Is there a simple reason, why the integrals of the form $$ C_1(s) =\int\limits_{0}^{\pi/2} \cos(2 \nu \theta) \cos^{2s-1}(\theta) \; d \theta $$ and $$ C_2(s)= \int\limits_{0}^{\infty} \cosh(2 \nu \theta) \cosh^{-1-2s}(\theta) \; d \theta,$$ can both be expressed in terms of the Beta function, and there quotient is $C_1(s) C_2(s) = D /s$ for some constant $D$?
With simple, I mean can we write the product of the integral and transform from polar coordinates to Cartesian coordinates and use some addition theorems for cos and cosh?
