Non-Elementary Integration · Superchallenge problem

Given $\displaystyle\int_0^{2\pi}\dfrac{dx}{b-\cos x}=\dfrac{2\pi}{\sqrt{b^2-1}}$ for $b>1$, differentiate with respect to $b$ to evaluate $\displaystyle\int_0^{2\pi}\dfrac{dx}{(2-\cos x)^2}$.