calculus - Computing the integral of $log(sin x)$

How to compute the following integral?

$$\int\log(\sin x)\,dx$$

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Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined?