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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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