Integrand size = 14, antiderivative size = 31 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=-\frac {1}{3} \cos \left (\frac {\pi }{12}-3 x\right )+\frac {1}{9} \cos ^3\left (\frac {\pi }{12}-3 x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2713} \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=\frac {1}{9} \cos ^3\left (\frac {\pi }{12}-3 x\right )-\frac {1}{3} \cos \left (\frac {\pi }{12}-3 x\right ) \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=-\frac {1}{4} \cos \left (\frac {\pi }{12}-3 x\right )+\frac {1}{36} \cos \left (3 \left (\frac {\pi }{12}-3 x\right )\right ) \]
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Time = 0.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {\sin \left (\frac {\pi }{4}+9 x \right )}{36}-\frac {\sin \left (\frac {5 \pi }{12}+3 x \right )}{4}\) | \(22\) |
parallelrisch | \(\frac {\sin \left (\frac {\pi }{4}+9 x \right )}{36}-\frac {\sin \left (\frac {5 \pi }{12}+3 x \right )}{4}\) | \(22\) |
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 | \(\frac {-\frac {4 \left (\tan ^{3}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{3}-\frac {2 \tan \left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )^{3}}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=-\frac {1}{9} \, {\left (\cos \left (\frac {5}{12} \, \pi + 3 \, x\right )^{2} + 2\right )} \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=- \frac {2 \sin ^{3}{\left (3 x + \frac {5 \pi }{12} \right )}}{9} - \frac {\sin {\left (3 x + \frac {5 \pi }{12} \right )} \cos ^{2}{\left (3 x + \frac {5 \pi }{12} \right )}}{3} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=\frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]
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Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=\frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx=\frac {\sin \left (\frac {5\,\Pi }{12}+3\,x\right )\,\left ({\sin \left (\frac {5\,\Pi }{12}+3\,x\right )}^2-3\right )}{9} \]
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