Good book on evaluating difficult definite integrals (without elementary antiderivatives)?

My favorite is a book of 1992 from Daniel Zwillinger : "Handbook of Integration" it is a "compilation of the most important methods" in 360 pages. Recommanded!

I've only skimmed it, but Irresistible Integrals by George Boros and Victor H. Moll seems worth a look.

As a first approximation, Whittaker and Watson's "... Modern Analysis" shows how to use the Gamma function, and other classical "special functions" (that one would not hear about in traditional calculus classes,...) to address such issues.

If Whittaker and Watson were Baroque, the standard reference "Gradshteyn and Ryzhik" (sp?) would be the Rococco. They do give serious references for all their formulas... and the whole thing is amazing...

A curiously small, tractable, readable, inexpensive, available source is Lebedev's book on special functions.

Vilenkin's AMS translation book about special functions "versus" representations is chock-full-of stuff.

(And, perhaps unsurprisingly, given that I'm responding, various notes of mine on my web site talk about "how to evaluate integrals". At http://www.math.umn.edu/~garrett/m/v/standard_integrals.pdf there is a straightforward discussion of certain not-really-elementary integrals that arise in number theory/autormophia... and might be of interest anyway. Other course notes on my various web pages address other aspects of similar things in that context... which may not be your main interest...)