The functional form is known already (as attached). But what is the solution for this integral?
The integral can apparently be done exactly and the answer is
which is consistent with the numerical evaluation that I mentioned doing in my comments on the question.
The value was apparently first derived in 2000 by Skeidov and Skvirsky in this paper:
As an explanation of their evaluation, they unfortunately have only this to say: "After some lengthy interactive session with Mathematica, we get...". I personally am unable to get Mathematica to produce this result, but I have no doubt that it is correct.