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$\ds{\int_{0}^{\infty}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x:\ {\large ?}}$.
\begin{align}&\color{#66f}{\large%
\int_{0}^{\infty}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x}
=\int_{0}^{1}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x\
+\ \overbrace{\int_{1}^{\infty}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x}
^{\ds{\dsc{x}\ \mapsto\ \dsc{1 \over x}}}
\\[5mm]&=2\ \overbrace{\int_{0}^{1}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x}
^{\ds{\dsc{x}\ \mapsto\ \dsc{x^{1/2}}}}\ =\
={1 \over 4}\int_{0}^{1}{\ln^{2}\pars{x} \over x^2 + x + 1}\,\dd x
=\ \overbrace{{1 \over 4}\int_{0}^{1}{\pars{1 - x}\ln^{2}\pars{x} \over 1 - x^{3}}\,\dd x}
^{\ds{\dsc{x}\ \mapsto\ \dsc{x^{1/3}}}}
\\[5mm]&={1 \over 4}\int_{0}^{1}{\pars{1 - x^{1/3}}\ln^{2}\pars{x^{1/3}} \over 1 - x}\,
{1 \over 3}\,x^{-2/3}\,\dd x
={1 \over 108}\int_{0}^{1}{\pars{x^{-2/3} - x^{-1/3}}\ln^{2}\pars{x} \over 1 - x}\,
\,\dd x
\\[5mm]&={1 \over 108}\lim_{\mu\ \to\ 0}\ \partiald[2]{}{\mu}
\int_{0}^{1}{x^{\mu - 2/3} - x^{\mu - 1/3} \over 1 - x}\,\,\dd x
\\[5mm]&={1 \over 108}\lim_{\mu\ \to\ 0}\ \partiald[2]{}{\mu}\pars{%
\int_{0}^{1}{1 - x^{\mu - 1/3} \over 1 - x}\,\,\dd x
-\int_{0}^{1}{1 - x^{\mu - 2/3} \over 1 - x}\,\,\dd x}
\\[5mm]&={1 \over 108}\lim_{\mu\ \to\ 0}\ \partiald[2]{}{\mu}\bracks{%
\Psi\pars{\mu + {2 \over 3}} - \Psi\pars{\mu + {1 \over 3}}}
\end{align}
donde $\ds{\Psi}$ es la
Función Digamma.
A continuación,
\begin{align}&\color{#66f}{\large%
\int_{0}^{\infty}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x}
={1 \over 108}\bracks{\Psi''\pars{2 \over 3} - \Psi''\pars{1 \over 3}}
\end{align}
Con
Euler Reflexión Fórmula
$\ds{\Psi"\pars{1 - z}
=\Psi"\pars{z} + 2\pi^{3}\cuna\pars{\pi z}\csc^{2}\pars{\pi z}}$:
\begin{align}&\color{#66f}{\large%
\int_{0}^{\infty}{x\ln^{2}\pars{x} \over x^4 + x^2 + 1}\,\dd x}
={1 \over 54}\,\pi^{3}\ \overbrace{\cot\pars{\pi \over 3}}^{\dsc{1 \over \root{3}}}
\ \overbrace{\csc^{2}\pars{\pi \over 3}}^{\dsc{4 \over 3}}
\ = \color{#66f}{\large{2\root{3} \over 243}\,\pi^{3}}
\end{align}
