# Evaluating $\int_{0}^{\infty} \frac{x}{\cosh (a x)} d x$ for a positive constant $a$

We first simplify the integral by a substitution and then convert the integrand into a power series as below: \begin{aligned} \int_{0}^{\infty} \frac{x}{\cosh (ax)} d x & \stackrel{ax\mapsto x}{=} \frac{1}{a^{2}} \int_{0}^{\infty} \frac{x}{\cosh x} d x \\ &=\frac{2}{a^{2}} \int_{0}^{\infty} \frac{x e^{-x}}{1+e^{-2 x}} d x \\ &=\frac{2}{a^{2}} \sum_{k=0}^{\infty}(-1)^{k} \int_{0}^{\infty} x e^{-(2 k+1) x} d x \end{aligned} Using integration by parts, we have $$\int_{0}^{\infty} x e^{-(2 k+1) x} d x=\frac{1}{(2 k+1)^{2}}$$ Hence we can now conclude that \begin{aligned} \int_{0}^{\infty} \frac{x}{\cosh (a x)} d x &=\frac{2}{a^{2}} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1)^{2}} =\frac{2G}{a^{2}} \end{aligned} where $$G$$ is the Catalan’s constant.

Are there methods other than power series?

• There is the other standard way of writing $\int_0^\infty\frac x{\cosh x}\,dx=-2\int_0^1\frac{\log u}{u^2+1}\,du$ which has an indefinite closed form involving dilogarithms, by writing $u^2+1=(u+i)(u-i)$. Jul 28, 2022 at 8:38
• Jul 28, 2022 at 9:39
• For what it’s worth, one can generalise this for $n \in \mathbb{N}$: $$\mathcal{M}\left[\operatorname{sech}^{2n}(t)\right](s)=\frac{2^{2n}(-1)^n \Gamma(s)}{\Gamma(2n)}\sum_{k=1}^{\infty} \frac{(-1)^{k}\Gamma(k+n)}{(2k)^s\Gamma(k-n+1)}$$ $$\mathcal{M}\left[\operatorname{sech}^{2n-1}(t)\right](s)=\frac{2^{2n-1}(-1)^{n}\Gamma(s)}{\Gamma(2n-1)}\sum_{k=1}^{\infty} \frac{(-1)^{k}\Gamma(k+n-1)}{(2k-1)^s\Gamma(k-n+1)}$$ where $\mathcal{M}$ denotes the Mellin transform. I’d be happy to provide a proof as an answer if interested. Jul 28, 2022 at 11:56

If you enjoy polylogarithms, let $$x=it$$ $$I=\int x \,\text{sech}(x)\,dx=-\int t \sec (t)\,dt$$ which, I think, you already solved.

Back to $$x$$ $$I=-i \left(\text{Li}_2\left(-i e^{-x}\right)-\text{Li}_2\left(i e^{-x}\right)+x \left(\log \left(1-i e^{-x}\right)-\log \left(1+i e^{-x}\right)\right)\right)$$

As suggested by TheSimpliFire, I transform the integral, by letting $$\tan \theta=e^{-x}$$, into \begin{aligned} \int_{0}^{\infty} \frac{x}{\cosh x} d x &=2 \int_{0}^{\infty} \frac{x}{e^{x}+e^{-x}} d \alpha \\ &=-2 \int_{0}^{\frac{\pi}{4}} \ln (\tan \theta) d \theta \end{aligned}

By my post, $$\int_{0}^{\frac{\pi}{4}} \ln (\tan \theta) d \theta=-G$$, therefore we can conclude that $$\boxed{\int_{0}^{\infty} \frac{x}{\cosh (a x)} d x =\frac{2G}{a^2} }$$

$$\newcommand{\d}{\,\mathrm{d}}$$Contour methods allow for a principal value transformation into a different integral. This isn't exactly efficient, but I enjoyed working it through.

First map $$x\mapsto x/a$$ - then it suffices to show that: $$I:=\int_0^\infty\frac{x}{\cosh x}\d x=2G$$

Consider the contour $$\gamma_{\rho,\varepsilon}$$ which runs from $$0\to\rho$$, $$\rho\to\rho+i\pi,\rho+i\pi\to i\pi$$ and then on the journey $$i\pi\to0$$ we take a small semicircular indent at $$\frac{i\pi}{2}$$, where the indent is specifically the map $$C_\varepsilon:[-\pi/2,\pi/2]\to\Bbb C$$, $$t\mapsto\frac{i\pi}{2}+\varepsilon e^{-it}$$, not enclosing the pole at $$\frac{i\pi}{2}$$.

The integral on the small strip $$\rho\to\rho+i\pi$$ asymptotically vanishes: \begin{align}\left|i\int_0^\pi\frac{\rho+it}{e^\rho e^{it}+e^{-\rho}e^{-it}}\d t\right|&\lt2(\rho+\pi)\int_0^{\pi/2}|(e^\rho+e^{-\rho})\cos t+(e^\rho-e^{-\rho})i\sin(t)|^{-1}\d t\\&=2(\rho+\pi)\int_0^{\pi/2}\frac{1}{\sqrt{(e^\rho+e^{-\rho})^2+2\cos(2t)}}\d t\\&\lt2(\rho+\pi)\frac{\pi}{2}\frac{1}{e^\rho-e^{-\rho}}\\&=2\pi\cdot\frac{\rho+\pi}{\sinh\rho}\overset{\rho\to\infty}{\longrightarrow}0\end{align}

When we return $$\rho+i\pi\to i\pi$$, there is a negative effect due to $$e^{i\pi}$$ which balances out the negative from running "the opposite way", with contribution: $$\int_0^\rho\frac{i\pi}{e^x+e^{-x}}\d x+I_\rho$$

The integral on $$C_\varepsilon$$ is rather like the half-residue theorem, since the pole is simple: $$\int_{C_\varepsilon}\sim-\frac{1}{2}\int_{-\pi/2}^{\pi/2}(i\pi/2+\varepsilon e^{-it})\d t\sim-\frac{i\pi^2}{2}$$As $$\varepsilon\to0^+$$. For the non-indented section ($$z=it$$): $$\frac{1}{2}\left(\int_0^{\pi/2-\varepsilon}+\int_{\pi/2+\varepsilon}^\pi\right)t\sec t\d t\sim0-2\int_0^{\pi/2}\ln(\sec t+\tan t)\d t$$Bringing it all together and taking limits: $$2\int_0^\infty\frac{x}{e^x+e^{-x}}\d x+\frac{1}{2}\cdot\operatorname{pv}\int_0^\pi t\sec t\d t-\frac{i\pi^2}{2}+i\pi[\arctan\tanh(x/2)]_0^\infty=0$$So: $$\int_0^\infty\frac{x}{\cosh x}\d x=-\frac{1}{2}\cdot\operatorname{pv}\int_0^\pi t\sec t\d t=\int_0^{\pi/2}\ln(\sec t+\tan t)\d t$$

And the last is known to be $$2G$$.