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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.

Pait
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    W|A says "$\text{ no result found in terms of standard mathematical functions }$". – K. Rmth Jan 06 '14 at 23:14
  • If you set $1+e^x=t$ the integral becomes $\int \frac{dt}{(t-1) \log t}=\int \frac{d \log (t-1)}{\log t}$ which probably doesn't have a closed-form solution – Alex Jan 06 '14 at 23:15
  • W|A is Wolfram alpha? Interesting, Mathematica merely did not give an answer. So it's likely that no closed-form exists in terms of any special function? Thanks! – Pait Jan 07 '14 at 01:56

1 Answers1

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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

Harry Peter
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