Mathematica/Maple/Sagemath don't have a freemium timeout mechanic and can solve a lot more. Truth be told I think that integration techniques are much less broadly crucial for everyone to learn than they used to be, although you need to have some clue of what's going on because you need to be able to guide yourself towards posing problems in such a way that the integrals that can be solved.
But there are integrals that you can easily solve by hand but both WolframAlpha and sage will (effectively) timeout on them. And I’m not even talking about something made deliberately hard for computers to symbolically analyze.
