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 = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {2320, 52, 65,
212} \begin {gather*} 2 \sqrt {1-e^x}-2 \tanh ^{-1}\left (\sqrt {1-e^x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 2320
Rubi steps
\begin {align*} \int \sqrt {1-e^x} \, dx &=\text {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,e^x\right )\\ &=2 \sqrt {1-e^x}+\text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,e^x\right )\\ &=2 \sqrt {1-e^x}-2 \text {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*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} 2 \sqrt {1-e^x}-2 \tanh ^{-1}\left (\sqrt {1-e^x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.79, size = 38, normalized size = 1.36 \begin {gather*} \text {Log}\left [-1+\sqrt {1-E^x}\right ]-\text {Log}\left [1+\sqrt {1-E^x}\right ]+2 \sqrt {1-E^x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 36, normalized size = 1.29
method | result | size |
risch | \(-\frac {2 \left (-1+{\mathrm e}^{x}\right )}{\sqrt {1-{\mathrm e}^{x}}}-2 \arctanh \left (\sqrt {1-{\mathrm e}^{x}}\right )\) | \(27\) |
derivativedivides | \(2 \sqrt {1-{\mathrm e}^{x}}+\ln \left (\sqrt {1-{\mathrm e}^{x}}-1\right )-\ln \left (\sqrt {1-{\mathrm e}^{x}}+1\right )\) | \(36\) |
default | \(2 \sqrt {1-{\mathrm e}^{x}}+\ln \left (\sqrt {1-{\mathrm e}^{x}}-1\right )-\ln \left (\sqrt {1-{\mathrm e}^{x}}+1\right )\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 35, normalized size = 1.25 \begin {gather*} 2 \, \sqrt {-e^{x} + 1} - \log \left (\sqrt {-e^{x} + 1} + 1\right ) + \log \left (\sqrt {-e^{x} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 35, normalized size = 1.25 \begin {gather*} 2 \, \sqrt {-e^{x} + 1} - \log \left (\sqrt {-e^{x} + 1} + 1\right ) + \log \left (\sqrt {-e^{x} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.79, size = 32, normalized size = 1.14 \begin {gather*} 2 \sqrt {1 - e^{x}} + \log {\left (\sqrt {1 - e^{x}} - 1 \right )} - \log {\left (\sqrt {1 - e^{x}} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 38, normalized size = 1.36 \begin {gather*} \ln \left (-\sqrt {-\mathrm {e}^{x}+1}+1\right )-\ln \left (\sqrt {-\mathrm {e}^{x}+1}+1\right )+2 \sqrt {-\mathrm {e}^{x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 40, normalized size = 1.43 \begin {gather*} 2\,\sqrt {1-{\mathrm {e}}^x}+\frac {2\,{\mathrm {e}}^{-\frac {x}{2}}\,\mathrm {asin}\left ({\mathrm {e}}^{-\frac {x}{2}}\right )\,\sqrt {1-{\mathrm {e}}^x}}{\sqrt {1-{\mathrm {e}}^{-x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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