Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
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Rubi [A] time = 0.0097277, antiderivative size = 5, 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, 3770} \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 3770
Rubi steps
\begin{align*} \int e^x \sec \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \sec (x) \, dx,x,e^x\right )\\ &=\tanh ^{-1}\left (\sin \left (e^x\right )\right )\\ \end{align*}
Mathematica [A] time = 0.005083, size = 5, normalized size = 1. \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 9, normalized size = 1.8 \begin{align*} \ln \left ( \sec \left ({{\rm e}^{x}} \right ) +\tan \left ({{\rm e}^{x}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0065, size = 11, normalized size = 2.2 \begin{align*} \log \left (\sec \left (e^{x}\right ) + \tan \left (e^{x}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.482003, size = 65, normalized size = 13. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (e^{x}\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (e^{x}\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.5652, size = 10, normalized size = 2. \begin{align*} \log{\left (\tan{\left (e^{x} \right )} + \sec{\left (e^{x} \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18458, size = 39, normalized size = 7.8 \begin{align*} \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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