Consider the following integral, and it's closed forms:
$$\displaystyle \int_{\mathbb{R}} \frac{\tan^{-1}\left(\frac{\sqrt{x^2+a^2}}{b}\right) \, \text{d}x}{(x^2+b^2) \left( \frac{\sqrt{x^2+a^2}}{b} \right)} = \begin{cases} \frac{\pi}{\sqrt{a^2-b^2}}\left(2\cos^{-1} \left(\frac{b}{a}\right) - \cos^{-1} \left(\frac{b^2}{a^2}\right) \right) & \text{ if } |a|>|b| \\ \frac{\pi}{\sqrt{b^2-a^2}}\left(2\cosh^{-1} \left(\frac{b}{a}\right) - \cosh^{-1} \left(\frac{b^2}{a^2}\right) \right) & \text{ if } |a|<|b| \end{cases}$$
Does there exist a meaningful physical/mechanical/geometric interpretation of the integral?
Ideally, such an interpretation would not be too contrived and artificial, but that isn't guaranteed.
Probability would also work but that doesn't seem like it would fit such an integral.
