, end}]`+"`"+` takes a substring of s."; +Attributes[StringTake] = {Protected}; +Tests`+"`"+`StringTake = { + ESimpleExamples[ + ESameTest["h", StringTake["hello", {1, 1}]], + ESameTest[StringTake["hello", {0, 1}], StringTake["hello", {0, 1}]], + ESameTest["hello", StringTake["hello", {1, StringLength["hello"]}]], + ESameTest["", StringTake["hello", {2, 1}]], + ESameTest[StringTake["hello", {2, 999}], StringTake["hello", {2, 999}]] + ] +}; + +StringReplace::usage = "`+"`"+`StringReplace[str, before->after]`+"`"+` replaces any occurrence of `+"`"+`before`+"`"+` with `+"`"+`after`+"`"+` in `+"`"+`str`+"`"+`."; +Attributes[StringReplace] = {Protected}; +Tests`+"`"+`StringReplace = { + ESimpleExamples[ + ESameTest["hello foo", StringReplace["hello world", "world"->"foo"]], + ] +}; + +ExportString::usage = "`+"`"+`ExportString[str, \"format\"]`+"`"+` encodes `+"`"+`str`+"`"+` into \"format\"."; +Attributes[ExportString] = {ReadProtected, Protected}; +Tests`+"`"+`ExportString = { + ESimpleExamples[ + ESameTest["SGVsbG8gV29ybGQ=\n", ExportString["Hello World","base64"]], + ], ETests[ + ESameTest["SGVsbG8gV29ybGQ=\n", ExportString["Hello World","Base64"]], + ESameTest[ExportString["kdkfdf"], ExportString["kdkfdf"]], + ESameTest[$Failed, ExportString["kdkfdf", "jkfdfd"]], + ] +}; +`) + +func resourcesStringMBytes() ([]byte, error) { + return _resourcesStringM, nil +} + +func resourcesStringM() (*asset, error) { + bytes, err := resourcesStringMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/string.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesSystemM = []byte(`Echo[expr_] := ( + Print[expr]; + expr +); + +ExpreduceSetLogging::usage = "`+"`"+`ExpreduceSetLogging[bool, level]`+"`"+` sets the logging state to `+"`"+`bool`+"`"+` and the level to `+"`"+`level`+"`"+`."; +Attributes[ExpreduceSetLogging] = {Protected}; + +ExpreduceDefinitionTimes::usage = "`+"`"+`ExpreduceDefinitionTimes[]`+"`"+` prints the time in seconds evaluating various definitions."; +Attributes[ExpreduceDefinitionTimes] = {Protected}; + +Attributes::usage = "`+"`"+`Attributes[sym]`+"`"+` returns a `+"`"+`List`+"`"+` of attributes for `+"`"+`sym`+"`"+`."; +Attributes[Attributes] = {HoldAll, Listable, Protected}; +Tests`+"`"+`Attributes = { + ESimpleExamples[ + ESameTest[{Protected, ReadProtected}, Attributes[Infinity]], + ESameTest[{HoldAll, Listable, Protected}, Attributes[Attributes]], + ESameTest[{Flat, Listable, NumericFunction, OneIdentity, Orderless, Protected}, Attributes[Plus]], + EComment["The default set of attributes is the empty list:"], + ESameTest[{}, Attributes[undefinedSym]] + ], EFurtherExamples[ + EComment["Only symbols can have attributes:"], + ESameTest[Attributes[2], Attributes[2]], + ESameTest[Attributes[a^2], Attributes[a^2]] + ] +}; + +Default::usage = "`+"`"+`Default[sym]`+"`"+` returns the default value of `+"`"+`sym`+"`"+` when used as an `+"`"+`Optional`+"`"+` pattern without a default specified."; +Attributes[Default] = {Protected}; +Tests`+"`"+`Default = { + ESimpleExamples[ + ESameTest[1, Default[Times]], + ESameTest[0, Default[Plus]] + ], ETests[ + ESameTest[Default[foo], Default[foo]] + ] +}; + +Clear::usage = "`+"`"+`Clear[sym1, sym2, ...]`+"`"+` clears the symbol definitions from the evaluation context."; +Attributes[Clear] = {HoldAll, Protected}; +Tests`+"`"+`Clear = { + ESimpleExamples[ + ESameTest[a, a], + ESameTest[5, a = 5], + ESameTest[6, b = 6], + ESameTest[7, c = 7], + ESameTest[5, a], + ESameTest[Null, Clear[a, 99, b]], + ESameTest[Symbol, Head[a]], + ESameTest[Symbol, Head[b]], + ESameTest[Integer, Head[c]], + ESameTest[Null, Clear[c]], + ESameTest[Symbol, Head[c]] + ] +}; + +Timing::usage = "`+"`"+`Timing[expr]`+"`"+` returns a `+"`"+`List`+"`"+` with the first element being the time in seconds for the evaluation of `+"`"+`expr`+"`"+`, and the second element being the result."; +Attributes[Timing] = {HoldAll, SequenceHold, Protected}; +Tests`+"`"+`Timing = { + ESimpleExamples[ + EExampleOnlyInstruction["{0.00167509, 5000000050000000}", "Timing[Sum[a, {a, 100000000}]]"] + ] +}; + +Print::usage = "`+"`"+`Print[expr1, expr2, ...]`+"`"+` prints the string representation of the expressions to the console and returns `+"`"+`Null`+"`"+`."; +Print[expressions___] := + WriteString[OutputStream["stdout", 1], + StringJoin[ToString /@ {expressions}] <> "\n"]; +Attributes[Print] = {Protected}; + +MessageName::usage = "`+"`"+`sym::msg`+"`"+` references a particular message for `+"`"+`sym`+"`"+`."; +Attributes[MessageName] = {HoldFirst, ReadProtected, Protected}; +Tests`+"`"+`MessageName = { + ESimpleExamples[ + EComment["`+"`"+`MessageName`+"`"+` is used to store the usage messages of built-in symbols:"], + ESameTest["`+"`"+`sym::msg`+"`"+` references a particular message for `+"`"+`sym`+"`"+`.", MessageName::usage] + ] +}; + +Trace::usage = "`+"`"+`Trace[expr]`+"`"+` traces the evaluation of `+"`"+`expr`+"`"+`."; +Attributes[Trace] = {HoldAll, Protected}; +Tests`+"`"+`Trace = { + ESimpleExamples[ + ESameTest[List[HoldForm[Plus[1, 2]], HoldForm[3]], 1 + 2 // Trace], + ESameTest[List[List[HoldForm[Plus[1, 3]], HoldForm[4]], HoldForm[Plus[4, 2]], HoldForm[6]], (1 + 3) + 2 // Trace], + ESameTest[List[List[HoldForm[Plus[1, 3]], HoldForm[4]], HoldForm[Plus[2, 4]], HoldForm[6]], 2 + (1 + 3) // Trace] + ], ETests[ + ESameTest[{}, Trace[a + b + c]], + ESameTest[{}, Trace[1]], + ESameTest[{HoldForm[2^2], HoldForm[4]}, Trace[2^2]], + ESameTest[{{HoldForm[2^2], HoldForm[4]}, HoldForm[4*5], HoldForm[20]}, Trace[2^2*5]], + ESameTest[{{{HoldForm[2^2], HoldForm[4]}, HoldForm[4*5], HoldForm[20]}, HoldForm[20 + 4], HoldForm[24]}, Trace[2^2*5+4]], + ESameTest[{{{HoldForm[2^2], HoldForm[4]}, {HoldForm[3^3], HoldForm[27]}, HoldForm[4*27*5], HoldForm[540]}, HoldForm[540 + 4], HoldForm[544]}, Trace[2^2*3^3*5+4]], + ESameTest[{HoldForm[b + a], HoldForm[a + b]}, Trace[b+a]], + ESameTest[{}, Trace[a+foo[a,b]]], + ESameTest[{HoldForm[foo[Sequence[a, b]]], HoldForm[foo[a, b]]}, Trace[foo[Sequence[a,b]]]] + ], EKnownFailures[ + ESameTest[{{{HoldForm[a*a], HoldForm[a^2]}, HoldForm[foo[a^2, b]]}, HoldForm[a + foo[a^2, b]]}, Trace[a+foo[a*a,b]]] + ] +}; + +N::usage = "`+"`"+`N[expr]`+"`"+` attempts to convert `+"`"+`expr`+"`"+` to a numeric value."; +Attributes[N] = {Protected}; +Tests`+"`"+`N = { + ETests[ + ESameTest[2., N[2]], + ESameTest[0.5, N[1/2]], + EStringTest["0.5", "(-1)^(1/6) // N // Im"], + ] +}; + +Listable::usage = "`+"`"+`Listable`+"`"+` is an attribute that calls for functions to automatically map over lists."; +Attributes[Listable] = {Protected}; +Tests`+"`"+`Listable = { + ESimpleExamples[ + ESameTest[{1, 1, 1, 0}, Boole[{True, True, True, False}]], + ESameTest[{False, True, True}, Positive[{-1, 4, 5}]], + ESameTest[{{False, True, True}}, Positive[{{-1, 4, 5}}]], + ESameTest[{{False, True, True}, {True, False}}, Positive[{{-1, 4, 5}, {6, -1}}]] + ], ETests[ + ESameTest[{Positive[-1, 2], Positive[4, 2], Positive[5, 2]}, Positive[{-1, 4, 5}, 2]], + ESameTest[Positive[{-1, 4, 5}, {1, 2}], Positive[{-1, 4, 5}, {1, 2}]], + ESameTest[{Positive[-1, 1], Positive[4, 2], Positive[5, 3]}, Positive[{-1, 4, 5}, {1, 2, 3}]] + ] +}; + +Get::usage = "`+"`"+`Get[file]`+"`"+` loads `+"`"+`file`+"`"+` and returns the last expression."; +Attributes[Get] = {Protected}; + +Save::usage = "`+"`"+`Save[filename, {sym1, sym2, ...}]`+"`"+` saves a list of symbols into a file." +Save[fn_String, pattern_String] := Save[fn, syms] /. syms -> Names[pattern]; +Attributes[Save] = {HoldRest, Protected}; + +Module::usage = "`+"`"+`Module[{locals}, expr]`+"`"+` evaluates `+"`"+`expr`+"`"+` with the local variables `+"`"+`locals`+"`"+`."; +Attributes[Module] = {HoldAll, Protected}; +Tests`+"`"+`Module = { + EKnownFailures[ + (*The numbers are off by N here because the symbols get marked as seen*) + (*upon parsing, probably.*) + ESameTest[{t$1,j$1,2}, $ModuleNumber=1;Module[{t,j},{t,j,$ModuleNumber}]], + ESameTest[{t$2,j$2,3}, $ModuleNumber=1;Module[{t,j},{t,j,$ModuleNumber}]], + ESameTest[{t$4,j$4,5}, $ModuleNumber=1;t$3=test;Module[{t,j},{t,j,$ModuleNumber}]], + ESameTest[{t$8,2,9}, $ModuleNumber=8;t$3=test;Module[{t,j=2},{t,j,$ModuleNumber}]], + ESameTest[{t$9,2,10}, $ModuleNumber=8;t$3=test;Module[{t,j:=2},{t,j,$ModuleNumber}]] + ] +}; + +(* TODO: Fix this hack. *) +Block[args___] := Module[args]; +Attributes[Block] = {HoldAll, Protected}; + +Hash::usage = "`+"`"+`Hash[expr]`+"`"+` returns an integer hash of `+"`"+`expr`+"`"+`."; +Attributes[Hash] = {Protected}; + +BeginPackage::usage = "`+"`"+`BeginPackage[context]`+"`"+` updates the context and sets the context path to only the current context and System."; +Attributes[BeginPackage] = {Protected}; +BeginPackage[c_String] := ( + $ExpreduceOldContext = $Context; + $Context = c; + $ExpreduceOldContextPath = $ContextPath; + $ContextPath = {c, "System`+"`"+`"}; + + $ExpreducePkgContext = c; +); + +EndPackage::usage = "`+"`"+`EndPackage[]`+"`"+` resets the contexts to the original values, but with the package context prepended."; +Attributes[EndPackage] = {Protected}; +EndPackage[] := ( + $Context = $ExpreduceOldContext; + $ExpreduceOldContext = Null; + $ContextPath = Prepend[$ExpreduceOldContextPath, $ExpreducePkgContext]; + $ExpreduceOldContextPath = Null; + + $ExpreducePkgContext = Null; +); + +Begin::usage = "`+"`"+`Begin[context]`+"`"+` updates the context."; +Attributes[Begin] = {Protected}; +Begin[c_String] := ( + $ExpreduceOldContext2 = $Context; + If[StringTake[c, {1, 1}] == "`+"`"+`", + $Context = $Context <> StringTake[c, {2, StringLength[c]}], + $Context = c + ]; + $Context +); + +End::usage = "`+"`"+`End[]`+"`"+` updates the context to rever the changes caused by `+"`"+`Begin`+"`"+`."; +Attributes[End] = {Protected}; +End[] := ( + expreduceToReturn = $Context; + $Context = $ExpreduceOldContext2; + expreduceToReturn +); + +PrintTemporary::usage = "`+"`"+`PrintTemporary[expr1, expr2, ...]`+"`"+` prints the string representation of the expressions to the console and returns `+"`"+`Null`+"`"+`."; +Attributes[PrintTemporary] = {Protected}; +PrintTemporary[exprs___] := Print[exprs]; + +SetAttributes::usage = "`+"`"+`SetAttributes[sym, attributes]`+"`"+` adds the `+"`"+`attributes`+"`"+` to `+"`"+`sym`+"`"+`."; +Attributes[SetAttributes] = {HoldFirst, Protected}; +SetAttributes[s_Symbol, attr_Symbol] := SetAttributes[s, {attr}]; +SetAttributes[s_Symbol, attrs_List] := ( + Attributes[s] = Union[Attributes[s], attrs]; +); +SetAttributes[l_List, attrs_] := Scan[SetAttributes[#, attrs]&, l]; + +ClearAttributes::usage = "`+"`"+`ClearAttributes[sym, attributes]`+"`"+` clears the `+"`"+`attributes`+"`"+` from `+"`"+`sym`+"`"+`."; +Attributes[ClearAttributes] = {HoldFirst, Protected}; +ClearAttributes[s_Symbol, attr_Symbol] := ClearAttributes[s, {attr}]; +ClearAttributes[s_Symbol, attrs_List] := ( + Attributes[s] = Complement[Attributes[s], attrs]; +); +ClearAttributes[l_List, attrs_] := Scan[ClearAttributes[#, attrs]&, l]; + +Protect::usage = "`+"`"+`Protect[sym]`+"`"+` adds the `+"`"+`Protected`+"`"+` attribute to `+"`"+`sym`+"`"+`."; +Attributes[Protect] = {HoldAll, Protected}; +Protect[s_Symbol] := SetAttributes[s, {Protected}]; + +Unprotect::usage = "`+"`"+`Unprotect[sym]`+"`"+` clears the `+"`"+`Protected`+"`"+` attribute from `+"`"+`sym`+"`"+`."; +Attributes[Unprotect] = {HoldAll, Protected}; +Unprotect[s_Symbol] := ClearAttributes[s, {Protected}]; + +Quiet::usage = "`+"`"+`Quiet[e]`+"`"+` runs `+"`"+`e`+"`"+` without printing any messages."; +Attributes[Quiet] = {HoldAll, Protected}; +Quiet[e_] := e; + +ReadList::usage = "`+"`"+`ReadList[file]`+"`"+` reads expressions in `+"`"+`file`+"`"+` into a list."; +Attributes[ReadList] = {Protected}; + +TimeConstrained::usage = "`+"`"+`TimeConstrained[expr, limit]`+"`"+` evaluates `+"`"+`expr`+"`"+` but expires after `+"`"+`limit`+"`"+`."; +(*TODO: Perhaps we can convert the checks to HasThrown to be more generic.*) +(*They can just check if we should abort and return a certain vailue*) +Attributes[TimeConstrained] = {HoldAll, Protected}; +TimeConstrained[e_, t_] := TimeConstrained[e, t, $Aborted]; +(* Hack just to get this function returning useful values for Rubi. *) +TimeConstrained[e_, t_, f_] := e; + +Throw::usage = "`+"`"+`Throw[e]`+"`"+` stops all execution, propagating the value down with the intention of it being caught. Only some of the flow control statements in the language actually support execution interruption right now."; +Attributes[Throw] = {Protected}; + +Catch::usage = "`+"`"+`Catch[e]`+"`"+` catches and returns any `+"`"+`Thrown`+"`"+` expressions, if any. Otherwise returns the result of `+"`"+`e`+"`"+`."; +Attributes[Catch] = {HoldFirst, Protected}; +Tests`+"`"+`Catch = { + ESimpleExamples[ + ESameTest[c, Catch[a; b; Throw[c]; d; e]], + ESameTest[c, Catch[{a, b, Throw[c], d}]] + ] +}; + +DownValues::usage = "`+"`"+`DownValues[sym]`+"`"+` returns a list of downvalues for `+"`"+`sym`+"`"+`."; +Attributes[DownValues] = {HoldAll, Protected}; +Tests`+"`"+`DownValues = { + ETests[ + ESameTest[Null, ClearAll[myfoo]], + ESameTest[Null, myfoo[_] := a /; b;], + ESameTest[1, Length[DownValues[myfoo]]], + (*Defining the exact same definition has no effect.*) + ESameTest[Null, myfoo[_] := a /; b;], + ESameTest[1, Length[DownValues[myfoo]]], + (*Defining the same condition but with a different RHS will overwrite.*) + ESameTest[Null, myfoo[_] := b /; b;], + ESameTest[1, Length[DownValues[myfoo]]], + (*While this new definition has the same LHS, the condition on the RHS*) + (*is different so it should count as a separate DownValue.*) + ESameTest[Null, myfoo[_] := a /; c;], + ESameTest[2, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{}, a /; c];], + ESameTest[3, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{}, b /; c];], + ESameTest[3, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{a=2}, b /; c];], + ESameTest[4, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{a=3}, b /; c];], + ESameTest[5, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{a=2}, c /; c];], + ESameTest[5, Length[DownValues[myfoo]]], + ESameTest[Null, myfoo[_] := With[{a=2}, c /; c] /; True;], + ESameTest[6, Length[DownValues[myfoo]]], + ESameTest[6, Length[DownValues[myfoo]]], + + (*Test assignment.*) + ESameTest[Null, DownValues[dvAssignTest] = { + HoldPattern[dvAssignTest[1]] :> 123, + HoldPattern[dvAssignTest[_]] :> 321 + };], + ESameTest[123, dvAssignTest[1]], + ESameTest[321, dvAssignTest[2]], + ] +}; + +ClearAll::usage = "`+"`"+`ClearAll[s1, s2, ...]`+"`"+` clears all definitions associated with the symbols."; +Attributes[ClearAll] = {HoldAll, Protected}; +ClearAll[syms___] := Scan[ + ( + Attributes[#] = {}; + Clear[#]; + )&, + {syms} +]; +Tests`+"`"+`ClearAll = { + ESimpleExamples[ + ESameTest[{HoldAll}, Attributes[mytestsym] = {HoldAll}], + ESameTest[Null, mytestsym[5] := 6], + ESameTest[Null, mytestsym[7] := 8], + ESameTest[1, Length[Attributes[mytestsym]]], + ESameTest[2, Length[DownValues[mytestsym]]], + ESameTest[Null, ClearAll[mytestsym]], + ESameTest[0, Length[Attributes[mytestsym]]], + ESameTest[0, Length[DownValues[mytestsym]]] + ] +}; + +ExpreduceMaskNonConditional::usage = "`+"`"+`ExpreduceMaskNonConditional[expr]`+"`"+` returns a version of `+"`"+`expr`+"`"+` with the nonconditional part masked out. Used internally for definitions."; +Attributes[ExpreduceMaskNonConditional] = {HoldAll, Protected}; +Tests`+"`"+`ExpreduceMaskNonConditional = { + ETests[ + ESameTest[Hold[ExpreduceNonConditional], ExpreduceMaskNonConditional[a+b]], + ESameTest[Hold[With[{}, ExpreduceNonConditional /; c]], ExpreduceMaskNonConditional[With[{}, a /; c]]], + ESameTest[Hold[With[{a=2}, ExpreduceNonConditional /; c] /; True], ExpreduceMaskNonConditional[With[{a=2}, c /; c] /; True]], + ESameTest[Hold[ExpreduceNonConditional], ExpreduceMaskNonConditional[With[{}, a]]], + ] +}; + +Unique::usage = "`+"`"+`Unique[]`+"`"+` returns a unique symbol."; +Attributes[Unique] = {Protected}; + +Attributes[Defer] = {HoldAll, Protected, ReadProtected}; + +Sow::usage = "`+"`"+`Sow[e]`+"`"+` sows a value `+"`"+`e`+"`"+` for `+"`"+`Reap[]`+"`"+`."; +Attributes[Sow] = {Protected}; + +Reap::usage = "`+"`"+`Reap[expr]`+"`"+` returns the result of `+"`"+`expr`+"`"+` and a list of all the sown values during evaluation."; +Attributes[Reap] = {HoldFirst, Protected}; + +Definition::usage = "`+"`"+`Definition[sym]`+"`"+` renders the attributes, downvalues, and default value of `+"`"+`sym`+"`"+`."; +Attributes[Definition] = {HoldAll, Protected}; + +Information::usage = "`+"`"+`Information[sym]`+"`"+` renders the usage, attributes, downvalues, and default value of `+"`"+`sym`+"`"+`."; +Information[sym_Symbol, LongForm->useLongForm_] := ( + Print[sym::usage]; + If[useLongForm, + Print[""]; + Print[Definition[sym]]; + ]; +); +Information[sym_Symbol] := Information[sym, LongForm->True]; +Attributes[Information] = {HoldAll, Protected, ReadProtected}; + +Attributes[OutputStream] = {Protected, ReadProtected}; +Attributes[WriteString] = {Protected}; + +Streams::usage = "`+"`"+`Streams[]`+"`"+` gets a list of all open streams."; +Attributes[Streams] = {Protected}; +Tests`+"`"+`Streams = { + ESimpleExamples[ + ESameTest[{OutputStream["stdout", 1], OutputStream["stderr", 2]}, Streams[]], + ] +}; + +Names::usage = "`+"`"+`Names[]`+"`"+` returns a list of all defined symbols. + +`+"`"+`Names[\"pattern\"]`+"`"+` returns a list of all defined symbols matching the regex pattern."; +Attributes[Names] = {Protected}; +Tests`+"`"+`Names = { + ETests[ + ESameTest[True, Length[Names[]] > 1], + ] +}; +`) + +func resourcesSystemMBytes() ([]byte, error) { + return _resourcesSystemM, nil +} + +func resourcesSystemM() (*asset, error) { + bytes, err := resourcesSystemMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/system.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesTestsM = []byte(` +Tests`+"`"+`ExpreduceMiscTests = { + ETests[ + ESameTest[Null, LoadRubiBundledSnapshot[]], + + ESameTest[1/2 E^((I Pi)/4), z=1/2 E^(\[ImaginaryJ]*Pi/4)], + ESameTest[1/(2 Sqrt[2]), Re[z]], + ESameTest[1/(2 Sqrt[2]), 1/2*Cos[Pi/4]], + ESameTest[1/(2 Sqrt[2]), Im[z]], + ESameTest[1/(2 Sqrt[2]), 1/2*Sin[Pi/4]], + ESameTest[1/2, Abs[z]], + ESameTest[Pi/4, Arg[z]], + ESameTest[-(1/(2 Sqrt[2])), Conjugate[z]//Im], + ESameTest[1/Sqrt[2], z+Conjugate[z]//FullSimplify], + ESameTest[1/Sqrt[2], Cos[Pi/4]], + ESameTest[{-(1/2),0}, ReIm[1/2 E^(\[ImaginaryJ]*Pi)]], + ESameTest[{-(1/2),0}, ReIm[1/2 E^(-\[ImaginaryJ]*Pi)]], + ESameTest[{0,1}, ReIm[E^(\[ImaginaryJ]*Pi/2)]], + ESameTest[{0,-1}, ReIm[E^(-\[ImaginaryJ]*Pi/2)]], + ESameTest[{0,1}, ReIm[E^(\[ImaginaryJ]*5Pi/2)]], + ESameTest[{5,0}, ReIm[5*E^(\[ImaginaryJ](0))]], + ESameTest[{-2,0}, ReIm[2*E^(\[ImaginaryJ](Pi))]], + ESameTest[{0,-3}, ReIm[3*E^(\[ImaginaryJ](-Pi/2))]], + ESameTest[{1/2,-(Sqrt[3]/2)}, ReIm[1*E^(\[ImaginaryJ](-2Pi/6))]], + ESameTest[{1,1}, ReIm[Sqrt[2]*E^(\[ImaginaryJ](Pi/4))]], + (* a,b,c *) + ESameTest[{-Im[E^(10 I t)],Re[E^(10 I t)]}, ReIm[\[ImaginaryJ]*E^(\[ImaginaryJ]*10*t)]], + (* Period: .6283 *) + ENearlySameTest[0.628319, 2Pi/10//N], + ESameTest[Null, myf[t_]:=E^((-1+\[ImaginaryJ])t);], + ESameTest[True, myf[t]==E^-t*E^(\[ImaginaryJ]*t)], + (* Always decreasing, not periodic *) + (* Period: .285*) + ENearlySameTest[0.285714, 2Pi/(7Pi)//N], + (* Period: 3.167 (maybe pi) *) + ENearlySameTest[0.628319, 2Pi/10//N], + ENearlySameTest[1.5708, 2Pi/4//N], + ENearlySameTest[3.14159, LCM[2/10,2/4]*Pi//N], + ENearlySameTest[3.14159, LCM[1/5,1/2]*Pi//N], + + ESameTest[Null, ClearAll[z, myf]], + + ESameTest[1/2 E^(-I t Subscript[\[Omega], 0])+1/2 E^(I t Subscript[\[Omega], 0]), TrigToExp[Cos[Subscript[\[Omega], 0]*t]]], + ESameTest[1/2 I E^(-I t Subscript[\[Omega], 0])-1/2 I E^(I t Subscript[\[Omega], 0]), TrigToExp[Sin[Subscript[\[Omega], 0]*t]]], + ESameTest[1/2 E^(-((I Pi)/4)-2 I t)+1/2 E^((I Pi)/4+2 I t), TrigToExp[Cos[2t+Pi/4]]], + ESameTest[1/2 E^(-4 I t)+1/2 E^(4 I t)+1/2 I E^(-6 I t)-1/2 I E^(6 I t), TrigToExp[Cos[4t]+Sin[6t]]], + ESameTest[1/2-1/4 E^(-2 I t)-1/4 E^(2 I t), TrigToExp[Sin[t]^2]], + ESameTest[-(I/2), 1/(2\[ImaginaryJ])//FullSimplify], + ESameTest[I/2, -1/(2\[ImaginaryJ])//FullSimplify], + ESameTest[A/2, 1/Subscript[T, 0]*Integrate[A*E^(-I*k*Subscript[\[Omega], 0]t)/.k->0,{t,0,Subscript[T, 0]/2}]], + ESameTest[(Sin[k Pi] Subscript[T, 0])/(2 k Pi), Integrate[Cos[(2Pi*k*t)/Subscript[T, 0]],{t,0,Subscript[T, 0]/2}]], + ESameTest[(I Sin[(k Pi)/2]^2 Subscript[T, 0])/(k Pi), Integrate[I*Sin[(2Pi*k*t)/Subscript[T, 0]],{t,0,Subscript[T, 0]/2}]], + ] +}; +`) + +func resourcesTestsMBytes() ([]byte, error) { + return _resourcesTestsM, nil +} + +func resourcesTestsM() (*asset, error) { + bytes, err := resourcesTestsMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/tests.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesTimeM = []byte(`UnixTime::usage = "`+"`"+`UnixTime[]`+"`"+` returns the integer seconds since the Unix epoch in UTC time."; +Attributes[UnixTime] = {ReadProtected, Protected}; +Tests`+"`"+`UnixTime = { + ESimpleExamples[ + EComment["Get the current Unix timestamp:"], + EExampleOnlyInstruction["1484805639", "UnixTime[]"], + EComment["`+"`"+`UnixTime`+"`"+` returns an Integer:"], + ESameTest[Integer, UnixTime[] // Head] + ] +}; + +Pause::usage = "`+"`"+`Pause[d]`+"`"+` sleeps for `+"`"+`d`+"`"+` seconds."; +Attributes[Pause] = {Protected}; +Tests`+"`"+`Pause = { + ESimpleExamples[ + EExampleOnlyInstruction["Null", "Pause[0.001]"], + ] +}; +`) + +func resourcesTimeMBytes() ([]byte, error) { + return _resourcesTimeM, nil +} + +func resourcesTimeM() (*asset, error) { + bytes, err := resourcesTimeMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/time.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +var _resourcesTrigM = []byte(`Sin::usage = "`+"`"+`Sin[x]`+"`"+` is the sine of `+"`"+`x`+"`"+`."; +Sin[0] := 0; +Sin[-x_] := -Sin[x]; +Sin[p_Plus] := -Sin[-p] /; (MatchQ[p[[1]], -_] || p[[1]] < 0); +Sin[x_Integer?Negative] := -Sin[-x]; +Sin[Pi] := 0; +Sin[n_Integer*Pi] := 0; +Sin[I*a_] := I*Sinh[a]; +Sin[(-5/2)*Pi] := -1; +Sin[(-3/2)*Pi] := 1; +Sin[(-2/3)*Pi] := -Sqrt[3]/2; +Sin[(-1/2)*Pi] := -1; +Sin[(-1/3)*Pi] := -(Sqrt[3]/2); +Sin[(1/4)*Pi] := 1/Sqrt[2]; +Sin[(1/3)*Pi] := Sqrt[3]/2; +Sin[(1/2)*Pi] := 1; +Sin[(2/3)*Pi] := Sqrt[3]/2; +Sin[(3/2)*Pi] := -1; +Sin[(5/2)*Pi] := 1; +Sin[Indeterminate] := Indeterminate; +Sin[ArcSin[a_]] := a; +Sin[ArcTan[1/2]] := 1/Sqrt[5]; +Attributes[Sin] = {Listable, NumericFunction, Protected}; + +Cos::usage = "`+"`"+`Cos[x]`+"`"+` is the cosine of `+"`"+`x`+"`"+`."; +Cos[0] := 1; +Cos[Pi] := -1; +Cos[n_Integer?EvenQ*Pi] := 1; +Cos[n_Integer?OddQ*Pi] := -1; +Cos[(-5/2)*Pi] := 0; +Cos[(-3/2)*Pi] := 0; +Cos[(-1/2)*Pi] := 0; +Cos[(-1/3)*Pi] := 1/2; +Cos[(1/4)*Pi] := 1/Sqrt[2]; +Cos[(1/3)*Pi] := 1/2; +Cos[(1/2)*Pi] := 0; +Cos[(2/3)*Pi] := -1/2; +Cos[(3/2)*Pi] := 0; +Cos[(5/2)*Pi] := 0; +Cos[I*a_] := Cosh[a]; +Cos[-x_] := Cos[x]; +Cos[x_Integer?Negative] := Cos[-x]; +Cos[inner : Verbatim[Plus][Repeated[_*I]]] := Cosh[-I*inner // Distribute] +Cos[Indeterminate] := Indeterminate; +Cos[ArcCos[a_]] := a; +Cos[ArcTan[1/2]] := 2/Sqrt[5]; +Attributes[Cos] = {Listable, NumericFunction, Protected}; + +Tan::usage = "`+"`"+`Tan[x]`+"`"+` is the tangent of `+"`"+`x`+"`"+`."; +Tan[x_]^(-1) := Cot[x]; +Tan[a_+Pi/2] := Cot[-a]; +(*Tan[a_-Pi/2] := (Print[a];-Cot[a]);*) +Tan[ArcTan[a_]] := a; +Attributes[Tan] = {Listable, NumericFunction, Protected}; + +Cot::usage = "`+"`"+`Cot[x]`+"`"+` is the cotangent of `+"`"+`x`+"`"+`."; +Cot[x_]^(-1) := Tan[x]; +Cot[x_ + Pi/2] := -Tan[x]; +Cot[Verbatim[Plus][-1*a_, b___]] := -Cot[a-b]; +Attributes[Cot] = {Listable, NumericFunction, Protected}; + +Sec[x_ - Pi/2] := Csc[x]; +Attributes[Sec] = {Listable, NumericFunction, Protected}; + +Csc[inner : Verbatim[Plus][Repeated[_*I]]] := -I*Csch[-I*inner // Distribute] +Attributes[Csc] = {Listable, NumericFunction, Protected}; + +Cosh[a_]*Csch[a_]^(b_Integer?Positive)*rest___ := Coth[a]*Csch[a]^(b - 1)*rest +Attributes[Cosh] = {Listable, NumericFunction, Protected}; + +ArcSin[p_Plus] := -ArcSin[-p] /; (MatchQ[p[[1]], -_] || p[[1]] < 0); +ArcSin[-x_] := -ArcSin[x]; +ArcSin[0] := 0; +Attributes[ArcSin] = {Listable, NumericFunction, Protected, ReadProtected}; + +Attributes[ArcCos] = {Listable, NumericFunction, Protected, ReadProtected}; + +ArcTan[-1] := -Pi/4; +ArcTan[0] := 0; +ArcTan[1] := Pi/4; +ArcTan[x_,y_] := Which[ + x > 0, ArcTan[y/x], + x < 0 && y >= 0, ArcTan[y/x] + Pi, + x < 0 && y < 0, ArcTan[y/x] - Pi, + x == 0 && y > 0, Pi/2, + x == 0 && y < 0, -Pi/2, + True, Indeterminate]; +Attributes[ArcTan] = {Listable, NumericFunction, Protected, ReadProtected}; + +TrigExpand[Cos[2*a_]] := Cos[a]^2-Sin[a]^2; +TrigExpand[Cos[a_]] := Cos[a]; +TrigExpand[a_] := (Print["Unsupported call to TrigExpand", a];a); +Attributes[TrigExpand] = {Protected}; + +TrigReduce[a_] := (Print["Unsupported call to TrigReduce", a];a); +Attributes[TrigReduce] = {Protected}; + +trigToExpInner[n_Integer] := n; +trigToExpInner[sym_Symbol] := sym; +trigToExpInner[Cos[inner_]] := E^(-I inner//Expand)/2+E^(I inner//Expand)/2; +trigToExpInner[Sin[inner_]] := 1/2 I E^(-I inner//Expand)-1/2 I E^(I inner//Expand); +trigToExpInner[Tan[inner_]] := (I (E^(-I inner)-E^(I inner)))/(E^(-I inner)+E^(I inner)); +trigToExpInner[a_] := (Print["Unsupported call to TrigToExp", a];a); +TrigToExp[exp_] := Map[trigToExpInner, exp, {0, Infinity}]//Expand; +Attributes[TrigToExp] = {Listable, Protected}; +Tests`+"`"+`TrigToExp = { + ESimpleExamples[ + ESameTest[1/2 E^(-I x y)+1/2 E^(I x y), TrigToExp[Cos[x*y]]], + ESameTest[b+1/2 E^(-I x-I y)+1/2 E^(I x+I y), TrigToExp[Cos[x+y]+b]], + ESameTest[1/2 I E^(-I x-I y)-1/2 I E^(I x+I y), TrigToExp[Sin[x+y]]], + ESameTest[x, TrigToExp[x]], + ], EKnownFailures[ + ESameTest[1/2-1/4 E^(-2 I x-2 I y)-1/4 E^(2 I x+2 I y), TrigToExp[Sin[x+y]^2]], + ESameTest[1/2+E^(-I x)/2+E^(I x)/2-1/4 E^(-2 I x-2 I y)-1/4 E^(2 I x+2 I y), TrigToExp[Sin[x+y]^2+Cos[x]]], + ESameTest[1/2 E^(1/2 (E^(-I x)-E^(I x)))+1/2 E^(1/2 (-E^(-I x)+E^(I x))), TrigToExp[Cos[Sin[x]]]], + ESameTest[(I (E^(-I x)-E^(I x)))/(E^(-I x)+E^(I x)), TrigToExp[Tan[x]]], + ] +}; + +Degree::usage = "`+"`"+`Degree`+"`"+` stands for Pi/180." +Degree = Pi/180; +Attributes[Degree] = {Constant,Protected,ReadProtected}; +Tests`+"`"+`Degree = { + ESimpleExamples[ + ESameTest[1, Sin[90 Degree]], + ] +}; + +RotationMatrix::usage = "`+"`"+`RotationMatrix[θ]`+"`"+` yields a rotation matrix for the angle `+"`"+`θ`+"`"+`."; +RotationMatrix[θ_] := {{Cos[θ],-Sin[θ]},{Sin[θ],Cos[θ]}}; +RotationMatrix[θ_, {x_, 0, 0}] := {{(x^3 Conjugate[x]^3)/Abs[x]^6,0,0},{0,Cos[\[Theta]],-((x^2 Conjugate[x] Sin[\[Theta]])/Abs[x]^3)},{0,(x Conjugate[x]^2 Sin[\[Theta]])/Abs[x]^3,(x^3 Conjugate[x]^3 Cos[\[Theta]])/Abs[x]^6}}; +RotationMatrix[θ_, {0, y_, 0}] := {{Cos[\[Theta]],0,(y^2 Conjugate[y] Sin[\[Theta]])/Abs[y]^3},{0,(y^3 Conjugate[y]^3)/Abs[y]^6,0},{-((y Conjugate[y]^2 Sin[\[Theta]])/Abs[y]^3),0,(y^3 Conjugate[y]^3 Cos[\[Theta]])/Abs[y]^6}}; +RotationMatrix[θ_, {0, 0, z_}] := {{Cos[\[Theta]],-((z Sin[\[Theta]])/Abs[z]),0},{(Conjugate[z] Sin[\[Theta]])/Abs[z],(z Conjugate[z] Cos[\[Theta]])/Abs[z]^2,0},{0,0,(z Conjugate[z])/Abs[z]^2}}; +Tests`+"`"+`RotationMatrix = { + ESimpleExamples[ + ESameTest[{{0, -1}, {1, 0}}, RotationMatrix[90 Degree]], + ] +}; +`) + +func resourcesTrigMBytes() ([]byte, error) { + return _resourcesTrigM, nil +} + +func resourcesTrigM() (*asset, error) { + bytes, err := resourcesTrigMBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "resources/trig.m", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +// Asset loads and returns the asset for the given name. +// It returns an error if the asset could not be found or +// could not be loaded. +func Asset(name string) ([]byte, error) { + cannonicalName := strings.Replace(name, "\\", "/", -1) + if f, ok := _bindata[cannonicalName]; ok { + a, err := f() + if err != nil { + return nil, fmt.Errorf("Asset %s can't read by error: %v", name, err) + } + return a.bytes, nil + } + return nil, fmt.Errorf("Asset %s not found", name) +} + +// MustAsset is like Asset but panics when Asset would return an error. +// It simplifies safe initialization of global variables. +func MustAsset(name string) []byte { + a, err := Asset(name) + if err != nil { + panic("asset: Asset(" + name + "): " + err.Error()) + } + + return a +} + +// AssetInfo loads and returns the asset info for the given name. +// It returns an error if the asset could not be found or +// could not be loaded. +func AssetInfo(name string) (os.FileInfo, error) { + cannonicalName := strings.Replace(name, "\\", "/", -1) + if f, ok := _bindata[cannonicalName]; ok { + a, err := f() + if err != nil { + return nil, fmt.Errorf("AssetInfo %s can't read by error: %v", name, err) + } + return a.info, nil + } + return nil, fmt.Errorf("AssetInfo %s not found", name) +} + +// AssetNames returns the names of the assets. +func AssetNames() []string { + names := make([]string, 0, len(_bindata)) + for name := range _bindata { + names = append(names, name) + } + return names +} + +// _bindata is a table, holding each asset generator, mapped to its name. +var _bindata = map[string]func() (*asset, error){ + "resources/arithmetic.m": resourcesArithmeticM, + "resources/atoms.m": resourcesAtomsM, + "resources/boolean.m": resourcesBooleanM, + "resources/calculus.m": resourcesCalculusM, + "resources/combinatorics.m": resourcesCombinatoricsM, + "resources/comparison.m": resourcesComparisonM, + "resources/equationdata.m": resourcesEquationdataM, + "resources/expression.m": resourcesExpressionM, + "resources/flowcontrol.m": resourcesFlowcontrolM, + "resources/functional.m": resourcesFunctionalM, + "resources/init.m": resourcesInitM, + "resources/list.m": resourcesListM, + "resources/manip.m": resourcesManipM, + "resources/matrix.m": resourcesMatrixM, + "resources/numbertheory.m": resourcesNumbertheoryM, + "resources/pattern.m": resourcesPatternM, + "resources/plot.m": resourcesPlotM, + "resources/power.m": resourcesPowerM, + "resources/random.m": resourcesRandomM, + "resources/replacement.m": resourcesReplacementM, + "resources/rubi/1.1.1 Linear binomial products.m": resourcesRubi111LinearBinomialProductsM, + "resources/rubi/1.1.3 General binomial products.m": resourcesRubi113GeneralBinomialProductsM, + "resources/rubi/1.1.4 Improper binomial products.m": resourcesRubi114ImproperBinomialProductsM, + "resources/rubi/1.2.1 Quadratic trinomial products.m": resourcesRubi121QuadraticTrinomialProductsM, + "resources/rubi/1.2.2 Quartic trinomial products.m": resourcesRubi122QuarticTrinomialProductsM, + "resources/rubi/1.2.3 General trinomial products.m": resourcesRubi123GeneralTrinomialProductsM, + "resources/rubi/1.2.4 Improper trinomial products.m": resourcesRubi124ImproperTrinomialProductsM, + "resources/rubi/1.3 Miscellaneous algebraic functions.m": resourcesRubi13MiscellaneousAlgebraicFunctionsM, + "resources/rubi/2 Exponentials.m": resourcesRubi2ExponentialsM, + "resources/rubi/3 Logarithms.m": resourcesRubi3LogarithmsM, + "resources/rubi/4.1 Sine.m": resourcesRubi41SineM, + "resources/rubi/4.2 Tangent.m": resourcesRubi42TangentM, + "resources/rubi/4.3 Secant.m": resourcesRubi43SecantM, + "resources/rubi/4.4 Miscellaneous trig functions.m": resourcesRubi44MiscellaneousTrigFunctionsM, + "resources/rubi/5 Inverse trig functions.m": resourcesRubi5InverseTrigFunctionsM, + "resources/rubi/6 Hyperbolic functions.m": resourcesRubi6HyperbolicFunctionsM, + "resources/rubi/7 Inverse hyperbolic functions.m": resourcesRubi7InverseHyperbolicFunctionsM, + "resources/rubi/8 Special functions.m": resourcesRubi8SpecialFunctionsM, + "resources/rubi/9.1 Integrand simplification rules.m": resourcesRubi91IntegrandSimplificationRulesM, + "resources/rubi/9.2 Derivative integration rules.m": resourcesRubi92DerivativeIntegrationRulesM, + "resources/rubi/9.3 Piecewise linear functions.m": resourcesRubi93PiecewiseLinearFunctionsM, + "resources/rubi/9.4 Miscellaneous integration rules.m": resourcesRubi94MiscellaneousIntegrationRulesM, + "resources/rubi/Integration Utility Functions.m": resourcesRubiIntegrationUtilityFunctionsM, + "resources/rubi/MakeRubiMxFile.m": resourcesRubiMakerubimxfileM, + "resources/rubi/README": resourcesRubiReadme, + "resources/rubi/Rubi.m": resourcesRubiRubiM, + "resources/rubi/Rubi4.12.nb": resourcesRubiRubi412Nb, + "resources/rubi/ShowStep Routines.m": resourcesRubiShowstepRoutinesM, + "resources/rubi.m": resourcesRubiM, + "resources/rubi_loader.m": resourcesRubi_loaderM, + "resources/simplify.m": resourcesSimplifyM, + "resources/solve.m": resourcesSolveM, + "resources/sort.m": resourcesSortM, + "resources/specialsyms.m": resourcesSpecialsymsM, + "resources/stats.m": resourcesStatsM, + "resources/string.m": resourcesStringM, + "resources/system.m": resourcesSystemM, + "resources/tests.m": resourcesTestsM, + "resources/time.m": resourcesTimeM, + "resources/trig.m": resourcesTrigM, +} + +// AssetDir returns the file names below a certain +// directory embedded in the file by go-bindata. +// For example if you run go-bindata on data/... and data contains the +// following hierarchy: +// data/ +// foo.txt +// img/ +// a.png +// b.png +// then AssetDir("data") would return []string{"foo.txt", "img"} +// AssetDir("data/img") would return []string{"a.png", "b.png"} +// AssetDir("foo.txt") and AssetDir("notexist") would return an error +// AssetDir("") will return []string{"data"}. +func AssetDir(name string) ([]string, error) { + node := _bintree + if len(name) != 0 { + cannonicalName := strings.Replace(name, "\\", "/", -1) + pathList := strings.Split(cannonicalName, "/") + for _, p := range pathList { + node = node.Children[p] + if node == nil { + return nil, fmt.Errorf("Asset %s not found", name) + } + } + } + if node.Func != nil { + return nil, fmt.Errorf("Asset %s not found", name) + } + rv := make([]string, 0, len(node.Children)) + for childName := range node.Children { + rv = append(rv, childName) + } + return rv, nil +} + +type bintree struct { + Func func() (*asset, error) + Children map[string]*bintree +} + +var _bintree = &bintree{nil, map[string]*bintree{ + "resources": &bintree{nil, map[string]*bintree{ + "arithmetic.m": &bintree{resourcesArithmeticM, map[string]*bintree{}}, + "atoms.m": &bintree{resourcesAtomsM, map[string]*bintree{}}, + "boolean.m": &bintree{resourcesBooleanM, map[string]*bintree{}}, + "calculus.m": &bintree{resourcesCalculusM, map[string]*bintree{}}, + "combinatorics.m": &bintree{resourcesCombinatoricsM, map[string]*bintree{}}, + "comparison.m": &bintree{resourcesComparisonM, map[string]*bintree{}}, + "equationdata.m": &bintree{resourcesEquationdataM, map[string]*bintree{}}, + "expression.m": &bintree{resourcesExpressionM, map[string]*bintree{}}, + "flowcontrol.m": &bintree{resourcesFlowcontrolM, map[string]*bintree{}}, + "functional.m": &bintree{resourcesFunctionalM, map[string]*bintree{}}, + "init.m": &bintree{resourcesInitM, map[string]*bintree{}}, + "list.m": &bintree{resourcesListM, map[string]*bintree{}}, + "manip.m": &bintree{resourcesManipM, map[string]*bintree{}}, + "matrix.m": &bintree{resourcesMatrixM, map[string]*bintree{}}, + "numbertheory.m": &bintree{resourcesNumbertheoryM, map[string]*bintree{}}, + "pattern.m": &bintree{resourcesPatternM, map[string]*bintree{}}, + "plot.m": &bintree{resourcesPlotM, map[string]*bintree{}}, + "power.m": &bintree{resourcesPowerM, map[string]*bintree{}}, + "random.m": &bintree{resourcesRandomM, map[string]*bintree{}}, + "replacement.m": &bintree{resourcesReplacementM, map[string]*bintree{}}, + "rubi": &bintree{nil, map[string]*bintree{ + "1.1.1 Linear binomial products.m": &bintree{resourcesRubi111LinearBinomialProductsM, map[string]*bintree{}}, + "1.1.3 General binomial products.m": &bintree{resourcesRubi113GeneralBinomialProductsM, map[string]*bintree{}}, + "1.1.4 Improper binomial products.m": &bintree{resourcesRubi114ImproperBinomialProductsM, map[string]*bintree{}}, + "1.2.1 Quadratic trinomial products.m": &bintree{resourcesRubi121QuadraticTrinomialProductsM, map[string]*bintree{}}, + "1.2.2 Quartic trinomial products.m": &bintree{resourcesRubi122QuarticTrinomialProductsM, map[string]*bintree{}}, + "1.2.3 General trinomial products.m": &bintree{resourcesRubi123GeneralTrinomialProductsM, map[string]*bintree{}}, + "1.2.4 Improper trinomial products.m": &bintree{resourcesRubi124ImproperTrinomialProductsM, map[string]*bintree{}}, + "1.3 Miscellaneous algebraic functions.m": &bintree{resourcesRubi13MiscellaneousAlgebraicFunctionsM, map[string]*bintree{}}, + "2 Exponentials.m": &bintree{resourcesRubi2ExponentialsM, map[string]*bintree{}}, + "3 Logarithms.m": &bintree{resourcesRubi3LogarithmsM, map[string]*bintree{}}, + "4.1 Sine.m": &bintree{resourcesRubi41SineM, map[string]*bintree{}}, + "4.2 Tangent.m": &bintree{resourcesRubi42TangentM, map[string]*bintree{}}, + "4.3 Secant.m": &bintree{resourcesRubi43SecantM, map[string]*bintree{}}, + "4.4 Miscellaneous trig functions.m": &bintree{resourcesRubi44MiscellaneousTrigFunctionsM, map[string]*bintree{}}, + "5 Inverse trig functions.m": &bintree{resourcesRubi5InverseTrigFunctionsM, map[string]*bintree{}}, + "6 Hyperbolic functions.m": &bintree{resourcesRubi6HyperbolicFunctionsM, map[string]*bintree{}}, + "7 Inverse hyperbolic functions.m": &bintree{resourcesRubi7InverseHyperbolicFunctionsM, map[string]*bintree{}}, + "8 Special functions.m": &bintree{resourcesRubi8SpecialFunctionsM, map[string]*bintree{}}, + "9.1 Integrand simplification rules.m": &bintree{resourcesRubi91IntegrandSimplificationRulesM, map[string]*bintree{}}, + "9.2 Derivative integration rules.m": &bintree{resourcesRubi92DerivativeIntegrationRulesM, map[string]*bintree{}}, + "9.3 Piecewise linear functions.m": &bintree{resourcesRubi93PiecewiseLinearFunctionsM, map[string]*bintree{}}, + "9.4 Miscellaneous integration rules.m": &bintree{resourcesRubi94MiscellaneousIntegrationRulesM, map[string]*bintree{}}, + "Integration Utility Functions.m": &bintree{resourcesRubiIntegrationUtilityFunctionsM, map[string]*bintree{}}, + "MakeRubiMxFile.m": &bintree{resourcesRubiMakerubimxfileM, map[string]*bintree{}}, + "README": &bintree{resourcesRubiReadme, map[string]*bintree{}}, + "Rubi.m": &bintree{resourcesRubiRubiM, map[string]*bintree{}}, + "Rubi4.12.nb": &bintree{resourcesRubiRubi412Nb, map[string]*bintree{}}, + "ShowStep Routines.m": &bintree{resourcesRubiShowstepRoutinesM, map[string]*bintree{}}, + }}, + "rubi.m": &bintree{resourcesRubiM, map[string]*bintree{}}, + "rubi_loader.m": &bintree{resourcesRubi_loaderM, map[string]*bintree{}}, + "simplify.m": &bintree{resourcesSimplifyM, map[string]*bintree{}}, + "solve.m": &bintree{resourcesSolveM, map[string]*bintree{}}, + "sort.m": &bintree{resourcesSortM, map[string]*bintree{}}, + "specialsyms.m": &bintree{resourcesSpecialsymsM, map[string]*bintree{}}, + "stats.m": &bintree{resourcesStatsM, map[string]*bintree{}}, + "string.m": &bintree{resourcesStringM, map[string]*bintree{}}, + "system.m": &bintree{resourcesSystemM, map[string]*bintree{}}, + "tests.m": &bintree{resourcesTestsM, map[string]*bintree{}}, + "time.m": &bintree{resourcesTimeM, map[string]*bintree{}}, + "trig.m": &bintree{resourcesTrigM, map[string]*bintree{}}, + }}, +}} + +// RestoreAsset restores an asset under the given directory +func RestoreAsset(dir, name string) error { + data, err := Asset(name) + if err != nil { + return err + } + info, err := AssetInfo(name) + if err != nil { + return err + } + err = os.MkdirAll(_filePath(dir, filepath.Dir(name)), os.FileMode(0755)) + if err != nil { + return err + } + err = ioutil.WriteFile(_filePath(dir, name), data, info.Mode()) + if err != nil { + return err + } + err = os.Chtimes(_filePath(dir, name), info.ModTime(), info.ModTime()) + if err != nil { + return err + } + return nil +} + +// RestoreAssets restores an asset under the given directory recursively +func RestoreAssets(dir, name string) error { + children, err := AssetDir(name) + // File + if err != nil { + return RestoreAsset(dir, name) + } + // Dir + for _, child := range children { + err = RestoreAssets(dir, filepath.Join(name, child)) + if err != nil { + return err + } + } + return nil +} + +func _filePath(dir, name string) string { + cannonicalName := strings.Replace(name, "\\", "/", -1) + return filepath.Join(append([]string{dir}, strings.Split(cannonicalName, "/")...)...) +} diff --git a/expreduce/rubi_snapshot/rubi_resources.go b/expreduce/rubi_snapshot/rubi_resources.go index 795d921..5b5c2b5 100644 --- a/expreduce/rubi_snapshot/rubi_resources.go +++ b/expreduce/rubi_snapshot/rubi_resources.go @@ -1,3 +1,219 @@ -version https://git-lfs.github.com/spec/v1 -oid sha256:23a208ce4f8d275ecbcbd073c80157248042a9c47e547e213a6c9a3b39c9f85c -size 28469128 +// Code generated for package rubi_snapshot by go-bindata DO NOT EDIT. (@generated) +// sources: +// rubi_snapshot/rubi_snapshot.expred +package rubi_snapshot + +import ( + "fmt" + "io/ioutil" + "os" + "path/filepath" + "strings" + "time" +) +type asset struct { + bytes []byte + info os.FileInfo +} + +type bindataFileInfo struct { + name string + size int64 + mode os.FileMode + modTime time.Time +} + +// Name return file name +func (fi bindataFileInfo) Name() string { + return fi.name +} + +// Size return file size +func (fi bindataFileInfo) Size() int64 { + return fi.size +} + +// Mode return file mode +func (fi bindataFileInfo) Mode() os.FileMode { + return fi.mode +} + +// Mode return file modify time +func (fi bindataFileInfo) ModTime() time.Time { + return fi.modTime +} + +// IsDir return file whether a directory +func (fi bindataFileInfo) IsDir() bool { + return fi.mode&os.ModeDir != 0 +} + +// Sys return file is sys mode +func (fi bindataFileInfo) Sys() interface{} { + return nil +} + +var _rubi_snapshotRubi_snapshotExpred = 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e\xb5\x17R)\xdb\x03b\xd8n\x8da}@L\f\xdb\x03b\xd9n\x8d\x8de\xbb5\x8e\xed\u05b88\xb6['\xb3]dXL\xef\x10Q\xac\xb7\x84\x1f\x19]\xa9\xfe\x01X\xf5\xef\xc1\xa5\xfeM\x94\x9f\xa9\xd0\xfb*a*\x92\x19U*~\x15:M\x10\xc2-\xdas\xb4M\xb4\xebDQZ\x8d\tq\xdf\xfbM3\u07af_\xcd\x03\x04\x9a\x11\xf1\x8d]7i\xdcz\x1a\u0542D\xa2\xdd%Y.\u0765\x17\x1f\u04c8\xa2\x1a\x8ayy\xa0\x88\xa2\xe4\x95\x7fJ\x18\x7fF\aV\xfd3\x9a\xcf\xfc\xb3r\xbe\xaa\xf8S\xe2\xaf\xf8\xf3\xff\x02\x00\x00\xff\xff\x15\xef4\xf0") + +func rubi_snapshotRubi_snapshotExpredBytes() ([]byte, error) { + return _rubi_snapshotRubi_snapshotExpred, nil +} + +func rubi_snapshotRubi_snapshotExpred() (*asset, error) { + bytes, err := rubi_snapshotRubi_snapshotExpredBytes() + if err != nil { + return nil, err + } + + info := bindataFileInfo{name: "rubi_snapshot/rubi_snapshot.expred", size: 0, mode: os.FileMode(0), modTime: time.Unix(0, 0)} + a := &asset{bytes: bytes, info: info} + return a, nil +} + +// Asset loads and returns the asset for the given name. +// It returns an error if the asset could not be found or +// could not be loaded. +func Asset(name string) ([]byte, error) { + cannonicalName := strings.Replace(name, "\\", "/", -1) + if f, ok := _bindata[cannonicalName]; ok { + a, err := f() + if err != nil { + return nil, fmt.Errorf("Asset %s can't read by error: %v", name, err) + } + return a.bytes, nil + } + return nil, fmt.Errorf("Asset %s not found", name) +} + +// MustAsset is like Asset but panics when Asset would return an error. +// It simplifies safe initialization of global variables. +func MustAsset(name string) []byte { + a, err := Asset(name) + if err != nil { + panic("asset: Asset(" + name + "): " + err.Error()) + } + + return a +} + +// AssetInfo loads and returns the asset info for the given name. +// It returns an error if the asset could not be found or +// could not be loaded. +func AssetInfo(name string) (os.FileInfo, error) { + cannonicalName := strings.Replace(name, "\\", "/", -1) + if f, ok := _bindata[cannonicalName]; ok { + a, err := f() + if err != nil { + return nil, fmt.Errorf("AssetInfo %s can't read by error: %v", name, err) + } + return a.info, nil + } + return nil, fmt.Errorf("AssetInfo %s not found", name) +} + +// AssetNames returns the names of the assets. +func AssetNames() []string { + names := make([]string, 0, len(_bindata)) + for name := range _bindata { + names = append(names, name) + } + return names +} + +// _bindata is a table, holding each asset generator, mapped to its name. +var _bindata = map[string]func() (*asset, error){ + "rubi_snapshot/rubi_snapshot.expred": rubi_snapshotRubi_snapshotExpred, +} + +// AssetDir returns the file names below a certain +// directory embedded in the file by go-bindata. +// For example if you run go-bindata on data/... and data contains the +// following hierarchy: +// data/ +// foo.txt +// img/ +// a.png +// b.png +// then AssetDir("data") would return []string{"foo.txt", "img"} +// AssetDir("data/img") would return []string{"a.png", "b.png"} +// AssetDir("foo.txt") and AssetDir("notexist") would return an error +// AssetDir("") will return []string{"data"}. +func AssetDir(name string) ([]string, error) { + node := _bintree + if len(name) != 0 { + cannonicalName := strings.Replace(name, "\\", "/", -1) + pathList := strings.Split(cannonicalName, "/") + for _, p := range pathList { + node = node.Children[p] + if node == nil { + return nil, fmt.Errorf("Asset %s not found", name) + } + } + } + if node.Func != nil { + return nil, fmt.Errorf("Asset %s not found", name) + } + rv := make([]string, 0, len(node.Children)) + for childName := range node.Children { + rv = append(rv, childName) + } + return rv, nil +} + +type bintree struct { + Func func() (*asset, error) + Children map[string]*bintree +} + +var _bintree = &bintree{nil, map[string]*bintree{ + "rubi_snapshot": &bintree{nil, map[string]*bintree{ + "rubi_snapshot.expred": &bintree{rubi_snapshotRubi_snapshotExpred, map[string]*bintree{}}, + }}, +}} + +// RestoreAsset restores an asset under the given directory +func RestoreAsset(dir, name string) error { + data, err := Asset(name) + if err != nil { + return err + } + info, err := AssetInfo(name) + if err != nil { + return err + } + err = os.MkdirAll(_filePath(dir, filepath.Dir(name)), os.FileMode(0755)) + if err != nil { + return err + } + err = ioutil.WriteFile(_filePath(dir, name), data, info.Mode()) + if err != nil { + return err + } + err = os.Chtimes(_filePath(dir, name), info.ModTime(), info.ModTime()) + if err != nil { + return err + } + return nil +} + +// RestoreAssets restores an asset under the given directory recursively +func RestoreAssets(dir, name string) error { + children, err := AssetDir(name) + // File + if err != nil { + return RestoreAsset(dir, name) + } + // Dir + for _, child := range children { + err = RestoreAssets(dir, filepath.Join(name, child)) + if err != nil { + return err + } + } + return nil +} + +func _filePath(dir, name string) string { + cannonicalName := strings.Replace(name, "\\", "/", -1) + return filepath.Join(append([]string{dir}, strings.Split(cannonicalName, "/")...)...) +}