Integrand size = 14, antiderivative size = 44 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {x}{\sqrt {2}}+\frac {\arctan \left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}-\cos (4+6 x)}\right )}{3 \sqrt {2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2736} \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {\arctan \left (\frac {\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}}+\frac {x}{\sqrt {2}} \]
[In]
[Out]
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 = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {\arctan \left (\sqrt {2} \tan (2+3 x)\right )}{3 \sqrt {2}} \]
[In]
[Out]
Time = 0.37 (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 | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}-3-2 \sqrt {2}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}-3+2 \sqrt {2}\right )}{12}\) | \(48\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) - \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {\sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (3 x + 2 \right )} \right )} + \pi \left \lfloor {\frac {3 x - \frac {\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - 2 \, \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - 2 \, \cos \left (6 \, x + 4\right ) + 2}\right ) + 2\right )} \]
[In]
[Out]
Time = 27.40 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \frac {2}{3-\cos (4+6 x)} \, dx=\frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]
[In]
[Out]