Optimal. Leaf size=4 \[ \sin ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0226324, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2249, 216} \[ \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 216
Rubi steps
\begin{align*} \int \frac{e^x}{\sqrt{1-e^{2 x}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,e^x\right )\\ &=\sin ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0041817, size = 4, normalized size = 1. \[ \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 4, normalized size = 1. \begin{align*} \arcsin \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4503, size = 4, normalized size = 1. \begin{align*} \arcsin \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.76985, size = 59, normalized size = 14.75 \begin{align*} -2 \, \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.671693, size = 3, normalized size = 0.75 \begin{align*} \operatorname{asin}{\left (e^{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36482, size = 4, normalized size = 1. \begin{align*} \arcsin \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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