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.

You asked to compute the Fourier cosine series of $x\cot x\cos x$ in your other question (http://math.stackexchange.com/questions/1168081/closed-form-for-int-01-2-fracx-cos-leftx-pi-right2-cos-left2-pi-kx), so it is just a matter of exploiting the orthogonality formulas. — Jack D'Aurizio, Feb 27 '15 at 18:14
However, direct computation through differentiation under the integral sign is way easier. — Jack D'Aurizio, Feb 27 '15 at 18:27

score 5 · Accepted Answer · edited Apr 13 '17 at 12:21

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.

1 Answers1