  We prove a Nekhoroshev type theorem for the nonlinear Schr\"odinger equation
$$ iu_t=-\Delta u+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$
where $V$ is a typical smooth potential and $g$ is analytic in both variables.
More precisely we prove that if the initial datum is analytic in a strip of
width $\rho>0$ with a bound on this strip equals to $\eps$ then, if $\eps$ is
small enough, the solution of the nonlinear Schr\"odinger equation above
remains analytic in a strip of width $\rho/2$ and bounded on this strip by
$C\eps$ during very long time of order $ \eps^{-\alpha|\ln \eps|^\beta}$ for
some constants $C> 0$, $\alpha>0$ and $\beta<1$.
