I have tried really hard to compute the following integrals, but got nowhere: Does anyone know a method that could work here ? The integrals are $$ \int_{1}^{x}\frac{\operatorname{Li}_{2}\left(t\right)\log\left(t + 1\right)}{t}\,{\rm d}t\quad \mbox{and}\quad \int_{1}^{x}\frac{\operatorname{Li}_{...

Empirically, i have obtained the following value: \begin{align}K&=\int_0^1 \frac{\arctan x\ln^2 x}{1+x^2}\,dx\\ &=\frac{151}{11520}\pi^4-\frac{1}{24}\ln^4 2-\text{Li}_4\left(\frac{1}{2}\right)+\frac{1}{24}\pi^2\ln^2 2-\frac{7}{8}\zeta(3)\ln 2\end{align} How to prove this? My attempt: Observe: \be...