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integral_test16.sage
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integral_test16.sage
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#!/usr/bin/env sage
##########################################################################
# Copyright (C) 2008 Tim Lahey <[email protected]>
#
# 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
# 14.325-14.338
# 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,r,p')
#
# Define the table of integral tests. Format is test #, [integrand,desired result]
int_table = { 'Schaum 14.325' : [1/(x*(x^n+a^n)),1/(n*a^n)*log(x^n/(x^n+a^n))],
'Schaum 14.326' : [x^(n-1)/(x^n+a^n),1/n*log(x^n+a^n)],
'Schaum 14.327' : [x^m/(x^n+a^n)^r,0],
'Schaum 14.328' : [1/(x^m*(x^n+a^n)^r),0],
'Schaum 14.329' : [1/(x*sqrt(x^n+a^n)),1/(n*sqrt(a^n))*log((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n)))],
'Schaum 14.330' : [1/(x*(x^n-a^n)),1/(n*a^n)*log((x^n-a^n)/x^n)],
'Schaum 14.331' : [x^(n-1)/(x^n-a^n),1/n*log(x^n-a^n)],
'Schaum 14.332' : [x^m/(x^n-a^n)^r,0],
'Schaum 14.333' : [1/(x^m*(x^n-a^n)^r),0],
'Schaum 14.334' : [1/(x*sqrt(x^n-a^n)),2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n))],
'Schaum 14.335' : [x^(p-1)/(x^(2*m)+a^(2*m)),0],
'Schaum 14.336' : [x^(p-1)/(x^(2*m)-a^(2*m)),0],
'Schaum 14.337' : [x^(p-1)/(x^(2*m+1)+a^(2*m+1)),0],
'Schaum 14.338' : [x^(p-1)/(x^(2*m+1)-a^(2*m+1)),0]
}
# 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)