@CarloBeenakker Yes you are right, $p=2$ for the integral equals the $arc \sin(x)/x$ as a special case and up until now that is the only one I have discovered too. I was more interested on how to treat relations like the one, on the second part of my question.

@CarloBeenakker I see. It is also interesting to see how this is derived, but still, I need to solve for $t$ :/