Calculate integral for the function :
$\displaystyle
\begin{array}{l}
\ln (\sin (x)) \\
\ln (\cos (x)) \\
\end{array}$
Maybe we can make that a tad more apparent? Let $\displaystyle x=\arcsin(z)\implies dx=\frac{dz}{\sqrt{1-z^2}}$ so our integral becomes $\displaystyle \int\frac{\ln(z)}{\sqrt{1-z^2}\ln\left(\sqrt{1-z^2}\right)}$. Let $\displaystyle \sqrt{1-z^2}=\tau\implies \frac{-\tau}{\sqrt{1-\tau^2}}d\tau=dz$. So then our integral becomes $\displaystyle \frac{-1}{2}\int\frac{\ln\left(1-\tau^2\right)}{\sqrt{1-\tau^2}\ln\left(\tau\right)}d\tau$. That looks almost doable. I am not feeling it right now. Maybe the OP can continue.