$\int_{0}^{1}{2n-x-x^3-x^5-\cdots-x^{4n-1}\over 1+x^2}\cdot{\mathrm dx\over \ln{x}}$

The first term in the LHS of your $(2)$ is a divergent integral: better to avoid them.
Instead, we may consider that:

$$ I(k)=\int_{0}^{1}\frac{1-x^{2k-1}}{1+x^2}\cdot\frac{dx}{\log(x)}=-\int_{0}^{+\infty}\frac{e^{-x}-e^{-2kx}}{1+e^{-2x}}\cdot\frac{dx}{x} $$ can be easily dealt with through a geometric series expansion and Frullani's theorem, leading to: $$ I(k) = \log\frac{2k}{1}-\log\frac{2k+2}{3}+\log\frac{2k+4}{5}-\log\frac{2k+6}{7}+\ldots $$ then to: $$ I(k) = -\log\left(\frac{2k}{1}\cdot\frac{3}{2k+2}\cdot\frac{2k+4}{5}\cdot\frac{7}{2k+6}\cdots\right) $$ and finally to: $$ I(k) = (-1)^{k+1}\log\sqrt{\pi} +\log\left(\frac{(k-2)!!\,\Gamma\left(\frac{3}{4}\right)}{2(k-1)!!\,\Gamma\left(\frac{5}{4}\right)}\right)$$ that is simple to sum over $k$.

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Basically, the whole problem is reduced to evaluate the following integral:

\begin{align} &\int_{0}^{1}{1 - x^{\mu} \over 1 + x^{2}}\,{\dd x \over \ln\pars{x}} = \int_{0}^{1}{1 - x^{2} - x^{\mu} + x^{\mu + 2}\over 1 - x^{4}} \,{\dd x \over \ln\pars{x}} \\[5mm] \stackrel{x^{4}\ \mapsto\ x}{=} & \int_{0}^{1}{x^{-3/4} - x^{-1/4} - x^{\mu/4 - 3/4} + x^{\mu/4 - 1/4} \over 1 - x}\,{\dd x \over \ln\pars{x}} \\[5mm] = &\ \int_{0}^{1}{x^{-3/4} - x^{-1/4} - x^{\mu/4 - 3/4} + x^{\mu/4 - 1/4} \over 1 - x}\pars{-\int_{0}^{\infty}x^{t}\,\dd t}\,\dd x \\[5mm] = &\ \int_{0}^{\infty}\int_{0}^{1} {-x^{t - 3/4} + x^{t - 1/4} + x^{t + \mu/4 - 3/4} - x^{t + \mu/4 - 1/4} \over 1 - x}\,\dd x\,\dd t \\[5mm] = &\ \int_{0}^{\infty}\bracks{\Psi\pars{t + {1 \over 4}} - \Psi\pars{t + {3 \over 4}} - \Psi\pars{t + {\mu + 1\over 4}} + \Psi\pars{t + {\mu + 3 \over 4}}}\dd t \\[5mm] = &\ \left. \ln\pars{\Gamma\pars{t + 1/4}\Gamma\pars{t + \bracks{\mu + 3}/4} \over \Gamma\pars{t + 3/4}\Gamma\pars{t + \bracks{\mu + 1}/4}} \right\vert_{\ t\ =\ 0}^{\ t\ \to\ \infty} = \bbx{\ds{-\ln\pars{{\root{2} \over 2\pi}\,\Gamma^{2}\pars{1 \over 4}\, {\Gamma\pars{\bracks{\mu + 3}/4} \over \Gamma\pars{\bracks{\mu + 1}/4}}}}} \end{align}