  In this paper, we show that total integrals and cointegrals are new sources
of stable anti Yetter-Drinfeld modules. We explicitly show that how special
types of total (co)integrals can be used to provide both (stable) anti
Yetter-Drinfeld and Yetter- Drinfeld modules. We use these modules to classify
total (co)integrals and (cleft) Hopf Galois (co)extensions for some examples of
the Connes-Moscovici Hopf algebra, universal enveloping algebras and polynomial
algebras.
