I can't remember the source of even the exact answer :( But I've seen integrals of that form done in some papers by Ramanujan where he expressed the answer in terms of the zeta function and something else, so it's not exactly an elementary integral.
This isn't how he did it, and I'm not sure if it will help, but you could express cos/sin in terms of an infinite product, split up the logs ( log [ab]= log [a] + log ) and then continue?