Integrand size = 7, antiderivative size = 17 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4368} \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]
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Rule 4368
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\) | \(14\) |
risch | \(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\) | \(14\) |
parallelrisch | \(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\) | \(14\) |
norman | \(\frac {-\frac {8 \tan \left (2 x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15}+\frac {2 \left (\tan ^{2}\left (2 x \right )\right ) \tan \left (\frac {x}{2}\right )}{15}+\frac {8 \tan \left (2 x \right )}{15}-\frac {2 \tan \left (\frac {x}{2}\right )}{15}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (2 x \right )\right )}\) | \(59\) |
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none
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{15} \, {\left (24 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]
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Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \cos (x) \cos (4 x) \, dx=- \frac {\sin {\left (x \right )} \cos {\left (4 x \right )}}{15} + \frac {4 \sin {\left (4 x \right )} \cos {\left (x \right )}}{15} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {\sin \left (3\,x\right )}{6}+\frac {\sin \left (5\,x\right )}{10} \]
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