In this paper we present the master integrals necessary for the
analytic calculation of the box diagrams with one electron loop
($N_{F}=1$) entering in the 2-loop ($\alpha^3$) QED virtual
corrections to the Bhabha scattering amplitude of the electron.
We consider on-shell electrons and positrons of finite mass $m$,
arbitrary squared c.m. energy $s$, and momentum transfer $t$;
both UV and soft IR divergences are regulated within the continuous
$D$-dimensional regularization scheme. After a brief overview of
the method employed in the calculation, we give the results,
for $s$ and $t$ in the Euclidean region, in terms of 1- and
2-dimensional harmonic polylogarithms, of maximum weight 3.
The corresponding results in the physical region can be
recovered by analytical continuation. For completeness, we also
provide the analytic expression of the 1-loop scalar box diagram
including the first order in $(D-4)$.
