TITLE: Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$
QUESTION [8 upvotes]: Evaluate $\displaystyle \int\limits_0^1 \dfrac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$

I was wondering if the above had some kind of a closed form, here some of the special cases have been discussed but this one is really a fascinating one. 
I guess there's no general taylor expansion for $\ln^m (1+x)$ and so transforming into a series wouldn't be that easy.

REPLY [3 votes]: Given the integral, $$I(n,p) = \int\limits_0^1 \dfrac{ \ln^{n-1}(x)\ln^p (1+x)}{x}\; dx$$
which is the notation consistent with Nielsen polylogs. Then closed-forms in terms of ordinary polylogarithms are known only for the following cases,
$$I(1,p) \\ I(n,1) \\ I(2k+1,2) \\ I(2,2) \\ I(2,3)$$
and no more. See this more general post on Nielsen polylogs.