Integrand size = 25, antiderivative size = 61 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=\frac {\log \left (\sqrt {2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt {2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 3245, 2738, 212} \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=\frac {\log \left (\sqrt {2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sin (3 x+2)+\sqrt {2} \cos (3 x+2)\right )}{6 \sqrt {2}} \]
[In]
[Out]
Rule 12
Rule 212
Rule 2738
Rule 3245
Rubi steps \begin{align*} \text {integral}& = -\left (2 \int \frac {\csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx\right ) \\ & = -\left (2 \int \frac {1}{1+3 \cos (4+6 x)} \, dx\right ) \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{4-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (4+6 x)\right )\right )\right ) \\ & = \frac {\log \left (\sqrt {2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt {2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.36 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=-\frac {\text {arctanh}\left (\frac {\tan (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
[In]
[Out]
Time = 0.91 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.30
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
default | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
risch | \(-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+\frac {1}{3}+\frac {2 i \sqrt {2}}{3}\right )}{12}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+\frac {1}{3}-\frac {2 i \sqrt {2}}{3}\right )}{12}\) | \(48\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=\frac {1}{24} \, \sqrt {2} \log \left (-\frac {7 \, \cos \left (6 \, x + 4\right )^{2} + 4 \, {\left (\sqrt {2} \cos \left (6 \, x + 4\right ) + 3 \, \sqrt {2}\right )} \sin \left (6 \, x + 4\right ) - 6 \, \cos \left (6 \, x + 4\right ) - 17}{9 \, \cos \left (6 \, x + 4\right )^{2} + 6 \, \cos \left (6 \, x + 4\right ) + 1}\right ) \]
[In]
[Out]
\[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=- 2 \int \frac {\csc {\left (6 x + 4 \right )}}{3 \cot {\left (6 x + 4 \right )} + \csc {\left (6 x + 4 \right )}}\, dx \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=\frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}{\sqrt {2} + \frac {\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}\right ) \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.64 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=\frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \]
[In]
[Out]
Time = 28.62 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.28 \[ \int -\frac {2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx=-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
[In]
[Out]