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integral_test11.sage
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integral_test11.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
# 14.237-14.264
# 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,p,q')
# Define the table of integral tests. Format is test #, [integrand,desired result]
int_table = { 'Schaum 14.237' : [1/(sqrt(a^2-x^2)),asin(x/a)],
'Schaum 14.238' : [x/(sqrt(a^2-x^2)),-sqrt(a^2-x^2)],
'Schaum 14.239' : [x^2/sqrt(a^2-x^2),-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a)],
'Schaum 14.240' : [x^3/sqrt(a^2-x^2),(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2)],
'Schaum 14.241' : [1/(x*sqrt(a^2-x^2)),-1/a*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.242' : [1/(x^2*sqrt(a^2-x^2)),-sqrt(a^2-x^2)/(a^2*x)],
'Schaum 14.243' : [1/(x^3*sqrt(a^2-x^2)),-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.244' : [sqrt(a^2-x^2),(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a)],
'Schaum 14.245' : [x*sqrt(a^2-x^2),-(a^2-x^2)^(3/2)/3],
'Schaum 14.246' : [x^2*sqrt(a^2-x^2),-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a)],
'Schaum 14.247' : [x^3*sqrt(a^2-x^2),(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3],
'Schaum 14.248' : [sqrt(a^2-x^2)/x,sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.249' : [sqrt(a^2-x^2)/x^2,-sqrt(a^2-x^2)/x-asin(x/a)],
'Schaum 14.250' : [sqrt(a^2-x^2)/x^3,-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.251' : [1/(a^2-x^2)^(3/2),x/(a^2*sqrt(a^2-x^2))],
'Schaum 14.252' : [x/(a^2-x^2)^(3/2),1/sqrt(a^2-x^2)],
'Schaum 14.253' : [x^2/(a^2-x^2)^(3/2),x/sqrt(a^2-x^2)-asin(x/a)],
'Schaum 14.254' : [x^3/(a^2-x^2)^(3/2),sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2)],
'Schaum 14.255' : [1/(x*(a^2-x^2)^(3/2)),1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.256' : [1/(x^2*(a^2-x^2)^(3/2)),-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2))],
'Schaum 14.257' : [1/(x^3*(a^2-x^2)^(3/2)),-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.258' : [(a^2-x^2)^(3/2),(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a)],
'Schaum 14.259' : [x*(a^2-x^2)^(3/2),-(a^2-x^2)^(5/2)/5],
'Schaum 14.260' : [x^2*(a^2-x^2)^(3/2),-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a)],
'Schaum 14.261' : [x^3*(a^2-x^2)^(3/2),(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5],
'Schaum 14.262' : [(a^2-x^2)^(3/2)/x,(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x)],
'Schaum 14.263' : [(a^2-x^2)^(3/2)/x^2,-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a)],
'Schaum 14.264' : [(a^2-x^2)^(3/2)/x^3,-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x)]
}
# 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]
# Note that some of the tests fail because Schaum's assumes a>=0.
assume(a>=0) # Additional assumption to match Schaum's.
# 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)