I am trying to find a solution/approximation in closed-form for following integral
\begin{equation} \int_{-\infty}^{\infty} \frac{[\phi(x)]^2}{\Phi(x)} \phi(a-bx) dx \end{equation}
where $a,b$ are real and positive. And $\phi(x)$ is the standard normal(Gaussian) pdf and $\Phi(x)$ is standard normal(Gaussian) CDF. I.e $\phi(x)=\frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}}$ and $\Phi(x)=\int_{-\infty}^{x} \phi(\alpha)d\alpha$. I have tried by part integral with different approaches but in no avail. I am wondering can Mathematica solve this? (I don't have mathematica)