TITLE: Some users are mind bogglingly skilled at integration. How did they get there?
QUESTION [185 upvotes]: Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.
Especially in a world where most scientists compute their integrals numerically, I find it astounding that people exist that can solve these problems analytically. Sometimes it is a truly bizarre contour, some examples use auxiliary functions and differentiation under the integral sign in a way that feels more like integration by wizardry.
Some of these users seem to have mastered an incredibly large number of special functions to a level of fluency that is almost unimaginable to me. Manipulations involving the error function almost looks like the work of an infant compared to some of these seemingly casual manipulations involving the Airy function, Barnes G function, the Legendre chi function and many more. Every time I read a post by them, it turns into an exercise in reading about yet another special function I have never heard of. 
My question is what sort of material do you have to study, or what kind of area do you have to work in to get so good? To me, it doesn't seem as obvious as saying something like, "oh, they know this material because they are a topologist".

REPLY [10 votes]: Well, I was surprised to see my name somewhere in this page. Anyways, I don't consider myself an integration guru but you may say inspired by them. Unfortunately, I don't have the same interest now as I did 4 years ago nevertheless, I will share my experience. I started my journey to learn about integration about 4 years ago in the second year of university. I thought I have some interest in integration and  might be good to push my knowledge from the basics to the advanced. My first step was to register in a forum and see how things work. I would post an integration question in the forum and someone said use the gamma function. To be frank, I faced all kinds of problems to collect the material needed to understand a single special function. This was the start of the journey. I suffered a lot to to understand all the basics and to move form one function to another. Actually sometimes you might find books on some special functions like the one by Lewin on polylogarithms and on the other hand suffer to collect some information about other functions like the theta function, loggamma integrals etc. I would say my process was slow but it eventually got me some way. I would suggest for you or anyone reading this post to start by the following materials because they are easy to understand 

Differentiation under the integral sign
Gamma function 
Beta function
Digamma function
Polylogarithms 

These are the easiest and you can find lots of materials on the internet. 
Here is a Advanced integration techniques of my journey, it talks about all the special functions I mastered.