  The classical Van Est theory relates the smooth cohomology of Lie groups with
the cohomology of the associated Lie algebra, or its relative versions. Some
aspects of this theory generalize to Lie groupoids and their Lie algebroids. In
this paper, continuing an idea from [18], we revisit the van Est theory using
the Perturbation Lemma from homological algebra. Using this technique, we
obtain precise results for the van Est differentiation and integrations maps at
the level of cochains. Specifically, we construct homotopy inverses to the van
Est differentiation maps that are right inverses at the cochain level.
