How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

Question

How can I prove that $$ \int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)\,? $$

The first thing to try is directly calculating the left hand side. However, I do not know how to find the antiderivative of the integrand: change of variables seems to be rather complicated. How can I go on with this problem.

Answer 1

Hint

I suppose that the trick is differentiation under the integral sign.

$$g(a)=\int\frac{\ln(1+\cos(a)\cos(x))}{\cos(x)}\,dx$$ $$g'(a)=-\int\frac{\sin (a)}{1+\cos (a) \cos (x)}\,dx$$