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Hay un conocido de la serie para $\ds{\ln\pars{\tan\pars{x}}}$. Es decir,
\begin{align}
\ln\pars{\tan\pars{x}} & =
-2\sum_{k = 0}^{\infty}{\cos\pars{2\bracks{2k + 1}x} \over 2k + 1}\,,\qquad
x\ \in\ \pars{0,{\pi \over 2}}
\end{align}
A continuación,
\begin{align}
\color{#f00}{\int_{\pi/20}^{3\pi/20}\ln\pars{\tan\pars{x}}\,\dd x} & =
-2\sum_{k = 0}^{\infty}{1 \over 2k + 1}
\int_{\pi/20}^{3\pi/20}\cos\pars{2\bracks{2k + 1}x}\,\dd x
\\[3mm] &=
-2\sum_{k = 0}^{\infty}{1 \over 2k + 1}
\int_{-\pi/20}^{\pi/20}\cos\pars{2\bracks{2k + 1}\pars{x + \pi/10}}\,\dd x
\\[3mm] &=
-4\sum_{k = 0}^{\infty}{\cos\pars{\bracks{2k + 1}\pi/5} \over 2k + 1}
\int_{0}^{\pi/20}\cos\pars{2\bracks{2k + 1}x}\,\dd x
\\[3mm] &=
-2\sum_{k = 0}^{\infty}{%
\sin\pars{\bracks{2k + 1}\pi/10}\cos\pars{\bracks{2k + 1}\pi/5} \over
\pars{2k + 1}^{2}}
\end{align}
Resultado preliminar:
$$
\color{#f00}{\int_{\pi/20}^{3\pi/20}\ln\pars{\tan\pars{x}}\,\dd x} =
-2\sum_{\omega_{k}}^{\infty}{%
\sin\pars{\omega_{k}\theta}\cos\pars{2\omega_{k}\theta} \over \omega_{k}^{2}}\,,\qquad
\left\lbrace\begin{array}{rcl}
\ds{\omega_{k}} & \ds{\equiv} & \ds{2k + 1\,,\quad k = 0,1,2,\ldots}
\\[1mm]
\ds{\theta} & \ds{\equiv} & \ds{\pi \over 10}
\end{array}\right.
$$
Por otra parte,
$$
\color{#f00}{\int_{\pi/20}^{3\pi/20}\ln\pars{\tan\pars{x}}\,\dd x} =
\sum_{\omega_{k}}^{\infty}{\sin\pars{\omega_{k}\theta} \over \omega_{k}^{2}} -
\sum_{\omega_{k}}^{\infty}{\sin\pars{3\omega_{k}\theta} \over \omega_{k}^{2}}
$$
Tenga en cuenta que
\begin{align}
\sum_{\omega_{k}}{\sin\pars{\omega_{k}t} \over \omega_{k}^{2}} & =
\Im\sum_{k = 1}^{\infty}{\expo{\ic\pars{2k + 1}t} \over \pars{2k + 1}^{2}} =
\half\,\Im\sum_{k = 1}^{\infty}{\pars{\expo{\ic t}}^{k} \over k^{2}} -
\half\,\Im\sum_{k = 1}^{\infty}{\pars{-\expo{\ic t}}^{k} \over k^{2}}
\\[3mm] & =
\half\,\Im\Li{2}\pars{\expo{\ic t}} - \half\,\Im\Li{2}\pars{-\expo{\ic t}}
\end{align}
Tenga en cuenta que $\ds{\Li{2}\pars{z} - \Li{2}\pars {z} = 2\,\chi_{2}\pars{z} =
2\sum_{k = 0}^{\infty}{z^{2k + 1} \\pars{2k + 1}^{2}}}$ where $\ds{\chi}$
es el Legendre Chi Función.
\begin{align}
\color{#f00}{\int_{\pi/20}^{3\pi/20}\ln\pars{\tan\pars{x}}\,\dd x}
& =
\color{#f00}{\half\,\Im\Li{2}\pars{\expo{\ic\pi/10}} -
\half\,\Im\Li{2}\pars{-\expo{\ic\pi/10}}}
\\
& - \color{#f00}{\half\,\Im\Li{2}\pars{\expo{3\ic\pi/10}} +
\half\,\Im\Li{2}\pars{-\expo{3\ic\pi/10}}}
\end{align}
Todavía estoy tratando de encontrar la relación con el catalán Constante $\ds{G}$.
