3.53 \(\int \frac{1}{(1-x^2) \coth ^{-1}(x)} \, dx\)

Optimal. Leaf size=3 \[ \log \left (\coth ^{-1}(x)\right ) \]

[Out]

Log[ArcCoth[x]]

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Rubi [A]  time = 0.0233692, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5947} \[ \log \left (\coth ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^2)*ArcCoth[x]),x]

[Out]

Log[ArcCoth[x]]

Rule 5947

Int[1/(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[Log[RemoveContent[a + b*A
rcCoth[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0250802, size = 3, normalized size = 1. \[ \log \left (\coth ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x^2)*ArcCoth[x]),x]

[Out]

Log[ArcCoth[x]]

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Maple [A]  time = 0.043, size = 4, normalized size = 1.3 \begin{align*} \ln \left ({\rm arccoth} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+1)/arccoth(x),x)

[Out]

ln(arccoth(x))

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Maxima [A]  time = 0.975901, size = 4, normalized size = 1.33 \begin{align*} \log \left (\operatorname{arcoth}\left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)/arccoth(x),x, algorithm="maxima")

[Out]

log(arccoth(x))

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Fricas [B]  time = 1.50668, size = 36, normalized size = 12. \begin{align*} \log \left (\log \left (\frac{x + 1}{x - 1}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)/arccoth(x),x, algorithm="fricas")

[Out]

log(log((x + 1)/(x - 1)))

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Sympy [A]  time = 0.411158, size = 3, normalized size = 1. \begin{align*} \log{\left (\operatorname{acoth}{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+1)/acoth(x),x)

[Out]

log(acoth(x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (x^{2} - 1\right )} \operatorname{arcoth}\left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)/arccoth(x),x, algorithm="giac")

[Out]

integrate(-1/((x^2 - 1)*arccoth(x)), x)