First of all, please forgive me for my poor English ability since it is not my first language. Truth be told, I am reluctant to answer this question because I don't think I am the right person to answer such question. But considering the one (if I may mention his name here: User) who asked me to answer this question has been very kind to me since I joined in Quora a few days ago, I will try my best to provide a good answer in order to honor him. I hope I am not making any immodest statements or showing narcissism on my part. Almost all parts of my answer here will just restate the obvious and re-quote the statements from Mr. Ron Gordon, one of my integration gurus on Mathematics StackExchange.

How do you get good at integrating?

First, you must have passion, a great passion. My interest in math is driven by my passion for solving complicated math problems specially calculus problems like integrals, series, or related topics. My heart will be filled with joy when learning a new cool integration technique or solving a daunting integral problem. So without passion, no matter how talented you are you will never be good at anything.

Second, of course practice. I love working out an enormous number of problems, completely and in great detail. Master every techniques such as integration by parts, substitution methods, partial fractions, algebra manipulations, reduction formulas, recurrence relations, differentiation under integral sign, double integration methods, series expansions, contour integration, residue methods, etc. Also, do not forget to learn related theorems and specials functions. This doesn't mean, however, that you sit with a textbook and merely work out the exercises or learn techniques and theorems in a boring, rote manner. What I mean by practice is the application of your knowledge to new, interesting problems. Find problems that look impossible to you right now. Maybe they are, but I guarantee you somebody somewhere has solved it. Here are some places to find interesting/challenging problems and amazing techniques:

1. Mathematics StackExchange
2. Integrals and Series
3. Putnam Mathematical Competition
4. The table of integrals by Gradshteyn and Ryzhik
5. Journals of Mathematics and Physics
6. College mathematics and physics textbooks

Third, learn from the best. Here are the list of people who I consider to be integration gurus:

1. Ron Gordon
2. Achille Hui
3. sos440
4. Olivier Oloa
   
5. Random Variable
6. Felix Marin
7. Tunk-Fey
8. Integrals and Series a.k.a. Shobhit
9. Galactus
10. Omran Kouba
11. Jack D'Aurizio
12. robjohn
    

Please forgive me if I left anyone out, but the complete list can be seen at Top Users of Integration.

I have no more to say and I hope this helps. (>‿◠)✌