  We present analytic expressions in terms of Goncharov Polylogarithms up to
transcendental weight four, for pentagon integrals with massless propagators
and with up to one external mass, in $d=4-2\epsilon$ space-time dimensions. A
pure basis of Master Integrals is constructed for the pentagon family with one
off-shell leg, satisfying a single-variable canonical differential equation in
the Simplified Differential Equations approach. The relevant boundary terms are
given in closed form, including a hypergeometric function which can be expanded
to arbitrary order in the dimensional regulator using the \texttt{Mathematica}
package \texttt{HypExp}. Thus one can obtain solutions of the canonical
differential equation to arbitrary order in the dimensional regulator. The
$x\to1$ limit can provide solutions for the massless pentagon family.
