Suppose we have three mutually independent random variables $U,V,W$ where $W \sim \mathcal{N}(0,1)$, $V \sim \mathcal{N}(0,c)$ and $E[U]=0$, $E[U^2]=1$.
Lets define $Z=U+V+W$.
Can we compute (or simplify) the following quantity \begin{align} E\left[f(Z,U)e^{\frac{V^2-(V+W)^2}{2}} \right] \end{align} where $f$ is some deterministic function. For now I don't want to make any assumptions about $f$ but if we need to here are some assumptions that we can make:
$f(Z,U)>0$
$0 \le E\left[f(Z,U)\right] \le 1$
I was thinking some thing like this
\begin{align} E\left[f(Z,U)e^{\frac{V^2-(V+W)^2}{2}} \right]&= E\left[ E\left[f(Z,U)e^{\frac{V^2-(V+W)^2}{2}} \Big| Z,U\right] \right]\\ &=E\left[ f(Z,U)E\left[e^{\frac{V^2-(V+W)^2}{2}} \Big| Z,U\right] \right] \end{align}
Now the question is how to compute the quantity $E\left[e^{\frac{V^2-(V+W)^2}{2}} \Big| Z,U\right]$. The good think is that $V$ and $W$ is are Gaussian.
Note that we can rewrite $E\left[e^{\frac{V^2-(V+W)^2}{2}} \Big| Z,U\right]=E\left[e^{\frac{-W^2-2VW}{2}} \Big| Z,U\right]$. The above looks like a moment generating function (but conditioned) and is related to this question.
Thank you in advance for any help.