Does $\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\right)\,\ln\left(1-\frac{\cos^2\phi}{C^2}\right)$ have a closed form? I am wondering if anyone has a nice way of approaching the following definite integral $\newcommand{\dilog}{\operatorname{Li}_2}$
$$\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\right)\,\ln\left(1-\frac{\cos^2\phi}{C^2}\right)\,.$$
Here $C$ is a positive, real constant that satisfies the constraint $C>1$. So far I have tried a simple $u$ substitution, $u = \cos\left(\phi\right)/C$. However, this doesn't get me anywhere. I have also tried performing a series expansion in small $1/C$ in the second log, performing the integration, and then summing in powers of $1/C$. However, the sum does not simplify nicely.
I have also tried relating the expression to $\dilog\left(1-\frac{\cos^2\phi}{C^2}\right)$ and $\dilog\left(\frac{\cos^2\phi}{C^2}\right)$ and performing the integration of these polylog functions using their series representation. However I have trouble performing the summation for the $\dilog\left(1-\frac{\cos^2\phi}{C^2}\right)$ term.

This integral has a closed form in terms of dilogarithms; the idea is that the series $$\ell(r,\phi)=\log(1-2r\cos\phi+r^2)=-2\sum_{n=1}^\infty\frac{r^n}{n}\cos n\phi\qquad(|r|<1)$$ (obtained from $1-2r\cos\phi+r^2=(1-re^{i\phi})(1-re^{-i\phi})$ and the power series of $\log$) may be considered as a Fourier series, giving $\int_0^{2\pi}\ell(r,2\phi)\,d\phi=0$ for $|r|\leqslant 1$ and, by Parseval's identity, $$\int_0^{2\pi}\ell(a,2\phi)\ell(b,2\phi)\,d\phi=4\pi\sum_{n=1}^\infty\frac{a^n}{n}\frac{b^n}{n}=4\pi\operatorname{Li}_2(ab)$$ for $|a|,|b|\leqslant 1$ (all the boundary cases are attainable in the limit). Now
$$
\log\left(\frac{\cos^2\phi}{C^2}\right)=\ell(-1,2\phi)-\log(4C^2),\\
\log\left(1-\frac{\cos^2\phi}{C^2}\right)=\ell(r,2\phi)-\log(4rC^2),\\
\color{blue}{r:=2C^2-1-2C\sqrt{C^2-1}},
$$
reducing the given integral to the above. The answer is $\color{blue}{2\operatorname{Li}_2(-r)+\log(4C^2)\log(4rC^2)}$.
