Evaluating an improper integral using

Evaluating an improper integral using

I am trying to evaluate the improper integral $I:=\int_{-\infty}^\infty
f(x)dx$, where $$ f(z) := \frac{\exp((1+i)z)}{(1+\exp z)^2}. $$
I tried to do this by using complex integration. Let $L,L^\prime>0$ be
real numbers, and $C_1, C_2, C_3, C_4$ be the line segments that go from
$-L^\prime$ to $L$, from $L$ to $L+2\pi i$, from $L + 2\pi i$ to
$-L^\prime+2\pi i$ and from $-L^\prime+2\pi i$ to $-L^\prime$,
respectively. Let $C = C_1 + C_2 + C_3 + C_4$.
Here we have (for sufficiently large $L$ and $L^\prime$) $$ \int_{C_2}f(z)
dz \le \int_0^{2\pi}\left|\frac{\exp((1+i)(L+iy))}{(\exp(L+iy)+1)}i\right|
dy \le \int\frac{1}{(1-e^{-L})(e^L -
1)}dy\rightarrow0\quad(L\rightarrow\infty), $$ $$
\int_{C_4}f(z)dz\le\int_0^{2\pi}\left|\frac{\exp((1+i)(-L^\prime+iy))}{(\exp(-L^\prime
+ iy) +
1))^2}(-i)\right|dy\le\int\frac{e^{-L^\prime}}{(1-e^{-L})^2}dy\rightarrow
0\quad(L^\prime\rightarrow\infty), $$ and $$ \int_{C_3}f(z)dz =
e^{-2\pi}\int_{C_1}f(z)dz. $$ Thus $$I =
\lim_{L,L^\prime\rightarrow\infty}\frac{1}{ (1 +
e^{-2\pi})}\oint_Cf(z)dz.$$
Within the perimeter $C$ of the rectangle, $f$ has only one pole: $z = \pi
i$. Around this point, $f$ has the expansion $$ f(z) =
\frac{O(1)}{(-(z-\pi i)(1 + O(z-\pi i)))^2} =\frac{O(1)(1+O(z-\pi
i))^2}{(z-\pi i)^2} = \frac{1}{(z-\pi i)^2} + O((z-\pi i)^{-1}), $$ and
thus the order of the pole is 2. Its residue is $$
\frac{1}{(2-1)!}\frac{d}{dz}\Big|_{z=\pi i}(z-\pi i)^2f(z) = -\pi
\exp(i\pi^2) $$ (after a long calculation) and we have finally
$I=-\exp(i\pi^2)/2i(1+\exp(-2\pi))$.
My question is whether this derivation is correct. I would also like to
know if there are easier ways to do this (especially, those of calculating
the residue). I would appreciate if you could help me work on this
problem.