How to find $\int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x$ Is there a closed-form formula for the following integral
$$
\int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x
$$
where $\alpha$ and $\beta$ are constants which assure that $\alpha+\beta\cos(x)>0$ for any $x\in[0,2\pi]$.

$$I-2\pi\ln(\alpha)=\int_0^{2\pi}\ln(1+s\cos(x))dx=-\int_0^{2\pi}\sum_{k=1}^\infty\frac{(-s)^k\cos^k(x)}kdx\\
=-\pi\sum_{k=1}^\infty\frac1{k4^k}\binom{2k}ks^{2k},$$ where $s=\frac\beta\alpha$, $|s|<1$.
