### 3.662 $$\int \frac{1}{(\coth (x)+\text{csch}(x))^4} \, dx$$

Optimal. Leaf size=26 $x-\frac{2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac{2 \sinh (x)}{\cosh (x)+1}$

[Out]

x - (2*Sinh[x])/(1 + Cosh[x]) - (2*Sinh[x]^3)/(3*(1 + Cosh[x])^3)

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Rubi [A]  time = 0.0848153, antiderivative size = 26, 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, 2680, 8} $x-\frac{2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac{2 \sinh (x)}{\cosh (x)+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^(-4),x]

[Out]

x - (2*Sinh[x])/(1 + Cosh[x]) - (2*Sinh[x]^3)/(3*(1 + Cosh[x])^3)

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 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0211706, size = 30, normalized size = 1.15 $x-\frac{8}{3} \tanh \left (\frac{x}{2}\right )+\frac{2}{3} \tanh \left (\frac{x}{2}\right ) \text{sech}^2\left (\frac{x}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^(-4),x]

[Out]

x - (8*Tanh[x/2])/3 + (2*Sech[x/2]^2*Tanh[x/2])/3

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Maple [A]  time = 0.06, size = 32, normalized size = 1.2 \begin{align*} -{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tanh \left ( x/2 \right ) -\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))^4,x)

[Out]

-2/3*tanh(1/2*x)^3-2*tanh(1/2*x)-ln(tanh(1/2*x)-1)+ln(tanh(1/2*x)+1)

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Maxima [A]  time = 1.28264, size = 51, normalized size = 1.96 \begin{align*} x - \frac{8 \,{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + 2\right )}}{3 \,{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.97454, size = 234, normalized size = 9. \begin{align*} \frac{3 \, x \cosh \left (x\right )^{2} + 3 \, x \sinh \left (x\right )^{2} + 4 \,{\left (3 \, x + 10\right )} \cosh \left (x\right ) + 2 \,{\left (3 \, x \cosh \left (x\right ) + 3 \, x + 4\right )} \sinh \left (x\right ) + 9 \, x + 24}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x*cosh(x)^2 + 3*x*sinh(x)^2 + 4*(3*x + 10)*cosh(x) + 2*(3*x*cosh(x) + 3*x + 4)*sinh(x) + 9*x + 24)/(cos
h(x)^2 + 2*(cosh(x) + 1)*sinh(x) + sinh(x)^2 + 4*cosh(x) + 3)

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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 )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x))**4,x)

[Out]

Integral((coth(x) + csch(x))**(-4), x)

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Giac [A]  time = 1.11941, size = 30, normalized size = 1.15 \begin{align*} x + \frac{8 \,{\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )}}{3 \,{\left (e^{x} + 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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