I need to evaluate \begin{equation} \int_a^\infty \mathrm{erfc}\left( \frac{b}{\sqrt{c\cdot h}} \right) e^{-d\cdot h} dh \end{equation} where $\mathrm{erfc}(s) = \frac{2}{\sqrt{\pi}} \int_{s}^{\infty} \exp(-t^2) dt$.
A closed-form expression is appreciated since ultimately, I need to do
\begin{equation} \int_0^\infty \left( \int_{k\cdot y}^\infty \mathrm{erfc}\left( \frac{b}{\sqrt{c\cdot h}} \right) e^{-d\cdot h} dh \right) e^{-m \cdot y} dy \end{equation}
I've noticed that a similar function - the Q-function - such that \begin{align} Q(s) &= \frac{1}{\sqrt{2\pi}} \int_s^\infty e^{-\frac{x^2}{2}}dx \\ &=\frac{1}{2} \mathrm{erfc}(\frac{x}{\sqrt{2}}) \end{align} and the Q-function has an alternative representation \begin{align} Q(s) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp{\left(\frac{-s^2}{2\sin^2{\phi}} \right)}d\phi \end{align} but I'm not sure if this helps.