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Given

$\int_0^1 \frac{x^3+x+2}{x^4+2x^2+1}dx=\frac{p}{q}+\frac{r\pi}{s}+\frac{\ln t}{u},$

where $\,t$ and $\,u$ are perfect squares. Find $\,p+q+r+s+t+u$.

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