3.20 \(\int \sqrt{1+e^{-x}} \text{csch}(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.129469, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + E^(-x)]*Csch[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7996, size = 26, normalized size = 1.04 \[ - 2 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{1 + e^{- x}}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.107541, size = 126, normalized size = 5.04 \[ \frac{\sqrt{2} e^{x/2} \sqrt{e^{-x}+1} \left (\log \left (1-e^{-x/2}\right )+\log \left (e^{-x/2}+1\right )-\log \left (e^{-x/2} \left (\sqrt{2} \sqrt{e^x+1}+e^{x/2}-1\right )\right )-\log \left (e^{-x/2} \left (\sqrt{2} \sqrt{e^x+1}+e^{x/2}+1\right )\right )\right )}{\sqrt{e^x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.197, size = 33, normalized size = 1.3 \[ -2\,\sqrt{2}\sqrt{ \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}}\sqrt{\tanh \left ( x/2 \right ) +1}{\it Artanh} \left ( \sqrt{\tanh \left ( x/2 \right ) +1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.5669, size = 50, normalized size = 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)/sinh(x),x, algorithm="maxima")

[Out]

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 = 0.232509, size = 74, normalized size = 2.96 \[ \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )}} - 3 \, \cosh \left (x\right ) - 3 \, \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log((2*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt((cosh(x) + sinh(x) + 1)/
(cosh(x) + sinh(x))) - 3*cosh(x) - 3*sinh(x) - 1)/(cosh(x) + sinh(x) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235928, size = 100, normalized size = 4. \[ -\sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right ) + \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*ln((sqrt(2) - 1)/(sqrt(2) + 1)) + 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))