[koningrobot]

Interesting example they chose there. It requires no fewer than three deus ex machinas: apply this trick, then apply this very particular substitution, then find another substitution. If you encountered this integral in the wild without knowing whether/how it can be solved, yes you might have tried the trick, but you wouldn't realize that it had gotten you anywhere. Would you continue on and find the just-so substitutions, or would you backtrack and try something else?

It's great that we have the tricks we have, but at the same time most nontrivial integrals are just impenetrable regardless. Any demonstration of integration techniques you find will be on an integrand that is amenable to these techniques, and will only show the straight path to the solution, not the process of finding that path. I hate integrals!