I need to compute Gaussian integrals of the form: $$ \mathcal{I}[f] = \int_{-\infty}^\infty \mathcal{N}\left(x\mid\mu,\sigma^2\right) f(x) \, dx, $$ where $\mathcal{N}\left(x\mid\mu,\sigma^2\right)$ is a normal PDF with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 \in \mathbb{R}_+$. (That is, the integral is the expectation of $f$ under a Gaussian PDF.)
I am wondering for which classes of $f$ - for arbitrary $\mu$ and $\sigma^2$ - the integral $\mathcal{I}[f]$ has known closed-form solutions.
For example, by looking here I could find the following cases:
- $f$ is a polynomial;
- $f$ is an exponential;
- $f$ is a squared exponential (e.g., another Gaussian PDF);
- $f$ is the error function (e.g., as in a Gaussian CDF), or some functions involving erf.
Are there other notable cases?
(Please note that I am not interested in numerical solutions, such as quadrature.)