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I have seen similar questions but they were quite dated (3+ years). When performing, say an Integration, it would be beneficial to see the major steps such as variable substitution, change the order of integration, perform by-parts, etc.
Thanks
$\begingroup$The Risch algorithm is the core method used for symbolic integration in most software. Searching for it in your favorite search engine should turn up at least a useful introduction.$\endgroup$
Integrate
in general does not do things in a way that would be recognizable from basic calculus. $\endgroup$