Inclusive $chi_{cJ}$ $(J=0,1,2)$ production from $Upsilon(1S)$ decay is studied within the framework of nonrelativistic QCD (NRQCD) factorization at leading order in $v_Q^2$, which includes the contributions of $bbar{b}({}^3S_1^{[1]})to cbar{c}(^3P_J^{[1]})+X$ and $bbar{b}({}^3S_1^{[1]})to cbar{c}(^3S_1^{[8]})+X$. For both channels, the short-distance coefficients are calculated through ${cal O}(alpha_s^5)$, which is next-to-leading order for the second one. By fitting to the measured $Upsilon(1S)$ branching fractions to $chi_{c1}$ and $chi_{c2}$, we obtain the color-octet long-distance matrix element (LDME) $langlemathcal{O}^{chi_{c0}}({}^3S_1^{[8]})rangle =(4.04pm0.47_{-0.34}^{+0.67})times10^{-3}$ GeV$^3$, where the first error is experimental and the second one due to the renormalization scale dependence, if we use as input $langlemathcal{O}^{chi_{c0}}({}^3P_0^{[1]})rangle=0.107$ GeV$^5$ as obtained via potential-model analysis. Previous LDME sets, extracted from data of prompt $chi_{cJ}$ hadroproduction, yield theoretical predictions that systematically undershoot or mildly overshoot the experimental values of $mathcal{B}(Upsilonto chi_{cJ}+X)$.
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