Integrand size = 4, antiderivative size = 14 \[ \int \sin ^2(x) \, dx=\frac {x}{2}-\frac {1}{2} \cos (x) \sin (x) \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2715, 8} \[ \int \sin ^2(x) \, dx=\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x) \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} \cos (x) \sin (x)+\frac {\int 1 \, dx}{2} \\ & = \frac {x}{2}-\frac {1}{2} \cos (x) \sin (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sin ^2(x) \, dx=\frac {x}{2}-\frac {1}{4} \sin (2 x) \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {x}{2}-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}\) | \(11\) |
risch | \(\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}\) | \(11\) |
parallelrisch | \(\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}\) | \(11\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {2 x}{\sqrt {\pi }}-\frac {\sin \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(22\) |
norman | \(\frac {\tan ^{3}\left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {x}{2}+\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(45\) |
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \, dx=-\frac {1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{2} \, x \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \, dx=\frac {x}{2} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \, dx=\frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \, dx=\frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \, dx=\frac {x}{2}-\frac {\sin \left (2\,x\right )}{4} \]
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