I believe that it is possible to get in closed-form the integral $$\int_0^1\int_0^1\int_0^1\frac{\log((x+z)(y+z))}{1+xyz}dxdydz,\tag{1}$$ but I think that it is tedious. My belief is from the calculations that I tried using a CAS. This CAS know how to get the corresponding indefinite integrals but it is tedious to me to evaluate the limits of integration. My last attempt was write our integral as $$\sum_{n=0}^\infty\int_0^1\int_0^1(-1)^k(yz)^k\int_0^1x^k\log((x+z)(y+z))dxdydz.\tag{2}$$ The CAS tells me what are the corresponding antiderivatives, but it is difficult/tedious to me calculate the expressions for the evaluation of the mentioned limits.
Question. Can you express as a series or well calculate a closed-form (if it is possible) our integral $(1)$? Many thanks.
If you think that is feasible to provide me a sketch of the required calculations instead of all details feel free to do it, or well if this is a known integral from the literature adding the reference.