Integrand size = 4, antiderivative size = 13 \[ \int \sin ^3(x) \, dx=-\cos (x)+\frac {\cos ^3(x)}{3} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \sin ^3(x) \, dx=\frac {\cos ^3(x)}{3}-\cos (x) \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right ) \\ & = -\cos (x)+\frac {\cos ^3(x)}{3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \sin ^3(x) \, dx=-\frac {3 \cos (x)}{4}+\frac {1}{12} \cos (3 x) \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}\) | \(11\) |
risch | \(-\frac {3 \cos \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{12}\) | \(12\) |
parallelrisch | \(-\frac {2}{3}-\frac {3 \cos \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{12}\) | \(13\) |
norman | \(\frac {-4 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {4}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\) | \(22\) |
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \sin ^3(x) \, dx=\frac {\cos ^{3}{\left (x \right )}}{3} - \cos {\left (x \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \sin ^3(x) \, dx=\frac {\cos \left (x\right )\,\left ({\cos \left (x\right )}^2-3\right )}{3} \]
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