We reconsider semiclassical quantization of folded string spinning in AdS_3
part of AdS_5 X S^5 using integrability-based (algebraic curve) method. We
focus on the "short string" (small spin S) limit with the angular momentum J in
S^5 scaled down according to \cal J = rho \sqrt \cal S in terms of the
variables \cal J = J/\sqrt\lambda, \cal S = S/\sqrt\lambda. The semiclassical
string energy in this particular scaling limit admits the double expansion E =
\sum_{n=0}^{\infty}\sum_{p=0}^{\infty} (\sqrt\lambda)^{1-n}\,a_{n,p}(rho)\,
\cal S^{p+1/2}. It behaves smoothly as J -> 0 and partially resums recent
results by Gromov and Valatka. We explicitly compute various one-loop
coefficients a_{1,p}(rho) by summing over the fluctuation frequencies for
integrable perturbations around the classical solution. For the simple folded
string, the result agrees with what could be derived exploiting a recent
conjecture of Basso. However, the method can be extended to more general
situations. As an example, we consider the m-folded string where Basso's
conjecture fails. For this classical solution, we present the exact values of
a_{1,0}(rho) and a_{1,1}(rho) for m=2, 3, 4, 5 and explain how to work out the
general case.Comment: 19 page