Optimal. Leaf size=33 \[ -\frac{2 x+1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.047321, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1805, 801, 635, 203, 260} \[ -\frac{2 x+1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1805
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx &=-\frac{1+2 x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{-2+4 x}{x \left (1+x^2\right )} \, dx\\ &=-\frac{1+2 x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \left (-\frac{2}{x}+\frac{2 (2+x)}{1+x^2}\right ) \, dx\\ &=-\frac{1+2 x}{2 \left (1+x^2\right )}+\log (x)-\int \frac{2+x}{1+x^2} \, dx\\ &=-\frac{1+2 x}{2 \left (1+x^2\right )}+\log (x)-2 \int \frac{1}{1+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=-\frac{1+2 x}{2 \left (1+x^2\right )}-2 \tan ^{-1}(x)+\log (x)-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0203084, size = 33, normalized size = 1. \[ \frac{-2 x-1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{{x}^{2}+1} \left ( x+{\frac{1}{2}} \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-2\,\arctan \left ( x \right ) +\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40388, size = 39, normalized size = 1.18 \begin{align*} -\frac{2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} - 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 = 1.90529, size = 130, normalized size = 3.94 \begin{align*} -\frac{4 \,{\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 ) + 2 \, x + 1}{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 = 0.131393, size = 27, normalized size = 0.82 \begin{align*} - \frac{2 x + 1}{2 x^{2} + 2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - 2 \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06265, size = 41, normalized size = 1.24 \begin{align*} -\frac{2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} - 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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