  We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2}
+M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R$$ where $M$ is
a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We
prove that such a Hamiltonian equation admits, in a neighborhood of the origin,
a Birkhoff normal form at any order and that, under generic conditions on $M$
related to the non resonance of the linear part, this normal form is integrable
when $d=1$ and gives rise to simple (in particular bounded) dynamics when
$d\geq 2$. As a consequence we prove the almost global existence for solutions
of the above equation with small Cauchy data. Furthermore we control the high
Sobolev norms of these solutions.

Analysis of PDEs;Dynamical Systems