I'd love to find out an expression for the indefinite integral
$$\int \frac{1}{\log(1+e^{\large x})} \mathrm{d}x$$
but I wasn't able to on my own. In fact I'll need to find the inverse of the resulting function. I suppose I could live with the series expansion but something more compact would be nice. Any help would be much appreciated.
Let $u=\log(1+e^x)$ ,
Then $x=\log(e^u-1)$
$dx=\dfrac{e^u}{e^u-1}du=\dfrac{1}{1-e^{-u}}du$
$\therefore\int\dfrac{1}{\log(1+e^x)}dx$
$=\int\dfrac{1}{u(1-e^{-u})}du$
Which relates to the incomplete polylogarithm function