Integrand size = 14, antiderivative size = 27 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {1}{2} \log (\cos (1+2 x))+\frac {1}{2} (1+2 x) \tan (1+2 x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 27, 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 = {4269, 3556} \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {1}{2} (2 x+1) \tan (2 x+1)+\frac {1}{2} \log (\cos (2 x+1)) \]
[In]
[Out]
Rule 3556
Rule 4269
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} (1+2 x) \tan (1+2 x)-\int \tan (1+2 x) \, dx \\ & = \frac {1}{2} \log (\cos (1+2 x))+\frac {1}{2} (1+2 x) \tan (1+2 x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {1}{2} \log (\cos (1+2 x))+\frac {1}{2} \tan (1+2 x)+x \tan (1+2 x) \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\cos \left (1+2 x \right )\right )}{2}+\frac {\left (1+2 x \right ) \tan \left (1+2 x \right )}{2}\) | \(24\) |
default | \(\frac {\ln \left (\cos \left (1+2 x \right )\right )}{2}+\frac {\left (1+2 x \right ) \tan \left (1+2 x \right )}{2}\) | \(24\) |
risch | \(-2 i x -i+\frac {i \left (1+2 x \right )}{{\mathrm e}^{2 i \left (1+2 x \right )}+1}+\frac {\ln \left ({\mathrm e}^{2 i \left (1+2 x \right )}+1\right )}{2}\) | \(43\) |
norman | \(\frac {-2 x \tan \left (x +\frac {1}{2}\right )-\tan \left (x +\frac {1}{2}\right )}{\tan \left (x +\frac {1}{2}\right )^{2}-1}+\frac {\ln \left (\tan \left (x +\frac {1}{2}\right )-1\right )}{2}+\frac {\ln \left (\tan \left (x +\frac {1}{2}\right )+1\right )}{2}-\frac {\ln \left (1+\tan \left (x +\frac {1}{2}\right )^{2}\right )}{2}\) | \(56\) |
parallelrisch | \(\frac {\ln \left (\tan \left (x +\frac {1}{2}\right )-1\right ) \cos \left (1+2 x \right )+\ln \left (\tan \left (x +\frac {1}{2}\right )+1\right ) \cos \left (1+2 x \right )-\ln \left (\sec \left (x +\frac {1}{2}\right )^{2}\right ) \cos \left (1+2 x \right )+2 x \sin \left (1+2 x \right )+\sin \left (1+2 x \right )}{2 \cos \left (1+2 x \right )}\) | \(70\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {\cos \left (2 \, x + 1\right ) \log \left (-\cos \left (2 \, x + 1\right )\right ) + {\left (2 \, x + 1\right )} \sin \left (2 \, x + 1\right )}{2 \, \cos \left (2 \, x + 1\right )} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {\left (2 x + 1\right ) \tan {\left (2 x + 1 \right )}}{2} + \frac {\log {\left (\cos {\left (2 x + 1 \right )} \right )}}{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.63 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {{\left (\cos \left (4 \, x + 2\right )^{2} + \sin \left (4 \, x + 2\right )^{2} + 2 \, \cos \left (4 \, x + 2\right ) + 1\right )} \log \left (\cos \left (4 \, x + 2\right )^{2} + \sin \left (4 \, x + 2\right )^{2} + 2 \, \cos \left (4 \, x + 2\right ) + 1\right ) + 4 \, {\left (2 \, x + 1\right )} \sin \left (4 \, x + 2\right )}{4 \, {\left (\cos \left (4 \, x + 2\right )^{2} + \sin \left (4 \, x + 2\right )^{2} + 2 \, \cos \left (4 \, x + 2\right ) + 1\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (23) = 46\).
Time = 0.40 (sec) , antiderivative size = 908, normalized size of antiderivative = 33.63 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int (1+2 x) \sec ^2(1+2 x) \, dx=\frac {\ln \left (\cos \left (2\,x+1\right )\right )}{2}+\frac {\mathrm {tan}\left (2\,x+1\right )\,\left (2\,x+1\right )}{2} \]
[In]
[Out]