The optimal design of a compound helicopter comprised of counterrotating coaxial rotors, a propeller, and optionally a fixed wing is investigated. The blade geometry, azimuthal blade pitch inputs, optimal shaft angle (rotor angle of attack), and division of propulsive and lifting forces among the components that minimize the total power for a given flight condition are determined. The optimal design problem is cast as a variational statement that minimizes the sum of induced and viscous power losses for a prescribed lift, propulsive force, and vehicle trim condition. The rotor, propeller, and wing geometry and control inputs are related to the far-field circulation through a lifting-line model that accounts for experimentally or computationally determined nonlinear lift and drag polars. The variational statement is discretized using a vortex lattice wake, and the resulting nonlinear constrained optimization problem is solved via Newton iteration. Results show that varying the prescribed propulsive force of the system affects the optimal shaft angle and rotor design, that higher harmonic control reduces the total vehicle power loss (inefficiency) in high-speed flight by as much as 15%, and that imposing a maximum allowable lateral lift offset can greatly increase the power loss of the coaxial rotors.