  The anomaly implies an obstruction to a fully chiral covariant calculation of
the effective action in the abnormal parity sector of chiral theories. The
standard approach then is to reconstruct the anomalous effective action from
its covariant current. In this work we use a recently introduced formulation
which allows to directly construct the non trivial chiral invariant part of the
effective action within a fully covariant formalism. To this end we develop an
appropriate version of Chan's approach to carry out the calculation within the
derivative expansion. The result to four derivatives, i.e., to leading order in
two and four dimensions and next-to-leading order in two dimensions, is
explicitly worked out. Fairly compact expressions are found for these terms.
