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\begin{align}
\int_{0}^{\infty}{\arctan\pars{x} \over x\pars{x^{2} + 1}}\,\dd x & \,\,\,\stackrel{\arctan\pars{x}\ \mapsto\ x}{=}\,\,\,
\int_{0}^{\pi/2}{x \over \tan\pars{x}}\dd x
\\[5mm] & =
\left.\Re\int_{x = 0}^{x = \pi/2}{-\ic\ln\pars{z} \over
\bracks{\pars{z - 1/z}/\pars{2\ic}}/\bracks{\pars{z + 1/z}/2}}
\,{\dd z \over \ic z}\,\right\vert_{\ z\ =\ \exp\pars{\ic x}}
\\[5mm] & =
\left.-\,\Im\int_{x = 0}^{x = \pi/2}{1 + z^{2} \over 1 - z^{2}}\,\ln\pars{z}\,
\,{\dd z \over z}\,\right\vert_{\ z\ =\ \exp\pars{\ic x}}
\\[1cm] & \stackrel{\mrm{as}\ \epsilon\ \to\ 0^{+}}{\sim}\,\,\,
\Im\int_{1}^{\epsilon}{1 - y^{2} \over 1 + y^{2}}
\bracks{\ln\pars{y} + {\pi \over 2}\,\ic}\,{\dd y \over y} +
\Im\int_{\pi/2}^{0}\bracks{\ln\pars{\epsilon} + \ic\theta}\ic\,\dd\theta
\\[2mm] & \phantom{\stackrel{\mrm{as}\ \epsilon\ \to\ 0^{+}}{\sim}\!\!\!}\ +\
\underbrace{\Im\int_{\epsilon}^{1 - \epsilon}
{1 + x^{2} \over 1 - x^{2}}\,\ln\pars{x}\,{\dd x \over x}}_{\ds{=\ 0}}
\\[1cm] & =
-\,{1 \over 2}\,\pi\int_{\epsilon}^{1}{1- x^{2} \over 1 + x^{2}}
\,{\dd x \over x} - {1 \over 2}\,\pi\ln\pars{\epsilon} =
{1 \over 2}\,\pi\int_{\epsilon}^{1}
\pars{1 - {1- x^{2} \over 1 + x^{2}}}\,{\dd x \over x}
\\[1cm] & \stackrel{\mrm{as}\ \epsilon\ \to\ 0^{+}}{\to}\,\,\,
\pi\int_{0}^{1}{x \over x^{2} + 1}\,\dd x = \bbx{{1 \over 2}\,\pi\ln\pars{2}}
\end{align}