  A new technique that utilizes surface integrals to find the force, torque and
potential energy between two non-spherical, rigid bodies is presented. The
method is relatively fast, and allows us to solve the full rigid two-body
problem for pairs of spheroids and ellipsoids with 12 degrees of freedom. We
demonstrate the method with two dimensionless test scenarios, one where
tumbling motion develops, and one where the motion of the bodies resemble
spinning tops. We also test the method on the asteroid binary (66391) 1999 KW4,
where both components are modelled either as spheroids or ellipsoids. The two
different shape models have negligible effects on the eccentricity and
semi-major axis, but have a larger impact on the angular velocity along the
$z$-direction. In all cases, energy and total angular momentum is conserved,
and the simulation accuracy is kept at the machine accuracy level.
