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)-1/2*coth(x)^2

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 11, normalized size = 1.00 \[ \coth (x)-\frac {\text {csch}^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(1 + Coth[x]),x]

[Out]

Coth[x] - Csch[x]^2/2

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fricas [B]  time = 0.40, size = 55, normalized size = 5.00 \[ -\frac {2}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]

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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giac [A]  time = 0.13, size = 10, normalized size = 0.91 \[ -\frac {2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]

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

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maple [B]  time = 0.10, size = 32, normalized size = 2.91 \[ -\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}} \]

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

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maxima [B]  time = 0.30, size = 41, normalized size = 3.73 \[ \frac {4 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]

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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mupad [B]  time = 1.18, size = 16, normalized size = 1.45 \[ -\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(coth(x) + 1)),x)

[Out]

-2/(exp(4*x) - 2*exp(2*x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{4}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]

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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