3.50 \(\int \frac{\coth ^{-1}(x)}{(a-a x^2)^{3/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}} \]

[Out]

-(1/(a*Sqrt[a - a*x^2])) + (x*ArcCoth[x])/(a*Sqrt[a - a*x^2])

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Rubi [A]  time = 0.0254153, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5959} \[ \frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x]/(a - a*x^2)^(3/2),x]

[Out]

-(1/(a*Sqrt[a - a*x^2])) + (x*ArcCoth[x])/(a*Sqrt[a - a*x^2])

Rule 5959

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcCoth[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx &=-\frac{1}{a \sqrt{a-a x^2}}+\frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0438355, size = 30, normalized size = 0.81 \[ \frac{\sqrt{a-a x^2} \left (1-x \coth ^{-1}(x)\right )}{a^2 \left (x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x]/(a - a*x^2)^(3/2),x]

[Out]

(Sqrt[a - a*x^2]*(1 - x*ArcCoth[x]))/(a^2*(-1 + x^2))

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Maple [A]  time = 0.226, size = 52, normalized size = 1.4 \begin{align*} -{\frac{{\rm arccoth} \left (x\right )-1}{ \left ( -2+2\,x \right ){a}^{2}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{{\rm arccoth} \left (x\right )+1}{ \left ( 2+2\,x \right ){a}^{2}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(x)/(-a*x^2+a)^(3/2),x)

[Out]

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

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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)/(-a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5362, size = 92, normalized size = 2.49 \begin{align*} -\frac{\sqrt{-a x^{2} + a}{\left (x \log \left (\frac{x + 1}{x - 1}\right ) - 2\right )}}{2 \,{\left (a^{2} x^{2} - a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(x)/(-a*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-a*x^2 + a)*(x*log((x + 1)/(x - 1)) - 2)/(a^2*x^2 - a^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(x)/(-a*x**2+a)**(3/2),x)

[Out]

Integral(acoth(x)/(-a*(x - 1)*(x + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (x\right )}{{\left (-a x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(x)/(-a*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate(arccoth(x)/(-a*x^2 + a)^(3/2), x)