Optimal. Leaf size=86 \[ 4 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+\frac {2}{3} \sqrt [3]{x} \cos ^3\left (\sqrt [3]{x}\right )-\frac {14}{3} \sin \left (\sqrt [3]{x}\right )+2 x^{2/3} \sin \left (\sqrt [3]{x}\right )+x^{2/3} \cos ^2\left (\sqrt [3]{x}\right ) \sin \left (\sqrt [3]{x}\right )+\frac {2}{9} \sin ^3\left (\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 86, 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 = {3443, 3392,
3377, 2717, 2713} \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2717
Rule 3377
Rule 3392
Rule 3443
Rubi steps
\begin {align*} \int \cos ^3\left (\sqrt [3]{x}\right ) \, dx &=3 \text {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} \text {Subst}\left (\int \cos ^3(x) \, dx,x,\sqrt [3]{x}\right )+2 \text {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} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (\sqrt [3]{x}\right )\right )-4 \text {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 \text {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*}
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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.77 \begin {gather*} \frac {1}{36} \left (162 \sqrt [3]{x} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \cos \left (3 \sqrt [3]{x}\right )+81 \left (-2+x^{2/3}\right ) \sin \left (\sqrt [3]{x}\right )+\left (-2+9 x^{2/3}\right ) \sin \left (3 \sqrt [3]{x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 58, normalized size = 0.67
method | result | size |
derivativedivides | \(x^{\frac {2}{3}} \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )-4 \sin \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\cos ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}-\frac {2 \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )}{9}\) | \(58\) |
default | \(x^{\frac {2}{3}} \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )-4 \sin \left (x^{\frac {1}{3}}\right )+4 x^{\frac {1}{3}} \cos \left (x^{\frac {1}{3}}\right )+\frac {2 x^{\frac {1}{3}} \left (\cos ^{3}\left (x^{\frac {1}{3}}\right )\right )}{3}-\frac {2 \left (2+\cos ^{2}\left (x^{\frac {1}{3}}\right )\right ) \sin \left (x^{\frac {1}{3}}\right )}{9}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 0.55 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 48, normalized size = 0.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 513 vs.
\(2 (85) = 170\).
time = 0.97, size = 513, normalized size = 5.97 \begin {gather*} \frac {54 x^{\frac {2}{3}} \tan ^{5}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {36 x^{\frac {2}{3}} \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {54 x^{\frac {2}{3}} \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {42 \sqrt [3]{x} \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {18 \sqrt [3]{x} \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {18 \sqrt [3]{x} \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} + \frac {42 \sqrt [3]{x}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {84 \tan ^{5}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {152 \tan ^{3}{\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} - \frac {84 \tan {\left (\frac {\sqrt [3]{x}}{2} \right )}}{9 \tan ^{6}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{4}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 27 \tan ^{2}{\left (\frac {\sqrt [3]{x}}{2} \right )} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 47, normalized size = 0.55 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.50, size = 62, normalized size = 0.72 \begin {gather*} 4\,x^{1/3}\,\cos \left (x^{1/3}\right )-\frac {2\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right )}{9}-\frac {40\,\sin \left (x^{1/3}\right )}{9}+2\,x^{2/3}\,\sin \left (x^{1/3}\right )+\frac {2\,x^{1/3}\,{\cos \left (x^{1/3}\right )}^3}{3}+x^{2/3}\,{\cos \left (x^{1/3}\right )}^2\,\sin \left (x^{1/3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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