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We examine a quantum Otto engine using both Bose-Einstein Condensation (BEC) and normal Bose gas as working medium trapped in generalized external potential. We treated the engine quasi-statically and endoreversibly. Since the expansion and compression in both quasi-static and endoreversible take place isentropic, the expression of efficiency is similar. However, the power output in the quasi-static cycle is zero due to infinite and long stroke time. In contrast, with an endoreversible cycle, thermalization with two reservoirs takes place at a finite time. We use Fourier's law in conduction to formulate the relation between temperature of medium and reservoir, making work depend on heating and cooling stroke time. Moreover, we maximized the power with respect to compression ratio $\kappa$ to obtain efficiency at maximum power (EMP). We found that EMP is significantly higher when using BEC as a working medium, meanwhile EMP with normal Bose gas is just Curzon-Ahlborn efficiency. We also investigate the effect of thermal contact time $\tau$ with hot ($\tau_{h}$) and cold ($\tau_{l}$) reservoir on EMP. We found that when $\tau_{h}=\tau_{l}$ stroke time occur, there are no significant differences. Nevertheless, adjusting various cooling and heating stroke time provide a significant result on EMP, which is much higher at $\tau_{h}\tau_{l}$ stroke time. We conclude this partial thermalization enhances the EMP of the engine due to residual coherence., Comment: 14 pages, 8 figures, 2 tables