Hyperbolic Hyperdrive!

$\displaystyle \begin{aligned} \int_0^\infty \dfrac{\sqrt{e^x-1}}{2{\rm coshx}-1}\; dx &= 2\int_0^\infty \dfrac{t^2}{t^4+t^2+1}\; dt\quad \color{#3D99F6}{e^x=1+t^2} \\ &= 2\int_0^\infty \dfrac{du}{u^4+u^2+1}\quad \color{#3D99F6}{t\to \dfrac{1}{u}} \\ &= \int_0^\infty \dfrac{x^{-1/2}}{x^2+x+1}\; dx \quad \color{#3D99F6}{u^2\to x} \\&= \dfrac{\pi}{\sqrt{3}}\end{aligned}$

In the last line we have used $\displaystyle F(a,s)=\int_{0}^{\infty} \frac{x^s}{x^2+2ax+1}dx = \frac{\pi\sin((1-s)\cos^{-1}a)}{\sin(\pi |s|)\sqrt{1-a^2}}$ which I have proved here