I am trying to evaluate the following $2$-dimensional integral:
$$I(x_1,x_2) = \int_{\mathbb{R}^2} d^2 x_3 \frac{\cos^{-1}\left( \left| \vec{x}_3 \right| / \left| \vec{x}_{13} \right| \right)}{\left| \vec{x}_3 \right| \sqrt{\vec{x}_{13}^2 -\vec{x}_3^2} \left(\vec{x}_{13}^2-\vec{x}_{23}^2\right)} \tag{1}$$
with $\vec{x} = (x,y)$, $d^2 x_3 = dx_3 dy_3$ and $\vec{x}_{ij} := \vec{x}_i - \vec{x}_j$. I tried doing the $y$-integral first with integration by parts, but that gave me an unpleasant $\tanh^{-1}(y')$ and I could not do it.
I also looked in the book "(Almost) Impossible Integrals, Sums and Series" by Nahin, but I did not find anything looking like that. It might be useful to note the following identity:
$$\cos^{-1}(x) = \frac{\pi}{2} + i \log \left(\sqrt{1-x^2}+i x \right), \tag{2}$$
but again I could not find what to do with that.
I am not sure whether the integral is fully solvable or not. In the case it is not, I would still be interested by either an integration of one variable only, or by a representation in Spencer's functions or something similar.
EDIT: I forgot to mention that the integral is divergent at $\left|\vec{x}_3\right| = 0$, so some regularization may be needed.