Question How to evaluate $$\int_0^1 \frac{\arctan(x)}{x} \ln^2(1 - x) \, dx$$ My attempt \begin{align} \int_0^1 \frac{\arctan(x)}{x} \ln^2(1 - x) \, dx &= \int_0^1 \int_0^1 \frac{\ln^2(1 - x)}{1 + x^2 y^2} \, dy \, dx \\ &= \int_0^1 \int_0^1 \ln^2(1 - x) \cdot \frac{i}{1 + x^2 y^2} \, dx \, d...

I am trying to prove that \begin{equation} \int_{0}^{1}\frac{\log\left(x\right) \log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}} \,\mathrm{d}x = \frac{\pi^{3}}{16} - 3\mathrm{G}\log\left(2\right) \tag{1} \end{equation} where $\mathrm{G}$ is Catalan's Constant. I was able to express it in terms of Eul...