Abel's Integral $\int_{0}^{\infty} \frac{x}{\sinh(\pi x)(1+x^2)} \,dx$

Note that $$\frac{b}{{{b^2} + 1}} = \int_0^\infty {{e^{ - bx}}\cos xdx} $$ Hence $$\int_0^\infty {\frac{x}{{{x^2} + 1}}{e^{ - ax}}dx} = \int_0^\infty {\int_0^\infty {{e^{ - xt}}(\cos t){e^{ - ax}}dt} dx} = \int_0^\infty {\frac{{\cos t}}{{t + a}}dt}$$ this gives $$\text{Ci}(\pi n) = {( - 1)^{n + 1}}\int_0^\infty {\frac{x}{{{x^2} + 1}}{e^{ - \pi nx}}dx}$$ Summing over $n=1,3,5...$ gives $$\sum_{n = 1,3,5...}\text{Ci}(\pi n) = \int_{0}^{\infty} \frac{x}{(x^2+1)(e^{\pi x}-e^{-\pi x})} dx$$

This is line 6.

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For the evaluation of the integral, consider

$$ \mathcal{I} = \int_{-\infty}^{\infty} \frac{x}{\sinh(\pi x)(1+x^2)}dx $$

Consider the rectangular contour with vertices $R, R+Ri, -R+Ri, -R$, where $R=2N+1/2$ with $N$ a very large integer. When $R$ tends to infinity, the following three integrals tend to 0 $$\int_{R}^{R+Ri} \frac{x}{\sinh(\pi x)(1+x^2)}dx \to 0$$ $$\int_{R+Ri}^{-R+Ri} \frac{x}{\sinh(\pi x)(1+x^2)}dx \to 0$$ $$\int_{-R+Ri}^{-R} \frac{x}{\sinh(\pi x)(1+x^2)}dx \to 0$$

Note that the function $\dfrac{z}{\sinh(\pi z)(1+z^2)}$ has simple pole at $z=ni$, with $n\geq 2$ and a double pole at $z=i$, the residue at $z=ni$ being $\dfrac{(-1)^n(ni)}{\pi(1-n^2)}$, and at $z=i$ being $\frac{i}{4\pi}$.

Hence $$\begin{aligned} \mathcal{I} &= 2\pi i\frac{i}{4\pi} + 2\pi i\sum_{n = 2}^{\infty}\frac{(-1)^n(ni)}{\pi(1-n^2)} \\ &= -\frac{1}{2}+2\sum_{n = 2}^{\infty}\frac{(-1)^nn}{n^2-1} \\ &= -\frac{1}{2}+\sum_{n = 2}^{\infty}\left(\frac{(-1)^n}{n-1}+\frac{(-1)^n}{n+1}\right) \\ &= 2\ln 2 -1 \end{aligned} $$

You can use Parseval's theorem $$ \int_{-\infty}^{\infty}f(x)\bar{g}(x)dx=\int_{-\infty}^{\infty}\hat{f}(s)\hat{\bar{g}}(s)ds $$ to evaluate this integral. Let $$ f(x)=\frac{x}{\sinh(\pi x)}, g(x)=\frac{1}{1+x^2}. $$ Clearly $$ \hat{g}(s)=\sqrt{\frac{\pi}{2}}e^{-|s|}. $$ From this, one has $$ \hat{f}(s)=\frac{1}{2\sqrt{2\pi}\cosh^2(\frac{s}{2})}. $$ So \begin{eqnarray} \int_{0}^{\infty}f(x)g(x)dx&=&\frac{1}{2}\int_{-\infty}^{\infty}f(x)\bar{g}(x)dx\\ &=&\frac{1}{2}\int_{-\infty}^{\infty}\hat{f}(s)\hat{\bar{g}}(s)ds\\ &=&\frac{1}{2}\int_{-\infty}^{\infty}\frac{1}{2\sqrt{2\pi}\cosh^2(\frac{s}{2})}\sqrt{\frac{\pi}{2}}e^{-|s|}ds\\ &=&\frac12(2\ln2-1). \end{eqnarray}