Unsolved Elementary Integrals I am currently in Integral Calculus, and I wondered if I could get a little creative with my practice. I was curious if there were any unsolved, but rather simple (solvable using methods taught in calc I/II), so that I could be practicing the integration tehniques I have learned while also solving a problem that hasn't been solved yet. 
I have consulted lists of integrals, but those are in a more general form (constants are represented by letters, etc.), and I was wondering if there was a place I could find lists of indefinite integrals that had never been calculated but weren't that difficult. 
Also, on a side note, if you happen to know any good resources for reading about more techniques of integration, I am currently looking to expand my "toolbox" of integration methods.
Thank you all, and I hope each and every one of you is having a nice day.
 A: Here's a relevant flow chart, because as Silvanus P Thompson said:

"Here note this very remarkable fact, that we could not have integrated in the above case [$y = a\log_c x + C$] if we had not happened to know the corresponding differentiation. If no one had found out that differentiating $\log_c x$ gave $x^{-1}$, we [would] have been utterly stuck by the problem [of] how to integrate $x^{-1} dx$. Indeed it should be frankly admitted that this is one of the curious features of the integral calculus :- that you can't integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate."
  Calculus Made Easy, p199.

Integration, being a inverse process, is much less systematic than differentiation.
EDIT Right after that quote the author gives an 'unsolved, but rather simple' integral. It may not be solvable though - that quality doesn't necessarily follow from simplicity when it comes to integration.
A: Some texts that cover the various techniques of integration at a level comparable to Calculus I and II (and a bit beyond) include:


*

*The Soviet text Problems in Mathematical Analysis edited by Boris Demidovich has many problems using the various techniques of integration to solve. A few of the techniques Demidovich gives are not usually found elsewhere.

*Joseph Edwards’ A Treatise on the Integral Calculus (Volume 1) is a particularly valuable source for many interesting integrals, and being published before 1923, is out of copyright meaning it can be readily found online.

*Michael Spivak’s Calculus contains many interesting questions that use integration, like the proof that $\pi$ is irrational for example.

*While it is a bit old, G. H. Hardy’s A Course of Pure Mathematics, is also a useful source for questions, and being published before 1923 means it too is no longer under copyright. 

*The recent text How to Integrate It: A Practical Guide to Solving Elementary Integrals by  Seán M. Stewart has individual chapters devoted to a particular technique of integration, with each being accompanied by a wealth of end-of-chapter exercises.
