3.6.79 \(\int \text {csch}^3(x) \, dx\) [579]

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

[Out]

1/2*arctanh(cosh(x))-1/2*coth(x)*csch(x)

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \begin {gather*} \frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^3,x]

[Out]

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

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \text {csch}^3(x) \, dx &=-\frac {1}{2} \coth (x) \text {csch}(x)-\frac {1}{2} \int \text {csch}(x) \, dx\\ &=\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x)\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(16)=32\).
time = 0.00, size = 36, normalized size = 2.25 \begin {gather*} -\frac {1}{8} \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {1}{8} \text {sech}^2\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^3,x]

[Out]

-1/8*Csch[x/2]^2 - Log[Tanh[x/2]]/2 - Sech[x/2]^2/8

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Maple [A]
time = 0.08, size = 11, normalized size = 0.69

method	result	size
default	\(-\frac {\coth \left (x \right ) \mathrm {csch}\left (x \right )}{2}+\arctanh \left ({\mathrm e}^{x}\right )\)	\(11\)
risch	\(-\frac {{\mathrm e}^{x} \left (1+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (12) = 24\).
time = 5.69, size = 45, normalized size = 2.81 \begin {gather*} \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 ) \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (12) = 24\).
time = 0.65, size = 211, normalized size = 13.19 \begin {gather*} -\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 )}} \end {gather*}

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)*sinh(x)^3 + sinh(x)^4 + 2*(3*co
sh(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)*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)*log(cosh(x) + sinh(x) - 1) + 2*(3*cosh(x)^2 + 1)*sinh(x) + 2*cosh(x))/(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {csch}^{3}{\left (x \right )}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (12) = 24\).
time = 0.83, size = 45, normalized size = 2.81 \begin {gather*} -\frac {e^{\left (-x\right )} + e^{x}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} + \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end {gather*}

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

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Mupad [B]
time = 0.29, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right )}{2}-\frac {\mathrm {cosh}\left (x\right )}{2\,{\mathrm {sinh}\left (x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(x)^3,x)

[Out]

- log(tanh(x/2))/2 - cosh(x)/(2*sinh(x)^2)

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