3.473 \(\int \frac{\tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=208 \[ \frac{4144 x}{3375 \sqrt{1-a^2 x^2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \]

[Out]

(2*x)/(125*(1 - a^2*x^2)^(5/2)) + (272*x)/(3375*(1 - a^2*x^2)^(3/2)) + (4144*x)/(3375*Sqrt[1 - a^2*x^2]) - (2*
ArcTanh[a*x])/(25*a*(1 - a^2*x^2)^(5/2)) - (8*ArcTanh[a*x])/(45*a*(1 - a^2*x^2)^(3/2)) - (16*ArcTanh[a*x])/(15
*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x]^2)/(15*(1 - a^2*x^2)^(3
/2)) + (8*x*ArcTanh[a*x]^2)/(15*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.152059, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 191, 192} \[ \frac{4144 x}{3375 \sqrt{1-a^2 x^2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2),x]

[Out]

(2*x)/(125*(1 - a^2*x^2)^(5/2)) + (272*x)/(3375*(1 - a^2*x^2)^(3/2)) + (4144*x)/(3375*Sqrt[1 - a^2*x^2]) - (2*
ArcTanh[a*x])/(25*a*(1 - a^2*x^2)^(5/2)) - (8*ArcTanh[a*x])/(45*a*(1 - a^2*x^2)^(3/2)) - (16*ArcTanh[a*x])/(15
*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x]^2)/(15*(1 - a^2*x^2)^(3
/2)) + (8*x*ArcTanh[a*x]^2)/(15*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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{2}{25} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{4}{5} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8}{125} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{45} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{15} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{16}{375} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{135} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{15} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{4144 x}{3375 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0849286, size = 94, normalized size = 0.45 \[ \frac{4144 a^5 x^5-8560 a^3 x^3+225 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \tanh ^{-1}(a x)^2-30 \left (120 a^4 x^4-260 a^2 x^2+149\right ) \tanh ^{-1}(a x)+4470 a x}{3375 a \left (1-a^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2),x]

[Out]

(4470*a*x - 8560*a^3*x^3 + 4144*a^5*x^5 - 30*(149 - 260*a^2*x^2 + 120*a^4*x^4)*ArcTanh[a*x] + 225*a*x*(15 - 20
*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]^2)/(3375*a*(1 - a^2*x^2)^(5/2))

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Maple [A]  time = 0.178, size = 118, normalized size = 0.6 \begin{align*} -{\frac{1800\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}+4144\,{x}^{5}{a}^{5}-3600\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) -4500\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}-8560\,{x}^{3}{a}^{3}+7800\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +3375\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}xa+4470\,ax-4470\,{\it Artanh} \left ( ax \right ) }{3375\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{3}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x)

[Out]

-1/3375/a*(-a^2*x^2+1)^(1/2)*(1800*arctanh(a*x)^2*x^5*a^5+4144*x^5*a^5-3600*a^4*x^4*arctanh(a*x)-4500*arctanh(
a*x)^2*x^3*a^3-8560*x^3*a^3+7800*a^2*x^2*arctanh(a*x)+3375*arctanh(a*x)^2*x*a+4470*a*x-4470*arctanh(a*x))/(a^2
*x^2-1)^3

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Maxima [B]  time = 1.7301, size = 694, normalized size = 3.34 \begin{align*} \frac{1}{15} \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{3 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\right )} \operatorname{artanh}\left (a x\right )^{2} + \frac{1}{3375} \, a{\left (\frac{9 \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} x +{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a}\right )}}{a} + \frac{9 \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} x -{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a}\right )}}{a} + \frac{100 \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x + \sqrt{-a^{2} x^{2} + 1} a}\right )}}{a} + \frac{100 \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x - \sqrt{-a^{2} x^{2} + 1} a}\right )}}{a} - \frac{1800 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac{1800 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac{1800 \, \log \left (a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{1800 \, \log \left (-a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{300 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} + \frac{300 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} - \frac{135 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2}} + \frac{135 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="maxima")

[Out]

1/15*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) + 3*x/(-a^2*x^2 + 1)^(5/2))*arctanh(a*x)^2 + 1/3375*a*
(9*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2*x + (-a^2*x^2 + 1)^(3/2)*a
))/a + 9*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2*x - (-a^2*x^2 + 1)^(
3/2)*a))/a + 100*(2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x + sqrt(-a^2*x^2 + 1)*a))/a + 100*(2*x/s
qrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x - sqrt(-a^2*x^2 + 1)*a))/a - 1800*sqrt(-a^2*x^2 + 1)/((a^2*x +
a)*a) - 1800*sqrt(-a^2*x^2 + 1)/((a^2*x - a)*a) - 1800*log(a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) + 1800*log(-a*x
+ 1)/(sqrt(-a^2*x^2 + 1)*a^2) - 300*log(a*x + 1)/((-a^2*x^2 + 1)^(3/2)*a^2) + 300*log(-a*x + 1)/((-a^2*x^2 + 1
)^(3/2)*a^2) - 135*log(a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2) + 135*log(-a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2))

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Fricas [A]  time = 1.72755, size = 329, normalized size = 1.58 \begin{align*} -\frac{{\left (16576 \, a^{5} x^{5} - 34240 \, a^{3} x^{3} + 225 \,{\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 17880 \, a x - 60 \,{\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{13500 \,{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="fricas")

[Out]

-1/13500*(16576*a^5*x^5 - 34240*a^3*x^3 + 225*(8*a^5*x^5 - 20*a^3*x^3 + 15*a*x)*log(-(a*x + 1)/(a*x - 1))^2 +
17880*a*x - 60*(120*a^4*x^4 - 260*a^2*x^2 + 149)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1)/(a^7*x^6 - 3*a^
5*x^4 + 3*a^3*x^2 - a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**2/(-a**2*x**2+1)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(7/2), x)