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

Optimal. Leaf size=25 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right ) \]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[1 + E^(-x)]/Sqrt[2]])

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Rubi [A]  time = 0.171012, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + E^(-x)]/(-E^(-x) + E^x),x]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[1 + E^(-x)]/Sqrt[2]])

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Rubi in Sympy [A]  time = 21.5085, size = 24, normalized size = 0.96 \[ - \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{1 + e^{- x}}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0853452, size = 112, normalized size = 4.48 \[ \frac{e^{x/2} \sqrt{e^{-x}+1} \left (\log \left (1-e^{x/2}\right )-\log \left (e^{x/2}+1\right )+\log \left (\sqrt{2} \sqrt{e^x+1}-e^{x/2}+1\right )-\log \left (\sqrt{2} \sqrt{e^x+1}+e^{x/2}+1\right )\right )}{\sqrt{2} \sqrt{e^x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + E^(-x)]/(-E^(-x) + E^x),x]

[Out]

(E^(x/2)*Sqrt[1 + E^(-x)]*(Log[1 - E^(x/2)] - Log[1 + E^(x/2)] + Log[1 - E^(x/2)
+ Sqrt[2]*Sqrt[1 + E^x]] - Log[1 + E^(x/2) + Sqrt[2]*Sqrt[1 + E^x]]))/(Sqrt[2]*
Sqrt[1 + E^x])

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Maple [B]  time = 0.031, size = 49, normalized size = 2. \[ -{\frac{{{\rm e}^{x}}\sqrt{2}}{2}\sqrt{{\frac{1+{{\rm e}^{x}}}{{{\rm e}^{x}}}}}{\it Artanh} \left ({\frac{ \left ( 1+3\,{{\rm e}^{x}} \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ({{\rm e}^{x}} \right ) ^{2}+{{\rm e}^{x}}}}}} \right ){\frac{1}{\sqrt{ \left ( 1+{{\rm e}^{x}} \right ){{\rm e}^{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+exp(-x))^(1/2)/(-exp(-x)+exp(x)),x)

[Out]

-1/2*((1+exp(x))/exp(x))^(1/2)*exp(x)/((1+exp(x))*exp(x))^(1/2)*2^(1/2)*arctanh(
1/4*(1+3*exp(x))*2^(1/2)/(exp(x)^2+exp(x))^(1/2))

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Maxima [A]  time = 1.61077, size = 51, normalized size = 2.04 \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{e^{\left (-x\right )} + 1}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (-x\right )} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2135, size = 46, normalized size = 1.84 \[ \frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{e^{x} + 1} e^{\left (\frac{1}{2} \, x\right )} - 3 \, e^{x} - 1}{e^{x} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*log((2*sqrt(2)*sqrt(e^x + 1)*e^(1/2*x) - 3*e^x - 1)/(e^x - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{1 + e^{- x}} e^{x}}{\left (e^{x} - 1\right ) \left (e^{x} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(1 + exp(-x))*exp(x)/((exp(x) - 1)*(exp(x) + 1)), x)

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GIAC/XCAS [A]  time = 0.236516, size = 101, normalized size = 4.04 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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