  We generalize the theory of Cooper pairing by spin excitations in the
metallic antiferromagnetic state to include situations with electron and/or
hole pockets. We show that Cooper pairing arises from transverse spin waves and
from gapped longitudinal spin fluctuations of comparable strength. However,
each of these interactions, projected on a particular symmetry of the
superconducting gap, acts primarily within one type of pocket. We find a
nodeless $d_{x^2-y^2}$-wave state is supported primarily by the longitudinal
fluctuations on the electron pockets, and both transverse and longitudinal
fluctuations support nodeless odd-parity spin singlet $p-$wave symmetry on the
hole pockets. Our results may be relevant to the asymmetry of the AF/SC
coexistence state in the cuprate phase diagram, as well as for the "nodal gap"
observed recently for strongly underdoped cuprates.
