If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression.
One possibility is that I learn about theories that are used to prove thinks like this, as discussed, for example, here, here, here.
But if I can take for granted that some functions are "famously" non-integrable in terms of elementary functions, for example $\int e^{x^2}\,\mathrm{d}x$, then I can use this fact when discussing integrals of other functions. (The argument would go like: If it was possible to integrate this function, then using the following steps, we would also be able to integrate $e^{x^2}$. Hence, the given function does not have elementary integral.)
(If I may make a comparison, this is somewhat similar to situation with set-theoretical results: Wise people, who have knowledge of forcing and other advanced stuff, have proven many results which are undecidable in ZFC. If I obtain some of these statements as a consequence of some conjecture, I get that this conjecture is also undecidable.)
So I wonder whether there is some book or website which gives some large list of integrals, which are known to be non-elementary. (And thus can be used in proofs that other functions do not have elementary integrals.)
Ideally, it would be good if there were also references to the proofs that the functions in the list do not have elementary integrals.
