Proving $\int_0^\pi \frac{\log(1+x\cos (y))}{\cos y}dy=\pi \arcsin x$ How would I go about proving that, $$\int_0^\pi \frac{\log(1+x\cos (y))}{\cos (y)}\,dy=\pi \arcsin (x)$$
I have tried to do this by computing the integral directly, but it appears to be too difficult. Maybe there is a better approach to this that I do not know of.

HINT:
Using Feynman's Trick (differentiating under the integral), we have for $|x|<1$
$$\begin{align}
\frac{d}{dx}\int_0^\pi \frac{\log(1+x\cos(y))}{\cos(y)}\,dy&=\int_0^\pi \frac{1}{1+x\cos(y)}\,dy\tag1
\end{align}$$
Use contour integration or apply the Weierstrass substitution to evaluate the integral on the right-hand side of $(1)$.  Then, integrate the result.
