Residue of $p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1$ 
Show that $$p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi
x)}dx=\text{sec}1$$ by integrating $\frac{e^{2z}}{\cosh(\pi z)}$ around
  rectangles with vertices at $z=\pm p,p+i,-p+i.$

I asked this similar question, and was recommended this link stackexchange, but the answer provided is confusing.
I am not sure how they got those vertices, but with those vertices I got the parameterization: 
$\begin{cases} \Gamma_1 = t, & -p\leq t\leq p \\\\ \Gamma_2= p+it, &0\leq t \leq 1 \\\\ \Gamma_3 = i-t, & -p\leq t\leq p \\\\ \Gamma_4=i(1-t)-p, & 0\leq t\leq 1. \end{cases}$
Thus I have $\Gamma_p =\Gamma_1 +\Gamma_2+\Gamma_3+\Gamma_4  .$
$f(z)=\frac{e^{2z}}{\cosh(\pi z)} = \frac{2e^{z(2+i\pi)}}{e^{2i\pi z}+1}$, where the denominator has a $\text{pole}=(2n+1)\frac{1}{2}$. 
And this is as far as I can get. 
 A: Consider the integral
$$\oint_C dz \frac{e^{2 z}}{\cosh{\pi z}}$$
where $C$ is the above-described rectangle.  On the one hand, this integral is equal to the integral about the individual legs of the contour, viz.:
$$\int_{-p}^p dx \frac{e^{2 x}}{\cosh{\pi x}} + i \int_0^1 dy \frac{e^{2 (p+i y)}}{\cos{\pi (p+i y)}} + \int_{p}^{-p} dx \frac{e^{2 (x+i)}}{\cosh{\pi (x+i)}} + i \int_1^0 dy \frac{e^{2 (-p+i y)}}{\cos{\pi (-p+i y)}}$$
Take the limit as $p \rightarrow \infty$.  It should be clear that the 2nd and 4th integrals - those over the vertical legs of $C$ - will vanish in this limit.  That leaves the 1st and 3rd integrals over the horizontal sections; these may be combined to produce
$$\left(1+e^{i 2}\right) \int_{-\infty}^{\infty} dx \frac{e^{2 x}}{\cosh{\pi x}}$$
This equals, by the residue theorem, the residue at the only pole within $C$, namely, $z=i/2$:
$$i 2 \pi \lim_{z->i/2} \frac{e^{2 z}}{\cosh{\pi z}} = i 2 \pi\frac{e^{i}}{\pi \sinh{(i \pi/2)}} = 2 e^{i}$$
Therefore
$$\int_{\infty}^{\infty} dx \frac{e^{2 x}}{\cosh{\pi x}} = \frac{2 e^{i}}{1+e^{i 2}}= \sec{1}$$
You should note that no $PV$ was needed here.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\large\mbox{A}\ \Re eal\mbox{-Approach}}$:
\begin{align}
&\bbox[5px,#ffd]
{\int_{-\infty}^{\infty}{\expo{2x} \over \cosh\pars{\pi x}}
\,\dd x} =
2\int_{0}^{\infty}{\cosh\pars{2x} \over \cosh\pars{\pi x}}
\,\dd x
\\[5mm] = &\
4\int_{0}^{\infty}{\sinh\pars{\pi x}\cosh\pars{2x} \over \sinh\pars{2\pi x}}\,\dd x =
{\tt J}\pars{\pi + 2} + {\tt J}\pars{\pi - 2}
\label{1}\tag{1}
\\[2mm] &\
\mbox{where}\ {\tt J}\pars{\alpha} \equiv
2\int_{0}^{\infty}{\sinh\pars{\alpha x} \over \sinh\pars{2\pi x}}
\,\dd x
\end{align}

\begin{align}
{\tt J}\pars{\alpha} & =
2\int_{0}^{\infty}
{\expo{-\pars{2\pi - \alpha}x} -
\expo{-\pars{2\pi + \alpha}x} \over 1 - \expo{-4\pi x}}\,\dd x
\\[5mm] \stackrel{4\pi x\ \mapsto\ x}{=}\,\,\, &\
{1 \over 2\pi}\int_{0}^{\infty}
{\expo{-\bracks{1/2 - \alpha/\pars{4\pi}}x}\,\,\, -
\expo{-\bracks{1/2 + \alpha/\pars{4\pi}}x} \over 1 - \expo{-x}}
\,\dd x
\\[5mm] = &\
{1 \over 2\pi}\int_{0}^{\infty}
{1 - \expo{-\bracks{1/2 + \alpha/\pars{4\pi}}x} \over
1 - \expo{-x}}\,\dd x
\\[2mm] - &\
{1 \over 2\pi}\int_{0}^{\infty}
{1 - \expo{-\bracks{1/2 - \alpha/\pars{4\pi}}x} \over
1 - \expo{-x}}\,\dd x
\\[5mm] = & {1 \over 2\pi}
\bracks{\Psi\pars{{1 \over 2} + {\alpha \over 4\pi}} -
\Psi\pars{{1 \over 2} - {\alpha \over 4\pi}}}
\\[5mm] = &\
{1 \over 2\pi}\,\pi
\cot\pars{\pi\bracks{{1 \over 2} - {\alpha \over 4\pi}}} =
{1 \over 2}\tan\pars{\alpha \over 4}\label{2}\tag{2}
\end{align}

With (\ref{1}) and (\ref{2}):
\begin{align}
&\bbox[5px,#ffd]
{\int_{-\infty}^{\infty}{\expo{2x} \over \cosh\pars{\pi x}}
\,\dd x} =
{1 \over 2}\tan\pars{\pi + 2 \over 4} +
{1 \over 2}\tan\pars{\pi - 2 \over 4}
\\[5mm] = &\
{1 \over 2}\,{1 + \tan\pars{1/2} \over 1 - \tan\pars{1/2}} +
{1 \over 2}\,{1 - \tan\pars{1/2} \over 1 + \tan\pars{1/2}} =
{1 + \tan^{2}\pars{1/2} \over 1 - \tan^{2}\pars{1/2}}
\\[5mm] = &\
{1 \over \cos^{2}\pars{1/2} - \sin^{2}\pars{1/2}} =
{1 \over \cos\pars{2\bracks{1/2}}}
\\[5mm] = & \bbx{\sec\pars{1}} \approx 1.8508 \\ &
\end{align}
