I am doing some research and got stuck in solving the following integral (which I am not sure whether it has a closed form solution or not, I hope it has:)) Here is the integral:
$$ \int_{l}^{h} {e^{-(x-a)^2}N(cx+d)dx}$$ where $N(x) =\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{x} {e^{-z^2/2}dz}$
Especially, I am particularly interested in the special case: $$ \int_{-\infty}^{h} {e^{-(x-a)^2}N(cx+d)dx}$$
Here are other two similar questions as far as I know: link1 and link2,they are special cases where $l = -\infty$ and $h = +\infty$.
Any help would greatly be appreciated. Thanks a lot!