TITLE: Evaluate the following definite integral
QUESTION [5 upvotes]: Evaluate
$$\int_0^{\frac{\pi}{4}}\arctan\sqrt{\frac{\cos 2x}{2\cos^2x}}dx$$
It seems that the anti-derivetive of the integrant cannot be written as elementary functions. How to evaluate this?

REPLY [4 votes]: This is an instance of the family of integrals so called Coxeter's integrals and is closely related to Ahmed's integrals.
You can find my solution here.