Optimal. Leaf size=35 \[ 6 \cos \left (\sqrt [3]{x}\right )-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3442, 3377,
2718} \begin {gather*} -3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )+6 \cos \left (\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3442
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=3 \text {Subst}\left (\int x^2 \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \text {Subst}\left (\int x \cos (x) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )-6 \text {Subst}\left (\int \sin (x) \, dx,x,\sqrt [3]{x}\right )\\ &=6 \cos \left (\sqrt [3]{x}\right )-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.83 \begin {gather*} -3 \left (-2+x^{2/3}\right ) \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 26, normalized size = 0.74
method | result | size |
derivativedivides | \(6 \cos \left (x^{\frac {1}{3}}\right )-3 x^{\frac {2}{3}} \cos \left (x^{\frac {1}{3}}\right )+6 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )\) | \(26\) |
default | \(6 \cos \left (x^{\frac {1}{3}}\right )-3 x^{\frac {2}{3}} \cos \left (x^{\frac {1}{3}}\right )+6 x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )\) | \(26\) |
meijerg | \(12 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{\frac {2}{3}}}{2}+1\right ) \cos \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}+\frac {x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}\right )\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, 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 = 0.59, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, 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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Sympy [A]
time = 0.17, size = 34, normalized size = 0.97 \begin {gather*} - 3 x^{\frac {2}{3}} \cos {\left (\sqrt [3]{x} \right )} + 6 \sqrt [3]{x} \sin {\left (\sqrt [3]{x} \right )} + 6 \cos {\left (\sqrt [3]{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 21, normalized size = 0.60 \begin {gather*} -3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, 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 = 0.21, size = 21, normalized size = 0.60 \begin {gather*} 6\,x^{1/3}\,\sin \left (x^{1/3}\right )-3\,\cos \left (x^{1/3}\right )\,\left (x^{2/3}-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}
Antiderivative was successfully verified.
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