I am trying to compute the following integral: $$\int_D \frac{\log|z-w|^2}{(z-z_1)(\bar{z}-\bar{w}_1)} d\bar{z}dz. $$ Here $D = \{z \in \mathbb{C}, |z|<1\} $ and $w,z_1,w_1 \in D$ are parameters. The integral converges if the parameters are pairwise distinct, which I assume.
I can compute integrals of this type if the numerator is holomorphic by expanding it in power series, and one can reduce to monomials $$\int_D \frac{z^k}{(z-z_1)(\bar{z}-\bar{w}_1)} d\bar{z}dz $$
that I know how to do. However, this method does not work for this integral. I have tried to use complex analysis by writing the absolute value as $(z-w)(\bar{z}-\bar{w})$, introducing a cut from $z_0 \in \partial D$ to $w$, but then one does not get a power series expansion valid on the disk.
I also tried to exploit the fact that $\log|z-w|^2$ is the 2D Green's function (up to a constant), but I was not successful with this idea.
Does anyone have an idea on how to tackle something like this? Any help is appreciated.