Is there an analytic solution for the following integral?
$$ I = \int_{-\infty}^{\infty} \Phi (a + b x) \phi (a + b x) \phi(x) \mathrm{d}x,$$
Wikipedia has a list of related integrals (https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions) but not the one I want to compute.
$\Phi(\cdot)$ and $\phi(\cdot)$ are the cdf and pdf respectively of the standard normal.
Does anyone have a way to solve this integral analytically? Thanks!