I wounder whether there are any Integrals that are known to have a solution in terms of elementary functions only (or that it is unknown whether they have an elementary solution), but couldn't be solved until now?
I have read about Risch algorithm already which can in principle answer this question, but there are cases where the algorithm fails because of the constant problem.
So, the question is, in these cases where the algorithm fails, is there (always) some way to get the result at least manually/approximatly? Or are there any integrals which haven't been solved where its unknown whether they have an elementary antiderivative?