Optimal. Leaf size=28 \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]
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Rubi [A] time = 0.0129139, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2282, 50, 63, 206} \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sqrt{1-e^x} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x} \, dx,x,e^x\right )\\ &=2 \sqrt{1-e^x}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,e^x\right )\\ &=2 \sqrt{1-e^x}-2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-e^x}\right )\\ &=2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right )\\ \end{align*}
Mathematica [A] time = 0.0087402, size = 28, normalized size = 1. \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 36, normalized size = 1.3 \begin{align*} 2\,\sqrt{1-{{\rm e}^{x}}}+\ln \left ( -1+\sqrt{1-{{\rm e}^{x}}} \right ) -\ln \left ( 1+\sqrt{1-{{\rm e}^{x}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951086, size = 47, normalized size = 1.68 \begin{align*} 2 \, \sqrt{-e^{x} + 1} - \log \left (\sqrt{-e^{x} + 1} + 1\right ) + \log \left (\sqrt{-e^{x} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02845, size = 95, normalized size = 3.39 \begin{align*} 2 \, \sqrt{-e^{x} + 1} - \log \left (\sqrt{-e^{x} + 1} + 1\right ) + \log \left (\sqrt{-e^{x} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24318, size = 32, normalized size = 1.14 \begin{align*} 2 \sqrt{1 - e^{x}} + \log{\left (\sqrt{1 - e^{x}} - 1 \right )} - \log{\left (\sqrt{1 - e^{x}} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06754, size = 50, normalized size = 1.79 \begin{align*} 2 \, \sqrt{-e^{x} + 1} - \log \left (\sqrt{-e^{x} + 1} + 1\right ) + \log \left (-\sqrt{-e^{x} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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