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$\ds{\int_{0}^{1}{1 - x \over 1 - x^{6}}\,\ln^{4}\pars{x}\,\dd x:\ {\large ?}}$
\begin{align}&\color{#c00000}{\int_{0}^{1}%
{1 - x \over 1 - x^{6}}\,\ln^{4}\pars{x}\,\dd x}
=\lim_{\mu \to 0}\partiald[4]{}{\mu}\int_{0}^{1}
{1 - x \over 1 - x^{6}}\,x^{\mu}\,\dd x
\\[3mm]&=\lim_{\mu \to 0}\partiald[4]{}{\mu}\int_{0}^{1}
{x^{\mu/6} - x^{\pars{\mu + 1}/6} \over 1 - x}\,{1 \over 6}\,x^{-5/6}\,\dd x
={1 \over 6}\,\lim_{\mu \to 0}\partiald[4]{}{\mu}\int_{0}^{1}
{x^{\pars{\mu - 5}/6} - x^{\pars{\mu - 4}/6} \over 1 - x^{6}}\,\dd x
\\[3mm]&={1 \over 6}\,\lim_{\mu \to 0}\partiald[4]{}{\mu}\bracks{%
\int_{0}^{1}{1 - x^{\pars{\mu - 4}/6} \over 1 - x^{6}}\,\dd x
-\int_{0}^{1}{1 - x^{\pars{\mu - 5}/6} \over 1 - x^{6}}\,\dd x}
\\[3mm]&={1 \over 6}\,\lim_{\mu \to 0}\partiald[4]{}{\mu}\bracks{%
\Psi\pars{\mu + 2 \over 6} - \Psi\pars{\mu + 1 \over 6}}
\end{align}
$$
\color{#66f}{\large\int_{0}^{1}%
{1 - x \a más de 1 - x^{6}}\,\ln^{4}\pars{x}\,\dd x
={1 \over 7776}\,\bracks{%
\Psi^{\tt\pars{IV}}\pars{1 \over 3} - \Psi^{\tt\pars{IV}}\pars{1 \over 6}}}
\aprox {\tt 23.2507}
$$
ANEXOS
\begin{align}
&\color{#00f}{\int_{0}^{1}{\ln^{4}\pars{x} \over x - a}\,\dd x}
=-\int_{0}^{1}{\ln^{4}\pars{a\bracks{x/a}} \over 1 - x/a}\,{\dd x \over a}
=-\int_{0}^{1/a}{\ln^{4}\pars{ax} \over 1 - x}\,\dd x
\\[3mm]&=-\int_{0}^{1/a}\ln\pars{1 - x}\,4\ln^{3}\pars{ax}\,{1 \over x}\,\dd x
=4\int_{0}^{1/a}{\rm Li}_{2}'\pars{x}\ln^{3}\pars{ax}\,\dd x
\\[3mm]&=-4\int_{0}^{1/a}{\rm Li}_{2}\pars{x}\,3\ln^{2}\pars{ax}\,{1 \over x}
\,\dd x
\\[3mm]&=-12\int_{0}^{1/a}{\rm Li}_{3}'\pars{x}\ln^{2}\pars{ax}\,\dd x
=12\int_{0}^{1/a}{\rm Li}_{3}\pars{x}2\ln\pars{ax}\,{1 \over x}\,\dd x
\\[3mm]&=24\int_{0}^{1/a}{\rm Li}_{4}'\pars{x}\ln\pars{ax}\,\dd x
=-24\int_{0}^{1/a}{\rm Li}_{4}\pars{x}\,{1 \over x}\,\dd x
=-24\int_{0}^{1/a}{\rm Li}_{5}'\pars{x}\,\dd x
\\[3mm]&=\color{#00f}{-24\,{\rm Li}_{5}\pars{1 \over a}}
\end{align}
Ahora, usted puede utilizar fracciones parciales. Por ejemplo:
$$
\int_{0}^{1}{\ln^{4}\pars{x} \más de 3\pars{x + 1}}= -8\,{\rm Li}_{5}\pars{-1}
={15 \over 2}\,\zeta\pars{5}
$$
