\begin{align} \mathcal I_1&=\int_0^1\frac{\ln^n(1-x)}{x}\text dx\\ &=\int_0^1\ln^n(1-x)\text d\ln x\\ &=-n\int_0^1\ln^{n-1}x\text d\text{Li}_2(x)\\ &=n(n-1)\int_0^1\ln^{n-2}x\text d\text{Li}_3(x)\\ &\cdots\\ &=(-1)^nn!\text{Li...