Optimal. Leaf size=31 \[ \frac{x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)+\frac{1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0389849, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {1805, 801, 635, 203, 260} \[ \frac{x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1805
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{1+x+x^2}{x \left (1+x^2\right )^2} \, dx &=\frac{x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{-2-x}{x \left (1+x^2\right )} \, dx\\ &=\frac{x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \left (-\frac{2}{x}+\frac{-1+2 x}{1+x^2}\right ) \, dx\\ &=\frac{x}{2 \left (1+x^2\right )}+\log (x)-\frac{1}{2} \int \frac{-1+2 x}{1+x^2} \, dx\\ &=\frac{x}{2 \left (1+x^2\right )}+\log (x)+\frac{1}{2} \int \frac{1}{1+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=\frac{x}{2 \left (1+x^2\right )}+\frac{1}{2} \tan ^{-1}(x)+\log (x)-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0102037, size = 28, normalized size = 0.9 \[ \frac{1}{2} \left (\frac{x}{x^2+1}-\log \left (x^2+1\right )+2 \log (x)+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 26, normalized size = 0.8 \begin{align*}{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46137, size = 34, normalized size = 1.1 \begin{align*} \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.993583, size = 117, normalized size = 3.77 \begin{align*} \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2 \,{\left (x^{2} + 1\right )} \log \left (x\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.135621, size = 24, normalized size = 0.77 \begin{align*} \frac{x}{2 x^{2} + 2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15628, size = 35, normalized size = 1.13 \begin{align*} \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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