Optimal. Leaf size=87 \[ -2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+\frac{14}{3} \cos \left (\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.0636996, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3361, 3311, 3296, 2638, 2633} \[ -2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \sin ^2\left (\sqrt [3]{x}\right ) \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+\frac{14}{3} \cos \left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3361
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^2 \sin ^3(x) \, dx,x,\sqrt [3]{x}\right )\\ &=-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )-\frac{2}{3} \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\sqrt [3]{x}\right )\right )+4 \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )-4 \operatorname{Subst}\left (\int \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{14}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac{2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0550805, size = 62, normalized size = 0.71 \[ \frac{1}{36} \left (-81 \left (x^{2/3}-2\right ) \cos \left (\sqrt [3]{x}\right )+\left (9 x^{2/3}-2\right ) \cos \left (3 \sqrt [3]{x}\right )-6 \sqrt [3]{x} \left (\sin \left (3 \sqrt [3]{x}\right )-27 \sin \left (\sqrt [3]{x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 59, normalized size = 0.7 \begin{align*} -{x}^{{\frac{2}{3}}} \left ( 2+ \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt [3]{x} \right ) +4\,\cos \left ( \sqrt [3]{x} \right ) +4\,\sqrt [3]{x}\sin \left ( \sqrt [3]{x} \right ) +{\frac{2}{3}\sqrt [3]{x} \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{3}}+{\frac{2}{9} \left ( 2+ \left ( \sin \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt [3]{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972562, size = 63, normalized size = 0.72 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (3 \, x^{\frac{1}{3}}\right ) - \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{1}{6} \, x^{\frac{1}{3}} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67498, size = 173, normalized size = 1.99 \begin{align*} \frac{1}{9} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right )^{3} - \frac{1}{3} \,{\left (9 \, x^{\frac{2}{3}} - 14\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{2}{3} \,{\left (x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{2} - 7 \, x^{\frac{1}{3}}\right )} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.2981, size = 80, normalized size = 0.92 \begin{align*} - \frac{9 x^{\frac{2}{3}} \cos{\left (\sqrt [3]{x} \right )}}{4} + \frac{x^{\frac{2}{3}} \cos{\left (3 \sqrt [3]{x} \right )}}{4} + \frac{9 \sqrt [3]{x} \sin{\left (\sqrt [3]{x} \right )}}{2} - \frac{\sqrt [3]{x} \sin{\left (3 \sqrt [3]{x} \right )}}{6} + \frac{9 \cos{\left (\sqrt [3]{x} \right )}}{2} - \frac{\cos{\left (3 \sqrt [3]{x} \right )}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09136, size = 63, normalized size = 0.72 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (3 \, x^{\frac{1}{3}}\right ) - \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right ) - \frac{1}{6} \, x^{\frac{1}{3}} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \sin \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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