\(\int \frac {x \text {arctanh}(a x)^3}{(1-a^2 x^2)^3} \, dx\) [316]

Optimal result

Integrand size = 20, antiderivative size = 188 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac {45 x}{256 a \left (1-a^2 x^2\right )}-\frac {45 \text {arctanh}(a x)}{256 a^2}+\frac {3 \text {arctanh}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {9 \text {arctanh}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 x \text {arctanh}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)^3}{32 a^2}+\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2} \] Output:

-3/128*x/a/(-a^2*x^2+1)^2-45/256*x/a/(-a^2*x^2+1)-45/256*arctanh(a*x)/a^2+ 
3/32*arctanh(a*x)/a^2/(-a^2*x^2+1)^2+9/32*arctanh(a*x)/a^2/(-a^2*x^2+1)-3/ 
16*x*arctanh(a*x)^2/a/(-a^2*x^2+1)^2-9/32*x*arctanh(a*x)^2/a/(-a^2*x^2+1)- 
3/32*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^3/a^2/(-a^2*x^2+1)^2
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.79 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-102 a x+90 a^3 x^3-48 \left (-4+3 a^2 x^2\right ) \text {arctanh}(a x)+48 a x \left (-5+3 a^2 x^2\right ) \text {arctanh}(a x)^2+\left (80+96 a^2 x^2-48 a^4 x^4\right ) \text {arctanh}(a x)^3+45 \left (-1+a^2 x^2\right )^2 \log (1-a x)-45 \log (1+a x)+90 a^2 x^2 \log (1+a x)-45 a^4 x^4 \log (1+a x)}{512 a^2 \left (-1+a^2 x^2\right )^2} \] Input:

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

Output:

(-102*a*x + 90*a^3*x^3 - 48*(-4 + 3*a^2*x^2)*ArcTanh[a*x] + 48*a*x*(-5 + 3 
*a^2*x^2)*ArcTanh[a*x]^2 + (80 + 96*a^2*x^2 - 48*a^4*x^4)*ArcTanh[a*x]^3 + 
 45*(-1 + a^2*x^2)^2*Log[1 - a*x] - 45*Log[1 + a*x] + 90*a^2*x^2*Log[1 + a 
*x] - 45*a^4*x^4*Log[1 + a*x])/(512*a^2*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6556, 6526, 215, 215, 219, 6518, 6556, 215, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx}{4 a}\)

\(\Big \downarrow \) 6526

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\)

Input:

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

Output:

ArcTanh[a*x]^3/(4*a^2*(1 - a^2*x^2)^2) - (3*(-1/8*ArcTanh[a*x]/(a*(1 - a^2 
*x^2)^2) + (x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) + (x/(4*(1 - a^2*x^2)^2) 
 + (3*(x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a)))/4)/8 + (3*((x*ArcTanh[a* 
x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTanh[a*x]/(2*a^2*(1 
 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a))))/4))/(4* 
a)
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6518

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

rule 6526

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] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p 
/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1 
)*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2))   Int 
[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], 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 6556

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] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.48 (sec) , antiderivative size = 848, normalized size of antiderivative = 4.51

\[\text {Expression too large to display}\]

Input:

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

Output:

1/a^2*(-9/128*I*arctanh(a*x)^2*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))*Pi-9/64*arctanh(a*x)^2*ln(a*x+1)+9/32*arctanh(a*x)^2*ln((a*x+1)/(-a^2 
*x^2+1)^(1/2))-3/32*arctanh(a*x)*(a*x+1)/(a*x-1)-3/32*arctanh(a*x)*(a*x-1) 
/(a*x+1)+9/64*arctanh(a*x)^2*ln(a*x-1)+9/128*I*arctanh(a*x)^2*csgn(I*(a*x+ 
1)^2/(a^2*x^2-1))^3*Pi+9/64*I*arctanh(a*x)^2*csgn(I/(-(a*x+1)^2/(a^2*x^2-1 
)+1))^2*Pi-9/64*I*arctanh(a*x)^2*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*Pi+9 
/128*I*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1) 
+1))^3*Pi-9/64*I*arctanh(a*x)^2*Pi-3/32*arctanh(a*x)^3-3/2048*(a*x+1)^2/(a 
*x-1)^2-3/64*arctanh(a*x)^2/(a*x-1)^2+3/64*arctanh(a*x)^2/(a*x+1)^2-3/64*( 
a*x-1)/(a*x+1)+3/64*(a*x+1)/(a*x-1)+3/2048*(a*x-1)^2/(a*x+1)^2+9/64*arctan 
h(a*x)^2/(a*x-1)+9/64*arctanh(a*x)^2/(a*x+1)+1/4/(a^2*x^2-1)^2*arctanh(a*x 
)^3-9/128*I*arctanh(a*x)^2*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))*Pi+9/64*I*arctanh(a*x)^2*csgn(I* 
(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi+9/128*I*arc 
tanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2 
-1))*Pi+9/128*I*arctanh(a*x)^2*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*Pi+3/512*arctanh(a*x)*( 
a*x+1)^2/(a*x-1)^2+3/512*(a*x-1)^2*arctanh(a*x)/(a*x+1)^2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.74 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {90 \, a^{3} x^{3} - 2 \, {\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 102 \, a x - 3 \, {\left (15 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 17\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \] Input:

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

Output:

1/512*(90*a^3*x^3 - 2*(3*a^4*x^4 - 6*a^2*x^2 - 5)*log(-(a*x + 1)/(a*x - 1) 
)^3 + 12*(3*a^3*x^3 - 5*a*x)*log(-(a*x + 1)/(a*x - 1))^2 - 102*a*x - 3*(15 
*a^4*x^4 - 6*a^2*x^2 - 17)*log(-(a*x + 1)/(a*x - 1)))/(a^6*x^4 - 2*a^4*x^2 
 + a^2)
 

Sympy [F]

\[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (163) = 326\).

Time = 0.06 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.24 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {3 \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2}}{64 \, a} + \frac {3 \, {\left (\frac {{\left (30 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - 34 \, a x - 3 \, {\left (5 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 5\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}} - \frac {4 \, {\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a \operatorname {artanh}\left (a x\right )}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}}\right )}}{512 \, a} + \frac {\operatorname {artanh}\left (a x\right )^{3}}{4 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \] Input:

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

Output:

3/64*(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)^2/a + 3/512*((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^2/(a^7*x^4 - 2*a^5*x 
^2 + a^3) - 4*(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)/(a^6*x^4 - 2*a^4*x^2 + a^2))/ 
a + 1/4*arctanh(a*x)^3/((a^2*x^2 - 1)^2*a^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (163) = 326\).

Time = 0.13 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.82 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{2048} \, {\left (4 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x - 1\right )}^{2} {\left (\frac {32 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {96 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} a \] Input:

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

Output:

-1/2048*(4*((a*x - 1)^2*(4*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) - (a 
*x + 1)^2/((a*x - 1)^2*a^3) + 4*(a*x + 1)/((a*x - 1)*a^3))*log(-(a*x + 1)/ 
(a*x - 1))^3 + 6*((a*x - 1)^2*(8*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3 
) + (a*x + 1)^2/((a*x - 1)^2*a^3) - 8*(a*x + 1)/((a*x - 1)*a^3))*log(-(a*x 
 + 1)/(a*x - 1))^2 + 6*((a*x - 1)^2*(16*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1 
)^2*a^3) - (a*x + 1)^2/((a*x - 1)^2*a^3) + 16*(a*x + 1)/((a*x - 1)*a^3))*l 
og(-(a*x + 1)/(a*x - 1)) + 3*(a*x - 1)^2*(32*(a*x + 1)/(a*x - 1) - 1)/((a* 
x + 1)^2*a^3) + 3*(a*x + 1)^2/((a*x - 1)^2*a^3) - 96*(a*x + 1)/((a*x - 1)* 
a^3))*a
 

Mupad [B] (verification not implemented)

Time = 5.55 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.20 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {48\,\ln \left (1-a\,x\right )-48\,\ln \left (a\,x+1\right )+45\,\mathrm {atanh}\left (a\,x\right )+51\,a\,x-5\,{\ln \left (a\,x+1\right )}^3+5\,{\ln \left (1-a\,x\right )}^3-15\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+15\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-45\,a^3\,x^3-6\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^3+6\,a^2\,x^2\,{\ln \left (1-a\,x\right )}^3-18\,a^3\,x^3\,{\ln \left (a\,x+1\right )}^2-18\,a^3\,x^3\,{\ln \left (1-a\,x\right )}^2+3\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^3-3\,a^4\,x^4\,{\ln \left (1-a\,x\right )}^3-90\,a^2\,x^2\,\mathrm {atanh}\left (a\,x\right )+45\,a^4\,x^4\,\mathrm {atanh}\left (a\,x\right )+30\,a\,x\,{\ln \left (a\,x+1\right )}^2+30\,a\,x\,{\ln \left (1-a\,x\right )}^2+36\,a^2\,x^2\,\ln \left (a\,x+1\right )-36\,a^2\,x^2\,\ln \left (1-a\,x\right )-60\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )-18\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+18\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+9\,a^4\,x^4\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-9\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+36\,a^3\,x^3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{256\,a^2\,{\left (a^2\,x^2-1\right )}^2} \] Input:

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

Output:

-(48*log(1 - a*x) - 48*log(a*x + 1) + 45*atanh(a*x) + 51*a*x - 5*log(a*x + 
 1)^3 + 5*log(1 - a*x)^3 - 15*log(a*x + 1)*log(1 - a*x)^2 + 15*log(a*x + 1 
)^2*log(1 - a*x) - 45*a^3*x^3 - 6*a^2*x^2*log(a*x + 1)^3 + 6*a^2*x^2*log(1 
 - a*x)^3 - 18*a^3*x^3*log(a*x + 1)^2 - 18*a^3*x^3*log(1 - a*x)^2 + 3*a^4* 
x^4*log(a*x + 1)^3 - 3*a^4*x^4*log(1 - a*x)^3 - 90*a^2*x^2*atanh(a*x) + 45 
*a^4*x^4*atanh(a*x) + 30*a*x*log(a*x + 1)^2 + 30*a*x*log(1 - a*x)^2 + 36*a 
^2*x^2*log(a*x + 1) - 36*a^2*x^2*log(1 - a*x) - 60*a*x*log(a*x + 1)*log(1 
- a*x) - 18*a^2*x^2*log(a*x + 1)*log(1 - a*x)^2 + 18*a^2*x^2*log(a*x + 1)^ 
2*log(1 - a*x) + 9*a^4*x^4*log(a*x + 1)*log(1 - a*x)^2 - 9*a^4*x^4*log(a*x 
 + 1)^2*log(1 - a*x) + 36*a^3*x^3*log(a*x + 1)*log(1 - a*x))/(256*a^2*(a^2 
*x^2 - 1)^2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.09 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-48 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+96 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}+80 \mathit {atanh} \left (a x \right )^{3}+144 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}-240 \mathit {atanh} \left (a x \right )^{2} a x -72 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+120 \mathit {atanh} \left (a x \right )+9 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}-18 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}+9 \,\mathrm {log}\left (a^{2} x -a \right )-9 \,\mathrm {log}\left (a^{2} x +a \right ) a^{4} x^{4}+18 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}-9 \,\mathrm {log}\left (a^{2} x +a \right )+90 a^{3} x^{3}-102 a x}{512 a^{2} \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 48*atanh(a*x)**3*a**4*x**4 + 96*atanh(a*x)**3*a**2*x**2 + 80*atanh(a*x 
)**3 + 144*atanh(a*x)**2*a**3*x**3 - 240*atanh(a*x)**2*a*x - 72*atanh(a*x) 
*a**4*x**4 + 120*atanh(a*x) + 9*log(a**2*x - a)*a**4*x**4 - 18*log(a**2*x 
- a)*a**2*x**2 + 9*log(a**2*x - a) - 9*log(a**2*x + a)*a**4*x**4 + 18*log( 
a**2*x + a)*a**2*x**2 - 9*log(a**2*x + a) + 90*a**3*x**3 - 102*a*x)/(512*a 
**2*(a**4*x**4 - 2*a**2*x**2 + 1))