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

Optimal. Leaf size=40 \[ \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{1}{a \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0252848, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5958} \[ \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{1}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(1 - a^2*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0312152, size = 27, normalized size = 0.68 \[ \frac{a x \tanh ^{-1}(a x)-1}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.22, size = 38, normalized size = 1. \begin{align*} -{\frac{ax{\it Artanh} \left ( ax \right ) -1}{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)/(-a^2*x^2+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.957387, size = 49, normalized size = 1.22 \begin{align*} \frac{x \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.98146, size = 101, normalized size = 2.52 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 2\right )}}{2 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(a*x*log(-(a*x + 1)/(a*x - 1)) - 2)/(a^3*x^2 - a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22904, size = 80, normalized size = 2. \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \,{\left (a^{2} x^{2} - 1\right )}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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