[kaba0]

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.

[whatshisface]

Really? I've never seen one, could you give some examples?

[bzbarsky]

For Wolfram Alpha, I just tried the integral from https://en.wikipedia.org/wiki/Contour_integration#Example_2_... and it times out. The by-hand solution is pretty straightforward, if you know contour integration.