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

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