3.579 \(\int \text{csch}^3(x) \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2} \tanh ^{-1}(\cosh (x))-\frac{1}{2} \coth (x) \text{csch}(x) \]

[Out]

ArcTanh[Cosh[x]]/2 - (Coth[x]*Csch[x])/2

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Rubi [A]  time = 0.02023, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{1}{2} \tanh ^{-1}(\cosh (x))-\frac{1}{2} \coth (x) \text{csch}(x) \]

Antiderivative was successfully verified.

[In]  Int[Csch[x]^3,x]

[Out]

ArcTanh[Cosh[x]]/2 - (Coth[x]*Csch[x])/2

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Rubi in Sympy [A]  time = 4.06, size = 17, normalized size = 1.06 \[ \frac{\operatorname{atanh}{\left (\cosh{\left (x \right )} \right )}}{2} + \frac{\cosh{\left (x \right )}}{2 \left (- \cosh ^{2}{\left (x \right )} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(csch(x)**3,x)

[Out]

atanh(cosh(x))/2 + cosh(x)/(2*(-cosh(x)**2 + 1))

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Mathematica [B]  time = 0.00619391, size = 47, normalized size = 2.94 \[ -\frac{1}{8} \text{csch}^2\left (\frac{x}{2}\right )-\frac{1}{8} \text{sech}^2\left (\frac{x}{2}\right )-\frac{1}{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )+\frac{1}{2} \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csch[x]^3,x]

[Out]

-Csch[x/2]^2/8 + Log[Cosh[x/2]]/2 - Log[Sinh[x/2]]/2 - Sech[x/2]^2/8

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Maple [A]  time = 0.059, size = 11, normalized size = 0.7 \[ -{\frac{{\rm coth} \left (x\right ){\rm csch} \left (x\right )}{2}}+{\it Artanh} \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(csch(x)^3,x)

[Out]

-1/2*coth(x)*csch(x)+arctanh(exp(x))

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Maxima [A]  time = 1.32647, size = 61, normalized size = 3.81 \[ \frac{e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x)^3,x, algorithm="maxima")

[Out]

(e^(-x) + e^(-3*x))/(2*e^(-2*x) - e^(-4*x) - 1) + 1/2*log(e^(-x) + 1) - 1/2*log(
e^(-x) - 1)

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Fricas [A]  time = 0.212668, size = 285, normalized size = 17.81 \[ -\frac{2 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{3} -{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right )}{2 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x)^3,x, algorithm="fricas")

[Out]

-1/2*(2*cosh(x)^3 + 6*cosh(x)*sinh(x)^2 + 2*sinh(x)^3 - (cosh(x)^4 + 4*cosh(x)*s
inh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^
3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) + 1) + (cosh(x)^4 + 4*cosh(x)*si
nh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3
- cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*(3*cosh(x)^2 + 1)*sinh(x
) + 2*cosh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1
)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \operatorname{csch}^{3}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x)**3,x)

[Out]

Integral(csch(x)**3, x)

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GIAC/XCAS [A]  time = 0.207146, size = 61, normalized size = 3.81 \[ -\frac{e^{\left (-x\right )} + e^{x}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} + \frac{1}{4} \,{\rm ln}\left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac{1}{4} \,{\rm ln}\left (e^{\left (-x\right )} + e^{x} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csch(x)^3,x, algorithm="giac")

[Out]

-(e^(-x) + e^x)/((e^(-x) + e^x)^2 - 4) + 1/4*ln(e^(-x) + e^x + 2) - 1/4*ln(e^(-x
) + e^x - 2)