Integrand size = 5, antiderivative size = 8 \[ \int \cos (x) \sin (x) \, dx=\frac {\sin ^2(x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2644, 30} \[ \int \cos (x) \sin (x) \, dx=\frac {\sin ^2(x)}{2} \]
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Rule 30
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int x \, dx,x,\sin (x)) \\ & = \frac {\sin ^2(x)}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin (x) \, dx=-\frac {1}{2} \cos ^2(x) \]
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Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\sin ^{2}\left (x \right )\right )}{2}\) | \(7\) |
default | \(\frac {\left (\sin ^{2}\left (x \right )\right )}{2}\) | \(7\) |
risch | \(-\frac {\cos \left (2 x \right )}{4}\) | \(7\) |
parallelrisch | \(\frac {1}{4}-\frac {\cos \left (2 x \right )}{4}\) | \(9\) |
norman | \(\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(19\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(19\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos (x) \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right )^{2} \]
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Time = 0.05 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62 \[ \int \cos (x) \sin (x) \, dx=\frac {\sin ^{2}{\left (x \right )}}{2} \]
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none
Time = 0.21 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos (x) \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right )^{2} \]
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none
Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos (x) \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right )^{2} \]
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Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cos (x) \sin (x) \, dx=\frac {{\sin \left (x\right )}^2}{2} \]
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