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

Optimal. Leaf size=203 \[ -\frac {45}{128 a \left (1-a^2 x^2\right )}-\frac {3}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {45 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}+\frac {45 \tanh ^{-1}(a x)^2}{128 a} \]

[Out]

-3/128/a/(-a^2*x^2+1)^2-45/128/a/(-a^2*x^2+1)+3/32*x*arctanh(a*x)/(-a^2*x^2+1)^2+45/64*x*arctanh(a*x)/(-a^2*x^
2+1)+45/128*arctanh(a*x)^2/a-3/16*arctanh(a*x)^2/a/(-a^2*x^2+1)^2-9/16*arctanh(a*x)^2/a/(-a^2*x^2+1)+1/4*x*arc
tanh(a*x)^3/(-a^2*x^2+1)^2+3/8*x*arctanh(a*x)^3/(-a^2*x^2+1)+3/32*arctanh(a*x)^4/a

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5964, 5956, 5994, 261, 5960} \[ -\frac {45}{128 a \left (1-a^2 x^2\right )}-\frac {3}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {45 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}+\frac {45 \tanh ^{-1}(a x)^2}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^3,x]

[Out]

-3/(128*a*(1 - a^2*x^2)^2) - 45/(128*a*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (45*x*ArcTan
h[a*x])/(64*(1 - a^2*x^2)) + (45*ArcTanh[a*x]^2)/(128*a) - (3*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)^2) - (9*ArcT
anh[a*x]^2)/(16*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x]^3)/(8*(1 - a^2*x
^2)) + (3*ArcTanh[a*x]^4)/(32*a)

Rule 261

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

Rule 5956

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTanh[c*x
])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + S
imp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] &&
GtQ[p, 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 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 5994

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

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{8} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {3}{4} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}+\frac {9}{32} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{8} (9 a) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {9 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {9 \tanh ^{-1}(a x)^2}{128 a}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}+\frac {9}{8} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{64} (9 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3}{128 a \left (1-a^2 x^2\right )^2}-\frac {9}{128 a \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {45 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {45 \tanh ^{-1}(a x)^2}{128 a}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}-\frac {1}{16} (9 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3}{128 a \left (1-a^2 x^2\right )^2}-\frac {45}{128 a \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {45 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {45 \tanh ^{-1}(a x)^2}{128 a}-\frac {3 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^4}{32 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 111, normalized size = 0.55 \[ \frac {\left (80 a x-48 a^3 x^3\right ) \tanh ^{-1}(a x)^3+\left (102 a x-90 a^3 x^3\right ) \tanh ^{-1}(a x)+45 a^2 x^2+12 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4+3 \left (15 a^4 x^4-6 a^2 x^2-17\right ) \tanh ^{-1}(a x)^2-48}{128 a \left (a^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48 + 45*a^2*x^2 + (102*a*x - 90*a^3*x^3)*ArcTanh[a*x] + 3*(-17 - 6*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^2 + (8
0*a*x - 48*a^3*x^3)*ArcTanh[a*x]^3 + 12*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)/(128*a*(-1 + a^2*x^2)^2)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 166, normalized size = 0.82 \[ \frac {3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 180 \, a^{2} x^{2} - 8 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 3 \, {\left (15 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 17\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (15 \, a^{3} x^{3} - 17 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 192}{512 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*(3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^4 + 180*a^2*x^2 - 8*(3*a^3*x^3 - 5*a*x)*log(-(a*x
+ 1)/(a*x - 1))^3 + 3*(15*a^4*x^4 - 6*a^2*x^2 - 17)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(15*a^3*x^3 - 17*a*x)*lo
g(-(a*x + 1)/(a*x - 1)) - 192)/(a^5*x^4 - 2*a^3*x^2 + a)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.92, size = 2646, normalized size = 13.03 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^
(1/2))*x^4+3/32*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^
2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^4-3/32*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(
a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x^4+3/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+
(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^2+3/8*I*a/(a*x-1)^2/(a*x
+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*x^2-3/16*I*a/(a*x-1
)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*
x^2+1)))^2*x^2+3/32*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^4*x^4+45/128*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*x
^4-195/1024/a/(a*x-1)^2/(a*x+1)^2+1/16/a*arctanh(a*x)^3/(a*x-1)^2-1/16/a*arctanh(a*x)^3/(a*x+1)^2+3/16*I*a^3/(
a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^4-3/32*I*a^3/(a*x-1)^2/(a*x+1)^2*a
rctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^4-3/32*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(
a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^4-3/16*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I
/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^4-3/8*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2
+1)))^3*x^2+3/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^2+3/16*I*a/(a*x-1
)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^2+3/8*I*a/(a*x-1)
^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^2-3/32*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctan
h(a*x)^3*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-3/16*I/
a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+3/3
2*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x
+1)^2/(-a^2*x^2+1)))^2-3/32*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csg
n(I*(a*x+1)^2/(a^2*x^2-1))-3/16/a*arctanh(a*x)^3/(a*x-1)-3/16/a*arctanh(a*x)^3*ln(a*x-1)-3/16/a*arctanh(a*x)^3
/(a*x+1)+3/16/a*arctanh(a*x)^3*ln(a*x+1)-3/8/a*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+3/16*I*a/(a*x-1)^
2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x^2-3/32*I*a^
3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*
x+1)^2/(-a^2*x^2+1)))^2*x^4+3/32*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1))
)*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*x^4-3/16*I*a/(a*x-1)^
2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^
2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*x^2+3/32*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)
^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))+3/16*
I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3+3/16*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)^2/(
-a^2*x^2+1)))^3-3/32*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-3/32*I/a/(a*x-1
)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3-3/16*I/a/(a*x-1)^2/
(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2+3/16*I*a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3
*Pi*x^4-3/8*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*x^2+189/1024*a^3/(a*x-1)^2/(a*x+1)^2*x^4-9/512*a/(a*x-1)
^2/(a*x+1)^2*x^2+51/64/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x+3/32/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^4-51/128/a/(
a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2-3/16*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^4*x^2-45/64*a^2/(a*x-1)^2/(a*x+1)^2*
arctanh(a*x)*x^3-9/64*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*x^2

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 663, normalized size = 3.27 \[ -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{3} + \frac {3 \, {\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a \operatorname {artanh}\left (a x\right )^{2}}{64 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} - \frac {3}{512} \, {\left (\frac {{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{4} - 60 \, a^{2} x^{2} + 3 \, {\left (5 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 5\right )} \log \left (a x + 1\right )^{2} + 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 64\right )} a^{2}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} + \frac {4 \, {\left (30 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - 34 \, a x - 3 \, {\left (5 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 5\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 3*log(a*x - 1)/a)*arctanh(a*x)^3 + 3
/64*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*
x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a*arctanh(a*x)^2/(a^6*x^4 - 2*a^4*x^2 + a^2) - 3/512
*(((a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^4 - 4*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3*log(a*x - 1) + (a^4*x
^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^4 - 60*a^2*x^2 + 3*(5*a^4*x^4 - 10*a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(
a*x - 1)^2 + 5)*log(a*x + 1)^2 + 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 2*(2*(a^4*x^4 - 2*a^2*x^2 + 1)*
log(a*x - 1)^3 + 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + 1) + 64)*a^2/(a^8*x^4 - 2*a^6*x^2 + a^4)
+ 4*(30*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*log
(a*x - 1) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 - 34*a*x - 3*(5*a^4*x^4 - 10*a^2*x^2 + 2*(a^4*x^4 - 2*a
^2*x^2 + 1)*log(a*x - 1)^2 + 5)*log(a*x + 1) + 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a*arctanh(a*x)/(a^7*
x^4 - 2*a^5*x^2 + a^3))*a

________________________________________________________________________________________

mupad [B]  time = 2.29, size = 736, normalized size = 3.63 \[ \frac {\frac {45\,a\,x^2}{2}-\frac {24}{a}}{64\,a^4\,x^4-128\,a^2\,x^2+64}-{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {3}{16\,a^2}-\frac {9\,x^2}{64}}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}-\frac {45}{512\,a}\right )-{\ln \left (1-a\,x\right )}^3\,\left (\frac {3\,\ln \left (a\,x+1\right )}{128\,a}+\frac {\frac {5\,x}{8}-\frac {3\,a^2\,x^3}{8}}{8\,a^4\,x^4-16\,a^2\,x^2+8}\right )-\ln \left (1-a\,x\right )\,\left (\frac {3\,{\ln \left (a\,x+1\right )}^3}{128\,a}+\ln \left (a\,x+1\right )\,\left (\frac {\frac {21\,x}{2}+9\,a\,x^2-\frac {12}{a}-\frac {15\,a^2\,x^3}{2}}{64\,a^4\,x^4-128\,a^2\,x^2+64}-\frac {\frac {21\,x}{2}-9\,a\,x^2+\frac {12}{a}-\frac {15\,a^2\,x^3}{2}}{64\,a^4\,x^4-128\,a^2\,x^2+64}+\frac {45\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}{4\,a\,\left (64\,a^4\,x^4-128\,a^2\,x^2+64\right )}\right )+\frac {\frac {9\,x}{2}+\frac {3\,a\,x^2}{2}-\frac {3}{2\,a}-\frac {9\,a^2\,x^3}{2}}{128\,a^4\,x^4-256\,a^2\,x^2+128}+\frac {21\,x+\frac {33\,a\,x^2}{2}-\frac {39}{2\,a}-18\,a^2\,x^3}{128\,a^4\,x^4-256\,a^2\,x^2+128}+\frac {\frac {51\,x}{2}-18\,a\,x^2+\frac {21}{a}-\frac {45\,a^2\,x^3}{2}}{128\,a^4\,x^4-256\,a^2\,x^2+128}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (30\,x-18\,a^2\,x^3\right )}{128\,a^4\,x^4-256\,a^2\,x^2+128}\right )+\frac {3\,{\ln \left (a\,x+1\right )}^4}{512\,a}+\frac {3\,{\ln \left (1-a\,x\right )}^4}{512\,a}+{\ln \left (1-a\,x\right )}^2\,\left (\frac {9\,{\ln \left (a\,x+1\right )}^2}{256\,a}+\frac {45}{512\,a}-\frac {\frac {21\,x}{2}-9\,a\,x^2+\frac {12}{a}-\frac {15\,a^2\,x^3}{2}}{128\,a^4\,x^4-256\,a^2\,x^2+128}+\frac {\frac {21\,x}{2}+9\,a\,x^2-\frac {12}{a}-\frac {15\,a^2\,x^3}{2}}{128\,a^4\,x^4-256\,a^2\,x^2+128}+\frac {\ln \left (a\,x+1\right )\,\left (30\,x-18\,a^2\,x^3\right )}{128\,a^4\,x^4-256\,a^2\,x^2+128}\right )+\frac {\ln \left (a\,x+1\right )\,\left (\frac {51\,x}{128\,a}-\frac {45\,a\,x^3}{128}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}+\frac {{\ln \left (a\,x+1\right )}^3\,\left (\frac {5\,x}{64\,a}-\frac {3\,a\,x^3}{64}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^3/(a^2*x^2 - 1)^3,x)

[Out]

((45*a*x^2)/2 - 24/a)/(64*a^4*x^4 - 128*a^2*x^2 + 64) - log(a*x + 1)^2*((3/(16*a^2) - (9*x^2)/64)/(1/a - 2*a*x
^2 + a^3*x^4) - 45/(512*a)) - log(1 - a*x)^3*((3*log(a*x + 1))/(128*a) + ((5*x)/8 - (3*a^2*x^3)/8)/(8*a^4*x^4
- 16*a^2*x^2 + 8)) - log(1 - a*x)*((3*log(a*x + 1)^3)/(128*a) + log(a*x + 1)*(((21*x)/2 + 9*a*x^2 - 12/a - (15
*a^2*x^3)/2)/(64*a^4*x^4 - 128*a^2*x^2 + 64) - ((21*x)/2 - 9*a*x^2 + 12/a - (15*a^2*x^3)/2)/(64*a^4*x^4 - 128*
a^2*x^2 + 64) + (45*(a^4*x^4 - 2*a^2*x^2 + 1))/(4*a*(64*a^4*x^4 - 128*a^2*x^2 + 64))) + ((9*x)/2 + (3*a*x^2)/2
- 3/(2*a) - (9*a^2*x^3)/2)/(128*a^4*x^4 - 256*a^2*x^2 + 128) + (21*x + (33*a*x^2)/2 - 39/(2*a) - 18*a^2*x^3)/
(128*a^4*x^4 - 256*a^2*x^2 + 128) + ((51*x)/2 - 18*a*x^2 + 21/a - (45*a^2*x^3)/2)/(128*a^4*x^4 - 256*a^2*x^2 +
128) + (log(a*x + 1)^2*(30*x - 18*a^2*x^3))/(128*a^4*x^4 - 256*a^2*x^2 + 128)) + (3*log(a*x + 1)^4)/(512*a) +
(3*log(1 - a*x)^4)/(512*a) + log(1 - a*x)^2*((9*log(a*x + 1)^2)/(256*a) + 45/(512*a) - ((21*x)/2 - 9*a*x^2 +
12/a - (15*a^2*x^3)/2)/(128*a^4*x^4 - 256*a^2*x^2 + 128) + ((21*x)/2 + 9*a*x^2 - 12/a - (15*a^2*x^3)/2)/(128*a
^4*x^4 - 256*a^2*x^2 + 128) + (log(a*x + 1)*(30*x - 18*a^2*x^3))/(128*a^4*x^4 - 256*a^2*x^2 + 128)) + (log(a*x
+ 1)*((51*x)/(128*a) - (45*a*x^3)/128))/(1/a - 2*a*x^2 + a^3*x^4) + (log(a*x + 1)^3*((5*x)/(64*a) - (3*a*x^3)
/64))/(1/a - 2*a*x^2 + a^3*x^4)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**3/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)

________________________________________________________________________________________