Integrand size = 21, antiderivative size = 61 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=\frac {\log \left (\sqrt {2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt {2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3756, 213} \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=\frac {\log \left (\sqrt {2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sin (3 x+2)+\sqrt {2} \cos (3 x+2)\right )}{6 \sqrt {2}} \]
[In]
[Out]
Rule 213
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\tan (2+3 x)\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 = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.36 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=-\frac {\text {arctanh}\left (\frac {\tan (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
[In]
[Out]
Time = 1.21 (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 = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=\frac {1}{24} \, \sqrt {2} \log \left (-\frac {7 \, \cos \left (3 \, x + 2\right )^{4} - 10 \, \cos \left (3 \, x + 2\right )^{2} + 4 \, {\left (\sqrt {2} \cos \left (3 \, x + 2\right )^{3} + \sqrt {2} \cos \left (3 \, x + 2\right )\right )} \sin \left (3 \, x + 2\right ) - 1}{9 \, \cos \left (3 \, x + 2\right )^{4} - 6 \, \cos \left (3 \, x + 2\right )^{2} + 1}\right ) \]
[In]
[Out]
\[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=\int \frac {\sec ^{2}{\left (3 x + 2 \right )}}{\tan ^{2}{\left (3 x + 2 \right )} - 2}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.52 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=\frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \tan \left (3 \, x + 2\right )}{\sqrt {2} + \tan \left (3 \, x + 2\right )}\right ) \]
[In]
[Out]
none
Time = 0.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.61 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \log \left ({\left | \sqrt {2} + \tan \left (3 \, x + 2\right ) \right |}\right ) + \frac {1}{12} \, \sqrt {2} \log \left ({\left | -\sqrt {2} + \tan \left (3 \, x + 2\right ) \right |}\right ) \]
[In]
[Out]
Time = 28.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.28 \[ \int \frac {\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx=-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
[In]
[Out]