Stable cohomology of $Aut(F_n)$ with bivariant twisted coefficients

We compute the cohomology groups of the automorphism group of the free group $F_n$, with coefficients in arbitrary tensor products of the standard representation $H_1(F_n,\mathbb{Q})$ and its dual, in a range where $n$ is sufficiently large compared to the cohomological degree and the number of tensor factors.