Integrand size = 8, antiderivative size = 7 \[ \int e^x \tan \left (e^x\right ) \, dx=-\log \left (\cos \left (e^x\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 7, 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 = {2320, 3556} \[ \int e^x \tan \left (e^x\right ) \, dx=-\log \left (\cos \left (e^x\right )\right ) \]
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Rule 2320
Rule 3556
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \tan (x) \, dx,x,e^x\right ) \\ & = -\log \left (\cos \left (e^x\right )\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int e^x \tan \left (e^x\right ) \, dx=-\log \left (\cos \left (e^x\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\ln \left (\cos \left ({\mathrm e}^{x}\right )\right )\) | \(7\) |
default | \(-\ln \left (\cos \left ({\mathrm e}^{x}\right )\right )\) | \(7\) |
norman | \(\frac {\ln \left (1+\tan \left ({\mathrm e}^{x}\right )^{2}\right )}{2}\) | \(11\) |
parallelrisch | \(\frac {\ln \left (1+\tan \left ({\mathrm e}^{x}\right )^{2}\right )}{2}\) | \(11\) |
risch | \(i {\mathrm e}^{x}-\ln \left ({\mathrm e}^{2 i {\mathrm e}^{x}}+1\right )\) | \(18\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.71 \[ \int e^x \tan \left (e^x\right ) \, dx=-\frac {1}{2} \, \log \left (\frac {1}{\tan \left (e^{x}\right )^{2} + 1}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.43 \[ \int e^x \tan \left (e^x\right ) \, dx=\frac {\log {\left (\tan ^{2}{\left (e^{x} \right )} + 1 \right )}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.57 \[ \int e^x \tan \left (e^x\right ) \, dx=\log \left (\sec \left (e^{x}\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int e^x \tan \left (e^x\right ) \, dx=-\log \left ({\left | \cos \left (e^{x}\right ) \right |}\right ) \]
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Time = 27.55 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.43 \[ \int e^x \tan \left (e^x\right ) \, dx=\frac {\ln \left ({\mathrm {tan}\left ({\mathrm {e}}^x\right )}^2+1\right )}{2} \]
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