The systematic approaches are nearly impossible to describe - it takes books and books and books, relying on mathematics well above standard calculus, especially when you consider that there are innumerable integrals that don't have solutions in terms of standard elementary functions.
There are several processes in mathematics where forward is easy but reverse is hard. That, in fact, is what allows encryption on the web to work. When you get into differential equations, often times the main point is to setup the problem, not solve it, or to use some method to come up with an estimated solution rather than an exact one.
The best way to be able to pick a "u" is to have your derivative table memorized, and be well-practiced with complex derivatives and what they look like. This helps you "see" what you should pick for a "u". If you apply the chain rule a lot, it is then much easier to see what it might look like in reverse.
But, as Batman said, it tends to be as much as an art as anything.
If you want to know more about how a computer might do it, you should pick up a book on computer algebra algorithms. However, there are integrals that are solvable that are beyond the reach of humans, too. As I said, forward is simple but reverse is hard.