Considering general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters, I will present the construction 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}. This is a joint work with Erwan Faou and Benoît Grébert.