Optimal. Leaf size=31 \[ -\frac{x^3}{3}-\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0354362, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1474, 773, 618, 204} \[ -\frac{x^3}{3}-\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1474
Rule 773
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{x^5 \left (1-x^3\right )}{1-x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(1-x) x}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x^3\right )\\ &=-\frac{x^3}{3}-\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0077299, size = 31, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3-1}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{3}}+{\frac{2\,\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 = 1.45651, size = 32, normalized size = 1.03 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80033, size = 76, normalized size = 2.45 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.126467, size = 32, normalized size = 1.03 \begin{align*} - \frac{x^{3}}{3} + \frac{2 \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 = 1.16666, size = 32, normalized size = 1.03 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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