The simple integral of an integer order Bessel function is given by Struve functions. You can find them in ch. 11 , "Integrals of Bessel Functions" of Abramowitz and Stegun. If you are serious about math, buy a copy (the Dover edition is inexpensive). For now you can browse an old online copy here
http://www.convertit.com/Go/Convertit/Reference/AMS55.ASP" [Broken]