# I would like some help to find a closed form for the following integral: int_0^1 ((log (1+x))^3)/(x)dx I was told it could be calculated in a closed form. I've already proved that int_0^1 (log (1+x))/(x)dx=(pi^2)/(12) using power series expansion. Thank you.

A closed form for ${\int }_{0}^{1}\frac{{\left(\mathrm{log}\left(1+x\right)\right)}^{3}}{x}dx$?
I would like some help to find a closed form for the following integral:
${\int }_{0}^{1}\frac{{\left(\mathrm{log}\left(1+x\right)\right)}^{3}}{x}dx$
I was told it could be calculated in a closed form. I've already proved that
${\int }_{0}^{1}\frac{\mathrm{log}\left(1+x\right)}{x}dx=\frac{{\pi }^{2}}{12}$
using power series expansion.
Thank you.
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RamPatWeese2w
Letting $u=\mathrm{log}\left(1+x\right)$

Then integrating by parts 3 times,

The answer could of course be simplified using the known values of ${\text{Li}}_{2}\left(\frac{1}{2}\right)$, ${\text{Li}}_{3}\left(\frac{1}{2}\right)$, and $\zeta \left(4\right)$