  We evaluate analytically the master integrals for one of two types of planar
families contributing to massive two-loop Bhabha scattering in QED. As in our
previous paper, we apply a recently suggested new strategy to solve
differential equations for master integrals for families of Feynman integrals.
The crucial point of this strategy is to use a new basis of the master
integrals where all master integrals are pure functions of uniform weight. This
allows to cast the differential equations into a simple canonical form, which
can straightforwardly be integrated order by order in epsilon in dimensional
regularization. The boundary conditions are also particularly transparent in
this setup. We identify the class of functions relevant to this problem to all
orders in epsilon. We present the results up to weight four for all except one
integrals in terms of a subset of Goncharov polylogarithms, which one may call
two-dimensional harmonic polylogarithms. For one integral, and more generally
at higher weight, the solution is written in terms of Chen iterated integrals.
