I came across this integral on Math Stack Exchange.
I’m trying to evaluate the integral
$$ I = \int_0^{\frac{\pi}{2}} \left( \text{chi}(\cot^2 x) + \text{Shi}(\cot^2 x) \right) \csc^2(x) e^{-\csc^2(x)} \, dx. $$
I’ve considered using substitution methods, but I'm stuck on how to simplify the integrand. Any ideas on how to approach this integral or if it converges?
Definitions $$\displaystyle \operatorname {Chi} (x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,$$ where $\displaystyle \gamma$ is the Euler–Mascheroni constant.
AND
The hyperbolic sine integral is defined as $$\displaystyle \operatorname {Shi} (x)=\int _{0}^{x}{\frac {\sinh(t)}{t}}\,dt$$ Thanks!
