3.1.53 \(\int \frac {1}{(1-x^2) \coth ^{-1}(x)} \, dx\) [53]

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

[Out]

ln(arccoth(x))

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Rubi [A]
time = 0.02, antiderivative size = 3, 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 = {6094} \begin {gather*} \log \left (\coth ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^2)*ArcCoth[x]),x]

[Out]

Log[ArcCoth[x]]

Rule 6094

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*}

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Mathematica [A]
time = 0.02, size = 3, normalized size = 1.00 \begin {gather*} \log \left (\coth ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x^2)*ArcCoth[x]),x]

[Out]

Log[ArcCoth[x]]

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Maple [A]
time = 0.09, size = 4, normalized size = 1.33

method	result	size
default	\(\ln \left (\mathrm {arccoth}\left (x \right )\right )\)	\(4\)
risch	\(\ln \left (\ln \left (1+x \right )-\ln \left (-1+x \right )\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(arccoth(x))

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Maxima [A]
time = 0.25, size = 3, normalized size = 1.00 \begin {gather*} \log \left (\operatorname {arcoth}\left (x\right )\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (3) = 6\).
time = 0.34, size = 11, normalized size = 3.67 \begin {gather*} \log \left (\log \left (\frac {x + 1}{x - 1}\right )\right ) \end {gather*}

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 = 0.12, size = 3, normalized size = 1.00 \begin {gather*} \log {\left (\operatorname {acoth}{\left (x \right )} \right )} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
time = 0.39, size = 12, normalized size = 4.00 \begin {gather*} \log \left ({\left | \log \left (\frac {x + 1}{x - 1}\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(log((x + 1)/(x - 1))))

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Mupad [B]
time = 0.30, size = 3, normalized size = 1.00 \begin {gather*} \ln \left (\mathrm {acoth}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(acoth(x)*(x^2 - 1)),x)

[Out]

log(acoth(x))

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