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Given

\[\begin{equation} \int_0^\infty \frac{x(1-x^2)}{(1+x^2)^2}\ln \left|\frac{x+1}{x-1} \right|\, dx=\frac{\pi^{a}}{b}-\frac{\pi^{c}}{d}, \end{equation}\]

find \(a+b+c+d.\)

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