Integrand size = 9, antiderivative size = 10 \[ \int (a \cos (x)+b \sin (x)) \, dx=-b \cos (x)+a \sin (x) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2717, 2718} \[ \int (a \cos (x)+b \sin (x)) \, dx=a \sin (x)-b \cos (x) \]
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Rule 2717
Rule 2718
Rubi steps \begin{align*} \text {integral}& = a \int \cos (x) \, dx+b \int \sin (x) \, dx \\ & = -b \cos (x)+a \sin (x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (a \cos (x)+b \sin (x)) \, dx=-b \cos (x)+a \sin (x) \]
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Time = 0.16 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(-b \cos \left (x \right )+a \sin \left (x \right )\) | \(11\) |
risch | \(-b \cos \left (x \right )+a \sin \left (x \right )\) | \(11\) |
parts | \(-b \cos \left (x \right )+a \sin \left (x \right )\) | \(11\) |
parallelrisch | \(a \sin \left (x \right )-b \cos \left (x \right )-b\) | \(14\) |
meijerg | \(a \sin \left (x \right )+b \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (x \right )}{\sqrt {\pi }}\right )\) | \(22\) |
norman | \(\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{1+\tan \left (\frac {x}{2}\right )^{2}}\) | \(23\) |
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (a \cos (x)+b \sin (x)) \, dx=-b \cos \left (x\right ) + a \sin \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int (a \cos (x)+b \sin (x)) \, dx=a \sin {\left (x \right )} - b \cos {\left (x \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (a \cos (x)+b \sin (x)) \, dx=-b \cos \left (x\right ) + a \sin \left (x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (a \cos (x)+b \sin (x)) \, dx=-b \cos \left (x\right ) + a \sin \left (x\right ) \]
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Time = 20.98 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (a \cos (x)+b \sin (x)) \, dx=a\,\sin \left (x\right )-b\,\cos \left (x\right ) \]
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