Integrand size = 6, antiderivative size = 35 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=6 \cos \left (\sqrt [3]{x}\right )-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3442, 3377, 2718} \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=-3 x^{2/3} \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right )+6 \cos \left (\sqrt [3]{x}\right ) \]
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Rule 2718
Rule 3377
Rule 3442
Rubi steps \begin{align*} \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{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=-3 \left (-2+x^{2/3}\right ) \cos \left (\sqrt [3]{x}\right )+6 \sqrt [3]{x} \sin \left (\sqrt [3]{x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 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 | \(12 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{\frac {2}{3}}}{2}+1\right ) \cos \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}+\frac {x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}\right )\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=-3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=- 3 x^{\frac {2}{3}} \cos {\left (\sqrt [3]{x} \right )} + 6 \sqrt [3]{x} \sin {\left (\sqrt [3]{x} \right )} + 6 \cos {\left (\sqrt [3]{x} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=-3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=-3 \, {\left (x^{\frac {2}{3}} - 2\right )} \cos \left (x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}} \sin \left (x^{\frac {1}{3}}\right ) \]
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Time = 16.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [3]{x}\right ) \, dx=6\,x^{1/3}\,\sin \left (x^{1/3}\right )-3\,\cos \left (x^{1/3}\right )\,\left (x^{2/3}-2\right ) \]
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