I'm looking for a closed form expression for
$$\mathbb{E}\left[\Phi\left(aZ+b\right)^{k}\right] = \int_{-\infty}^{\infty}\Phi\left(az+b\right)^{k}\phi(z)\,dz$$
where $a$ and $b$ are real numbers, $k>0$ is an integer, $Z\sim\mathcal{N}(0,1)$, and $\phi(\cdot)$ and $\Phi(\cdot)$ are the density and distribution functions of a standard normal random variable. A special case has been treated here but I was wondering whether a general expression for all $k$ would be available.
