3.78 \(\int \frac{\text{csch}^6(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{5} \coth ^5(x)+\frac{\coth ^4(x)}{4}+\frac{\coth ^3(x)}{3}-\frac{\coth ^2(x)}{2} \]

[Out]

-Coth[x]^2/2 + Coth[x]^3/3 + Coth[x]^4/4 - Coth[x]^5/5

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Rubi [A]  time = 0.0536724, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3516, 848, 75} \[ -\frac{1}{5} \coth ^5(x)+\frac{\coth ^4(x)}{4}+\frac{\coth ^3(x)}{3}-\frac{\coth ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^6/(1 + Tanh[x]),x]

[Out]

-Coth[x]^2/2 + Coth[x]^3/3 + Coth[x]^4/4 - Coth[x]^5/5

Rule 3516

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

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int \frac{\text{csch}^6(x)}{1+\tanh (x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x^6 (1+x)} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{(-1+x)^2 (1+x)}{x^6} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{x^6}-\frac{1}{x^5}-\frac{1}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{2} \coth ^2(x)+\frac{\coth ^3(x)}{3}+\frac{\coth ^4(x)}{4}-\frac{\coth ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0507342, size = 27, normalized size = 0.82 \[ \frac{1}{120} \text{csch}^5(x) (30 \sinh (x)-20 \cosh (x)-5 \cosh (3 x)+\cosh (5 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^6/(1 + Tanh[x]),x]

[Out]

(Csch[x]^5*(-20*Cosh[x] - 5*Cosh[3*x] + Cosh[5*x] + 30*Sinh[x]))/120

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Maple [B]  time = 0.033, size = 80, normalized size = 2.4 \begin{align*} -{\frac{1}{160} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}+{\frac{1}{96} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{16}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{1}{96} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{1}{160} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}}+{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^6/(1+tanh(x)),x)

[Out]

-1/160*tanh(1/2*x)^5+1/64*tanh(1/2*x)^4+1/96*tanh(1/2*x)^3-1/16*tanh(1/2*x)^2+1/16*tanh(1/2*x)-1/16/tanh(1/2*x
)^2+1/16/tanh(1/2*x)+1/96/tanh(1/2*x)^3-1/160/tanh(1/2*x)^5+1/64/tanh(1/2*x)^4

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Maxima [B]  time = 1.10543, size = 201, normalized size = 6.09 \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac{8 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-6 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1} - \frac{4}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(1+tanh(x)),x, algorithm="maxima")

[Out]

4/3*e^(-2*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) - 8/3*e^(-4*x)/(5*e^(-2*x)
- 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) + 8*e^(-6*x)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x)
- 5*e^(-8*x) + e^(-10*x) - 1) - 4/15/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1)

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Fricas [B]  time = 1.98475, size = 626, normalized size = 18.97 \begin{align*} -\frac{4 \,{\left (19 \, \cosh \left (x\right )^{2} + 42 \, \cosh \left (x\right ) \sinh \left (x\right ) + 19 \, \sinh \left (x\right )^{2} + 5\right )}}{15 \,{\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} +{\left (28 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{6} - 5 \, \cosh \left (x\right )^{6} + 2 \,{\left (28 \, \cosh \left (x\right )^{3} - 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \,{\left (14 \, \cosh \left (x\right )^{4} - 15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 10 \, \cosh \left (x\right )^{4} + 4 \,{\left (14 \, \cosh \left (x\right )^{5} - 25 \, \cosh \left (x\right )^{3} + 10 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} +{\left (28 \, \cosh \left (x\right )^{6} - 75 \, \cosh \left (x\right )^{4} + 60 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{2} - 11 \, \cosh \left (x\right )^{2} + 2 \,{\left (4 \, \cosh \left (x\right )^{7} - 15 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(1+tanh(x)),x, algorithm="fricas")

[Out]

-4/15*(19*cosh(x)^2 + 42*cosh(x)*sinh(x) + 19*sinh(x)^2 + 5)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + (2
8*cosh(x)^2 - 5)*sinh(x)^6 - 5*cosh(x)^6 + 2*(28*cosh(x)^3 - 15*cosh(x))*sinh(x)^5 + 5*(14*cosh(x)^4 - 15*cosh
(x)^2 + 2)*sinh(x)^4 + 10*cosh(x)^4 + 4*(14*cosh(x)^5 - 25*cosh(x)^3 + 10*cosh(x))*sinh(x)^3 + (28*cosh(x)^6 -
75*cosh(x)^4 + 60*cosh(x)^2 - 11)*sinh(x)^2 - 11*cosh(x)^2 + 2*(4*cosh(x)^7 - 15*cosh(x)^5 + 20*cosh(x)^3 - 9
*cosh(x))*sinh(x) + 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{6}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**6/(1+tanh(x)),x)

[Out]

Integral(csch(x)**6/(tanh(x) + 1), x)

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Giac [A]  time = 1.21643, size = 32, normalized size = 0.97 \begin{align*} -\frac{4 \,{\left (20 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}}{15 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^6/(1+tanh(x)),x, algorithm="giac")

[Out]

-4/15*(20*e^(4*x) + 5*e^(2*x) - 1)/(e^(2*x) - 1)^5