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*(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)

________________________________________________________________________________________

Rubi [A]  time = 0.177304, antiderivative size = 203, normalized size of antiderivative = 1., 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 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 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 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]

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 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 )^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 = 0.153703, size = 111, normalized size = 0.55 \[ \frac{45 a^2 x^2+12 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4+\left (80 a x-48 a^3 x^3\right ) \tanh ^{-1}(a x)^3+3 \left (15 a^4 x^4-6 a^2 x^2-17\right ) \tanh ^{-1}(a x)^2+\left (102 a x-90 a^3 x^3\right ) \tanh ^{-1}(a x)-48}{128 a \left (a^2 x^2-1\right )^2} \]

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)

________________________________________________________________________________________

Maple [C]  time = 0.452, size = 2646, normalized size = 13. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*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)+1))^2*x^2-3/32*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1
)+1))^2+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))*csgn(I*(a*x+1)^2/(a^2*x^2
-1)/((a*x+1)^2/(-a^2*x^2+1)+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))^2-3/32*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*csgn(I*(a*x+1)^2/(a^2*x^2-1))+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)+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)/((a*x+1)^2
/(-a^2*x^2+1)+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))^3*x^4-3
/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)+1))^2*x^4-3/8*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)+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)/((a*x+1)^2/(-a^2*x^2+1)+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+3/32*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))-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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^
2/(-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)/((a*x+1
)^2/(-a^2*x^2+1)+1))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*x^4-3/16*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)/(-a^2*x^2+1)^(1/2))^2*x^4+3/16*I*a/(a*x-1)^2/(a*x+1)^2*arct
anh(a*x)^3*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-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)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*cs
gn(I*(a*x+1)^2/(a^2*x^2-1))*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)/(-a^2*x^2+1)^(1/2))^2*x^2-195/1024/a/(a*x-1)^2/(a*x+1)^2+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))+1/16/a*arctanh(a*x)^3/(a*x-1)^2-3/16/a*arctanh(a*x)^3/(a*x
-1)-3/16/a*arctanh(a*x)^3*ln(a*x-1)-1/16/a*arctanh(a*x)^3/(a*x+1)^2-3/16/a*arctanh(a*x)^3/(a*x+1)+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^4-3/16*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+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+3/
16*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))^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)/((a*x+1)^2/(-a^2*x^2+1)+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))^3-3/16*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))^2+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))*csgn(
I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(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/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)
+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*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*I/a/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3

________________________________________________________________________________________

Maxima [B]  time = 1.06578, size = 895, normalized size = 4.41 \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)^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

________________________________________________________________________________________

Fricas [A]  time = 2.12718, size = 375, normalized size = 1.85 \begin{align*} \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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}

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)

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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