Optimal. Leaf size=13 \[ \frac{\cos ^3(x)}{3}-\cos (x) \]
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Rubi [A] time = 0.0054989, antiderivative size = 13, 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} \[ \frac{\cos ^3(x)}{3}-\cos (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \sin ^3(x) \, dx &=-\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)+\frac{\cos ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0017633, size = 15, normalized size = 1.15 \[ \frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928092, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97143, size = 31, normalized size = 2.38 \begin{align*} \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06428, size = 8, normalized size = 0.62 \begin{align*} \frac{\cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07697, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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