I've come across the following integral involving the Bessel Functions $K_\nu$ that would be great if it were possible to evaluate.
$$\int_{y_1}^{\infty}dy\frac{y}{\sqrt{a^2+y^2}} \left(\left[K_0\left(m \sqrt{a^2+y^2}\right)+K_2\left(m \sqrt{a^2+y^2}\right)\right] \left[y-\mathrm{arctan}\left(\frac{y}{a}\right)\right] +2 y K_1\left(m \sqrt{a^2+y^2}\right)\right)$$
Here m is a real parameter greater than 0.
I'm having trouble using any table of integrals or Mathematica, since the bounds of integration are not from $0 \rightarrow \infty$, and I can't find how to get closed form expressions for any of the parts' integrals.
Even being able to evaluate any term in this expression could be very helpful, and any resources on where these have been done would be appreciated.