Optimal. Leaf size=29 \[ -\frac{1}{2 x^2}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]
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Rubi [A] time = 0.0160911, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 266, 44} \[ -\frac{1}{2 x^2}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (x-x^3\right )} \, dx &=\int \frac{1}{x^5 \left (1-x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1-x) x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{1-x}+\frac{1}{x^3}+\frac{1}{x^2}+\frac{1}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 x^4}-\frac{1}{2 x^2}+\log (x)-\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0034672, size = 29, normalized size = 1. \[ -\frac{1}{2 x^2}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 26, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{2\,{x}^{2}}}+\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( -1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14142, size = 36, normalized size = 1.24 \begin{align*} -\frac{2 \, x^{2} + 1}{4 \, x^{4}} - \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 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 = 1.40528, size = 78, normalized size = 2.69 \begin{align*} -\frac{2 \, x^{4} \log \left (x^{2} - 1\right ) - 4 \, x^{4} \log \left (x\right ) + 2 \, x^{2} + 1}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111545, size = 22, normalized size = 0.76 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} - 1 \right )}}{2} - \frac{2 x^{2} + 1}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23591, size = 45, normalized size = 1.55 \begin{align*} -\frac{3 \, x^{4} + 2 \, x^{2} + 1}{4 \, x^{4}} + \frac{1}{2} \, \log \left (x^{2}\right ) - \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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