I would like to compute the following integral $$ f(a) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \mathrm{sech}(ax)~e^{-x^2/2} \, dx, \quad a > 0. $$ It is the expectation of $\mathrm{sech}(aX)$ where $X \sim N(0, 1)$.
One bound I got uses the fact that $\mathrm{sech}(ax) \geq e^{-a^2 x^2/2}$. Working through the algebra, you will obtain $$ f(a) \geq \sqrt{\frac{1}{1 + a^2}} . $$ Are there better upper and lower bounds for this integral?
$$f(a) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \text{sech}(a x)\,\,e^{-\frac{x^2}{2}} \, dx=\frac{1}{a\sqrt{2\pi}} \int_{-\infty}^{+\infty} \text{sech}(y)\,\,e^{-\frac{y^2}{2 a^2}}\,dy$$ Using $$ \text{sech}(y) < \frac{1}{\left(1+\frac{y^2}{3}\right)^{3/2}}$$ we have $$a\,f(a) < \sqrt{\frac{3}{2}}\,\, U\left(\frac{1}{2},0,\frac{3}{2 a^2}\right) $$ where appears Tricomi's confluent hypergeometric function. This can also write in terms of Bessel functions $$a\,f(a) < \frac{3}{2} \sqrt{\frac{3}{2 \pi }}\frac 1{a^2}e^{\frac{3}{4 a^2}} \left(K_1\left(\frac{3}{4 a^2}\right)-K_0\left(\frac{3}{4 a^2}\right)\right)$$ Just a few numbers $$\left( \begin{array}{ccc} a & a\,f(a) & \text{upper bound} \\ 1 & 0.74126 & 0.74384 \\ 2 & 1.01374 & 1.03235 \\ 3 & 1.12100 & 1.15955 \\ 4 & 1.17114 & 1.22666 \\ 5 & 1.19788 & 1.26662 \\ 6 & 1.21361 & 1.29247 \\ 7 & 1.22357 & 1.31024 \\ 8 & 1.23024 & 1.32302 \\ 9 & 1.23491 & 1.33255 \\ 10 & 1.23831 & 1.33986 \end{array} \right)$$
Edit
If we make $$ \text{sech}(y) \sim {\left(1+\frac{y^2}{b}\right)^{-c}}$$ for suitable $(b,c)$ $$a\,f(a) \sim \sqrt{\frac{b}{2}}\,\,U\left(\frac{1}{2},\frac{3}{2}-c,\frac{b}{2 a^2}\right)$$ Minimizing $$\Phi(b,c)=\int_0^\infty \Bigg[\text{sech}(y) - {\left(1+\frac{y^2}{b}\right)^{-c}}\Bigg]^2 \,dy$$ making $b,c$ rational lead to $$b=\frac{659}{143} \qquad \qquad c=\frac{69}{32}\qquad \qquad \Phi_{\text{min}}=1.42\times 10^{-4}$$
$$\left( \begin{array}{cccc} a & a\, f(a) & \text{approximation} & \Delta \\ 1 & 0.74126 & 0.74431 & 0.00304 \\ 2 & 1.01374 & 1.01575 & 0.00201 \\ 3 & 1.12100 & 1.12392 & 0.00291 \\ 4 & 1.17114 & 1.17608 & 0.00494 \\ 5 & 1.19788 & 1.20488 & 0.00699 \\ 6 & 1.21361 & 1.22236 & 0.00875 \\ 7 & 1.22357 & 1.23373 & 0.01016 \\ 8 & 1.23024 & 1.24153 & 0.01130 \\ 9 & 1.23491 & 1.24711 & 0.01220 \\ 10 & 1.23831 & 1.25123 & 0.01292 \end{array} \right)$$