TITLE: Integral $\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx$
QUESTION [10 upvotes]: How can I show that
$$\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx = 2\pi \cos \left( r\log(1+a)\right)$$
where $a \in \mathbb{R}^+$ and $r \in \mathbb{R}$?
The relatively simple result suggests that it might be possible to evaluate the integral using contour integration if one could find the right function and contour combination.

REPLY [4 votes]: $\newcommand{\+}{^{\dagger}}
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\begin{align}
&\int_{-\infty}^{\infty}{%
\exp\pars{r\arctan\pars{ax}} + \exp\pars{-r \arctan(ax)} \over 1+x^2}\,
\cos\pars{{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}\,\dd x
\\[3mm]&=2\pi\cos\pars{r\ln\pars{1 + a}}:\ {\large }?.\qquad\qquad\qquad
a \in {\mathbb R}^{+}\,,\quad r \in {\mathbb R}.
\end{align}

Note that
  \begin{align}
&\bracks{\exp\pars{r\arctan\pars{ax}} + \exp\pars{-r \arctan(ax)}}\,
\cos\pars{{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}
\\[3mm]&=2\cosh\pars{r\arctan\pars{ax}}
\cosh\pars{\ic\,{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}
\\[3mm]&=\cosh\pars{r\arctan\pars{ax} +\ic\,{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}
\\[3mm]&+\cosh\pars{r\arctan\pars{ax} -\ic\,{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}
\\[3mm]&=2\Re
\cosh\pars{r\arctan\pars{ax} +\ic\,{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}}
\\[3mm]&=2\Re
\cosh\pars{r\,{\ic \over 2}\,\ln\pars{1 - \ic ax \over 1 + \ic ax}
+\ic\,{r \over 2}\,\bracks{\ln\pars{1 - \ic ax} + \ln\pars{1 + \ic ax}}}
\\[3mm]&=2\Re\cosh\pars{\ic r\ln\pars{1 - \ic ax}}
\end{align}

such that
\begin{align}
&\!\!\!\!\!\!\!\color{#66f}{\large\int_{-\infty}^{\infty}\!\!{%
\exp\pars{r\arctan\pars{ax}} + \exp\pars{-r \arctan(ax)} \over 1+x^2}\,
\cos\pars{\!{r \over 2}\,\ln\pars{1 + a^{2}x^{2}}\!}\,\dd x}
\\[3mm]&=2\Re\int_{-\infty}^{\infty}
{\cosh\pars{\ic r\ln\pars{1 - \ic\verts{a}x}} \over 1 + x^{2}}\,\dd x
=2\Re\braces{2\pi\ic\,
{\cosh\pars{\ic r\ln\pars{1 - \ic\verts{a}\bracks{\ic}}} \over \ic + \ic}}
\\[3mm]&=2\pi\,\Re\bracks{\cosh\pars{\ic r\ln\pars{1 + \verts{a}}}}
=\color{#66f}{\large 2\pi\cos\pars{r\ln\pars{1 + \verts{a}}}}
\end{align}