Recently I have calculated the long resisting indefinite integral $\int \frac{1}{x} \log (1-x) \log (x) \log (x+1) \, dx$ (https://math.stackexchange.com/a/3535943/198592).
Similar cases, but for definite integrals with fixed constant limits (mostly between $0$ and $1$), are abundant in this forum, and this led me to the generalizing
Question
for which $k$ can you evaluate the indefinite integral
$$f_k(x) = \int x^k \log (1-x) \log (x) \log (x+1) \, dx\tag{1}$$
I have found that $(1)$ can be solved for any integer $k$, and I have also found a solution for $k=\pm\frac{1}{2}$.
In order not to spoil your fun I shall give my proofs later. Also I am interested seeing investigations and proofs of other contributors.
