All Planar Two-Loop Amplitudes in Maximally Supersymmetric Yang-Mills Theory

We calculate the general planar dual-conformally invariant double-pentagon and pentabox integrals in four dimensions. Concretely, we derive onefold integral representations for these elliptic integrals over polylogarithms of weight three. These integral representations allow us to determine the respective symbols using consistency conditions alone. Together with the previously calculated double-box integral, these integrals suffice to express all two-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory in four dimensions-which we thus calculate for any number of particles and any helicity configuration.