$$\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} e^{\frac{-1}{1-(x^2+y^2)}}\log\Big(\log\big(\frac{2}{\sqrt{(1-x)^2+(1-y)^2}}\big)\Big)\,dy\,dx.$$
I got this super hard integral while doing an exercise on distributions. The only thing I could think of is to transform it into polar coordinates. It changes into
$$\int_{0}^{2\pi} \int_{0}^{1} e^{\frac{-1}{1-r^2}}\log\Big(\log\big(\frac{2}{\sqrt{r^2-2r\cos(\theta)-2r\sin(\theta)+2}}\big)\Big)\,r\,dr\,d\theta.$$
But even WolframAlpha couldn't compute this. Does anyone know how to evaluate this?
Edit:- There is no closed form of this integral. If anyone has any suggestions regarding approximations, please comment below.