Is there a closed-form solution for the following integral?

Question

The integral under consideration is as follows: $$ F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx, $$ where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the notation $\Phi(\cdot)$ denotes the standard normal distribution function given by $$ \Phi(z) = \mathbb{P}(Z\leq z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{u^2}{2}}\; \mathrm du. $$

Answer 1

Thanks to Gordon's help, we have that \begin{eqnarray*} F=exp\Big\{d + \frac{{c}^2}{2}\Big\}\Big[ \Phi\Big(\Phi^{-1}\Big(1+b\Big)-{c}\Big)- \Phi\Big(\Phi^{-1}\Big(a+b\Big)-{c}\Big)\Big] \end{eqnarray*}