Basically, this question can be considered to be an extenstion to my other question.
So using Mathematica's
Integrate function the answer was
Apparently, after rigorous substitutions and transformations the answer was found to be correct.
What I wanted to know was how Mathematica integrates these functions that require a human tons of intuition to compute, within seconds, and often in the most simple way and also presents them in the most humanly computable form.
(Even differentiation for that matter)