Is there a possibility to extend this result
$$\int_0^\infty \frac{\exp\left(-\sqrt{x^2+y^2}\right)}{x^2+y^2}dx = \frac{\pi}{2}\left(\frac{1}{y} - K_0(y)L_{-1}(y) - K_1(y)L_{0}(y)\right).$$
to the integral $$\int_0^\infty \frac{\exp\left(-\sqrt{x^2+y^2}\right)}{x^2\,(1+a^2)+y^2}dx$$
with $0\le a^2 \le 1$, $y \in \Bbb C$, $K$ the modfied Bessel function and $L$ the modified Sturve function.
If this is not straightforward, could somebody show me, how to transform the given integral to a different integral using contour integration and the poles at $\pm i \sqrt{\frac{y^2}{(1+a^2)}}$ and the branch points at $\pm i y$. Thank you.