I think you want to break the integral into two parts by the sum. Then make the u substitution u=1/2cos(x). That will get you the sin(x)dx you need. so you will have a

May need to break this integral up again to deal with that absolute value.

Then I think just do the integration by parts twice trick and you will get back something familar and solve that way. I assume you know what you are doing if you were assigned this kind of problem, so I left out a lot of steps. Let me know if you need me to fill in details.