Interesting integral! #Challenge
$${\displaystyle {\int }_0^1{\displaystyle \frac{\ln (x)\ln (1-x)}{x(1-x)}}{\mathrm{Li}}_2(x)dx=5\zeta (2)\zeta (3)-8\zeta (5)}$$
Where ${\mathrm{Li}}_2(x)$ denotes the dilogarithm (or Spence's function), and $\zeta (x)$ denotes the Riemann zeta function.