Optimal. Leaf size=86 \[ 2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.0682848, antiderivative size = 86, 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 = {3362, 3311, 3296, 2637, 2633} \[ 2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \sin \left (\sqrt [3]{x}\right ) \cos ^2\left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3362
Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rubi steps
\begin{align*} \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^2 \cos ^3(x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )-\frac{2}{3} \operatorname{Subst}\left (\int \cos ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \operatorname{Subst}\left (\int x^2 \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{3} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (\sqrt [3]{x}\right )\right )-4 \operatorname{Subst}\left (\int x \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac{2}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )-4 \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac{2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac{14}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac{2}{9} \sin ^3\left (\sqrt [3]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0992082, size = 66, normalized size = 0.77 \[ \frac{1}{36} \left (81 \left (x^{2/3}-2\right ) \sin \left (\sqrt [3]{x}\right )+\left (9 x^{2/3}-2\right ) \sin \left (3 \sqrt [3]{x}\right )+162 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \cos \left (3 \sqrt [3]{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 58, normalized size = 0.7 \begin{align*}{x}^{{\frac{2}{3}}} \left ( 2+ \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \sin \left ( \sqrt [3]{x} \right ) -4\,\sin \left ( \sqrt [3]{x} \right ) +4\,\sqrt [3]{x}\cos \left ( \sqrt [3]{x} \right ) +{\frac{2}{3}\sqrt [3]{x} \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{3}}-{\frac{2}{9} \left ( 2+ \left ( \cos \left ( \sqrt [3]{x} \right ) \right ) ^{2} \right ) \sin \left ( \sqrt [3]{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78985, size = 63, normalized size = 0.73 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \sin \left (x^{\frac{1}{3}}\right ) + \frac{1}{6} \, x^{\frac{1}{3}} \cos \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \cos \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 = 1.90552, size = 166, normalized size = 1.93 \begin{align*} \frac{2}{3} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right )^{3} + \frac{1}{9} \,{\left ({\left (9 \, x^{\frac{2}{3}} - 2\right )} \cos \left (x^{\frac{1}{3}}\right )^{2} + 18 \, x^{\frac{2}{3}} - 40\right )} \sin \left (x^{\frac{1}{3}}\right ) + 4 \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.29064, size = 513, normalized size = 5.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15009, size = 63, normalized size = 0.73 \begin{align*} \frac{1}{36} \,{\left (9 \, x^{\frac{2}{3}} - 2\right )} \sin \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{4} \,{\left (x^{\frac{2}{3}} - 2\right )} \sin \left (x^{\frac{1}{3}}\right ) + \frac{1}{6} \, x^{\frac{1}{3}} \cos \left (3 \, x^{\frac{1}{3}}\right ) + \frac{9}{2} \, x^{\frac{1}{3}} \cos \left (x^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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