Compute $\mathbb{E}[\Phi^n(\mu-\sigma X)]$, where $\Phi$ is the standard normal CDF and $X\sim\Phi$

Question

For $X\sim\mathcal{N}(0,1)$, $\mu\in\mathbb{R}$, $\sigma>0$ and $n\in\mathbb{N}$ I would like to compute the following expectation: $$ \mathbb{E}[\Phi^n(\mu-\sigma X)] = \int_{\mathbb{R}} \Phi^n(\mu-\sigma x) \varphi(x) dx, $$ where $\Phi$ is the standard normal CDF and $\varphi$ is the corresponding density. For $n=1$ one can derive $\mathbb{E}[\Phi(\mu-\sigma X)] =\Phi(\mu/\sqrt{1+\sigma^2})$ with the approach shown here: https://quant.stackexchange.com/questions/7505/calculate-the-expectation-of-a-shift-cdf , but I'm stuck for $n=2,3,...$ since the approach does not generalize nicely (as far as I see). I would be happy for pointers into other directions...