Integrand size = 13, antiderivative size = 10 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos (3 x)+\log (\sin (2 x)) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3556, 2718} \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos (3 x)+\log (\sin (2 x)) \]
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Rule 2718
Rule 3556
Rubi steps \begin{align*} \text {integral}& = 2 \int \cot (2 x) \, dx-3 \int \sin (3 x) \, dx \\ & = \cos (3 x)+\log (\sin (2 x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.50 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos (3 x)+\log (\cos (2 x))+\log (\tan (2 x)) \]
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Time = 0.63 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.70
method | result | size |
default | \(-\frac {\ln \left (1+\cot \left (2 x \right )^{2}\right )}{2}+\cos \left (3 x \right )\) | \(17\) |
parts | \(-\frac {\ln \left (1+\cot \left (2 x \right )^{2}\right )}{2}+\cos \left (3 x \right )\) | \(17\) |
risch | \(-2 i x +\ln \left ({\mathrm e}^{4 i x}-1\right )+\cos \left (3 x \right )\) | \(18\) |
parallelrisch | \(\ln \left (\tan \left (2 x \right )\right )+\ln \left (\frac {1}{\sqrt {\sec \left (2 x \right )^{2}}}\right )+1+\cos \left (3 x \right )\) | \(21\) |
norman | \(\frac {2}{1+\tan \left (\frac {3 x}{2}\right )^{2}}-\frac {\ln \left (1+\tan \left (2 x \right )^{2}\right )}{2}+\ln \left (\tan \left (2 x \right )\right )\) | \(30\) |
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none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right ) + \log \left (-\frac {1}{2} \, \cos \left (x\right ) \sin \left (x\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\log {\left (\sin {\left (2 x \right )} \right )} + \cos {\left (3 x \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos \left (3 \, x\right ) + \log \left (\sin \left (2 \, x\right )\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos \left (3 \, x\right ) + \log \left ({\left | \sin \left (2 \, x\right ) \right |}\right ) \]
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Time = 27.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.40 \[ \int (2 \cot (2 x)-3 \sin (3 x)) \, dx=\cos \left (3\,x\right )+\ln \left (\cos \left (\frac {x}{2}\right )\,\left (\sin \left (\frac {x}{2}\right )-2\,{\sin \left (\frac {x}{2}\right )}^3\right )\right ) \]
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