Integrand size = 9, antiderivative size = 21 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=\frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4367} \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=\frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \]
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Rule 4367
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=\frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {2 \sin \left (\frac {3 x}{4}\right )}{3}-\frac {2 \sin \left (\frac {5 x}{4}\right )}{5}\) | \(14\) |
parallelrisch | \(\frac {2 \sin \left (\frac {3 x}{4}\right )}{3}-\frac {2 \sin \left (\frac {5 x}{4}\right )}{5}\) | \(14\) |
derivativedivides | \(-\frac {32 \left (\sin ^{5}\left (\frac {x}{4}\right )\right )}{5}+\frac {16 \left (\sin ^{3}\left (\frac {x}{4}\right )\right )}{3}\) | \(18\) |
default | \(-\frac {32 \left (\sin ^{5}\left (\frac {x}{4}\right )\right )}{5}+\frac {16 \left (\sin ^{3}\left (\frac {x}{4}\right )\right )}{3}\) | \(18\) |
norman | \(\frac {-\frac {8 \tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{8}\right )\right )}{15}+\frac {32 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{8}\right )}{15}+\frac {8 \tan \left (\frac {x}{2}\right )}{15}-\frac {32 \tan \left (\frac {x}{8}\right )}{15}}{\left (1+\tan ^{2}\left (\frac {x}{8}\right )\right ) \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\) | \(59\) |
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none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=-\frac {16}{15} \, {\left (6 \, \cos \left (\frac {1}{4} \, x\right )^{4} - 7 \, \cos \left (\frac {1}{4} \, x\right )^{2} + 1\right )} \sin \left (\frac {1}{4} \, x\right ) \]
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Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=- \frac {16 \sin {\left (\frac {x}{4} \right )} \cos {\left (x \right )}}{15} + \frac {4 \sin {\left (x \right )} \cos {\left (\frac {x}{4} \right )}}{15} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=-\frac {2}{5} \, \sin \left (\frac {5}{4} \, x\right ) + \frac {2}{3} \, \sin \left (\frac {3}{4} \, x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=-\frac {32}{5} \, \sin \left (\frac {1}{4} \, x\right )^{5} + \frac {16}{3} \, \sin \left (\frac {1}{4} \, x\right )^{3} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx=\frac {2\,\sin \left (\frac {3\,x}{4}\right )}{3}-\frac {2\,\sin \left (\frac {5\,x}{4}\right )}{5} \]
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