3.474 \(\int \frac{\tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{9/2}} \, dx\)

Optimal. Leaf size=277 \[ \frac{413312 x}{385875 \sqrt{1-a^2 x^2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]

[Out]

(2*x)/(343*(1 - a^2*x^2)^(7/2)) + (888*x)/(42875*(1 - a^2*x^2)^(5/2)) + (30256*x)/(385875*(1 - a^2*x^2)^(3/2))
+ (413312*x)/(385875*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(49*a*(1 - a^2*x^2)^(7/2)) - (12*ArcTanh[a*x])/(17
5*a*(1 - a^2*x^2)^(5/2)) - (16*ArcTanh[a*x])/(105*a*(1 - a^2*x^2)^(3/2)) - (32*ArcTanh[a*x])/(35*a*Sqrt[1 - a^
2*x^2]) + (x*ArcTanh[a*x]^2)/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x]^2)/(35*(1 - a^2*x^2)^(5/2)) + (8*x*Ar
cTanh[a*x]^2)/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x]^2)/(35*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.228388, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 191, 192} \[ \frac{413312 x}{385875 \sqrt{1-a^2 x^2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2),x]

[Out]

(2*x)/(343*(1 - a^2*x^2)^(7/2)) + (888*x)/(42875*(1 - a^2*x^2)^(5/2)) + (30256*x)/(385875*(1 - a^2*x^2)^(3/2))
+ (413312*x)/(385875*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(49*a*(1 - a^2*x^2)^(7/2)) - (12*ArcTanh[a*x])/(17
5*a*(1 - a^2*x^2)^(5/2)) - (16*ArcTanh[a*x])/(105*a*(1 - a^2*x^2)^(3/2)) - (32*ArcTanh[a*x])/(35*a*Sqrt[1 - a^
2*x^2]) + (x*ArcTanh[a*x]^2)/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x]^2)/(35*(1 - a^2*x^2)^(5/2)) + (8*x*Ar
cTanh[a*x]^2)/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x]^2)/(35*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 )^{9/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{2}{49} \int \frac{1}{\left (1-a^2 x^2\right )^{9/2}} \, dx+\frac{6}{7} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{12}{343} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{12}{175} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{24}{35} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{48 \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1715}+\frac{48}{875} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{105} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{35} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{32 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{1715}+\frac{32}{875} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{32}{315} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{32}{35} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{413312 x}{385875 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.110049, size = 120, normalized size = 0.43 \[ \frac{2 a x \left (-206656 a^6 x^6+635096 a^4 x^4-654220 a^2 x^2+226905\right )-11025 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)^2+210 \left (1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right ) \tanh ^{-1}(a x)}{385875 a \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2),x]

[Out]

(2*a*x*(226905 - 654220*a^2*x^2 + 635096*a^4*x^4 - 206656*a^6*x^6) + 210*(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4
+ 1680*a^6*x^6)*ArcTanh[a*x] - 11025*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x]^2)/(385875*
a*(1 - a^2*x^2)^(7/2))

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Maple [A]  time = 0.191, size = 152, normalized size = 0.6 \begin{align*} -{\frac{176400\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{7}{a}^{7}+413312\,{a}^{7}{x}^{7}-352800\,{\it Artanh} \left ( ax \right ){x}^{6}{a}^{6}-617400\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}-1270192\,{x}^{5}{a}^{5}+1117200\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) +771750\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+1308440\,{x}^{3}{a}^{3}-1202460\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -385875\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}xa-453810\,ax+453810\,{\it Artanh} \left ( ax \right ) }{385875\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{4}}\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)^(9/2),x)

[Out]

-1/385875/a*(-a^2*x^2+1)^(1/2)*(176400*arctanh(a*x)^2*x^7*a^7+413312*a^7*x^7-352800*arctanh(a*x)*x^6*a^6-61740
0*arctanh(a*x)^2*x^5*a^5-1270192*x^5*a^5+1117200*a^4*x^4*arctanh(a*x)+771750*arctanh(a*x)^2*x^3*a^3+1308440*x^
3*a^3-1202460*a^2*x^2*arctanh(a*x)-385875*arctanh(a*x)^2*x*a-453810*a*x+453810*arctanh(a*x))/(a^2*x^2-1)^4

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Maxima [B]  time = 1.79396, size = 1014, normalized size = 3.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(9/2),x, algorithm="maxima")

[Out]

1/35*(16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) + 6*x/(-a^2*x^2 + 1)^(5/2) + 5*x/(-a^2*x^2 + 1)^(7/2)
)*arctanh(a*x)^2 + 1/385875*a*(225*(16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) - 5/((-a^2*x^2 + 1)^(5/
2)*a^2*x + (-a^2*x^2 + 1)^(5/2)*a) + 6*x/(-a^2*x^2 + 1)^(5/2))/a + 225*(16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^
2 + 1)^(3/2) - 5/((-a^2*x^2 + 1)^(5/2)*a^2*x - (-a^2*x^2 + 1)^(5/2)*a) + 6*x/(-a^2*x^2 + 1)^(5/2))/a + 882*(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 +
882*(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 + 9800*(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 + 9800*(2*x/sqr
t(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x - sqrt(-a^2*x^2 + 1)*a))/a - 176400*sqrt(-a^2*x^2 + 1)/((a^2*x +
a)*a) - 176400*sqrt(-a^2*x^2 + 1)/((a^2*x - a)*a) - 176400*log(a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) + 176400*log
(-a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) - 29400*log(a*x + 1)/((-a^2*x^2 + 1)^(3/2)*a^2) + 29400*log(-a*x + 1)/((-a
^2*x^2 + 1)^(3/2)*a^2) - 13230*log(a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2) + 13230*log(-a*x + 1)/((-a^2*x^2 + 1)^(
5/2)*a^2) - 7875*log(a*x + 1)/((-a^2*x^2 + 1)^(7/2)*a^2) + 7875*log(-a*x + 1)/((-a^2*x^2 + 1)^(7/2)*a^2))

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Fricas [A]  time = 1.70749, size = 428, normalized size = 1.55 \begin{align*} -\frac{{\left (1653248 \, a^{7} x^{7} - 5080768 \, a^{5} x^{5} + 5233760 \, a^{3} x^{3} + 11025 \,{\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 1815240 \, a x - 420 \,{\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{1543500 \,{\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, 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)^(9/2),x, algorithm="fricas")

[Out]

-1/1543500*(1653248*a^7*x^7 - 5080768*a^5*x^5 + 5233760*a^3*x^3 + 11025*(16*a^7*x^7 - 56*a^5*x^5 + 70*a^3*x^3
- 35*a*x)*log(-(a*x + 1)/(a*x - 1))^2 - 1815240*a*x - 420*(1680*a^6*x^6 - 5320*a^4*x^4 + 5726*a^2*x^2 - 2161)*
log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1)/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*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)**(9/2),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{9}{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)^(9/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(9/2), x)