Optimal. Leaf size=13 \[ \frac {\cos ^3(x)}{3}-\cos (x) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.41, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 11, normalized size = 0.85 \[ -\frac {\left (\sin ^{2}\relax (x )+2\right ) \cos \relax (x )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\cos \relax (x)\,\left ({\cos \relax (x)}^2-3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 8, normalized size = 0.62 \[ \frac {\cos ^{3}{\relax (x )}}{3} - \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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