Optimal. Leaf size=7 \[ -\log \left (\cos \left (e^x\right )\right ) \]
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Rubi [A] time = 0.0101972, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 3475} \[ -\log \left (\cos \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 3475
Rubi steps
\begin{align*} \int e^x \tan \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \tan (x) \, dx,x,e^x\right )\\ &=-\log \left (\cos \left (e^x\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0081558, size = 7, normalized size = 1. \[ -\log \left (\cos \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 7, normalized size = 1. \begin{align*} -\ln \left ( \cos \left ({{\rm e}^{x}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994762, size = 5, normalized size = 0.71 \begin{align*} \log \left (\sec \left (e^{x}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.474122, size = 41, normalized size = 5.86 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{\tan \left (e^{x}\right )^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.643082, size = 10, normalized size = 1.43 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (e^{x} \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20248, size = 9, normalized size = 1.29 \begin{align*} -\log \left ({\left | \cos \left (e^{x}\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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