Evaluating $\int_{\frac{-1}{2}}^{\frac{1}{2}} \int_{\frac{-1}{2}}^{\frac{1}{2}} \frac{x^2}{(x^2+y^2)^2 \log^2(\frac{2}{\sqrt{x^2+y^2}})} \,dx\,dy.$ 
$$\int_{\frac{-1}{2}}^{\frac{1}{2}} \int_{\frac{-1}{2}}^{\frac{1}{2}} \frac{x^2}{(x^2+y^2)^2 \log^2(\frac{2}{\sqrt{x^2+y^2}})} \,dx\,dy.$$

I encountered this integral while evaluating norm of a function. The first attempt was to change it into polar coordinates, which didn't work well, since the region of integration is rectangular.
I found a similar integral here, but even WolframAlpha  couldn't calculate this.
Does anybody know how to evaluate this?

We may tackle the equivalent integral
$$ \mathcal{J}\stackrel{\text{def}}{=}\iint_{(0,1)^2}\frac{dx\,dy}{(x^2+y^2)\log^2\left(\frac{1}{4}\sqrt{x^2+y^2}\right)} $$
by writing it as
$$ \int_{0}^{1}\frac{\frac{\pi}{2}\rho}{\rho^2\log^2\left(\rho/4\right)}\,d\rho +\int_{1}^{\sqrt{2}}\frac{\left(\frac{\pi}{2}-2\arccos\frac{1}{\rho}\right)\rho}{\rho^2\log^2(\rho/4)}\,d\rho$$
which equals
$$ \frac{\pi}{3\log 2}-2\int_{1}^{\sqrt{2}}\frac{\arccos\frac{1}{\rho}}{\rho \log^2(\rho/4)}\,d\rho=\frac{\pi}{3\log 2}-2\int_{1/\sqrt{2}}^{1}\frac{\arccos\rho}{\rho\log^2(4\rho)}\,d\rho $$
or
$$ \frac{\pi}{6\log 2}+2\int_{1/\sqrt{2}}^{1}\frac{\arcsin\rho}{\rho\log^2(4\rho)}\,d\rho = \frac{\pi}{3\log 2}-\int_{0}^{\pi/4}\frac{\theta\tan\theta}{\log^2(4\cos\theta)}\,d\theta.$$
The last integral is very far from being elementary or expressible through a simple hypergeometric series, but $\frac{\theta\tan\theta}{\log^2(4\cos\theta)}$ approximately behaves like $C\theta^4$ on $(0,\pi/4)$, hence the composite Boole's rule provides excellent numerical approximations with few computations. We get
$$ \mathcal{J}\approx 1.239117616448\approx \frac{35725}{28831}. $$
