  We present analytic results of the two-loop master integrals for hadronic
$tW$ production that contain two massive propagators. For the planar integral
family, we succeed in constructing the canonical basis, so the results are
written in terms of multiple polylogarithms. The master integrals in the
non-planar integral families have been calculated by Taylor series expansion
around $m_W^2=0$. Even with this simplification, there are multiple square
roots involved in the differential equations, which can not be rationalized
simultaneously. We find that a proper linear combination of the integral basis
alleviates the problem so that the results are expressed by multiple
polylogarithms or single integrals over multiple polylogarithms up to weight
four. These single integrals can be evaluated using the classic Newton-Cotes
formulas. Our analytic results of master integrals would be used in the
computation of the two-loop amplitudes for $tW$ production.
