Optimal. Leaf size=11 \[ \sin (x)-\frac{\sin ^3(x)}{3} \]
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Rubi [A] time = 0.0063308, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2633} \[ \sin (x)-\frac{\sin ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \cos ^3(x) \, dx &=-\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac{\sin ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.001777, size = 15, normalized size = 1.36 \[ \frac{3 \sin (x)}{4}+\frac{1}{12} \sin (3 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 1. \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938879, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68354, size = 36, normalized size = 3.27 \begin{align*} \frac{1}{3} \,{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061609, size = 8, normalized size = 0.73 \begin{align*} - \frac{\sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10237, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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