Optimal. Leaf size=39 \[ \frac{1}{6} \log \left (x^6-x^3+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0354415, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1357, 634, 618, 204, 628} \[ \frac{1}{6} \log \left (x^6-x^3+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^5}{1-x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{1-x+x^2} \, dx,x,x^3\right )\\ &=\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}{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{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{6} \log \left (1-x^3+x^6\right )\\ \end{align*}
Mathematica [A] time = 0.0100902, size = 39, normalized size = 1. \[ \frac{1}{6} \log \left (x^6-x^3+1\right )+\frac{\tan ^{-1}\left (\frac{2 x^3-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 0.9 \begin{align*}{\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 = 1.59843, size = 43, normalized size = 1.1 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43572, size = 95, normalized size = 2.44 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.129845, size = 37, normalized size = 0.95 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09055, size = 43, normalized size = 1.1 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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