3.95 \(\int \frac{\text{csch}^3(x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=8 \[ \tanh ^{-1}(\cosh (x))-\text{csch}(x) \]

[Out]

ArcTanh[Cosh[x]] - Csch[x]

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Rubi [A]  time = 0.0400128, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3501, 3770} \[ \tanh ^{-1}(\cosh (x))-\text{csch}(x) \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^3/(1 + Coth[x]),x]

[Out]

ArcTanh[Cosh[x]] - Csch[x]

Rule 3501

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3770

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

Rubi steps

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

Mathematica [A]  time = 0.0373046, size = 14, normalized size = 1.75 \[ -\text{csch}(x)-\log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^3/(1 + Coth[x]),x]

[Out]

-Csch[x] - Log[Tanh[x/2]]

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Maple [B]  time = 0.019, size = 23, normalized size = 2.9 \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^3/(1+coth(x)),x)

[Out]

1/2*tanh(1/2*x)-1/2/tanh(1/2*x)-ln(tanh(1/2*x))

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Maxima [B]  time = 1.0878, size = 42, normalized size = 5.25 \begin{align*} \frac{2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.59187, size = 306, normalized size = 38.25 \begin{align*} \frac{{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**3/(1+coth(x)),x)

[Out]

Integral(csch(x)**3/(coth(x) + 1), x)

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Giac [B]  time = 1.14436, size = 35, normalized size = 4.38 \begin{align*} -\frac{2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="giac")

[Out]

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