3.292 \(\int \frac{1}{(a-a x^2) (b-2 b \coth ^{-1}(x))} \, dx\)

Optimal. Leaf size=17 \[ -\frac{\log \left (1-2 \coth ^{-1}(x)\right )}{2 a b} \]

[Out]

-Log[1 - 2*ArcCoth[x]]/(2*a*b)

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Rubi [A]  time = 0.0443519, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5947} \[ -\frac{\log \left (1-2 \coth ^{-1}(x)\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - a*x^2)*(b - 2*b*ArcCoth[x])),x]

[Out]

-Log[1 - 2*ArcCoth[x]]/(2*a*b)

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

Mathematica [A]  time = 0.0524332, size = 17, normalized size = 1. \[ -\frac{\log \left (2 \coth ^{-1}(x)-1\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - a*x^2)*(b - 2*b*ArcCoth[x])),x]

[Out]

-Log[-1 + 2*ArcCoth[x]]/(2*a*b)

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Maple [A]  time = 0.072, size = 19, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( 2\,b{\rm arccoth} \left (x\right )-b \right ) }{2\,ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a*ln(2*b*arccoth(x)-b)/b

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Maxima [A]  time = 1.12692, size = 28, normalized size = 1.65 \begin{align*} -\frac{\log \left (\log \left (x + 1\right ) - \log \left (x - 1\right ) - 1\right )}{2 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(log(x + 1) - log(x - 1) - 1)/(a*b)

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Fricas [A]  time = 1.64775, size = 57, normalized size = 3.35 \begin{align*} -\frac{\log \left (\log \left (\frac{x + 1}{x - 1}\right ) - 1\right )}{2 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(log((x + 1)/(x - 1)) - 1)/(a*b)

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Sympy [A]  time = 1.31523, size = 14, normalized size = 0.82 \begin{align*} - \frac{\log{\left (\operatorname{acoth}{\left (x \right )} - \frac{1}{2} \right )}}{2 a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(acoth(x) - 1/2)/(2*a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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