  We derive the integrated forms of specific initial-final tree-level
four-parton antenna functions involving a massless initial-state parton and a
massive final-state fermion as hard radiators. These antennae are needed in the
subtraction terms required to evaluate the double real corrections to
$t\bar{t}$ hadronic production at the NNLO level stemming from the partonic
processes $q\bar{q}\to t\bar{t}q'\bar{q}'$ and $gg\to t\bar{t}q\bar{q}$.
