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

Optimal. Leaf size=88 \[ -\frac{6}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \]

[Out]

-6/(a*Sqrt[1 - a^2*x^2]) + (6*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (
x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2]

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Rubi [A]  time = 0.0707551, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5962, 5958} \[ -\frac{6}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(a*Sqrt[1 - a^2*x^2]) + (6*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (
x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2]

Rule 5962

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[(b*p*(a + b*ArcTa
nh[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2
)^(3/2), x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ
[c^2*d + e, 0] && GtQ[p, 1]

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)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac{3 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+6 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0503811, size = 45, normalized size = 0.51 \[ \frac{a x \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2+6 a x \tanh ^{-1}(a x)-6}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6 + 6*a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + a*x*ArcTanh[a*x]^3)/(a*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.224, size = 56, normalized size = 0.6 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}ax+6\,ax{\it Artanh} \left ( ax \right ) -3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-6}{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)^3/(-a^2*x^2+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.968363, size = 116, normalized size = 1.32 \begin{align*} \frac{x \operatorname{artanh}\left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1}} + 6 \, a{\left (\frac{x \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2}}\right )} - \frac{3 \, \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.24759, size = 196, normalized size = 2.23 \begin{align*} -\frac{{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 24 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 6 \, \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 48\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a*x*log(-(a*x + 1)/(a*x - 1))^3 + 24*a*x*log(-(a*x + 1)/(a*x - 1)) - 6*log(-(a*x + 1)/(a*x - 1))^2 - 48)
*sqrt(-a^2*x^2 + 1)/(a^3*x^2 - a)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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