Integrand size = 7, antiderivative size = 18 \[ \int \cos (a+x) \sin (x) \, dx=-\frac {1}{4} \cos (a+2 x)-\frac {1}{2} x \sin (a) \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4670, 2718} \[ \int \cos (a+x) \sin (x) \, dx=-\frac {1}{2} x \sin (a)-\frac {1}{4} \cos (a+2 x) \]
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Rule 2718
Rule 4670
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sin (a)}{2}+\frac {1}{2} \sin (a+2 x)\right ) \, dx \\ & = -\frac {1}{2} x \sin (a)+\frac {1}{2} \int \sin (a+2 x) \, dx \\ & = -\frac {1}{4} \cos (a+2 x)-\frac {1}{2} x \sin (a) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \cos (a+x) \sin (x) \, dx=\frac {1}{4} (-\cos (a+2 x)-2 x \sin (a)) \]
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\cos \left (a +2 x \right )}{4}-\frac {x \sin \left (a \right )}{2}\) | \(15\) |
risch | \(-\frac {\cos \left (a +2 x \right )}{4}-\frac {x \sin \left (a \right )}{2}\) | \(15\) |
parallelrisch | \(-\frac {\cos \left (a +2 x \right )}{4}+\frac {\cos \left (a \right )}{4}-\frac {x \sin \left (a \right )}{2}\) | \(19\) |
meijerg | \(\frac {\cos \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{4}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \left (\frac {2 x}{\sqrt {\pi }}-\frac {\sin \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(45\) |
norman | \(\frac {x \tan \left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {a}{2}+\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right ) \tan \left (\frac {a}{2}+\frac {x}{2}\right )-x \tan \left (\frac {a}{2}+\frac {x}{2}\right )-x \tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {a}{2}+\frac {x}{2}\right )\right )}\) | \(91\) |
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \cos (a+x) \sin (x) \, dx=-\frac {1}{2} \, \cos \left (a + x\right )^{2} \cos \left (a\right ) - \frac {1}{2} \, \cos \left (a + x\right ) \sin \left (a + x\right ) \sin \left (a\right ) - \frac {1}{2} \, x \sin \left (a\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \cos (a+x) \sin (x) \, dx=\frac {x \sin {\left (x \right )} \cos {\left (a + x \right )}}{2} - \frac {x \sin {\left (a + x \right )} \cos {\left (x \right )}}{2} - \frac {\cos {\left (x \right )} \cos {\left (a + x \right )}}{2} \]
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Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \cos (a+x) \sin (x) \, dx=-\frac {1}{2} \, x \sin \left (a\right ) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \cos (a+x) \sin (x) \, dx=-\frac {1}{2} \, x \sin \left (a\right ) - \frac {1}{4} \, \cos \left (a + 2 \, x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \cos (a+x) \sin (x) \, dx=-\frac {\cos \left (a+2\,x\right )}{4}-\frac {x\,\sin \left (a\right )}{2} \]
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