We review the calculation of the two-loop QCD amplitudes that contribute at NLO to the decay of a Higgs boson to three partons (or equivalently the production of a Higgs boson with a jet in hadronic collisions). We concentrate on planar corrections, we discuss the structure of the amplitude in terms of dimensionally regularized Feynman integrals, the reduction to the master integrals and their analytic calculation with the differential equations method. Beside the more common case in which the solution can be expressed in terms of generalized polylogarithms, we discuss also the case of four-point functions whose differential equations can be solved in terms of elliptic integrals.
