I am grateful to Cornel Ioan Valean for sharing this nice problem.
Find a closed form in terms of the values of the $\zeta$ function for the following triple integral: $$ \iiint_{(0,1)^3}\frac{\text{Li}_2(1-xyz)}{1-xyz}\,dx\,dy\,dz.$$
My approach has been the following one
- By exploiting the dilogarithm reflection formula, it is enough to compute the above integral where $\text{Li}_2(1-xyz)$ is replaced by $1,\log(xyz)$ and $\text{Li}_2(xyz)$;
- By Fubini's theorem these integrals boil down to Euler sums of weight $5$, which can be computed in terms of values of the $\zeta$ function through Theorems $2.2$ and $3.1$ of Flajolet-Salvy.
I am looking for an alternative/more elementary approach.