Let $\Phi(x)$ be the standard normal CDF, and $\phi(x)$ be the PDF. Any ideas on how to evaluate the indefinite integral $$\int \Phi(x^2) \cdot \phi(x) \,dx$$
I tried integration by parts, and got the following form: $$\int \Phi(x) \cdot \phi(x^2)\cdot x \,dx.$$
Also tried simple substitutions, which didn't help.