# definite normal integral, need closed form

can any one help me with following integral? I need closed-form

$$\int_{-\infty}^{\infty} \frac{(\phi(x))^2}{\Phi(x)} \phi(ax+b) dx$$

where $\phi(x)= \frac{1}{\sqrt{(2 \pi)}} e^{-x^2/2}$ is the pdf of a standard normal distribution and $\Phi(x)= \int_{-\infty}^x \phi(t)dt$ is the CDF of a standard normal distribution.

• That $\frac{1}{\Phi(x)}$ term in the integrand function is highly suspicious. Which probability are you trying to compute? My bet is that $\Phi(x)$ is in the wrong place. – Jack D'Aurizio Nov 13 '17 at 13:05
• @JackD'Aurizio no sir, it is in right place. I am sure. I am calculating a Fisher information. – Alireza Nov 14 '17 at 4:40