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)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = 0.01, size = 12, normalized size = 1.00 \[ \frac {\coth ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x]^n/(1 - x^2),x]

[Out]

ArcCoth[x]^(1 + n)/(1 + n)

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 62, normalized size = 5.17 \[ \frac {\cosh \left (n \log \left (\frac {1}{2} \, \log \left (\frac {x + 1}{x - 1}\right )\right )\right ) \log \left (\frac {x + 1}{x - 1}\right ) + \log \left (\frac {x + 1}{x - 1}\right ) \sinh \left (n \log \left (\frac {1}{2} \, \log \left (\frac {x + 1}{x - 1}\right )\right )\right )}{2 \, {\left (n + 1\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 1, normalized size = 0.08 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

________________________________________________________________________________________

maple [A]  time = 0.06, size = 13, normalized size = 1.08 \[ \frac {\mathrm {arccoth}\relax (x )^{1+n}}{1+n} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.30, size = 12, normalized size = 1.00 \[ \frac {\operatorname {arcoth}\relax (x)^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccoth(x)^(n + 1)/(n + 1)

________________________________________________________________________________________

mupad [B]  time = 1.36, size = 22, normalized size = 1.83 \[ \left \{\begin {array}{cl} \ln \left (\mathrm {acoth}\relax (x)\right ) & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {acoth}\relax (x)}^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-acoth(x)^n/(x^2 - 1),x)

[Out]

piecewise(n == -1, log(acoth(x)), n ~= -1, acoth(x)^(n + 1)/(n + 1))

________________________________________________________________________________________

sympy [A]  time = 1.72, size = 15, normalized size = 1.25 \[ \begin {cases} \frac {\operatorname {acoth}^{n + 1}{\relax (x )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {acoth}{\relax (x )} \right )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________