Optimal. Leaf size=13 \[ -\cos (x)+\frac {\cos ^3(x)}{3} \]
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Rubi [A]
time = 0.00, 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 = {2713}
\begin {gather*} \frac {\cos ^3(x)}{3}-\cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \sin ^3(x) \, dx &=-\text {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 \begin {gather*} -\frac {3 \cos (x)}{4}+\frac {1}{12} \cos (3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 11, normalized size = 0.85
method | result | size |
default | \(-\frac {\left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}\) | \(11\) |
risch | \(-\frac {3 \cos \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{12}\) | \(12\) |
norman | \(\frac {-4 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {4}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.09, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 8, normalized size = 0.62 \begin {gather*} \frac {\cos ^{3}{\left (x \right )}}{3} - \cos {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end {gather*}
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 \begin {gather*} \frac {\cos \left (x\right )\,\left ({\cos \left (x\right )}^2-3\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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