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The nonlinear motion of electrons in a quadrupole free-electron laser is analyzed. If the betatron frequency is close to the mismatch frequency, the three-dimensional nonlinear equations can be reduced to an integrable one-dimensional equation. To enhance the efficiency, a tapered wiggler is introduced and the electron orbits in a tapered quadrupole free-electron laser are analyzed. The conditions for the particles to remain trapped as the heat wave decelerates are found.