calculus - How do you integrate $int_{0}^{infty}frac{acos{(cx)}}{a^2+x^2}dx$?

This integral has been haunting me for a while now, as it's eluded every method of integration I could come up with (u-substitution, integration by parts, trigonometric substitution, and even Feynman's method). I realize it's non-elementary, but I can't figure out how to find the definite integral. I know you'll need to use Feynman's method, but I'm at a loss.

I don't know if this will help, but a and c are both positive.

To be clear, I want to know how to integrate it, not what the value is.