TITLE: Quadirlogarithm value $\operatorname{Li}_4 \left( \frac{1}{2}\right)$
QUESTION [9 upvotes]: Is there a known closed form for the following 
$$\operatorname{Li}_4 \left( \frac{1}{2}\right)$$
I know that we can derive the closed of $\operatorname{Li}_1 \left( \frac{1}{2}\right),\operatorname{Li}_2 \left( \frac{1}{2}\right),\operatorname{Li}_3 \left( \frac{1}{2}\right)$
To put it in an integral representation, the problem asks to solve  
$$\int^1_0 \frac{\log(x)^3}{2-x}\, dx$$

REPLY [3 votes]: $\newcommand{\+}{^{\dagger}}
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$\ds{\int_{0}^{1}{\ln^{3}\pars{x} \over 2 - x}\,\dd x:\ {\large ?}}$

\begin{align}&\overbrace{\color{#c00000}{\int_{0}^{1}
{\ln^{3}\pars{x} \over 2 - x}\,\dd x}}
^{\ds{\mbox{Set}\ x \equiv \expo{-t}\ \imp\ t = -\ln\pars{x}}}\ =\
\half\int_{\infty}^{0}{-t^{3} \over 1 - \expo{-t}/2}\,\pars{-\expo{-t}\,\dd t}
\\[3mm]&=-\,\half\int_{0}^{\infty}
t^{3}\expo{-t}\sum_{n = 0}^{\infty}\pars{\half}^{n}\expo{-nt}\,\dd t
=-\,\half\sum_{n = 0}^{\infty}\pars{\half}^{n}
\int_{0}^{\infty}t^{3}\expo{-\pars{n + 1}t}\,\dd t
\\[3mm]&=-\,\half\sum_{n = 0}^{\infty}{\pars{1/2}^{n} \over \pars{n + 1}^{4}}\
\overbrace{\int_{0}^{\infty}t^{3}\expo{-t}\,\dd t}^{\ds{=\ 3!\ = 6}}\ =\
-6\sum_{n = 1}^{\infty}{\pars{1/2}^{n} \over n^{4}}
\end{align}

$$
\color{#66f}{\large%
\int_{0}^{1}{\ln^{3}\pars{x} \over 2 - x}\,\dd x
=-6\,{\rm Li}_{4}\pars{1 \over 2}} \approx -3.1049
$$

$\ds{{\rm Li_{s}}\pars{z}}$ is a
  PolyLogarithm Function.