IDK if this is the best way to get the answer, but it's how I found it. Fist, cos2x = cos^2x-sin^2x so

multiply by

Mess with the denominator:

And now we can use these rules:

Simplify

Simplify

Make substitution

Distribute the 2 into the denominator

Integrate

Anti-substitute

I'm pretty sure that's right, at the very least, I like all the steps I used to get there.