TITLE: A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$
QUESTION [27 upvotes]: We already know that 

\begin{align} 
\displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x  = -\frac{7}{16} \zeta(3), 
\\\\ &  \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x  = \frac{11 \pi}{16} \zeta(4).  \end{align} 

Does the following integral admit a closed form?

\begin{align} \displaystyle &  \int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x  \end{align}

REPLY [14 votes]: Proposition.
  $$
\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:{\mathrm{d}}x  = \frac{45}{512} \zeta(7)-\frac{3\pi^2}{16} \zeta(5)-\frac{5\pi^4}{64} \zeta(3)-\frac{9}{4}\zeta(\bar{5},1,1)
$$

where $\zeta(\bar{p},1,1)$ is the colored MZV (Multi Zeta Values) function of depth 3 and weight $p+2$ given by
$$
\zeta(\bar{p},1,1) : = \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{p}}\sum_{k=1}^{n-1}\frac{H_{k-1}}{k} 
$$
belonging to a family of functions introduced by L. Euler and also called Euler(-Zagier) sums.
We have a general result.

Theorem. Let $\ell$ be any positive integer. 
  Then
  $$
\int_{0}^{\pi/2}\! \! x^{2\ell+1}\! \ln^{2\ell+1}(2 \cos x){\mathrm{d}}x  \in \mathbb{Q} \!\left( \zeta(4\ell+3),\zeta(2)\zeta(4\ell+1), ... ,\zeta(4\ell)\zeta(3), \zeta(\overline{2\ell+3},\{1\}_{2\ell})\right) 
$$ 

It is remarkable that there is only one constant 
$$ \zeta(\overline{2\ell+3},\{1\}_{2\ell})=\sum_{n_{1}> ...>n_{2\ell+1}>0} \frac{(-1)^{n_1}}{n_1^{2\ell+3} n_2\cdots n_{2\ell+1}} $$ 
for each integral of the considered form. The question of whether one can reduce this constant to colored MZVs/MZVs of lower depths is still subject to a conjecture (Zagier).
This paper may be useful, another one and of course Hoffman's site which has a list of many related references.