$\newcommand{\sech}{\operatorname{sech}}$I am solving a larger problem, to which a smaller part is the following improper integral
$$\int\limits_{x}^{\infty}e^{-2\kappa s- s}\sech s\,\mathrm ds\qquad\;\kappa\in(0,1)$$
I am unable to solve it by hand, and MapleSoft for example, doesn't return a closed form expression. I tried the Gradshtein-Ryzhik, table of integrals textbook, in which I found the following elementary form
$$\int e^{ax}\cosh(bx + c)\,\mathrm dx = \frac{e^{ax}}{ a^2 − b^2} [a\cosh(bx + c) − b\sinh(bx + c)]$$
where $a^2\neq b^2,$ but not a combination $e^{ax}\sech(bx)$. Does anyone know, where I might find this integral? Or which software I could use as an alternative.
There is an antiderivative in terms of the Gaussian hypergeometric function $$\int e^{-(2 \kappa +1) s}\, \text{sech}(s)\,ds=-\frac{e^{-2 \kappa s} }{\kappa}\,\, _2F_1\left(1,-\kappa ;1-\kappa ;-e^{2 s}\right)$$ and for $\kappa\in(0,1)$ $$\int_x^\infty e^{-(2 \kappa +1) s}\, \text{sech}(s)\,ds=\frac{e^{-2 \kappa x} }{\kappa }\, _2F_1\left(1,-\kappa ;1-\kappa ;-e^{2 x}\right)-\pi \csc (\pi \kappa )$$
Using the expansion @KStarGamer gave in comments, you will obtain another Gaussian hypergeometric function.