Integrand size = 10, antiderivative size = 4 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos (\cos (\sin (x))) \]
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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 = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4419, 4420, 2718} \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos (\cos (\sin (x))) \]
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Rule 2718
Rule 4419
Rule 4420
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int \sin (x) \sin (\cos (x)) \, dx,x,\sin (x)) \\ & = -\text {Subst}(\int \sin (x) \, dx,x,\cos (\sin (x))) \\ & = \cos (\cos (\sin (x))) \\ \end{align*}
Time = 9.25 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos (\cos (\sin (x))) \]
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Time = 8.37 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
default | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
risch | \(\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(5\) |
parallelrisch | \(-1+\cos \left (\cos \left (\sin \left (x \right )\right )\right )\) | \(7\) |
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (4) = 8\).
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 11.00 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos \left (\frac {\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} - 1}{\tan \left (\frac {\tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )^{2} + 1}\right ) \]
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Time = 0.90 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos {\left (\cos {\left (\sin {\left (x \right )} \right )} \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos \left (\cos \left (\sin \left (x\right )\right )\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos \left (\cos \left (\sin \left (x\right )\right )\right ) \]
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Time = 14.97 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin (\cos (\sin (x))) \sin (\sin (x)) \, dx=\cos \left (\cos \left (\sin \left (x\right )\right )\right ) \]
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