Is there an closed form solution for the integral $$I(x,y,a,b)=\int_{-b}^{b} \mathrm{K}_0 \left( a\,\sqrt{(x-x_0)^2+y^2}\right) \, \mathrm{d}x_0$$ with $x>0$ and $y>0$. I am assuming that the solution is different for $x>l$ and $x<l$, since I know the solution for $y=0$ and there it is actually the case. I computed the solution for $y=0$ through a complete different approach though, which is not valid for $y>0$.
My first idea was to use the Basset Integral representation of the modified Bessel function. see here
$$ \mathrm{K}_0(x\,z) = \int_{0}^{\infty}\frac{\mathrm{cos}(x\,t)}{\sqrt{t^2+z^2}}\,\mathrm{d}t$$
Since $a$ is not necessary a real number, I used $x=1$ and $z=a\,\sqrt{(x-x_0)^2+y^2}$
Using this relations lead to the integral (I changed the order of integration):
$$\int_{0}^{\infty}\int_{-b}^{b}\frac{\mathrm{cos}(t)}{\sqrt{t^2+a^2\,((x-x_0)^2+y^2)}}\,\mathrm{d}x_0\,\mathrm{d}t$$
According to Mathematica the integral with respect to $x_0$ leads to $$\int_{0}^{\infty}\left[\mathrm{ln}\left(a(a\,(x+l)+\sqrt{t^2+a^2((x+l)^2+y^2)})\right)-\\ \mathrm{ln}\left(a(a\,(x-l)+\sqrt{t^2+a^2((x-l)^2+y^2)})\right)\right]\mathrm{cos}(t)\,\mathrm{d}t$$
Any idea how to integrate this? I was thinking about integration by parts. I also assume that the result is a combination of modified Bessel functions and modified Struve Functions for the same reasions as mentioned above.
Thank you.