We will review the calculation of the two-loop QCD amplitudes that
contribute to the NLO cross section for the production of a Higgs boson with a jet (or
to Higgs decay to three partons). In particular we concentrate on planar
corrections, we discuss the structure of the amplitude in terms of helicity
amplitudes, their expression in terms of dimensionally regularized Feynman
integrals and, finally, the reduction to the master integrals and their
calculation with the differential equations method. We show a class of
four-point functions that show an elliptic behaviour, i.e. whose
differential equation can be solved in terms of complete elliptic integrals.