  Decomposition of the free rigid body Hamiltonian into a "main problem" and a
perturbation term provides an efficient integration scheme that avoids the use
of elliptic functions and integrals. In the case of short-axis-mode rotation,
it is shown that the use of complex variables converts the integration of the
torque-free motion by perturbations into a simple exercise of polynomial
algebra that can also accommodate the gravity-gradient perturbation when the
rigid body rotation is close enough to the axis of maximum inertia.
