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$\ds{\int_{0}^{1}\ln\pars{x}\,{\pars{1 + x^{2}}x^{c - 2} \over 1 - x^{2c}}\,\dd x
=-\pars{\pi \over 2c}^{2}\sec^{2}\pars{\pi \over 2c}:\ {\large ?}}$
With $\ds{x \equiv t^{1/\pars{2c}}\quad\imp\quad t = x^{2c}}$:
\begin{align}
&\color{#c00000}{%
\int_{0}^{1}\ln\pars{x}\,{\pars{1 + x^{2}}x^{c - 2} \over 1 - x^{2c}}\,\dd x}
=\int_{0}^{1}
\ln\pars{t^{1/\bracks{2c}}}\,{\pars{1 + t^{1/c}}t^{1/2 - 1/c} \over 1 - t}
\,{1 \over 2c}\,t^{1/\pars{2c} - 1}\,\dd t
\\[3mm]&={1 \over 4c^{2}}\bracks{%
\int_{0}^{1}{\ln\pars{t}t^{-\pars{1 + 1/c}/2} \over 1 - t}\,\dd t
+\int_{0}^{1}{\ln\pars{t}t^{-\pars{1 - 1/c}/2} \over 1 - t}\,\dd t}
\\[3mm]&=-\,{1 \over 4c^{2}}
\lim_{\mu \to -\pars{1 + 1/c}/2}
\partiald{}{\mu}\int_{0}^{1}{1 - t^{\mu} \over 1 - t}\,\dd t + \pars{~c \to -c~}
\\[3mm]&=\color{#c00000}{-\,{1 \over 4c^{2}}
\lim_{\mu \to -\pars{1 + 1/c}/2}
\partiald{\bracks{\Psi\pars{\mu + 1} + \gamma}}{\mu} + \pars{~c \to -c~}}
\end{align}
where we used the A&S table identity ${\bf\mbox{6.3.22}}$. $\ds{\Psi\pars{z}}$ is the Digamma Function and $\ds{\gamma}$ is the
Euler-Mascheroni Constant.
\begin{align}
&\color{#c00000}{%
\int_{0}^{1}\ln\pars{x}\,{\pars{1 + x^{2}}x^{c - 2} \over 1 - x^{2c}}\,\dd x}
=
-\,{1 \over 4c^{2}}\bracks{\Psi'\pars{\half + {1 \over 2c}} + \Psi'\pars{\half - {1 \over 2c}}}
\end{align}
With Euler Reflection Formula
${\bf\mbox{6.4.7}}$:
\begin{align}
&\color{#c00000}{%
\int_{0}^{1}\ln\pars{x}\,{\pars{1 + x^{2}}x^{c - 2} \over 1 - x^{2c}}\,\dd x}
=
-\,{1 \over 4c^{2}}\bracks{-\pi\,\totald{\cot\pars{\pi z}}{z}}
_{z\ =\ 1/2\ -\ 1/\pars{2c}}
\\[3mm]&=-\,{\pi^{2} \over 4c^{2}}\,\csc^{2}\pars{\pi\bracks{\half - {1 \over 2c}}}
=-\,{\pi^{2} \over 4c^{2}}\,\sec^{2}\pars{\pi \over 2c}
\end{align}
$$
\color{#00f}{\large%
\int_{0}^{1}\ln\pars{x}\,{\pars{1 + x^{2}}x^{c - 2} \over 1 - x^{2c}}\,\dd x}
=\color{#00f}{\large-\,\pars{\pi \over 2c}^{2}\sec^{2}\pars{\pi \over 2c}}\,,\qquad
\verts{c} > 1
$$
