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If $$\int_{-\pi/2}^{\pi/2} \frac{8\sqrt{2}\cos x\,dx}{(1+e^{\sin x})(1+\sin^4 x)} = \alpha\pi + \beta\log_e(3+2\sqrt{2})$$, where $$\alpha, \beta$$ are integers, then $$\alpha^2 + \beta^2$$ equals:
Correct Answer: 8
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