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$I=\int\limits_{0}^{\frac{\pi}{4}}\dfrac{\tan^2x\left(x^2+1\right)+x^2}{1+\tan^2x}dx=\int\limits_{0}^{\frac{\pi}{4}}\left (x^2+ \dfrac{\tan^2x}{1+\tan^2x} \right )dx=\int\limits_{0}^{\frac{\pi}{4}}\left (x^2+1- \dfrac{1}{1+\tan^2x} \right )dx =\left[ {\frac{x^3}{3}+x-\arctan x} \right]_{0}^{\frac{\pi}{4}}$
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