Integrand size = 8, antiderivative size = 16 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3842, 3852, 8} \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \]
[In]
[Out]
Rule 8
Rule 3842
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sec ^2(x)-\frac {1}{2} \int \sec ^2(x) \, dx \\ & = \frac {1}{2} x \sec ^2(x)+\frac {1}{2} \text {Subst}(\int 1 \, dx,x,-\tan (x)) \\ & = \frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {x}{2 \cos \left (x \right )^{2}}-\frac {\tan \left (x \right )}{2}\) | \(13\) |
parallelrisch | \(\frac {x \left (\sec ^{2}\left (x \right )\right )}{2}-\frac {\tan \left (x \right )}{2}\) | \(13\) |
risch | \(\frac {2 x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}-i}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}\) | \(30\) |
norman | \(\frac {\tan ^{3}\left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {x}{2}+\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{2}}\) | \(45\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int x \sec ^2(x) \tan (x) \, dx=-\frac {\cos \left (x\right ) \sin \left (x\right ) - x}{2 \, \cos \left (x\right )^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (12) = 24\).
Time = 0.41 (sec) , antiderivative size = 128, normalized size of antiderivative = 8.00 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 2} + \frac {2 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 2} + \frac {x}{2 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 2} + \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 2} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (12) = 24\).
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 8.25 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {4 \, x \cos \left (2 \, x\right )^{2} + 4 \, x \sin \left (2 \, x\right )^{2} + {\left (2 \, x \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (4 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) + {\left (2 \, x \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1\right )} \sin \left (4 \, x\right ) - \sin \left (2 \, x\right )}{2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {x \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, x \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac {1}{2} \, x\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int x \sec ^2(x) \tan (x) \, dx=\frac {2\,x-\sin \left (2\,x\right )}{4\,{\cos \left (x\right )}^2} \]
[In]
[Out]