This response is posted after I got the integral directly by email.
regarding all the constraints you posted with your question to the forum that you need a real result and not deal with the complex plane. Under this assumption, the integrand is ill-posed. If you check for example the expression under the Sqrt that is the argument of the ArcSin function, you will find out that as x approaches c the expression of the Sqrt function approaches - infinity. This points that under my first assumption that you need a closed form real result (in terms of a,b,c, and r) that your constraints are not sufficient. Mathematica complains on convergence are justified in this case