Integrand size = 6, antiderivative size = 3 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin (\sin (x)) \]
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Time = 0.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4419, 2717} \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin (\sin (x)) \]
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Rule 2717
Rule 4419
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int \cos (x) \, dx,x,\sin (x)) \\ & = \sin (\sin (x)) \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin (\sin (x)) \]
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Time = 0.26 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\sin \left (\sin \left (x \right )\right )\) | \(4\) |
default | \(\sin \left (\sin \left (x \right )\right )\) | \(4\) |
risch | \(\sin \left (\sin \left (x \right )\right )\) | \(4\) |
parallelrisch | \(\sin \left (\sin \left (x \right )\right )\) | \(4\) |
norman | \(\frac {2 \tan \left (\frac {x}{2}\right )^{2} \tan \left (\frac {\tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )+2 \tan \left (\frac {\tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {\tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )^{2}\right )}\) | \(77\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (3) = 6\).
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 5.67 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin \left (\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin {\left (\sin {\left (x \right )} \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin \left (\sin \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin \left (\sin \left (x\right )\right ) \]
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Time = 26.96 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \, dx=\sin \left (\sin \left (x\right )\right ) \]
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