  We analytically evaluate the master integrals for the second type of planar
contributions to the massive two-loop Bhabha scattering in QED using
differential equa- tions with canonical bases. We obtain results in terms of
multiple polylogarithms for all the master integrals but one, for which we
derive a compact result in terms of elliptic mul- tiple polylogarithms. As a
byproduct, we also provide a compact analytic result in terms of elliptic
multiple polylogarithms for an integral belonging to the first family of planar
Bhabha integrals, whose computation in terms of polylogarithms was addressed
previously in the literature.
