  We present the analytic computation of a family of non-planar master
integrals which contribute to the two-loop scattering amplitudes for Higgs plus
one jet production, with full heavy-quark mass dependence. These are relevant
for the NNLO corrections to inclusive Higgs production and for the NLO
corrections to Higgs production in association with a jet, in QCD. The
computation of the integrals is performed with the method of differential
equations. We provide a choice of basis for the polylogarithmic sectors, that
puts the system of differential equations in canonical form. Solutions up to
weight 2 are provided in terms of logarithms and dilogarithms, and 1-fold
integral solutions are provided at weight 3 and 4. There are two elliptic
sectors in the family, which are computed by solving their associated set of
differential equations in terms of generalized power series. The resulting
series may be truncated to obtain numerical results with high precision. The
series solution renders the analytic continuation to the physical region
straightforward. Moreover, we show how the series expansion method can be used
to obtain accurate numerical results for all the master integrals of the family
in all kinematic regions.
