Integrand size = 4, antiderivative size = 8 \[ \int x \sin (x) \, dx=-x \cos (x)+\sin (x) \]
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Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3377, 2717} \[ \int x \sin (x) \, dx=\sin (x)-x \cos (x) \]
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Rule 2717
Rule 3377
Rubi steps \begin{align*} \text {integral}& = -x \cos (x)+\int \cos (x) \, dx \\ & = -x \cos (x)+\sin (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \sin (x) \, dx=-x \cos (x)+\sin (x) \]
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Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12
method | result | size |
default | \(-x \cos \left (x \right )+\sin \left (x \right )\) | \(9\) |
risch | \(-x \cos \left (x \right )+\sin \left (x \right )\) | \(9\) |
parallelrisch | \(-x \cos \left (x \right )+\sin \left (x \right )\) | \(9\) |
parts | \(-x \cos \left (x \right )+\sin \left (x \right )\) | \(9\) |
meijerg | \(2 \sqrt {\pi }\, \left (-\frac {x \cos \left (x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (x \right )}{2 \sqrt {\pi }}\right )\) | \(22\) |
norman | \(\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-x +2 \tan \left (\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(30\) |
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none
Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \sin (x) \, dx=-x \cos \left (x\right ) + \sin \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int x \sin (x) \, dx=- x \cos {\left (x \right )} + \sin {\left (x \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \sin (x) \, dx=-x \cos \left (x\right ) + \sin \left (x\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \sin (x) \, dx=-x \cos \left (x\right ) + \sin \left (x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int x \sin (x) \, dx=\sin \left (x\right )-x\,\cos \left (x\right ) \]
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