  While it is well known that constant rotation induces linear dispersive
effects in various fluid models, we study here its effect on long time
nonlinear dynamics in the inviscid setting. More precisely, we investigate
stability in the 3d rotating Euler equations in $\mathbb{R}^3$ with a fixed
speed of rotation. We show that for any $M>0$, axisymmetric initial data of
sufficiently small size $\varepsilon$ lead to solutions that exist for a long
time at least $\varepsilon^{-M}$ and disperse. This is a manifestation of the
stabilizing effect of rotation, regardless of its speed.
  To achieve this we develop an anisotropic framework that naturally builds on
the available symmetries. This allows for a precise quantification and control
of the geometry of nonlinear interactions, while at the same time giving enough
information to obtain dispersive decay via adapted linear dispersive estimates.
