Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard deviation $\sigma$.
Is it possible to give a closed form solution for the integral:
$$\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw?$$
I realized that this is related to this. I tried using integration by substitution and by parts but didn't get to a solution. Is there a closed form expression for this integral?