$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{{\displaystyle #1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\on}[1]{\operatorname{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\sr}[2]{\,\,\,\stackrel{{#1}}{{#2}}\,\,\,}
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\begin{align}
& \color{#44f}{\int_{-\infty}^{\infty}
{\cos\pars{x -1/x}\over 1 + x^{2}}\,\dd x} =
2\int_{0}^{\infty}
{\cos\pars{x -1/x}\over 1 + x^{2}}\,\dd x
\\[5mm] \sr{x\ =\ \expo{\theta}}{=} &
2\int_{-\infty}^{\infty}
{\cos\pars{2\sinh\pars{\theta}}\over 1 + \expo{2\theta}}\expo{\theta}\,\dd\theta
\\[5mm] = & \
2\int_{0}^{\infty}
{\cos\pars{2\sinh\pars{\theta}}\over 1 + \sinh^{2}{\theta}}\cosh\pars{\theta}\,\dd\theta
\\[5mm] \sr{\sinh\pars{\theta}\ =\ t}{=} & \
\int_{-\infty}^{\infty}{\cos\pars{2t}\over 1 + t^{2}}\,\dd t
=
\Re\int_{-\infty}^{\infty}{\expo{2t\ic}\over \pars{t + \ic}\pars{t - \ic}}\,\dd t
\\[5mm] = & \
\Re\pars{2\pi\ic\,{\expo{-2} \over 2\ic}} = \bbx{\color{#44f}{\pi \over \expo{2}}} \approx 0.4252\\ &
\end{align}