The above is, I think, the most difficult integral I've ever evaluated. Usually the strategy on such integrals is to figure out which of the variables gives an easier integration, and do it first, but in this case I had to handle both at once, performing the following substitutions one after the other:
Doing just one leaves a mess that looks just as bad as before, but doing both produces an amazing cancellation. Not only that, I defined the second new variable as an implicit function of the remaining old variable, only later solving for the old variable explicitly. That's a trick I've never used before, and it entails some messiness with the limits of integration. The entire problem (involving conservation of momentum with a moving charge) took me about two and a half hours, and now I'm thoroughly proud of myself.
Problems like this are why it is said: differentiation is a skill, but integration is an art.