How to compute a special double integral on a finite domain

Question

For any given integer $n>2$ and real constants $0\leq a\leq 1$ and $c>0$, I want to compute the integral:

$$ \int _0^a\int _0^a\frac{(1+ c\,x\, y)^{n-2} (1+n\,c\, x\, y)}{(1+x) (1+y)}\, dx\,dy $$

Wolfram Mathematica 11.3 does not give an answer.

Answer 1

we have: $$I=\int_0^a\int_0^a\frac{(1+cxy)^{n-2}(1+ncxy)}{(1+x)(1+y)}dxdy$$ $$=\int_0^a\frac{1}{(1+y)}\int_0^a\frac{(1+cyx)^{n-2}+ncyx(1+cyx)^{n-2}}{(1+x)}dxdy$$ if we focus on the first one: $$I_1=\int_0^a\frac{1}{(1+y)}\int_0^a\frac{(1+cyx)^{n-2}}{(1+x)}dxdy$$ this inside integral has no elementary derivative and neither do the others involved, so it appears that if there is a solution it will involve a double summation potentially