Three integral involving polylogarithm function

Question

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_4(x)}{x}dx .$$

Answer 1

There's a closed form because all integrals of this type can be expressed using multiple zeta values. I found numerically, that the closed forms are, in order: $$ -9 \zeta (5) \zeta (2)+\tfrac{15}{2} \zeta (3) \zeta (4)+\tfrac{51}{8} \zeta (7) $$ $$ \tfrac{5}{2} s_h (2,6)-\tfrac{3}{2} \zeta (3)^2 \zeta (2)+10 \zeta (3) \zeta (5)-\tfrac{74}{9} \zeta (8) $$ $$ -18 \zeta (7) \zeta (2)+\tfrac{19}{2} \zeta (5) \zeta (4)+\tfrac{7}{2} \zeta (3) \zeta (6)+\tfrac{559}{36} \zeta (9) $$ Here $s_h(2,6)$ is an Euler sum without a simpler closed form.

These can be obtained if you write the integrals as 7-, 8-, and 9-fold iterated integrals using $$ \mathrm{Li}_2(x) = \int_0^x \frac{du}{u} \int_0^u \frac{dv}{1-v}, $$ and rearrange the domain of integration appropriately to bring the integral to a form that is expressible using multiple zeta values. These can then be simplified to ordinary zeta values using known formulas. There are more relevant links here.