Definite integral of a product of normal pdf and cdf

Solution 1:

As already explained, when $a\gt0$ the full integral is $1-\Phi\left(b/\sqrt{a^2+1}\right)$. The same approach shows that the integral considered here is $$ I(y)=P(Y\leqslant(X-b)/a,X\leqslant y), $$ where $(X,Y)$ are i.i.d. standard normal, that is, $$ I(y)=P(aY+b\leqslant X\leqslant y). $$ I see no reason to expect more explicit formulas.

Solution 2:

$$\int_{-\infty}^y \phi(x) \Phi(\frac{x-b}{a})dx = BvN\left[\frac{-b}{\sqrt{a^2+1}}, y; \rho= \frac{-1}{\sqrt{a^2+1}}\right]$$

where $BvN(w, z; \rho)$ is the bivariate normal cumulative with upper bounds $w$ and $z$, and correlation $\rho$.

For reference, see equation (10,010.1) in Owen (Comm. in Stat., 1980).