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

Direct construction of the effective action of chiral gauge fermions in the anomalous sector