Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $

Question

Prove the following

$$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$

where

$$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$

Answer 1

Integrating once by parts, we find \begin{align} I=\int_0^{1}\frac{\mathrm{Li}_2^2(x)\,dx}{x}&=\int_0^{1}\mathrm{Li}_2(x)\,d\left(\mathrm{Li}_3(x)\right)=\\ &=\mathrm{Li}_2(1)\mathrm{Li}_3(1)+\int_0^{1}\frac{\mathrm{Li}_3(x)\,\ln(1-x)\,dx}{x}=\\ &=\zeta(2)\zeta(3)-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{1}{n^3k(n+k)},\tag{1} \end{align} where the last line is obtained by expanding $\mathrm{Li}_3(x)$ and $\ln(1-x)$ into series and integrating. Now if we denote $H_n=\sum_{k=1}^n\frac{1}{k}$ the $n$th harmonic number, the sum with respect to $k$ in (1) can be written as $$\sum_{k=1}^{\infty}\frac{1}{k(n+k)}=\frac{1}{n}\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{n+k}\right)=\frac{H_n}{n},$$ so that $$I=\zeta(2)\zeta(3)-\sum_{n=1}^{\infty}\frac{H_n}{n^4}.\tag{2}$$ Now using that $\zeta(2)=\frac{\pi^2}{6}$ and the formula (20) here (it is a particular case of a more general Euler sum (24)), namely, $$ \sum_{n=1}^{\infty}\frac{H_n}{n^4} = 3\zeta(5)-\frac{\pi^2}{6}\zeta(3),$$ we finally obtain the desired result for $I$. $\blacksquare$