We consider integrable superstring theory on AdS × S × M where M = T or M = S × S with generic ratio of the radii of the two 3-spheres. We compute the one-loop energy of a short folded string spinning in AdS and rotating in S. The computation is performed by world-sheet small spin perturbation theory as well as by quantizing the classical algebraic curve characterizing the finite-gap equations. The two methods give equal results up to regularization contributions that are under control. One important byproduct of the calculation is the part of the energy which is due to the dressing phase in the Bethe Ansatz. Remarkably, this contribution E turns out to be independent on the radii ratio. In the M = T limit, we discuss how E relates to a recent proposal for the dressing phase tested in the su(2) sector. We point out some difficulties suggesting that quantization of the AdS classical finite-gap equations could be subtler than the easier AdS × S case.