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We study the phase diagram of the Hubbard model in the weak-coupling limitfor coexisting spin-density-wave order and spin-fluctuation-mediatedsuperconductivity. Both longitudinal and transverse spin fluctuationscontribute significantly to the effective interaction potential, which createsCooper pairs of the quasi-particles of the antiferromagnetic metallic state. Wefind a dominant $d_{x^2-y^2}$-wave solution in both electron- and hole-dopedcases. In the quasi-spin triplet channel, the longitudinal fluctuations giverise to an effective attraction supporting a $p$-wave gap, but are overcome byrepulsive contributions from the transverse fluctuations which disfavor$p$-wave pairing compared to $d_{x^2-y^2}$. The sub-leading pair instability isfound to be in the $g$-wave channel, but complex admixtures of $d$ and $g$ arenot energetically favored since their nodal structures coincide. Inclusion ofinterband pairing, in which each fermion in the Cooper pair belongs to adifferent spin-density-wave band, is considered for a range of electron dopingsin the regime of well-developed magnetic order. We demonstrate that theseinterband pairing gaps, which are non-zero in the magnetic state, must have thesame parity under inversion as the normal intraband gaps. The self-consistentsolution to the full system of five coupled gap equations give intraband andinterband pairing gaps of $d_{x^2-y^2}$ structure and similar gap magnitude. Inconclusion, the $d_{x^2-y^2}$ gap dominates for both hole and electron dopinginside the spin-density-wave phase.

Superconducting phase diagram of itinerant antiferromagnets