3.248 \(\int \frac{\sqrt{\tanh ^{-1}(a x)}}{1-a^2 x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac{2 \tanh ^{-1}(a x)^{3/2}}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.024674, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {5948} \[ \frac{2 \tanh ^{-1}(a x)^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0080679, size = 15, normalized size = 1. \[ \frac{2 \tanh ^{-1}(a x)^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.036, size = 12, normalized size = 0.8 \begin{align*}{\frac{2}{3\,a} \left ({\it Artanh} \left ( ax \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctanh(a*x)^(3/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.42955, size = 63, normalized size = 4.2 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{a x + 1}{a x - 1}\right )^{\frac{3}{2}}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(2)*log(-(a*x + 1)/(a*x - 1))^(3/2)/a

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Sympy [A]  time = 1.76586, size = 14, normalized size = 0.93 \begin{align*} \begin{cases} \frac{2 \operatorname{atanh}^{\frac{3}{2}}{\left (a x \right )}}{3 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*atanh(a*x)**(3/2)/(3*a), Ne(a, 0)), (0, True))

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Giac [B]  time = 1.15293, size = 34, normalized size = 2.27 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{a x + 1}{a x - 1}\right )^{\frac{3}{2}}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(2)*log(-(a*x + 1)/(a*x - 1))^(3/2)/a