Integrand size = 12, antiderivative size = 42 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}+\cos (4+6 x)}\right )}{3 \sqrt {2}} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2736} \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}} \]
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Rule 12
Rule 2736
Rubi steps \begin{align*} \text {integral}& = 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 = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\arctan \left (\frac {\tan (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
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Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.43
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
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 = 31, normalized size of antiderivative = 0.74 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\sqrt {2} \left (\operatorname {atan}{\left (\frac {\sqrt {2} \tan {\left (3 x + 2 \right )}}{2} \right )} + \pi \left \lfloor {\frac {3 x - \frac {\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \]
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Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{2 \, {\left (\cos \left (6 \, x + 4\right ) + 1\right )}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.36 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \]
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Time = 26.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
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