Integration Technique

Today I was helping out at the Buchholz calculus team practice and there was only one problem that they had no idea how to solve. The problem was similar to the one below.

I have left of the limits of integration because for this kind of problem they are not very important. There are several ways to proceed when solving this, first you can write the denominator as one sine or cosine function. This is the most obvious way to proceed but is certainly not the best, I am not sure where it will lead but just with a little mental computation you should quickly realize that you have not made the problem any easier. The method I suggest is that you make an appropriate substitution. When I first mentioned to them that there existed a convenient substitution they looked at me in disbelief. I suggest that you try and think of the substitution before reading any further.
The substitution I am referring to is,

After some algebra and trigonometry you will arrive at the following,
After you replace each trigonometric expression with its corresponding algebraic one you should have an integral that is much easier to solve.