  The $tW$ associated production has the second-largest cross section among
three single top production channels. A complete study of NNLO QCD corrections
to $tW$ production is still missing in literature. We present the calculation
of part of the requisite two-loop master integrals for the process $b+g \to
t+W$ at NNLO QCD. It turns out that the 80 master integrals in two families,
involving up to three massive internal lines, can be expressed by Goncharov
polylogarithms. The canonical differential equations that these integrals obey
contain at most three square roots. By separating the canonical master
integrals into groups according to their dependences on roots, we succeed in
achieving optimization of final expressions, which are validated by independent
numerical checks.
