An integral trick I picked up from a lecturer at university: if you know the result has to be of the form ax^n for some a that's probably rational and some integer n but you're feeling really lazy and/or it's annoying to simplify (even for mathematica), just plug in a transcendental value for x like Zeta[3].

Then just divide by powers of that irrational number until you have something that looks rational. That'll give you a and n. It's more or less numerical dimensional analysis.

It's not that useful for complicated integrals, but when you're feeling lazy it's a fucking godsend to know what the answer should be before you've proven it.

EDIT: s/irrational/transcendental/