Optimal. Leaf size=48 \[ -\frac{1}{3 x^3}-\frac{1}{6} \log \left (x^6-x^3+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\log (x) \]
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Rubi [A] time = 0.0520241, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1357, 709, 800, 634, 618, 204, 628} \[ -\frac{1}{3 x^3}-\frac{1}{6} \log \left (x^6-x^3+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\log (x) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 709
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (1-x^3+x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1-x}{x \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{x}-\frac{x}{1-x+x^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\log (x)-\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\log (x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,x^3\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\log (x)-\frac{1}{6} \log \left (1-x^3+x^6\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x^3\right )\\ &=-\frac{1}{3 x^3}+\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\log (x)-\frac{1}{6} \log \left (1-x^3+x^6\right )\\ \end{align*}
Mathematica [C] time = 0.0136032, size = 51, normalized size = 1.06 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\& \right ]-\frac{1}{3 x^3}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 40, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,{x}^{3}}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{6}-{x}^{3}+1 \right ) }{6}}-{\frac{\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56136, size = 58, normalized size = 1.21 \begin{align*} -\frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{3 \, x^{3}} - \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \frac{1}{3} \, \log \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44726, size = 143, normalized size = 2.98 \begin{align*} -\frac{2 \, \sqrt{3} x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + 3 \, x^{3} \log \left (x^{6} - x^{3} + 1\right ) - 18 \, x^{3} \log \left (x\right ) + 6}{18 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.174397, size = 48, normalized size = 1. \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{6} - x^{3} + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{3}}{3} - \frac{\sqrt{3}}{3} \right )}}{9} - \frac{1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12789, size = 61, normalized size = 1.27 \begin{align*} -\frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{x^{3} + 1}{3 \, x^{3}} - \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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