### 3.663 $$\int \frac{1}{(\coth (x)+\text{csch}(x))^5} \, dx$$

Optimal. Leaf size=22 $\frac{4}{\cosh (x)+1}-\frac{2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1)$

[Out]

-2/(1 + Cosh[x])^2 + 4/(1 + Cosh[x]) + Log[1 + Cosh[x]]

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Rubi [A]  time = 0.0631212, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {4392, 2667, 43} $\frac{4}{\cosh (x)+1}-\frac{2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^(-5),x]

[Out]

-2/(1 + Cosh[x])^2 + 4/(1 + Cosh[x]) + Log[1 + Cosh[x]]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0216541, size = 32, normalized size = 1.45 $-\frac{1}{2} \text{sech}^4\left (\frac{x}{2}\right )+2 \text{sech}^2\left (\frac{x}{2}\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^(-5),x]

[Out]

2*Log[Cosh[x/2]] + 2*Sech[x/2]^2 - Sech[x/2]^4/2

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Maple [A]  time = 0.041, size = 36, normalized size = 1.6 \begin{align*} -{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)+csch(x))^5,x)

[Out]

-1/2*tanh(1/2*x)^4-tanh(1/2*x)^2-ln(tanh(1/2*x)-1)-ln(tanh(1/2*x)+1)

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Maxima [B]  time = 1.10317, size = 70, normalized size = 3.18 \begin{align*} x + \frac{8 \,{\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^5,x, algorithm="maxima")

[Out]

x + 8*(e^(-x) + e^(-2*x) + e^(-3*x))/(4*e^(-x) + 6*e^(-2*x) + 4*e^(-3*x) + e^(-4*x) + 1) + 2*log(e^(-x) + 1)

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Fricas [B]  time = 2.00276, size = 925, normalized size = 42.05 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} + 4 \,{\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \,{\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} + 6 \,{\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} + 4 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} + 3 \,{\left (x - 2\right )} \cosh \left (x\right )^{2} +{\left (3 \, x - 4\right )} \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^5,x, algorithm="fricas")

[Out]

-(x*cosh(x)^4 + x*sinh(x)^4 + 4*(x - 2)*cosh(x)^3 + 4*(x*cosh(x) + x - 2)*sinh(x)^3 + 2*(3*x - 4)*cosh(x)^2 +
2*(3*x*cosh(x)^2 + 6*(x - 2)*cosh(x) + 3*x - 4)*sinh(x)^2 + 4*(x - 2)*cosh(x) - 2*(cosh(x)^4 + 4*(cosh(x) + 1)
*sinh(x)^3 + sinh(x)^4 + 4*cosh(x)^3 + 6*(cosh(x)^2 + 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 +
3*cosh(x)^2 + 3*cosh(x) + 1)*sinh(x) + 4*cosh(x) + 1)*log(cosh(x) + sinh(x) + 1) + 4*(x*cosh(x)^3 + 3*(x - 2)*
cosh(x)^2 + (3*x - 4)*cosh(x) + x - 2)*sinh(x) + x)/(cosh(x)^4 + 4*(cosh(x) + 1)*sinh(x)^3 + sinh(x)^4 + 4*cos
h(x)^3 + 6*(cosh(x)^2 + 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 + 3*cosh(x)^2 + 3*cosh(x) + 1)*s
inh(x) + 4*cosh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{5}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))**5,x)

[Out]

Integral((coth(x) + csch(x))**(-5), x)

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Giac [A]  time = 1.11155, size = 41, normalized size = 1.86 \begin{align*} -x + \frac{8 \,{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} + 1\right )}^{4}} + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))^5,x, algorithm="giac")

[Out]

-x + 8*(e^(3*x) + e^(2*x) + e^x)/(e^x + 1)^4 + 2*log(e^x + 1)