Integration of $\frac{1}{\log(x)}$?

Question

How to integrate $\frac{1}{\log(x)}$?

I have tried integration by parts, but it is a never ending series with no specific general term.

PS: It is indefinite integration.

Answer 1

There is no indefinite integral expressible in terms of elementary functions. A special function has to be defined for this purpose, and it is known as the Logarithmic Integral Function.

Note that many, many seemingly elementary functions do not have indefinite integrals that can be expressed in terms of elementary functions. Another famous example is $e^{-x^2}$.