### 3.825 $$\int \frac{1}{(\coth ^2(x)-\text{csch}^2(x))^3} \, dx$$

Optimal. Leaf size=1 $x$

[Out]

x

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Rubi [A]  time = 0.0160287, antiderivative size = 1, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {4382, 8} $x$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x]^2 - Csch[x]^2)^(-3),x]

[Out]

x

Rule 4382

Int[((a_.) + cot[(d_.) + (e_.)*(x_)]^2*(b_.) + csc[(d_.) + (e_.)*(x_)]^2*(c_.))^(p_.)*(u_.), x_Symbol] :> Dist
[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{\left (\coth ^2(x)-\text{csch}^2(x)\right )^3} \, dx &=\int 1 \, dx\\ &=x\\ \end{align*}

Mathematica [A]  time = 0.0004154, size = 1, normalized size = 1. $x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x]^2 - Csch[x]^2)^(-3),x]

[Out]

x

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Maple [C]  time = 0.027, size = 8, normalized size = 8. \begin{align*} 2\,{\it Artanh} \left ( \tanh \left ( x/2 \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)^2-csch(x)^2)^3,x)

[Out]

2*arctanh(tanh(1/2*x))

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Maxima [A]  time = 1.10683, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)^2-csch(x)^2)^3,x, algorithm="maxima")

[Out]

x

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Fricas [A]  time = 1.8898, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)^2-csch(x)^2)^3,x, algorithm="fricas")

[Out]

x

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)**2-csch(x)**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.1213, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)^2-csch(x)^2)^3,x, algorithm="giac")

[Out]

x