3.575 \(\int \text{csch}(x) \, dx\)

Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cosh (x)) \]

[Out]

-ArcTanh[Cosh[x]]

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Rubi [A]  time = 0.00719738, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]  Int[Csch[x],x]

[Out]

-ArcTanh[Cosh[x]]

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Rubi in Sympy [A]  time = 1.65586, size = 5, normalized size = 1. \[ - \operatorname{atanh}{\left (\cosh{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(csch(x),x)

[Out]

-atanh(cosh(x))

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Mathematica [B]  time = 0.00729465, size = 17, normalized size = 3.4 \[ \log \left (\sinh \left (\frac{x}{2}\right )\right )-\log \left (\cosh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csch[x],x]

[Out]

-Log[Cosh[x/2]] + Log[Sinh[x/2]]

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Maple [A]  time = 0.013, size = 6, normalized size = 1.2 \[ \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(csch(x),x)

[Out]

ln(tanh(1/2*x))

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Maxima [A]  time = 1.33546, size = 7, normalized size = 1.4 \[ \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x),x, algorithm="maxima")

[Out]

log(tanh(1/2*x))

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Fricas [A]  time = 0.20445, size = 23, normalized size = 4.6 \[ -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x),x, algorithm="fricas")

[Out]

-log(cosh(x) + sinh(x) + 1) + log(cosh(x) + sinh(x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \operatorname{csch}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x),x)

[Out]

Integral(csch(x), x)

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GIAC/XCAS [A]  time = 0.19852, size = 19, normalized size = 3.8 \[ -{\rm ln}\left (e^{x} + 1\right ) +{\rm ln}\left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x),x, algorithm="giac")

[Out]

-ln(e^x + 1) + ln(abs(e^x - 1))