Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$

Question

I would like to know if there exists a closed form for this integral

$$\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx.$$

I tried the relation

$$\log\left(\sin\left(x\right)\right)=-\log\left(2\right)-\sum_{n=1}^{\infty}\frac{\cos\left(2nx\right)}{n}$$

but it seems useless.

Answer 1

There exist a closed-form for the integral. It is

$$\frac{\pi^3}{32}+\frac{\pi}{8}\ln^2 2 - 2G-4\,\Im\left[\operatorname{Li}_3\left(\frac{1+i}2\right)\right]+2,$$

where $G$ is Catalan's contant, and $\operatorname{Li}_3$ is the trilogarithm function.

Using Cleo's result you can express it in term of a hypergeometric function, but finding a closed-form of $\Im\left[\operatorname{Li}_3\left(\frac{1+i}2\right)\right]$ is a well-known open problem on math.se.