Integrand size = 6, antiderivative size = 12 \[ \int x \sin (a+x) \, dx=-x \cos (a+x)+\sin (a+x) \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3377, 2717} \[ \int x \sin (a+x) \, dx=\sin (a+x)-x \cos (a+x) \]
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Rule 2717
Rule 3377
Rubi steps \begin{align*} \text {integral}& = -x \cos (a+x)+\int \cos (a+x) \, dx \\ & = -x \cos (a+x)+\sin (a+x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sin (a+x) \, dx=-x \cos (a+x)+\sin (a+x) \]
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Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-x \cos \left (a +x \right )+\sin \left (a +x \right )\) | \(13\) |
parallelrisch | \(-x \cos \left (a +x \right )+\sin \left (a +x \right )\) | \(13\) |
parts | \(-x \cos \left (a +x \right )+\sin \left (a +x \right )\) | \(13\) |
derivativedivides | \(a \cos \left (a +x \right )+\sin \left (a +x \right )-\left (a +x \right ) \cos \left (a +x \right )\) | \(21\) |
default | \(a \cos \left (a +x \right )+\sin \left (a +x \right )-\left (a +x \right ) \cos \left (a +x \right )\) | \(21\) |
norman | \(\frac {x \left (\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right )-x +2 \tan \left (\frac {a}{2}+\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )}\) | \(42\) |
meijerg | \(2 \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (x \right )}{2 \sqrt {\pi }}\right )+2 \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {x \cos \left (x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (x \right )}{2 \sqrt {\pi }}\right )\) | \(53\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sin (a+x) \, dx=-x \cos \left (a + x\right ) + \sin \left (a + x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int x \sin (a+x) \, dx=- x \cos {\left (a + x \right )} + \sin {\left (a + x \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67 \[ \int x \sin (a+x) \, dx=-{\left (a + x\right )} \cos \left (a + x\right ) + a \cos \left (a + x\right ) + \sin \left (a + x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sin (a+x) \, dx=-x \cos \left (a + x\right ) + \sin \left (a + x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \sin (a+x) \, dx=\sin \left (a+x\right )-x\,\cos \left (a+x\right ) \]
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