Optimal. Leaf size=33 \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{x}+\log (x)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.0447255, antiderivative size = 33, 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} \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{x}+\log (x)-\tan ^{-1}(x) \]
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^2 \left (1+x^2\right )^2} \, dx &=\frac{1}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{-2-2 x}{x^2 \left (1+x^2\right )} \, dx\\ &=\frac{1}{2 \left (1+x^2\right )}-\frac{1}{2} \int \left (-\frac{2}{x^2}-\frac{2}{x}+\frac{2 (1+x)}{1+x^2}\right ) \, dx\\ &=-\frac{1}{x}+\frac{1}{2 \left (1+x^2\right )}+\log (x)-\int \frac{1+x}{1+x^2} \, dx\\ &=-\frac{1}{x}+\frac{1}{2 \left (1+x^2\right )}+\log (x)-\int \frac{1}{1+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=-\frac{1}{x}+\frac{1}{2 \left (1+x^2\right )}-\tan ^{-1}(x)+\log (x)-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0161544, size = 33, normalized size = 1. \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{x}+\log (x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 30, normalized size = 0.9 \begin{align*} -{x}^{-1}+{\frac{1}{2\,{x}^{2}+2}}-\arctan \left ( x \right ) +\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 = 1.50181, size = 46, normalized size = 1.39 \begin{align*} -\frac{2 \, x^{2} - x + 2}{2 \,{\left (x^{3} + x\right )}} - \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 = 0.986168, size = 138, normalized size = 4.18 \begin{align*} -\frac{2 \, x^{2} + 2 \,{\left (x^{3} + x\right )} \arctan \left (x\right ) +{\left (x^{3} + x\right )} \log \left (x^{2} + 1\right ) - 2 \,{\left (x^{3} + x\right )} \log \left (x\right ) - x + 2}{2 \,{\left (x^{3} + x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137438, size = 31, normalized size = 0.94 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \operatorname{atan}{\left (x \right )} - \frac{2 x^{2} - x + 2}{2 x^{3} + 2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18142, size = 47, normalized size = 1.42 \begin{align*} -\frac{2 \, x^{2} - x + 2}{2 \,{\left (x^{3} + x\right )}} - \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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