Integrand size = 8, antiderivative size = 9 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]
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Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4419, 2717} \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]
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Rule 2717
Rule 4419
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int \cos (2 x) \, dx,x,\sin (x)) \\ & = \frac {1}{2} \sin (2 \sin (x)) \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \sin (2 \sin (x)) \]
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Time = 0.44 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\) | \(8\) |
default | \(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\) | \(8\) |
risch | \(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\) | \(8\) |
parallelrisch | \(\frac {\sin \left (2 \sin \left (x \right )\right )}{2}\) | \(8\) |
norman | \(\frac {\tan \left (\frac {x}{2}\right )^{2} \tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )+\tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )^{2}\right )}\) | \(77\) |
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.11 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (\frac {4 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {\sin {\left (2 \sin {\left (x \right )} \right )}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {1}{2} \, \sin \left (2 \, \sin \left (x\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos (x) \cos (2 \sin (x)) \, dx=\frac {\sin \left (2\,\sin \left (x\right )\right )}{2} \]
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