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

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

[Out]

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

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Rubi [A]  time = 0.0245407, 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 = {5958} \[ \frac{x \tanh ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[x]/(a - a*x^2)^(3/2),x]

[Out]

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

Rule 5958

Int[((a_.) + ArcTanh[(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*ArcTanh[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{\tanh ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx &=-\frac{1}{a \sqrt{a-a x^2}}+\frac{x \tanh ^{-1}(x)}{a \sqrt{a-a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0420467, size = 30, normalized size = 0.81 \[ \frac{\sqrt{a-a x^2} \left (1-x \tanh ^{-1}(x)\right )}{a^2 \left (x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.217, size = 52, normalized size = 1.4 \begin{align*} -{\frac{{\it Artanh} \left ( x \right ) -1}{ \left ( -2+2\,x \right ){a}^{2}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{1+{\it Artanh} \left ( x \right ) }{ \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(arctanh(x)/(-a*x^2+a)^(3/2),x)

[Out]

-1/2*(arctanh(x)-1)*(-(-1+x)*(1+x)*a)^(1/2)/(-1+x)/a^2-1/2*(1+arctanh(x))*(-(-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(arctanh(x)/(-a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01893, size = 93, normalized size = 2.51 \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(arctanh(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{atanh}{\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(atanh(x)/(-a*x**2+a)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.19783, size = 73, normalized size = 1.97 \begin{align*} -\frac{\sqrt{-a x^{2} + a} x \log \left (-\frac{x + 1}{x - 1}\right )}{2 \,{\left (a x^{2} - a\right )} a} - \frac{1}{\sqrt{-a x^{2} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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