Gaussian functions often appear in conditional expectation integrals.
I am interested to know if analytic integrals of the following forms are available
$$ \int \exp(-\exp(a x)) \exp(\frac{-(x-\mu)^2}{\sigma^2}) \; dx \quad\quad\quad (*) $$
or \begin{equation} \int \exp( P(a x)) \exp(\frac{-(x-\mu)^2}{\sigma^2}) \; dx \end{equation} where $P$ is a polynomial in $x$, and $a$, $\mu$ and $\sigma$ are real constants.
While the first integral is the one I would like to solve, I can rewrite the double exponential as a series
\begin{eqnarray}
\text{Series}[\exp (-\exp (a x)),\{x,0,8\}] &=& \exp\left( -1-a x-\frac{a^2 x^2}{2}-\frac{a^3 x^3}{6}-\frac{a^4 x^4}{24}-\frac{a^5 x^5}{120}-\frac{a^6 x^6}{720}-\frac{a^7 x^7}{5040}+O\left(x^8\right) \right) \\
&=& \frac{1}{e}-\frac{a x}{e}+\frac{a^3 x^3}{6 e}+\frac{a^4 x^4}{24 e}-\frac{a^5 x^5}{60 e}-\frac{a^6 x^6}{80 e}-\frac{a^7 x^7}{560 e}+O\left(x^8\right)
\end{eqnarray}
and this allows the integral in equation $(*)$ to be written as a sum of integrals of the form.
\begin{equation}
\sum_{n=1}^{\infty} c_n \int(ax)^n \exp(\frac{-(x-\mu)^2}{\sigma^2}) \; dx
\end{equation}
so that each term looks like it is related to a moment-like integral (as listed on
https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions).
$\int \phi(x) \; dx=\Phi(x)+C$
$\int x \phi(x) \;dx=−\phi(x)+C$
$\int x^2 \phi(x)dx=Φ(x)−x\phi(x)+C$
$\int x^{2k+1} \phi(x)\;dx=−\phi(x)\sum_{j=0}^{k}\frac{(2k)!!}{(2j)!!} x^{2j}+C$
$\int x^{2k+2} \phi(x)\;dx=−\phi(x)\sum_{j=0}^{k}\frac{(2k+1)!!}{(2j+1)!!}x^{2j+1}+(2k+1)!!\Phi(x)+C$
While practically it may be possible to truncate the series and obtain numerical results, it would be much nicer if the definite integral for the exponent of exponent form exists.
I would be grateful if anyone recognises the form of the integral in equation $(*)$ and could point me to its definite integrated form. I have not been able to locate it in some of the standard references.