  We calculate analytically the master integrals of a planar double-box family
for top-quark pair production using the method of differential equations. With
a proper choice of the bases, the differential equations can be transformed to
the $d$-log form. The square roots of the kinematic variables in the
differential equations can be rationalized by defining two dimensionless
variables. We find that all the boundary conditions can be fully fixed either
by simple integrals or regularity conditions at some special kinematic points.
The analytic results for thirty-three master integrals at general kinematics
are all expressed in terms of multiple polylogarithms up to transcendental
weight four.
