Could we find another representation for this integral when Wolfram Alpha provide no answer for that? Could we find another representation for this integral when Wolfram Alpha provide no answer for that? If we cannot, is it possible for we to prove that there are other representation for this?
$$\int\frac{1}{x-\sin x}\ dx$$
 A: The short answer is that it is possible to either find the indefinite integral or prove that it cannot be expressed in terms of elementary functions. But the methodology is well beyond by-hand practicality.
The problem of finding a well-defined algorithm to attempt to determine the indefinite integral of an expression involving elementary functions (trig, hyperbolic, exponential, log, roots, but not things like erf o or elliptic functions or Bessel functions), such that if the algorithm fails to find the integral then the integral cannot be expressed using only elementary functions, was solved in an MIT PhD thesis in the 90's.  Wolfram, in a colloquium at Fermi National Accelerator Laboratory, mentioned that the full algorithm had not (at that time) been implemented in Mathematica, and I would not be surprised if it were by now implemented, but I also would not be shocked to learn that it never was implemented.
Still, as a rule, if Mathka fails to find the integral (the real Mathematica, not Wolfram Alpha) then you won't be able to either.  The particular integral you present feels like it might be expressible using elliptic integrals, but I'd be surprised if it can be expressed in elementary functions.
