  We investigate cubic-in-spin effects for inspiralling compact objects
binaries, both in the dynamics and the energy flux emitted in gravitational
waves, at the leading post-Newtonian order. We use a Lagrangian formalism to
implement finite-size effects, and extend it at cubic order in the spins, which
corresponds to the octupolar order in a multipolar decomposition. This
formalism allows us to derive the equation of motion, equations of precession
for the spin, and stress-energy tensor of each body in covariant form, and
admits a formal generalization to any multipolar order. For spin-induced
multipoles, i.e. in the case where the rotation of the compact object is sole
responsible for the additional multipole moments, we find a unique structure
for the octupolar moment representing cubic-in-spin effects. We apply these
results to compute the associated effects in the dynamics of compact binary
systems, and deduce the corresponding terms in the energy loss rate due to
gravitational waves. These effects enter at the third-and-a-half post-Newtonian
order, and can be important for binaries involving rapidly spinning black
holes. We provide simplified results for spin-aligned, circular orbits, and
discuss the quantitative importance of the new contributions.
