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

Optimal. Leaf size=12 \[ \frac{\coth ^{-1}(x)^{n+1}}{n+1} \]

[Out]

ArcCoth[x]^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0263976, antiderivative size = 12, 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 = {5949} \[ \frac{\coth ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]^n/(1 - x^2),x]

[Out]

ArcCoth[x]^(1 + n)/(1 + n)

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

Mathematica [A]  time = 0.0089221, size = 12, normalized size = 1. \[ \frac{\coth ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x]^n/(1 - x^2),x]

[Out]

ArcCoth[x]^(1 + n)/(1 + n)

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Maple [A]  time = 0.059, size = 13, normalized size = 1.1 \begin{align*}{\frac{ \left ({\rm arccoth} \left (x\right ) \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccoth(x)^(1+n)/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.61418, size = 182, normalized size = 15.17 \begin{align*} \frac{\cosh \left (n \log \left (\frac{1}{2} \, \log \left (\frac{x + 1}{x - 1}\right )\right )\right ) \log \left (\frac{x + 1}{x - 1}\right ) + \log \left (\frac{x + 1}{x - 1}\right ) \sinh \left (n \log \left (\frac{1}{2} \, \log \left (\frac{x + 1}{x - 1}\right )\right )\right )}{2 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(n*log(1/2*log((x + 1)/(x - 1))))*log((x + 1)/(x - 1)) + log((x + 1)/(x - 1))*sinh(n*log(1/2*log((x +
1)/(x - 1)))))/(n + 1)

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Sympy [A]  time = 4.82176, size = 15, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\operatorname{acoth}^{n + 1}{\left (x \right )}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (\operatorname{acoth}{\left (x \right )} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((acoth(x)**(n + 1)/(n + 1), Ne(n, -1)), (log(acoth(x)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arccoth(x)^n/(x^2 - 1), x)