Can any one suggest me a direction to attack following integral? I need an analytical solution for that.
$$ \begin{equation} \int_{-\infty}^{\infty} \frac{\phi(x)}{(a^2x^2+b^2)^3} \left[ \frac{\{\phi(\frac{x}{\sqrt{a^2x^2+b^2}})\}^2}{\Phi(\frac{x}{\sqrt{a^2x^2+b^2}}) \Phi(\frac{-x}{\sqrt{a^2x^2+b^2}})} \right] dx \end{equation} $$ 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.