Hint1: let 2a=n and 2b=m.
Hint2: Beta function.
I was looking at a book on Mathematical Physics today and found the following interesting identity involving sine and cosine integrals. But I am not going to tell you what the identity is, to make it more difficult to find it.
Here it is below. In the special case where n,m are integers it solves nicely.
But is also works for real numbers as well. Probably also for complex numbers but let us not go there.