Using the data sets taken at center-of-mass energies above 4 GeV by the
BESIII detector at the BEPCII storage ring, we search for the reaction e+e- ->
gamma_ISR X(3872) -> gamma_ISR pi+pi-J/psi via the Initial State Radiation
technique. The production of a resonance with quantum numbers J^PC = 1^++ such
as the X(3872) via single photon e+e- annihilation is forbidden, but is allowed
by a next-to-leading order box diagram. We do not observe a significant signal
of X(3872), and therefore give an upper limit for the electronic width times
the branching fraction Gamma_ee^X(3872)Br(X(3872) -> pi+pi-Jpsi) < 0.13 eV at
the 90% confidence level. This measurement improves upon existing limits by a
factor of 46. Using the same final state, we also measure the electronic width
of the psi(3686) to be Gamma_ee^psi(3686) = 2231 +- 18 +- 99 eV.
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