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 = 0.02, antiderivative size = 1, normalized size of antiderivative = 1.00, 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 8

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 1, normalized size = 1.00 \[ x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x

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fricas [A]  time = 0.38, size = 1, normalized size = 1.00 \[ x \]

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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giac [A]  time = 0.11, size = 1, normalized size = 1.00 \[ x \]

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

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maple [C]  time = 0.12, size = 8, normalized size = 8.00 \[ 2 \arctanh \left (\tanh \left (\frac {x}{2}\right )\right ) \]

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 = 0.32, size = 1, normalized size = 1.00 \[ x \]

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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mupad [B]  time = 1.54, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)^2 - 1/sinh(x)^2)^3,x)

[Out]

x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\coth {\relax (x )} - \operatorname {csch}{\relax (x )}\right )^{3} \left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((coth(x) - csch(x))**3*(coth(x) + csch(x))**3), x)

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