Does anyone know, how to solve the following integral:
$$\int_{-\infty}^{\infty} e^{ax}\varphi(x)\Phi(bx)T(x,b)\ \mathrm{d}x,$$
where $\varphi(x):={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}$ and $\Phi(x):=\int_{-\infty}^x\varphi(t)\ \mathrm{d}t$ denote the PDF and CDF of the standard normal distribution, $T$ denotes the Owen's-T function and $a,b \in \mathbb{R}$ are fixed number.
I already considered the list of Gaussian integrals https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions as well as the list provided by Owen https://doi.org/10.1080%2F03610918008812164, but I did not really progress ... Does a closed form solution even exist?