Is it possible to evaluate this integral in a closed form?
It also can be represented as
Okay, finally I was able to prove it.
Step 0. Observations. In view of the following identity
Vladimir’s result suggests that there may exists a general formula connecting
and the Legendre chi function χ2. Indeed, inspired by Vladimir’s result, I conjectured that
I succeeded in proving this identity, so I post a solution here.
Step 1. Proof of the identity (1). It is easy to check that the following identity holds:
So it follows that
For the convenience of notation, we put
Then it is easy to check that arsinh(1/r)=−logα and likewise for s and β. Thus with the substitution x↦sinhx and y↦sinhy, we have
Applying the substitution e−x↦x and e−y↦y, it follows that
as desired, proving the identity (1).
EDIT. I found a much simpler and intuitive proof of (1). We first observe that (1) is equivalent to the following identity
Now we first observe that from the addition formula for the hyperbolic tangent, we obtain the following formula
which holds for sufficiently small x,y. Thus
We readily check this holds for any |r|<1. Therefore