Is rigid body rotation an integrable problem?

Yes - rigid body motion in 3 space can be modeled by the Euler equations, which because of cross-axis coupling are nonlinear - but they can be integrated. A complete reference on how to model rigid body dynamics is given in these MIT lecture Notes. Solve equations (9), (10) and (11) for the rate of change of angular velocity around each principle axis, and integrate each to solve for the angular velocities. In this form you would have the set of equations needed for simulation, and you would need to specify the initial conditions of angular momentum and velocity - as well as the body's moments of inertia.

The lecture notes also provide a good discussion on the stability of these equations which are dependent on the ratios of the moments of inertia.