Optimal. Leaf size=8 \[ \frac {\sin ^3(x)}{3} \]
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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2644, 30}
\begin {gather*} \frac {\sin ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rubi steps
\begin {align*} \int \cos (x) \sin ^2(x) \, dx &=\text {Subst}\left (\int x^2 \, dx,x,\sin (x)\right )\\ &=\frac {\sin ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sin ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.76, size = 6, normalized size = 0.75 \begin {gather*} \frac {\text {Sin}\left [x\right ]^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\sin ^{3}\left (x \right )\right )}{3}\) | \(7\) |
default | \(\frac {\left (\sin ^{3}\left (x \right )\right )}{3}\) | \(7\) |
risch | \(\frac {\sin \left (x \right )}{4}-\frac {\sin \left (3 x \right )}{12}\) | \(12\) |
norman | \(\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \sin \left (x\right )^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 10, normalized size = 1.25 \begin {gather*} -\frac {1}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 5, normalized size = 0.62 \begin {gather*} \frac {\sin ^{3}{\left (x \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \frac {\sin ^{3}x}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\sin \left (x\right )}^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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