Integrand size = 10, antiderivative size = 4 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin (\sin (\sin (x))) \]
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Time = 0.03 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4419, 2717} \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin (\sin (\sin (x))) \]
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Rule 2717
Rule 4419
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int \cos (x) \cos (\sin (x)) \, dx,x,\sin (x)) \\ & = \text {Subst}(\int \cos (x) \, dx,x,\sin (\sin (x))) \\ & = \sin (\sin (\sin (x))) \\ \end{align*}
Time = 3.76 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin (\sin (\sin (x))) \]
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Time = 10.41 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\sin \left (\sin \left (\sin \left (x \right )\right )\right )\) | \(5\) |
default | \(\sin \left (\sin \left (\sin \left (x \right )\right )\right )\) | \(5\) |
risch | \(\sin \left (\sin \left (\sin \left (x \right )\right )\right )\) | \(5\) |
parallelrisch | \(\sin \left (\sin \left (\sin \left (x \right )\right )\right )\) | \(5\) |
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (4) = 8\).
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 10.25 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin \left (\frac {2 \, \tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )}{\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} + 1}\right ) \]
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Time = 0.88 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin {\left (\sin {\left (\sin {\left (x \right )} \right )} \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin \left (\sin \left (\sin \left (x\right )\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin \left (\sin \left (\sin \left (x\right )\right )\right ) \]
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Time = 27.63 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx=\sin \left (\sin \left (\sin \left (x\right )\right )\right ) \]
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