Integrand size = 8, antiderivative size = 16 \[ \int (1+\tan (2 x))^2 \, dx=-\log (\cos (2 x))+\frac {1}{2} \tan (2 x) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3558, 3556} \[ \int (1+\tan (2 x))^2 \, dx=\frac {1}{2} \tan (2 x)-\log (\cos (2 x)) \]
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Rule 3556
Rule 3558
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \tan (2 x)+2 \int \tan (2 x) \, dx \\ & = -\log (\cos (2 x))+\frac {1}{2} \tan (2 x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int (1+\tan (2 x))^2 \, dx=x-\frac {1}{2} \arctan (\tan (2 x))-\log (\cos (2 x))+\frac {1}{2} \tan (2 x) \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
default | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
norman | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
parallelrisch | \(\frac {\tan \left (2 x \right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(19\) |
parts | \(x +\frac {\tan \left (2 x \right )}{2}-\frac {\arctan \left (\tan \left (2 x \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (2 x \right )\right )}{2}\) | \(27\) |
risch | \(2 i x +\frac {i}{{\mathrm e}^{4 i x}+1}-\ln \left ({\mathrm e}^{4 i x}+1\right )\) | \(28\) |
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int (1+\tan (2 x))^2 \, dx=-\frac {1}{2} \, \log \left (\frac {1}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int (1+\tan (2 x))^2 \, dx=\frac {\log {\left (\tan ^{2}{\left (2 x \right )} + 1 \right )}}{2} + \frac {\tan {\left (2 x \right )}}{2} \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (1+\tan (2 x))^2 \, dx=\log \left (\sec \left (2 \, x\right )\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int (1+\tan (2 x))^2 \, dx=-\frac {1}{2} \, \log \left (\frac {4}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac {1}{2} \, \tan \left (2 \, x\right ) \]
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Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (1+\tan (2 x))^2 \, dx=\frac {\mathrm {tan}\left (2\,x\right )}{2}+\frac {\ln \left ({\mathrm {tan}\left (2\,x\right )}^2+1\right )}{2} \]
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