I am trying to evaluate the integral
$$F(x,y) = \int_0^1 du_1\, \int_0^{1-u_1} du_2\, \frac{\log f(x,y|u_1,u_2)}{f(x,y|u_1,u_2)}\,, \tag{1}$$
with
$$f(x,y|u_1,u_2) := u_1(1-u_1)+y\, u_2(1-u_2) + (x-y-1)u_1 u_2\,, \tag{2}$$
where $0<x<1, 0<y<1$. My question is simple: is it possible to evaluate this integral for general $x$ and $y$?
Mathematica returns a result for the first integral, which takes the form
$$F(x,y) = \int_0^1 du_1 \frac{1}{\sqrt{\ldots}} (\text{arctanh} (\ldots) + \log (\ldots) + \text{Li}_2 (\ldots))\,. \tag{3}$$
I cannot specify the $\ldots$ here because the expressions are very lengthy. I have been told in the past that $1/\sqrt{\ldots}$ can be a sign of the function being elliptic. Is it possible to say whether $F(x,y)$ is an elliptic function or not?