9.2.20 Integrals depending on a parameter

Problem: Determine the domain of existence and express the following integral in terms of the Euler integrals: \[ I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)^{\cos 2 \alpha} d x \]