  An analytic formula is given for the total scattering cross section of an
electron and a photon at order $\alpha^3$. This includes both the
double-Compton scattering real-emission contribution as well as the virtual
Compton scattering part. When combined with the recent analytic result for the
pair-production cross section, the complete $\alpha^3$ cross section is now
known. Both the next-to-leading order calculation as well as the
pair-production cross section are computed using modern multiloop calculation
techniques, where cut diagrams are decomposed into a set of master integrals
that are then computed using differential equations.
