  We consider general classes of nonlinear Schr\"odinger equations on the
circle with nontrivial cubic part and without external parameters. We construct
a new type of normal forms, namely rational normal forms, on open sets
surrounding the origin in high Sobolev regularity. With this new tool we prove
that, given a large constant $M$ and a sufficiently small parameter
$\varepsilon$, for generic initial data of size $\varepsilon$, the flow is
conjugated to an integrable flow up to an arbitrary small remainder of order
$\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control
the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$.
Furthermore this property is locally stable: if $v(0)$ is sufficiently close to
$u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also
controled for time of order $\varepsilon^{-M}$.
