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$\int_{0}^{^\pi/_4} \frac{x^{2} (\sin 2x - \cos 2x) }{(1+\sin 2x)\cos^2 x} dx$ can be expressed as $\frac{\pi^{\alpha}}{\beta} - \frac{\pi}{\gamma} \ln \lambda$.What is the value of $\alpha + \beta + \gamma + \lambda$?

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