3.476 \(\int \frac{\tanh ^{-1}(a x)^3}{(1-a^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{40}{9 a \sqrt{1-a^2 x^2}}-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

-2/(27*a*(1 - a^2*x^2)^(3/2)) - 40/(9*a*Sqrt[1 - a^2*x^2]) + (2*x*ArcTanh[a*x])/(9*(1 - a^2*x^2)^(3/2)) + (40*
x*ArcTanh[a*x])/(9*Sqrt[1 - a^2*x^2]) - ArcTanh[a*x]^2/(3*a*(1 - a^2*x^2)^(3/2)) - (2*ArcTanh[a*x]^2)/(a*Sqrt[
1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(3*(1 - a^2*x^2)^(3/2)) + (2*x*ArcTanh[a*x]^3)/(3*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.164848, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 5958, 5960} \[ -\frac{40}{9 a \sqrt{1-a^2 x^2}}-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^(5/2),x]

[Out]

-2/(27*a*(1 - a^2*x^2)^(3/2)) - 40/(9*a*Sqrt[1 - a^2*x^2]) + (2*x*ArcTanh[a*x])/(9*(1 - a^2*x^2)^(3/2)) + (40*
x*ArcTanh[a*x])/(9*Sqrt[1 - a^2*x^2]) - ArcTanh[a*x]^2/(3*a*(1 - a^2*x^2)^(3/2)) - (2*ArcTanh[a*x]^2)/(a*Sqrt[
1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(3*(1 - a^2*x^2)^(3/2)) + (2*x*ArcTanh[a*x]^3)/(3*Sqrt[1 - a^2*x^2])

Rule 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
+ 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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
]

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{4}{9} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+4 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}-\frac{40}{9 a \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0927984, size = 87, normalized size = 0.46 \[ \frac{120 a^2 x^2-9 a x \left (2 a^2 x^2-3\right ) \tanh ^{-1}(a x)^3+9 \left (6 a^2 x^2-7\right ) \tanh ^{-1}(a x)^2-6 a x \left (20 a^2 x^2-21\right ) \tanh ^{-1}(a x)-122}{27 a \left (1-a^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(5/2),x]

[Out]

(-122 + 120*a^2*x^2 - 6*a*x*(-21 + 20*a^2*x^2)*ArcTanh[a*x] + 9*(-7 + 6*a^2*x^2)*ArcTanh[a*x]^2 - 9*a*x*(-3 +
2*a^2*x^2)*ArcTanh[a*x]^3)/(27*a*(1 - a^2*x^2)^(3/2))

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Maple [A]  time = 0.178, size = 105, normalized size = 0.6 \begin{align*} -{\frac{18\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{3}{a}^{3}+120\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -54\,{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-27\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}ax-120\,{a}^{2}{x}^{2}-126\,ax{\it Artanh} \left ( ax \right ) +63\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+122}{27\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{2}}\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)^(5/2),x)

[Out]

-1/27/a*(-a^2*x^2+1)^(1/2)*(18*arctanh(a*x)^3*x^3*a^3+120*a^3*x^3*arctanh(a*x)-54*a^2*x^2*arctanh(a*x)^2-27*ar
ctanh(a*x)^3*a*x-120*a^2*x^2-126*a*x*arctanh(a*x)+63*arctanh(a*x)^2+122)/(a^2*x^2-1)^2

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Maxima [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{5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(5/2), x)

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Fricas [A]  time = 1.6996, size = 305, normalized size = 1.6 \begin{align*} \frac{{\left (960 \, a^{2} x^{2} - 9 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 18 \,{\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 24 \,{\left (20 \, a^{3} x^{3} - 21 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 976\right )} \sqrt{-a^{2} x^{2} + 1}}{216 \,{\left (a^{5} x^{4} - 2 \, 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)^(5/2),x, algorithm="fricas")

[Out]

1/216*(960*a^2*x^2 - 9*(2*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 18*(6*a^2*x^2 - 7)*log(-(a*x + 1)/(a*
x - 1))^2 - 24*(20*a^3*x^3 - 21*a*x)*log(-(a*x + 1)/(a*x - 1)) - 976)*sqrt(-a^2*x^2 + 1)/(a^5*x^4 - 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}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{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)**(5/2),x)

[Out]

Integral(atanh(a*x)**3/(-(a*x - 1)*(a*x + 1))**(5/2), 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{5}{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)^(5/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(5/2), x)