# Evaluating $\int^1_0 \frac{\operatorname{Li}_2(-x)}{x} \log(1+x)\, \mathrm dx$

How would you solve the following?

$$\int^1_0 \frac{\operatorname{Li}_2(-x)}{x} \log(1+x)\, \mathrm dx$$

Hint

$$\int \frac{\log(1+x)}{x} \, \mathrm dx=-\text{Li}_2(-x)$$

Using the trick given in the hint, then $$\int \frac{\operatorname{Li}_2(-x)}{x} \log(1+x)\, \mathrm dx=-\frac{\text{Li}_2(-x){}^2}{2}$$ and so $$\int^1_0 \frac{\operatorname{Li}_2(-x)}{x} \log(1+x)\, \mathrm dx=-\frac{\pi ^4}{288}$$