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integral_test15.sage
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integral_test15.sage
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#!/usr/bin/env sage
##########################################################################
# Copyright (C) 2008 Tim Lahey <tim.lahey@gmail.com>
#
# Distributed under the terms of the BSD License:
#
# http://www.opensource.org/licenses/bsd-license.php
##########################################################################
# The source of the integrals for comparison are from:
# Spiegel, Murray R.
# Mathematical Handbook of Formulas and Tables
# Schaum's Outline Series McGraw-Hill 1968
# 14.311-14.324
# Original Inspiration for this from:
# http://axiom-developer.org/axiom-website/CATS/
#
# Thanks to Tim Daly.
# Define the necessary variables
var('x,a,b,n,m,c')
#
# Define the table of integral tests. Format is test #, [integrand,desired result]
int_table = { 'Schaum 14.311' : [1/(x^4+a^4),1/(4*a^3*sqrt(2))*log((x^2+a*x*sqrt(2)+a^2)/(x^2-a*x*sqrt(2)+a^2))-1/(2*a^3*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))],
'Schaum 14.312' : [x/(x^4+a^4),1/(2*a^2)*atan(x^2/a^2)],
'Schaum 14.313' : [x^2/(x^4+a^4),1/(4*a*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))-1/(2*a*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))],
'Schaum 14.314' : [x^3/(x^4+a^4),1/4*log(x^4+a^4)],
'Schaum 14.315' : [1/(x*(x^4+a^4)),1/(4*a^4)*log(x^4/(x^4+a^4))],
'Schaum 14.316' : [1/(x^2*(x^4+a^4)),-1/(a^4*x)-1/(4*a^5*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))+1/(2*a^5*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))],
'Schaum 14.317' : [1/(x^3*(x^4+a^4)),-1/(2*a^4*x^2)-1/(2*a^6)*atan(x^2/a^2)],
'Schaum 14.318' : [1/(x^4-a^4),1/(4*a^3)*log((x-a)/(x+a))-1/(2*a^3)*atan(x/a)],
'Schaum 14.319' : [x/(x^4-a^4),1/(4*a^2)*log((x^2-a^2)/(x^2+a^2))],
'Schaum 14.320' : [x^2/(x^4-a^4),1/(4*a)*log((x-a)/(x+a))+1/(2*a)*atan(x/a)],
'Schaum 14.321' : [x^3/(x^4-a^4),1/4*log(x^4-a^4)],
'Schaum 14.322' : [1/(x*(x^4-a^4)),1/(4*a^4)*log((x^4-a^4)/x^4)],
'Schaum 14.323' : [1/(x^2*(x^4-a^4)),1/(a^4*x)+1/(4*a^5)*log((x-a)/(x+a))+1/(2*a^5)*atan(x/a)],
'Schaum 14.324' : [1/(x^3*(x^4-a^4)),1/(2*a^4*x^2)+1/(4*a^6)*log((x^2-a^2)/(x^2+a^2))]
}
# Check to see if test passed and print result.
def test_eval(test, test_int, desired_result):
try:
test_cmp = (desired_result.simplify_full()-test_int.simplify_full()).simplify_full()
except:
print "Test", test,": Test failed. Unable to compare results."
print "Calculated Integral: ", test_int
return
if (test_cmp == 0):
print "Test", test,": Test Passed."
else:
print "Test", test," Difference in Results:", test_cmp
# If the difference is constant, the result is valid within a constant of integration.
if (test_cmp.diff(x) == 0):
print "Correct within a constant of integration."
print "Test Passed."
else:
div_cmp = (desired_result.simplify_full()/test_int.simplify_full()).simplify_full()
if (div_cmp.diff(x) == 0):
print "Division of Results:", div_cmp
print "Correct within a constant multiple."
else:
print "Test Failed."
print "Calculated Integral: ", test_int
print "Comparison Integral: ", desired_result
# Time integration of Maxima and FriCAS for integral.
def time_Maxima_friCAS(integrand):
mx_time = timeit.eval('integrand.integrate(x)')
fCAS_time= timeit.eval('axiom.integrate(integrand,x)')
print "Maxima Time:", mx_time.stats[3], mx_time.stats[4]
print "FriCAS Time:", fCAS_time.stats[3], fCAS_time.stats[4]
# Loop over tests
for test in int_table.keys():
test_set = int_table[test]
integrand = test_set[0]
desired_result = test_set[1]
try:
test_int = integrand.integrate(x)
except:
print "Test", test,": Test failed due to exception."
else:
test_eval(test,test_int,desired_result)
time_Maxima_friCAS(integrand)