Optimal. Leaf size=11 \[ \frac{1}{2} \log \left (\cos \left (e^{-2 x}\right )\right ) \]
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Rubi [A] time = 0.0117125, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 3475} \[ \frac{1}{2} \log \left (\cos \left (e^{-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 3475
Rubi steps
\begin{align*} \int e^{-2 x} \tan \left (e^{-2 x}\right ) \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \tan (x) \, dx,x,e^{-2 x}\right )\right )\\ &=\frac{1}{2} \log \left (\cos \left (e^{-2 x}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0080568, size = 11, normalized size = 1. \[ \frac{1}{2} \log \left (\cos \left (e^{-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 9, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( \cos \left ({{\rm e}^{-2\,x}} \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958688, size = 11, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (\sec \left (e^{\left (-2 \, x\right )}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06475, size = 46, normalized size = 4.18 \begin{align*} \frac{1}{4} \, \log \left (\frac{1}{\tan \left (e^{\left (-2 \, x\right )}\right )^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.37184, size = 15, normalized size = 1.36 \begin{align*} - \frac{\log{\left (\tan ^{2}{\left (e^{- 2 x} \right )} + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07857, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \log \left ({\left | \cos \left (e^{\left (-2 \, x\right )}\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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