Integrand size = 27, antiderivative size = 44 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {x}{\sqrt {2}}+\frac {\arctan \left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}-\cos (4+6 x)}\right )}{3 \sqrt {2}} \]
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Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 3245, 2736} \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {\arctan \left (\frac {\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}}+\frac {x}{\sqrt {2}} \]
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Rule 12
Rule 2736
Rule 3245
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {\csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx \\ & = 2 \int \frac {1}{3-\cos (4+6 x)} \, dx \\ & = \frac {x}{\sqrt {2}}+\frac {\arctan \left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}-\cos (4+6 x)}\right )}{3 \sqrt {2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {\arctan \left (\sqrt {2} \tan (2+3 x)\right )}{3 \sqrt {2}} \]
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Time = 0.61 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.39
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \arctan \left (\tan \left (2+3 x \right ) \sqrt {2}\right )}{6}\) | \(17\) |
default | \(\frac {\sqrt {2}\, \arctan \left (\tan \left (2+3 x \right ) \sqrt {2}\right )}{6}\) | \(17\) |
risch | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}-3-2 \sqrt {2}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}-3+2 \sqrt {2}\right )}{12}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) - \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]
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\[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=- 2 \int \frac {\csc {\left (6 x + 4 \right )}}{\cot {\left (6 x + 4 \right )} - 3 \csc {\left (6 x + 4 \right )}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - 2 \, \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - 2 \, \cos \left (6 \, x + 4\right ) + 2}\right ) + 2\right )} \]
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Time = 27.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.36 \[ \int \frac {2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]
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