$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove armada]{\displaystyle{#1}}\,}
\newcommand{\llaves}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\parcial #3^{#1}}}
\newcommand{\raíz}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
De ahora en adelante, voy a usar
Iverson Corchetes $\ds{\pars{~\mbox{namely,}\ \bracks{\cdots}~}}$ que son muy eficientes cuando un engorroso restricción está presente.
\begin{align}
I & \,\,\,\substack{\mbox{def.} \\[1mm] \ds{\equiv}}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\int_{-\infty}^{\infty}
\braces{x \over y}\braces{y \over z}\braces{z \over w}\braces{w \over x}
\bracks{0 < x < y < z < w < 1}\dd x\,\dd y\,\dd z\,\dd w
\\[5mm] & =
\int_{-\infty}^{1}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\int_{0}^{\infty}{x \over y}\,{y \over z}\,{z \over w}\,\braces{w \over x}
\bracks{x < y < z < w}\dd x\,\dd y\,\dd z\,\dd w
\\[5mm] & =
\int_{-\infty}^{\infty}
\int_{-\infty}^{\infty}\int_{0}^{\infty}\int_{-\infty}^{1}
{1 \over w/x}\,\braces{w \over x}
\bracks{x < y < z < x\,{w \over x}}x\,{\dd w \over x}\,\dd x\,\dd y\,\dd z
\\[5mm] & =
\int_{-\infty}^{\infty}
\int_{-\infty}^{\infty}\int_{0}^{\infty}x\int_{-\infty}^{1/x}
{\braces{w} \over w}\bracks{x < y < z < xw}\dd w\,\dd x\,\dd y\,\dd z
\\[5mm] & =
\int_{-\infty}^{\infty}
\int_{-\infty}^{\infty}\int_{0}^{\infty}x\int_{-\infty}^{\infty}
\bracks{w < {1 \over x}}{\braces{w} \over w}\bracks{x < y < z}\bracks{z < xw}
\dd w\,\dd x\,\dd y\,\dd z
\\[5mm] & =
\int_{0}^{\infty}x\int_{-\infty}^{\infty}\bracks{w < {1 \over x}}
{\braces{w} \over w}\int_{-\infty}^{\infty}\bracks{z < xw}
\int_{-\infty}^{\infty}\bracks{x < y < z}\dd y\,\dd z\,\dd w\,\dd x
\\[5mm] & =
\int_{0}^{\infty}x\int_{-\infty}^{\infty}\bracks{w < {1 \over x}}
{\braces{w} \over w}\int_{-\infty}^{\infty}\bracks{z < xw}
\bracks{x < z}\int_{x}^{z}\dd y\,\dd z\,\dd w\,\dd x
\\[5mm] & =
\int_{0}^{\infty}x\int_{-\infty}^{\infty}\bracks{w < {1 \over x}}
{\braces{w} \over w}\int_{-\infty}^{\infty}\bracks{x < z < xw}
\pars{z - x}\dd z\,\dd w\,\dd x
\\[5mm] & =
\int_{0}^{\infty}x\int_{-\infty}^{\infty}\bracks{w < {1 \over x}}
{\braces{w} \over w}\bracks{x < xw}\int_{x}^{xw}
\pars{z - x}\dd z\,\dd w\,\dd x
\\[5mm] & =
\int_{0}^{\infty}x\int_{-\infty}^{\infty}\bracks{1 < w < {1 \over x}}
{\braces{w} \over w}
\pars{{1 \over 2}\,w^{2}x^{2} - {1 \over 2}\,x^{2} - wx^{2} + x^{2}}\dd w\,\dd x
\\[5mm] & =
{1 \over 2}\int_{0}^{\infty}x^{3}\int_{-\infty}^{\infty}
\bracks{1 < w < {1 \over x}}{\braces{w} \over w}\pars{w - 1}^{2}
\,\dd w\,\dd x
\\[5mm] & =
{1 \over 2}\int_{-\infty}^{\infty}
{\braces{w} \over w}\pars{w - 1}^{2}\int_{0}^{\infty}x^{3}
\bracks{x < wx < 1}\,\dd x\,\dd w
\\[5mm] & =
{1 \over 2}\int_{-\infty}^{\infty}
{\braces{w} \over w}\pars{w - 1}^{2}\bracks{w > 1}\int_{0}^{1/w}x^{3}
\,\dd x\,\dd w =
{1 \over 8}\int_{1}^{\infty}
{\pars{w - 1}^{2} \over w^{5}}\,\pars{w - \left\lfloor w\right\rfloor}\dd w
\\[5mm] & =
{ 1 \over 24} -
{1 \over 8}\sum_{n = 1}^{\infty}\int_{n}^{n + 1}{\pars{w - 1}^{2} \over w^{5}}\,n\,\dd w =
\bbx{{1 \over 24} - {1 \over 16}\,\zeta\pars{2} + {1 \over 12}\,\zeta\pars{3} -
{1 \over 32}\,\zeta\pars{4}}
\\[5mm] & =
\bbx{{1 \over 24} - {\pi^{2} \over 96} + {1 \over 12}\,\zeta\pars{3} -
{\pi^{4} \over 2880}} \approx 0.00520709503181230\ldots
\end{align}
Ya he realizado una simulación, hasta el $\ds{16,777,215}$ cuatrillizos
$\ds{\pars{x,y,z,w}}$, lo cual está de acuerdo con el anterior resultado analítico. Simulación de este tipo 'genera' el valor de $\ds{0.00520\color{#f00}{943}}$.
