Optimal. Leaf size=31 \[ -\frac {1}{3} \cos \left (\frac {\pi }{12}-3 x\right )+\frac {1}{9} \cos ^3\left (\frac {\pi }{12}-3 x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2713}
\begin {gather*} \frac {1}{9} \cos ^3\left (\frac {\pi }{12}-3 x\right )-\frac {1}{3} \cos \left (\frac {\pi }{12}-3 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx &=-\left (\frac {1}{3} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\frac {\pi }{12}-3 x\right )\right )\right )\\ &=-\frac {1}{3} \cos \left (\frac {\pi }{12}-3 x\right )+\frac {1}{9} \cos ^3\left (\frac {\pi }{12}-3 x\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \cos \left (\frac {\pi }{12}-3 x\right )+\frac {1}{36} \cos \left (3 \left (\frac {\pi }{12}-3 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 23, normalized size = 0.74
method | result | size |
risch | \(\frac {\sin \left (\frac {\pi }{4}+9 x \right )}{36}-\frac {\sin \left (\frac {5 \pi }{12}+3 x \right )}{4}\) | \(22\) |
derivativedivides | \(-\frac {\left (2+\cos ^{2}\left (\frac {5 \pi }{12}+3 x \right )\right ) \sin \left (\frac {5 \pi }{12}+3 x \right )}{9}\) | \(23\) |
default | \(-\frac {\left (2+\cos ^{2}\left (\frac {5 \pi }{12}+3 x \right )\right ) \sin \left (\frac {5 \pi }{12}+3 x \right )}{9}\) | \(23\) |
norman | \(\frac {-\frac {4 \left (\tan ^{3}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{3}-\frac {2 \tan \left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )^{3}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.54, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 22, normalized size = 0.71 \begin {gather*} -\frac {1}{9} \, {\left (\cos \left (\frac {5}{12} \, \pi + 3 \, x\right )^{2} + 2\right )} \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 39, normalized size = 1.26 \begin {gather*} - \frac {2 \sin ^{3}{\left (3 x + \frac {5 \pi }{12} \right )}}{9} - \frac {\sin {\left (3 x + \frac {5 \pi }{12} \right )} \cos ^{2}{\left (3 x + \frac {5 \pi }{12} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 22, normalized size = 0.71 \begin {gather*} \frac {\sin \left (\frac {5\,\Pi }{12}+3\,x\right )\,\left ({\sin \left (\frac {5\,\Pi }{12}+3\,x\right )}^2-3\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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