I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training courses from many years ago. We have $m,n \in \mathbb{N}$, $\sigma,\omega \in \mathbb{C}$, $|\arg \sigma| <\pi, |\arg \omega| <\pi $. I am stuck as to where to start. Thanks. The polylogarithm function is given by $$ Li_n(z)=\sum_{j=1}^\infty \frac{z^j}{j^n}, \ (|z| \leq 1). $$
For $|z|>1$ we can use the analytical continuation result given by $$ Li_n(z)=(-1)^{n+1}Li_n(z^{-1})-\frac{1}{n!}\ln^n(-z)-\sum_{j=0}^{n-2}\frac{1}{j!}(1+(-1)^{n-j})(1-2^{1-n+j})\zeta(n-j)\ln^j(-z) $$