I'm trying to evaluate the following integral (or at least get a bound involving $s_x$ and $s_y$): $$ \int_0^\infty\int_0^\infty \frac{xy}{(x^2+y^2)^{3/2}}\exp\left\{-\frac{1}{2}\left(\frac{x^2}{s_x^2}+\frac{y^2}{s_y^2}\right)\right\}dy dx $$
Numerical methods suggest that it is finite (This integral is related to the the expectation of the absolute value of $\partial^2 \sqrt{x^2+y^2}/\partial x\partial y$ when $x$ and $y$ are independent normal r.v.s)
Any hint on how to proceed? Thank's.
Converting to polar coordinated reveals
$$\begin{align} \int_0^\infty \int_0^\infty \frac{xy}{(x^2+y^2)^{3/2}}e^{-x^2/2s_x^2}e^{-x^2/2s_x^2}\,dx\,dy&=\int_0^{\pi/2}\int_0^\infty \sin(\theta)\cos(\theta)e^{-\frac12 r^2 \left(\frac{\cos^2(\theta)}{s_x^2}+\frac{\sin^2(\theta)}{s_y^2}\right)}\,dr\,d\theta\\\\ &=\frac{\sqrt{\pi}}2 |s_xs_y|\int_0^{\pi/2}\frac{\sin(\theta)\cos(\theta)}{\sqrt{\left(s_y^2\cos^2(\theta)+s_x^2\sin^2(\theta)\right)}} \,d\theta\\\\ &=\frac12\sqrt{\pi} |s_xs_y|\int_0^{\pi/2}\frac{\sin(2\theta)}{\sqrt{(s_x^2+s_y^2)+(s_y^2-s_x^2)\cos(2\theta)}}\,d\theta\\\\ &=\sqrt{\frac\pi 2}\frac{|s_xs_y|}{|s_x|+|s_y|} \end{align}$$
And we are done!
