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

Optimal. Leaf size=8 \[ \frac{1}{2} \coth ^{-1}(x)^2 \]

[Out]

ArcCoth[x]^2/2

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Rubi [A]  time = 0.0138965, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5949} \[ \frac{1}{2} \coth ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]/(1 - x^2),x]

[Out]

ArcCoth[x]^2/2

Rule 5949

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

Rubi steps

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

Mathematica [A]  time = 0.0039697, size = 8, normalized size = 1. \[ \frac{1}{2} \coth ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x]/(1 - x^2),x]

[Out]

ArcCoth[x]^2/2

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Maple [A]  time = 0.029, size = 13, normalized size = 1.6 \begin{align*}{\it Artanh} \left ( x \right ){\rm arccoth} \left (x\right )-{\frac{ \left ({\it Artanh} \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(x)*arccoth(x)-1/2*arctanh(x)^2

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Maxima [A]  time = 0.953711, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \operatorname{arcoth}\left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arccoth(x)^2

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Fricas [B]  time = 1.55847, size = 38, normalized size = 4.75 \begin{align*} \frac{1}{8} \, \log \left (\frac{x + 1}{x - 1}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log((x + 1)/(x - 1))^2

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Sympy [A]  time = 0.817232, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{acoth}^{2}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

acoth(x)**2/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arccoth(x)/(x^2 - 1), x)