[gohrt]

The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.

[amelius]

This makes me wonder. Is Mathematica also applying a bag of tricks (I suppose in a breadth first search), or does it have a more structural approach?

[qubex]

As I recall, Mathematica embodies a (perhaps incomplete) implementation of the Risch algorithm.