Optimal. Leaf size=31 \[ \frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0451354, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1474, 800, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 1474
Rule 800
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1-x}{x^2 \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{1}{-1+x-x^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x-x^2} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 x^3\right )\\ &=-\frac{1}{3 x^3}+\frac{2 \tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0128113, size = 45, normalized size = 1.45 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\& \right ]-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.8 \begin{align*} -{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52609, size = 32, normalized size = 1.03 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35235, size = 84, normalized size = 2.71 \begin{align*} -\frac{2 \, \sqrt{3} x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) + 3}{9 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.158256, size = 36, normalized size = 1.16 \begin{align*} - \frac{2 \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 = 1.10625, size = 32, normalized size = 1.03 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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