Hmmm...well I have fiddled around with the indefinite integral. I tried a bunch of tricks but to no avail. So I am going to say that this has no closed form answer. But for the definite part, I am not completely done yet, but letting then using differentation under the integral sign is looking promising.

EDIT: Also, based upon human nature I am going to guess that a sub of will be in here somewhere, I got this based upon human being's propensity for nice solutions, e.g. look at the limits of integration