3.241 \(\int \frac{1}{\sqrt{1+e^x}} \, dx\)

Optimal. Leaf size=12 \[ -2 \tanh ^{-1}\left (\sqrt{e^x+1}\right ) \]

[Out]

-2*ArcTanh[Sqrt[1 + E^x]]

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Rubi [A]  time = 0.0177997, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -2 \tanh ^{-1}\left (\sqrt{e^x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + E^x],x]

[Out]

-2*ArcTanh[Sqrt[1 + E^x]]

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Rubi in Sympy [A]  time = 1.35893, size = 12, normalized size = 1. \[ - 2 \operatorname{atanh}{\left (\sqrt{e^{x} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(exp(x) + 1))

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Mathematica [A]  time = 0.00726841, size = 12, normalized size = 1. \[ -2 \tanh ^{-1}\left (\sqrt{e^x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + E^x],x]

[Out]

-2*ArcTanh[Sqrt[1 + E^x]]

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Maple [A]  time = 0.006, size = 10, normalized size = 0.8 \[ -2\,{\it Artanh} \left ( \sqrt{1+{{\rm e}^{x}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+exp(x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.38464, size = 28, normalized size = 2.33 \[ -\log \left (\sqrt{e^{x} + 1} + 1\right ) + \log \left (\sqrt{e^{x} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211383, size = 28, normalized size = 2.33 \[ -\log \left (\sqrt{e^{x} + 1} + 1\right ) + \log \left (\sqrt{e^{x} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.573748, size = 26, normalized size = 2.17 \[ \log{\left (-1 + \frac{1}{\sqrt{e^{x} + 1}} \right )} - \log{\left (1 + \frac{1}{\sqrt{e^{x} + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+exp(x))**(1/2),x)

[Out]

log(-1 + 1/sqrt(exp(x) + 1)) - log(1 + 1/sqrt(exp(x) + 1))

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GIAC/XCAS [A]  time = 0.204356, size = 28, normalized size = 2.33 \[ -{\rm ln}\left (\sqrt{e^{x} + 1} + 1\right ) +{\rm ln}\left (\sqrt{e^{x} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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