I'm looking for a good & unquestionably complete reference on solving basic integrals. Out of all the books in my library I've checked they're all more or less just Thomas calculus with the odd frill attached here & there. The best books I've come across are Piskunov's calculus & a few Indian textbooks on google books but I'm hoping for something better, something more complete & systematic but is more than an integral table with no/few/incomplete/unsystematic methods (i.e. something that would go through the trouble of mentioning that three techniques apply to some integral, systematizing special non-obvious cases etc... ). To give you an example of what I'm talking about, & why I feel someone will mention a book I haven't come across yet that is my saviour, just pick up every ordinary differential equations book you can & count how many mention first order Abel's Equations of the first or second kind, the Isobaric equation, first order Jacobi & Darboux equations etc... etc... The integral analogue of what I'm looking for would be something like either of Murphy, Ordinary Differential Equations and Their Solutions http://www.amazon.com/Ordinary-Differential-Equations-Solutions-Mathematics/dp/0486485919/ or Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations http://www.amazon.com/Handbook-Solutions-Ordinary-Differential-Equations/dp/1584882972/
Note that the last book you reference was written with Polyanin. This last author has a whole collection of books of this kind and a very useful website EqWorld allowing to identify your ODE or PDE equation (with Zaitsev and twelve other people in the board)!
One of his books is a 'condensation' of all (from my point of view) : it is the 1500 pages 'Handbook of Mathematics for Engineers and Scientists' (index here). Of course there is some 'overlay' with your ODE book!
In the board appears too Daniel Zwillinger that produced in 1992 a fine book : 'Handbook of Integration' that I recommend warmly (he is associated with the G&R tables and has too a book on ODEs). It is a "compilation of the most important methods" in 360 pages.
Should you, after all that, be interested by the 'computer point of view' for indefinite integration (differential algebra : another subject I'll admit...) then there is the Geddes & all or the Bronstein.