Can any one suggest me a direction to attack following integral? I need an analytical solution for that.
\begin{equation} \int_{-\infty}^{\infty} \phi(x) \left[ \frac{\{\phi(\frac{x}{\sqrt{a^2x^2+b^2}})\}^2}{\Phi(\frac{x}{\sqrt{a^2x^2+b^2}}) \Phi(\frac{-x}{\sqrt{a^2x^2+b^2}})} \right] dx \end{equation}
where $\phi(\alpha)=\frac{1}{\sqrt{2\pi}} e^{-\alpha^2/2}$ is Standard Gaussian probability distribution function and $\Phi(x)=\int_{-\infty}^{x} \phi(\alpha)d\alpha$ ~is standard Gaussian Cumulative distribution function,
For the special case $b=0$, the answer is $$ \frac{2 e^{-1/a^2}}{\pi \left(1 - \text{erf}(1/(\sqrt{2}a))^2\right)}$$ In general, it seems unlikely that there should be a closed form, though stranger things have happened. Do you have any reason to think there is one?