Integrand size = 15, antiderivative size = 11 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{2} \text {arctanh}(2 \cos (x) \sin (x)) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3756, 212} \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{2} \text {arctanh}(2 \sin (x) \cos (x)) \]
[In]
[Out]
Rule 212
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {arctanh}(2 \cos (x) \sin (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{2} \text {arctanh}(\sin (2 x)) \]
[In]
[Out]
Time = 0.95 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.36
method | result | size |
default | \(\operatorname {arctanh}\left (\tan \left (x \right )\right )\) | \(4\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i x}-i\right )}{2}+\frac {\ln \left (i+{\mathrm e}^{2 i x}\right )}{2}\) | \(24\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (9) = 18\).
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{4} \, \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
[In]
[Out]
\[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=- \int \frac {\sec ^{2}{\left (x \right )}}{\tan ^{2}{\left (x \right )} - 1}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{2} \, \log \left (\tan \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (\tan \left (x\right ) - 1\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\frac {1}{2} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) \]
[In]
[Out]
Time = 26.55 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.27 \[ \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx=\mathrm {atanh}\left (\mathrm {tan}\left (x\right )\right ) \]
[In]
[Out]