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\int \frac{x-arctanx}{x^3}dx

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\frac{d}{dx}( x-arctanx ) = 1-\frac{1}{1+x^2}=\frac{x^2}{x^2+1}

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= x^2 \sum_{n=0}^{\infty}(-1)^nx^{2n} = \sum_{n=0}^{\infty}(-1)^nx^{2n+2}

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\int \sum_{n=0}^{\infty}(-1)^nx^{2n+2} dx = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}+C

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C=0?

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\int \frac{\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}}{x^3} dx

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\int \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+3}}dx

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\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+3)(2n+1)}}

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\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n+2)(2n)}}+C

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here i took d/dx converted to geomtric then integrated divided by x^3 then integrated again not sure how to deal with the ontants of integration in this case took c=0 on the first integral