I wrote the integral as .
Then I switched the order of integration and used Weierstrass substitution to get .
Finally I let and got
Sometimes Wolfram Alpha returns which reduces to my solution if is real and positive.
But other times it returns which reduces to if is real and positive. On another forum this was stated as the solution. But this cannot be the solution for when and it's not even the solution for when . Is it the solution for ?
EDIT: It's not the solution for either.
When faced with difficulties of the kind we have with this integral I always ask myself why do I need to know? There may be ways of approximating this that meet your needs without having a closed form expression for the integral itself.
For instance for small we may expand the log as a power series and we end up with a simpler integrals.
CB