Integral $\int \frac{\tan x}{x} dx$

Question

Evaluate the integral: $$\int \frac{\tan x}{x} dx$$

I tried integration by parts, got stuck. Ideas/ suggestions please.

Answer 1

There is no solution, based on this link.

Answer 2

If you want to prove things like the inexpressibility of various integrands in terms of some set of functions, you want to look at Liouville's theorem and possibly some differential Galois theory. Check out https://mathoverflow.net/questions/58966/solvability-in-differential-galois-theory for some reading in this area.

As others have said, it is likely no elementary integral exists, since this holds for similar integrals like $\int \sin x/x$.