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This is known as rule-based integration. Here's a powerful system of such rules: https://rulebasedintegration.org/integrationRules.html

Does one know if mathematica uses that too ? I rememebr an HN post where one could see that sympy, maxima and the likes where way behind for a lot of more peculiar integrals..

ah the comparison is here : https://www.12000.org/my_notes/CAS_integration_tests/index.h...

(and posted by other on this very thread...)

There is a go implementation of a CAS which syntax similar to Mathematica which uses this rules based integration library.

https://github.com/corywalker/expreduce

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