Where to learn advanced integral calculus?
I would recommend:
Inside Interesting Integrals by Paul Nahin. It contains many of the tricks that you mentioned.
The Handbook of Integration by Daniel Zwillinger. It is no longer in print but used copies can be found.
I have not read the following books but all are on my "wish list".
Integration for Engineers and Scientists by William Squire. It is cited repeatedly by Zwillinger. It is out of print but used copies are available.
Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering: A Self-Study Approach by C.C. Maican
While Nahin covers basic contour integration and part of Zwillinger uses contour integration, all of these books are suitable for those who have yet to study complex analysis.
Addendum
One of the members here, @Zaid Alyafeai, has written a freely available book on integration methods. I am currently looking at it and it is in the same vein as the books listed above. The link to the book can be found here.
Another good free resource: Integral Kokeboken (Integral Cookbook [Norwegian]).