3.478 \(\int \frac {\tanh ^{-1}(a x)^3}{(1-a^2 x^2)^{9/2}} \, dx\)

Optimal. Leaf size=385 \[ -\frac {413312}{128625 a \sqrt {1-a^2 x^2}}-\frac {30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)^3}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {48 \tanh ^{-1}(a x)^2}{35 a \sqrt {1-a^2 x^2}}-\frac {8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {413312 x \tanh ^{-1}(a x)}{128625 \sqrt {1-a^2 x^2}}+\frac {30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac {2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}} \]

[Out]

-6/2401/a/(-a^2*x^2+1)^(7/2)-2664/214375/a/(-a^2*x^2+1)^(5/2)-30256/385875/a/(-a^2*x^2+1)^(3/2)+6/343*x*arctan
h(a*x)/(-a^2*x^2+1)^(7/2)+2664/42875*x*arctanh(a*x)/(-a^2*x^2+1)^(5/2)+30256/128625*x*arctanh(a*x)/(-a^2*x^2+1
)^(3/2)-3/49*arctanh(a*x)^2/a/(-a^2*x^2+1)^(7/2)-18/175*arctanh(a*x)^2/a/(-a^2*x^2+1)^(5/2)-8/35*arctanh(a*x)^
2/a/(-a^2*x^2+1)^(3/2)+1/7*x*arctanh(a*x)^3/(-a^2*x^2+1)^(7/2)+6/35*x*arctanh(a*x)^3/(-a^2*x^2+1)^(5/2)+8/35*x
*arctanh(a*x)^3/(-a^2*x^2+1)^(3/2)-413312/128625/a/(-a^2*x^2+1)^(1/2)+413312/128625*x*arctanh(a*x)/(-a^2*x^2+1
)^(1/2)-48/35*arctanh(a*x)^2/a/(-a^2*x^2+1)^(1/2)+16/35*x*arctanh(a*x)^3/(-a^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.48, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5964, 5962, 5958, 5960} \[ -\frac {413312}{128625 a \sqrt {1-a^2 x^2}}-\frac {30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)^3}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {48 \tanh ^{-1}(a x)^2}{35 a \sqrt {1-a^2 x^2}}-\frac {8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {413312 x \tanh ^{-1}(a x)}{128625 \sqrt {1-a^2 x^2}}+\frac {30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac {2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2),x]

[Out]

-6/(2401*a*(1 - a^2*x^2)^(7/2)) - 2664/(214375*a*(1 - a^2*x^2)^(5/2)) - 30256/(385875*a*(1 - a^2*x^2)^(3/2)) -
413312/(128625*a*Sqrt[1 - a^2*x^2]) + (6*x*ArcTanh[a*x])/(343*(1 - a^2*x^2)^(7/2)) + (2664*x*ArcTanh[a*x])/(4
2875*(1 - a^2*x^2)^(5/2)) + (30256*x*ArcTanh[a*x])/(128625*(1 - a^2*x^2)^(3/2)) + (413312*x*ArcTanh[a*x])/(128
625*Sqrt[1 - a^2*x^2]) - (3*ArcTanh[a*x]^2)/(49*a*(1 - a^2*x^2)^(7/2)) - (18*ArcTanh[a*x]^2)/(175*a*(1 - a^2*x
^2)^(5/2)) - (8*ArcTanh[a*x]^2)/(35*a*(1 - a^2*x^2)^(3/2)) - (48*ArcTanh[a*x]^2)/(35*a*Sqrt[1 - a^2*x^2]) + (x
*ArcTanh[a*x]^3)/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x]^3)/(35*(1 - a^2*x^2)^(5/2)) + (8*x*ArcTanh[a*x]^3
)/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x]^3)/(35*Sqrt[1 - a^2*x^2])

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]

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 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]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx &=-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx+\frac {6}{7} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {36}{343} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {36}{175} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {24}{35} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {144 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1715}+\frac {144}{875} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {16}{35} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {16}{35} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac {30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {48 \tanh ^{-1}(a x)^2}{35 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \tanh ^{-1}(a x)^3}{35 \sqrt {1-a^2 x^2}}+\frac {96 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{1715}+\frac {96}{875} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {32}{105} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {96}{35} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac {30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac {413312}{128625 a \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac {413312 x \tanh ^{-1}(a x)}{128625 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {48 \tanh ^{-1}(a x)^2}{35 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \tanh ^{-1}(a x)^3}{35 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 151, normalized size = 0.39 \[ \frac {43397760 a^6 x^6-131252240 a^4 x^4+132479032 a^2 x^2-385875 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)^3-210 a x \left (206656 a^6 x^6-635096 a^4 x^4+654220 a^2 x^2-226905\right ) \tanh ^{-1}(a x)+11025 \left (1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right ) \tanh ^{-1}(a x)^2-44658302}{13505625 a \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2),x]

[Out]

(-44658302 + 132479032*a^2*x^2 - 131252240*a^4*x^4 + 43397760*a^6*x^6 - 210*a*x*(-226905 + 654220*a^2*x^2 - 63
5096*a^4*x^4 + 206656*a^6*x^6)*ArcTanh[a*x] + 11025*(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6)*ArcTa
nh[a*x]^2 - 385875*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x]^3)/(13505625*a*(1 - a^2*x^2)^
(7/2))

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fricas [A]  time = 0.52, size = 214, normalized size = 0.56 \[ \frac {{\left (347182080 \, a^{6} x^{6} - 1050017920 \, a^{4} x^{4} + 1059832256 \, a^{2} x^{2} - 385875 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 22050 \, {\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 840 \, {\left (206656 \, a^{7} x^{7} - 635096 \, a^{5} x^{5} + 654220 \, a^{3} x^{3} - 226905 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 357266416\right )} \sqrt {-a^{2} x^{2} + 1}}{108045000 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="fricas")

[Out]

1/108045000*(347182080*a^6*x^6 - 1050017920*a^4*x^4 + 1059832256*a^2*x^2 - 385875*(16*a^7*x^7 - 56*a^5*x^5 + 7
0*a^3*x^3 - 35*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 22050*(1680*a^6*x^6 - 5320*a^4*x^4 + 5726*a^2*x^2 - 2161)*lo
g(-(a*x + 1)/(a*x - 1))^2 - 840*(206656*a^7*x^7 - 635096*a^5*x^5 + 654220*a^3*x^3 - 226905*a*x)*log(-(a*x + 1)
/(a*x - 1)) - 357266416)*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(9/2), x)

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maple [A]  time = 0.57, size = 201, normalized size = 0.52 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (6174000 \arctanh \left (a x \right )^{3} x^{7} a^{7}[phone] \arctanh \left (a x \right ) x^{7} a^{7}-18522000 \arctanh \left (a x \right )^{2} x^{6} a^{6}-21609000 \arctanh \left (a x \right )^{3} x^{5} a^{5}-43397760 x^{6} a^{6}-133370160 \arctanh \left (a x \right ) x^{5} a^{5}[phone] a^{4} x^{4} \arctanh \left (a x \right )^{2}[phone] \arctanh \left (a x \right )^{3} x^{3} a^{3}[phone] x^{4} a^{4}[phone] a^{3} x^{3} \arctanh \left (a x \right )-63129150 a^{2} x^{2} \arctanh \left (a x \right )^{2}-13505625 \arctanh \left (a x \right )^{3} a x -132479032 a^{2} x^{2}-47650050 a x \arctanh \left (a x \right )[phone] \arctanh \left (a x \right )^{2}[phone]\right )}{13505625 a \left (a^{2} x^{2}-1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x)

[Out]

-1/13505625/a*(-a^2*x^2+1)^(1/2)*(6174000*arctanh(a*x)^3*x^7*a^7+43397760*arctanh(a*x)*x^7*a^7-18522000*arctan
h(a*x)^2*x^6*a^6-21609000*arctanh(a*x)^3*x^5*a^5-43397760*x^6*a^6-133370160*arctanh(a*x)*x^5*a^5+58653000*a^4*
x^4*arctanh(a*x)^2+27011250*arctanh(a*x)^3*x^3*a^3+131252240*x^4*a^4+137386200*a^3*x^3*arctanh(a*x)-63129150*a
^2*x^2*arctanh(a*x)^2-13505625*arctanh(a*x)^3*a*x-132479032*a^2*x^2-47650050*a*x*arctanh(a*x)[phone]*arctanh
(a*x)^2+44658302)/(a^2*x^2-1)^4

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="maxima")

[Out]

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^3/(1 - a^2*x^2)^(9/2),x)

[Out]

int(atanh(a*x)^3/(1 - a^2*x^2)^(9/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**3/(-a**2*x**2+1)**(9/2),x)

[Out]

Integral(atanh(a*x)**3/(-(a*x - 1)*(a*x + 1))**(9/2), x)

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