Optimal. Leaf size=31 \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]
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Rubi [A] time = 0.0090875, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 44} \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{-2 x^3+x^4} \, dx &=\int \frac{1}{(-2+x) x^3} \, dx\\ &=\int \left (\frac{1}{8 (-2+x)}-\frac{1}{2 x^3}-\frac{1}{4 x^2}-\frac{1}{8 x}\right ) \, dx\\ &=\frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8}\\ \end{align*}
Mathematica [A] time = 0.0018443, size = 31, normalized size = 1. \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 22, normalized size = 0.7 \begin{align*}{\frac{1}{4\,{x}^{2}}}+{\frac{1}{4\,x}}-{\frac{\ln \left ( x \right ) }{8}}+{\frac{\ln \left ( -2+x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937358, size = 26, normalized size = 0.84 \begin{align*} \frac{x + 1}{4 \, x^{2}} + \frac{1}{8} \, \log \left (x - 2\right ) - \frac{1}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13007, size = 66, normalized size = 2.13 \begin{align*} \frac{x^{2} \log \left (x - 2\right ) - x^{2} \log \left (x\right ) + 2 \, x + 2}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.100686, size = 19, normalized size = 0.61 \begin{align*} - \frac{\log{\left (x \right )}}{8} + \frac{\log{\left (x - 2 \right )}}{8} + \frac{x + 1}{4 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06752, size = 28, normalized size = 0.9 \begin{align*} \frac{x + 1}{4 \, x^{2}} + \frac{1}{8} \, \log \left ({\left | x - 2 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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