Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate $$\begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*}$$ where $\Phi\left(\cdot\right)$ is the standard normal CDF, $X\sim\mathcal{N}\left(0,1\right)$, and $\mu\left(\cdot\right)$ is the unit step function.
What I have been able to do is to take the derivative with respect to $a$ and invoke the Stein's lemma to find the derivative explicitly. However, integrating the resulting derivative doesn't seem to be easy. I wonder if at all the expectation in $(1)$ has a simple expression in terms of $\Phi$.
My work: Define $f(a) := \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right]$. Then $f'(a) = \mathbb{E}\left[X\phi\left(aX\right)\mu\left(X+T\right)\right]$ where $\phi(t) = \Phi'(t)$ is the standard normal PDF. Therefore, $$f'(a)=\frac{s}{\sqrt{2\pi}}\mathbb{E}\left[Z\mu\left(Z+T\right)\right],$$ with $Z\sim\mathcal{N}(0,s^2:=a^2/(a^2+1))$. Invoking the Stein's lemma then yields \begin{align*} f'(a)&=\frac{s^3}{\sqrt{2\pi}}\mathbb{E}\left[\delta\left(Z+T\right)\right]\\ &=\frac{s^2}{2\pi}\exp\left(-\frac{T^2}{2s^2}\right), \end{align*} where $\delta(\cdot)$ denotes the Dirac delta function. At this point, I'm not sure if the integral of $f'(a)$ can be written as a simple function which perhaps includes something in terms of the normal CDF.