I am wondering if it is possible to obtain a closed-form formula for $$ f(\alpha) = \frac{1}{{\sqrt{2 \pi } \; \alpha }} \int^\infty_{-\infty} x^2 \cosh(x) \; e^{-\frac{\sinh ^2(x)}{2 \alpha ^2}} \mathop{}\!\mathrm{d}x \; . $$ This integral came up when I tried to calculate the second moment of the random variable $$ \DeclareMathOperator\arsinh{arsinh} X = \arsinh(Z \cdot \alpha) $$ where $Z \sim \mathcal{N}(0, 1)$ is a normally distributed random variable and $\arsinh$ denotes the inverse hyperbolic sine.

*Motivation/Context:* The above came up as I was investigating whether $\arsinh$ could be a useful variance-stabilizing, non-saturating activation function for artificial neural networks. For more details, see this Reddit thread and the original research cited there.