One thing I've often wondered: Is there a reason to learn all of these methods in the modern era, when Wolfram Alpha or Mathematica can apply hundreds of methods automatically?

It's good to understand things conceptually. But once I got the concept of integrals as the area under a curve, it felt like a lot of grunt work to learn so many tactics for solving them. But most of my focus has been on computers rather than pure mathematics. For pure math, it probably makes more sense to learn as many different methods as possible.