Have you tried to read any of the literature on the Risch algorithm? If you haven't, you might want to get started by taking a look at the paper "Integration in Finite Terms" by Rosenlicht [1] and chasing down some of the references mentioned in [2].

Of course, in the real world we don't give up on integrals just because they can't be expressed in terms of elementary functions. Usually we also check if the result happens to be a hypergeometric function, such as a Bessel function. If you want to get started on understanding hypergeometric functions, maybe try reading [3] (as well as the tangentially related book "A = B" [4]).

[1] https://www.cs.ru.nl/~freek/courses/mfocs-2012/risch/Integra... [2] https://mathoverflow.net/questions/374089/does-there-exist-a... [3] https://www.math.ru.nl/~heckman/tsinghua.pdf [4] https://www2.math.upenn.edu/~wilf/AeqB.html

These might of interest to you:

https://link.springer.com/book/10.1007/978-3-030-43788-6

https://link.springer.com/book/10.1007/978-3-031-24228-1

https://link.springer.com/book/10.1007/978-3-030-02462-8

https://link.springer.com/book/10.1007/978-3-031-21262-8