Optimal. Leaf size=87 \[ \frac {14}{3} \cos \left (\sqrt [3]{x}\right )-2 x^{2/3} \cos \left (\sqrt [3]{x}\right )-\frac {2}{9} \cos ^3\left (\sqrt [3]{x}\right )+4 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-x^{2/3} \cos \left (\sqrt [3]{x}\right ) \sin ^2\left (\sqrt [3]{x}\right )+\frac {2}{3} \sqrt [3]{x} \sin ^3\left (\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 = {3442, 3392,
3377, 2718, 2713} \begin {gather*} -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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2718
Rule 3377
Rule 3392
Rule 3442
Rubi steps
\begin {align*} \int \sin ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \text {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} \text {Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \text {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} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\sqrt [3]{x}\right )\right )+4 \text {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 \text {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*}
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Mathematica [A]
time = 0.04, size = 62, normalized size = 0.71 \begin {gather*} \frac {1}{36} \left (-81 \left (-2+x^{2/3}\right ) \cos \left (\sqrt [3]{x}\right )+\left (-2+9 x^{2/3}\right ) \cos \left (3 \sqrt [3]{x}\right )-6 \sqrt [3]{x} \left (-27 \sin \left (\sqrt [3]{x}\right )+\sin \left (3 \sqrt [3]{x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 59, normalized size = 0.68
method | result | size |
derivativedivides | \(-x^{\frac {2}{3}} \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )+4 \cos \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\sin ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}+\frac {2 \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )}{9}\) | \(59\) |
default | \(-x^{\frac {2}{3}} \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )+4 \cos \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\sin ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}+\frac {2 \left (2+\sin ^{2}\left (x^{\frac {1}{3}}\right )\right ) \cos \left (x^{\frac {1}{3}}\right )}{9}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 0.54 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 51, normalized size = 0.59 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.96, size = 80, normalized size = 0.92 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.37, size = 47, normalized size = 0.54 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.78, size = 58, normalized size = 0.67 \begin {gather*} \frac {14\,\cos \left (x^{1/3}\right )}{3}-3\,x^{2/3}\,\cos \left (x^{1/3}\right )+\frac {14\,x^{1/3}\,\sin \left (x^{1/3}\right )}{3}-\frac {2\,{\cos \left (x^{1/3}\right )}^3}{9}+x^{2/3}\,{\cos \left (x^{1/3}\right )}^3-\frac {2\,x^{1/3}\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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