Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du.$$ The parameters are possibly complex, and satisfy $$\Re(c)>-1, \qquad |a|\leq1, \qquad |b|\leq1. $$
The best I could do was $$ f(a,b,c) = \frac{1}{1+c}\frac{\partial^2}{\partial s \partial t}\Big|_{s=t=0}F_1(1+c; -s,-t; 2+c; a, b),$$ where $F_1$ is an Appell function, but this doesn't help me much. Also, when $a=b$, I know that $f$ is the second derivative of the beta function.
Is it possible to get a nicer expression for $f$?