Optimal. Leaf size=11 \[ \sin (x)-\frac {\sin ^3(x)}{3} \]
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Rubi [A] time = 0.01, antiderivative size = 11, 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} \[ \sin (x)-\frac {\sin ^3(x)}{3} \]
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*}
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Mathematica [A] time = 0.00, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.84, size = 10, normalized size = 0.91 \[ \frac {1}{3} \, {\left (\cos \relax (x)^{2} + 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 9, normalized size = 0.82 \[ -\frac {1}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 11, normalized size = 1.00
method | result | size |
default | \(\frac {\left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{3}\) | \(11\) |
risch | \(\frac {3 \sin \relax (x )}{4}+\frac {\sin \left (3 x \right )}{12}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 9, normalized size = 0.82 \[ -\frac {1}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 9, normalized size = 0.82 \[ \sin \relax (x)-\frac {{\sin \relax (x)}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 8, normalized size = 0.73 \[ - \frac {\sin ^{3}{\relax (x )}}{3} + \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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