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We complete the analytic evaluation of the master integrals for the two-loopnon-planar box diagrams contributing to the top-pair production in thequark-initiated channel, at next-to-next-to-leading order in QCD. The integralsare determined from their differential equations, which are cast into acanonical form using the Magnus exponential. The analytic expressions of theLaurent series coefficients of the integrals are expressed as combinations ofgeneralized polylogarithms, which we validate with several numerical checks. Wediscuss the analytic continuation of the planar and the non-planar masterintegrals, which contribute to $q {\bar q} \to t {\bar t}$ in QCD, as well asto the companion QED scattering processes $ e e \to \mu \mu$ and $e \mu \to e\mu$.

Master integrals for the NNLO virtual corrections to $q \bar{q} \rightarrow t \bar{t}$ scattering in QCD: the non-planar graphs