I would like to get a simple formula for the complex valued integral $$ \frac{1}{\sqrt{2\pi\sigma_Z^2}}\int_{-\infty}^\infty\exp\left(iux+\delta d(t)\vert x\vert\right)\exp\left(-\frac{(x-\mu_Z)^2}{2\sigma_Z^2}\right)\ dx. $$ This formula is also allowed to contain the cdf $\Phi$ of a standard normal RV. The above integral appears in an ODE I am trying to solve and I would like to avoid solving an integral in each step of my ODE solver. Hence, I would like to have a simple formula for the integral only containing functions that are available in most standard programming libraries. If I replace $iu$ with $u$, i.e. only consider the real valued case I can simplify the above integral to $$ -\exp\left((u+\delta d(t))\mu_Z+(u+\delta d(t))^2\frac{\sigma_Z^2}{2}\right)\Phi\left(\frac{(u+\delta d(t))\sigma_Z^2+\mu_Z}{\sigma_Z}\right)\\ -\exp\left((u-\delta d(t))\mu_Z+(u-\delta d(t))^2\frac{\sigma_Z^2}{2}\right)\Phi\left(-\frac{(u-\delta d(t))\sigma_Z^2 + \mu_Z}{\sigma_Z}\right) $$
However, in the complex case I would need to evaluate the cdf $\Phi$ at a complex number which is implemented in most computer packages like MATLAB. Is there a way to get around this?