3.242 \(\int \sqrt{1-e^x} \, dx\)

Optimal. Leaf size=28 \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]

[Out]

2*Sqrt[1 - E^x] - 2*ArcTanh[Sqrt[1 - E^x]]

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Rubi [A]  time = 0.031239, 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 \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - E^x],x]

[Out]

2*Sqrt[1 - E^x] - 2*ArcTanh[Sqrt[1 - E^x]]

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Rubi in Sympy [A]  time = 2.0329, size = 20, normalized size = 0.71 \[ 2 \sqrt{- e^{x} + 1} - 2 \operatorname{atanh}{\left (\sqrt{- e^{x} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-exp(x))**(1/2),x)

[Out]

2*sqrt(-exp(x) + 1) - 2*atanh(sqrt(-exp(x) + 1))

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Mathematica [A]  time = 0.0162804, size = 28, normalized size = 1. \[ 2 \sqrt{1-e^x}-2 \tanh ^{-1}\left (\sqrt{1-e^x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - E^x],x]

[Out]

2*Sqrt[1 - E^x] - 2*ArcTanh[Sqrt[1 - E^x]]

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Maple [A]  time = 0.008, size = 36, normalized size = 1.3 \[ 2\,\sqrt{1-{{\rm e}^{x}}}+\ln \left ( -1+\sqrt{1-{{\rm e}^{x}}} \right ) -\ln \left ( 1+\sqrt{1-{{\rm e}^{x}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-exp(x))^(1/2),x)

[Out]

2*(1-exp(x))^(1/2)+ln(-1+(1-exp(x))^(1/2))-ln(1+(1-exp(x))^(1/2))

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Maxima [A]  time = 1.36187, size = 47, normalized size = 1.68 \[ 2 \, \sqrt{-e^{x} + 1} - \log \left (\sqrt{-e^{x} + 1} + 1\right ) + \log \left (\sqrt{-e^{x} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^x + 1),x, algorithm="maxima")

[Out]

2*sqrt(-e^x + 1) - log(sqrt(-e^x + 1) + 1) + log(sqrt(-e^x + 1) - 1)

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Fricas [A]  time = 0.211196, size = 47, normalized size = 1.68 \[ 2 \, \sqrt{-e^{x} + 1} - \log \left (\sqrt{-e^{x} + 1} + 1\right ) + \log \left (\sqrt{-e^{x} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^x + 1),x, algorithm="fricas")

[Out]

2*sqrt(-e^x + 1) - log(sqrt(-e^x + 1) + 1) + log(sqrt(-e^x + 1) - 1)

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Sympy [A]  time = 0.658805, size = 32, normalized size = 1.14 \[ 2 \sqrt{- e^{x} + 1} + \log{\left (\sqrt{- e^{x} + 1} - 1 \right )} - \log{\left (\sqrt{- e^{x} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-exp(x))**(1/2),x)

[Out]

2*sqrt(-exp(x) + 1) + log(sqrt(-exp(x) + 1) - 1) - log(sqrt(-exp(x) + 1) + 1)

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GIAC/XCAS [A]  time = 0.205491, size = 50, normalized size = 1.79 \[ 2 \, \sqrt{-e^{x} + 1} -{\rm ln}\left (\sqrt{-e^{x} + 1} + 1\right ) +{\rm ln}\left (-\sqrt{-e^{x} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^x + 1),x, algorithm="giac")

[Out]

2*sqrt(-e^x + 1) - ln(sqrt(-e^x + 1) + 1) + ln(-sqrt(-e^x + 1) + 1)