I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would lead to an exact solution of my power series if I can find an exact anti-derivative.

$\int^x_0 \frac{u^{r-1}}{1-u}\, du$

I am only really interested in values of $r$ between $0$ and $1$ as larger values could essentially be handled by a reindexing of the series. So far the only value of $r$ which I have found an exact anti-derivative for is $r = 1/2$ in which case the integral evaluates to $2 \mathrm{arctanh}(\sqrt{x})$. Can anyone think of any other values of $r$ for which the integral can be evaluated exactly? Or prove that there are no others?

If it is not possible to get any other exact solutions would approximating the integral via Simpson's rule or something similar generally be quicker then approximating a power series?

Any advice on the problem would be greatly appreciated, thanks in advance.