3.96 \(\int \frac{\text{csch}^4(x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=11 \[ \coth (x)-\frac{\coth ^2(x)}{2} \]

[Out]

Coth[x] - Coth[x]^2/2

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Rubi [A]  time = 0.0343565, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3487} \[ \coth (x)-\frac{\coth ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(1 + Coth[x]),x]

[Out]

Coth[x] - Coth[x]^2/2

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rubi steps

\begin{align*} \int \frac{\text{csch}^4(x)}{1+\coth (x)} \, dx &=\operatorname{Subst}(\int (1-x) \, dx,x,\coth (x))\\ &=\coth (x)-\frac{\coth ^2(x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0245922, size = 11, normalized size = 1. \[ \coth (x)-\frac{\text{csch}^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(1 + Coth[x]),x]

[Out]

Coth[x] - Csch[x]^2/2

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Maple [B]  time = 0.023, size = 32, normalized size = 2.9 \begin{align*} -{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(1+coth(x)),x)

[Out]

-1/8*tanh(1/2*x)^2+1/2*tanh(1/2*x)-1/8/tanh(1/2*x)^2+1/2/tanh(1/2*x)

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Maxima [B]  time = 1.10184, size = 55, normalized size = 5. \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+coth(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.37182, size = 181, normalized size = 16.45 \begin{align*} -\frac{2}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+coth(x)),x, algorithm="fricas")

[Out]

-2/(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. \begin{align*} \int \frac{\operatorname{csch}^{4}{\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)**4/(1+coth(x)),x)

[Out]

Integral(csch(x)**4/(coth(x) + 1), x)

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Giac [A]  time = 1.15129, size = 14, normalized size = 1.27 \begin{align*} -\frac{2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(1+coth(x)),x, algorithm="giac")

[Out]

-2/(e^(2*x) - 1)^2