I've been struggling working on the following double integration. Given that $U$ is truncated normal distribution where $0<U<\inf$ and $f_U(u)=\frac{2}{\sqrt{2\pi}}\exp{\frac{-U^2}{2}}$. I need to simplify $$\int_{u=0}^{\inf}\int_{t=-\inf}^{hu+k}2\frac{1}{\sqrt{2\pi}}\exp^{\frac{-U^2}{2}}\frac{1}{\sqrt{2\pi}}\exp^{\frac{-t^2}{2}}dt.du$$
I am interested in simplifying the double integration into one integration of some truncated standard normal distribution.
Any help is greatly appreciated.