Since these integrals are run in a batch mode, using an automated script, and by using sagemath (SageMath uses Maxima), then any integral where Maxima needs an interactive response from the user to answer a question during evaluation of the integral in order to complete the integration, will fail and is counted as failed.
The exception raised is ValueError. Therefore Maxima result below is lower than what could result if Maxima was run directly and each question Maxima asks was answered correctly.
The percentage of such failures were not counted for each test file, but for an example, for the Timofeev test file, there were about 14 such integrals out of total 705, or about 2 percent. This pecrentage can be higher or lower depending on the specific input test file.
Such integrals can be indentified by looking at the output of the integration in each section for Maxima. The exception message will indicate of the error is due to the interactive question being asked or not.
Maxima integrate was run using SageMath with the following settings set by default
'besselexpand : true' 'display2d : false' 'domain : complex' 'keepfloat : true' 'load(to_poly_solve)' 'load(simplify_sum)' 'load(abs_integrate)' 'load(diag)'
SageMath loading of Maxima abs_integrate was found to cause some problem. So the following code was added to disable this effect.
from sage.interfaces.maxima_lib import maxima_lib maxima_lib.set('extra_definite_integration_methods', '[]') maxima_lib.set('extra_integration_methods', '[]')
See https://ask.sagemath.org/question/43088/integrate-results-that-are-different-from-using-maxima/ for reference.
There are Few integrals which failed due to SageMath not able to translate the result back to SageMath syntax and not because these CAS system were not able to do the integrations.
These will fail With error Exception raised: NotImplementedError
The number of such cases seems to be very small. About 1 or 2 percent of all integrals.
Hopefully the next version of SageMath will have complete translation of FriCAS and XCAS syntax and I will re-run all the tests again when this happens.
For Mathematica, Rubi and Maple, the buildin system function LeafSize is used to find the leaf size of each antiderivative.
The other CAS systems (SageMath and Sympy) do not have special buildin function for this purpose at this time. Therefore the leaf size for Fricas and Sympy and Giac antiderivatives is determined using the following function, thanks to user slelievre at https://ask.sagemath.org/question/57123/could-we-have-a-leaf_count-function-in-base-sagemath/
For Sympy, which is called directly from Python, the following code is used to obtain the leafsize of its result
Matlab’s symbolic toolbox does not have a leaf count function to measure the size of the antiderivative, Maple was used to determine the leaf size of Mupad output by post processing.
Currently no grading of the antiderivative for Mupad is implemented. If it can integrate the problem, it was assigned a B grade automatically as a placeholder. In the future, when grading function is implemented for Mupad, the tests will be rerun again.
The following is an example of using Matlab’s symbolic toolbox (Mupad) to solve an integral
Which gives sin(x)^2/2