\(\int \frac {\text {arctanh}(a x)^3}{(1-a^2 x^2)^2} \, dx\) [276]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {3}{8 a \left (1-a^2 x^2\right )}+\frac {3 x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3 \text {arctanh}(a x)^2}{8 a}-\frac {3 \text {arctanh}(a x)^2}{4 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a} \] Output:

-3/8/a/(-a^2*x^2+1)+3*x*arctanh(a*x)/(-4*a^2*x^2+4)+3/8*arctanh(a*x)^2/a-3 
/4*arctanh(a*x)^2/a/(-a^2*x^2+1)+x*arctanh(a*x)^3/(-2*a^2*x^2+2)+1/8*arcta 
nh(a*x)^4/a
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.62 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {3-6 a x \text {arctanh}(a x)+3 \left (1+a^2 x^2\right ) \text {arctanh}(a x)^2-4 a x \text {arctanh}(a x)^3+\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^4}{8 a \left (-1+a^2 x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6518, 6556, 6518, 241}

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.

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 241

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

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 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 [A] (verified)

Time = 24.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75

method	result	size
parallelrisch	\(-\frac {-\operatorname {arctanh}\left (a x \right )^{4} a^{2} x^{2}-3 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+4 \operatorname {arctanh}\left (a x \right )^{3} a x -3 a^{2} x^{2}+\operatorname {arctanh}\left (a x \right )^{4}+6 a x \,\operatorname {arctanh}\left (a x \right )-3 \operatorname {arctanh}\left (a x \right )^{2}}{8 \left (a^{2} x^{2}-1\right ) a}\)	\(86\)
risch	\(\frac {\ln \left (a x +1\right )^{4}}{128 a}-\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x -\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{32 \left (a^{2} x^{2}-1\right ) a}+\frac {3 \left (a^{2} x^{2} \ln \left (-a x +1\right )^{2}+2 a^{2} x^{2}+4 a x \ln \left (-a x +1\right )-\ln \left (-a x +1\right )^{2}+2\right ) \ln \left (a x +1\right )^{2}}{64 a \left (a x -1\right ) \left (a x +1\right )}-\frac {\left (a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}+6 a \ln \left (-a x +1\right )^{2} x -\ln \left (-a x +1\right )^{3}+12 a x +6 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{32 a \left (a x -1\right ) \left (a x +1\right )}+\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{4}+12 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+8 a x \ln \left (-a x +1\right )^{3}-\ln \left (-a x +1\right )^{4}+48 a x \ln \left (-a x +1\right )+12 \ln \left (-a x +1\right )^{2}+48}{128 a \left (a x -1\right ) \left (a x +1\right )}\)	\(336\)
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \operatorname {arctanh}\left (a x \right )^{3} {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \pi }{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \operatorname {arctanh}\left (a x \right )^{3} {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \pi }{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{8}+\frac {3 \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \left (a x -1\right )}{16 \left (a x +1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}+\frac {\frac {3 a x}{32}-\frac {3}{32}}{a x +1}}{a}\)	\(767\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{8}+\frac {i \operatorname {arctanh}\left (a x \right )^{3} {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \pi }{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{8}-\frac {i \operatorname {arctanh}\left (a x \right )^{3} {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \pi }{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{8}+\frac {3 \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \left (a x -1\right )}{16 \left (a x +1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}+\frac {\frac {3 a x}{32}-\frac {3}{32}}{a x +1}}{a}\)	\(767\)
parts	\(\text {Expression too large to display}\)	\(833\)

Input:

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

Output:

-1/8*(-arctanh(a*x)^4*a^2*x^2-3*a^2*x^2*arctanh(a*x)^2+4*arctanh(a*x)^3*a* 
x-3*a^2*x^2+arctanh(a*x)^4+6*a*x*arctanh(a*x)-3*arctanh(a*x)^2)/(a^2*x^2-1 
)/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {8 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - {\left (a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 48 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 12 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 48}{128 \, {\left (a^{3} x^{2} - a\right )}} \] Input:

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

Output:

-1/128*(8*a*x*log(-(a*x + 1)/(a*x - 1))^3 - (a^2*x^2 - 1)*log(-(a*x + 1)/( 
a*x - 1))^4 + 48*a*x*log(-(a*x + 1)/(a*x - 1)) - 12*(a^2*x^2 + 1)*log(-(a* 
x + 1)/(a*x - 1))^2 - 48)/(a^3*x^2 - a)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (99) = 198\).

Time = 0.04 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.99 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{3} - \frac {3 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname {artanh}\left (a x\right )^{2}}{16 \, {\left (a^{4} x^{2} - a^{2}\right )}} - \frac {1}{128} \, {\left (\frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{4} + 6 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right )^{2} + 12 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) - 48\right )} a^{2}}{a^{6} x^{2} - a^{4}} - \frac {8 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{5} x^{2} - a^{3}}\right )} a \] Input:

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

Output:

-1/4*(2*x/(a^2*x^2 - 1) - log(a*x + 1)/a + log(a*x - 1)/a)*arctanh(a*x)^3 
- 3/16*((a^2*x^2 - 1)*log(a*x + 1)^2 - 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a* 
x - 1) + (a^2*x^2 - 1)*log(a*x - 1)^2 - 4)*a*arctanh(a*x)^2/(a^4*x^2 - a^2 
) - 1/128*(((a^2*x^2 - 1)*log(a*x + 1)^4 - 4*(a^2*x^2 - 1)*log(a*x + 1)^3* 
log(a*x - 1) + (a^2*x^2 - 1)*log(a*x - 1)^4 + 6*(2*a^2*x^2 + (a^2*x^2 - 1) 
*log(a*x - 1)^2 - 2)*log(a*x + 1)^2 + 12*(a^2*x^2 - 1)*log(a*x - 1)^2 - 4* 
((a^2*x^2 - 1)*log(a*x - 1)^3 + 6*(a^2*x^2 - 1)*log(a*x - 1))*log(a*x + 1) 
 - 48)*a^2/(a^6*x^2 - a^4) - 8*((a^2*x^2 - 1)*log(a*x + 1)^3 - 3*(a^2*x^2 
- 1)*log(a*x + 1)^2*log(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^3 - 12*a*x + 
 3*(2*a^2*x^2 + (a^2*x^2 - 1)*log(a*x - 1)^2 - 2)*log(a*x + 1) - 6*(a^2*x^ 
2 - 1)*log(a*x - 1))*a*arctanh(a*x)/(a^5*x^2 - a^3))*a
 

Giac [A] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {1}{32} \, a^{2} {\left (\frac {{\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3}}{{\left (a x + 1\right )} a^{4}} + \frac {3 \, {\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x + 1\right )} a^{4}} + \frac {6 \, {\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{{\left (a x + 1\right )} a^{4}} + \frac {6 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{4}}\right )} \] Input:

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

Output:

1/32*a^2*((a*x - 1)*log(-(a*x + 1)/(a*x - 1))^3/((a*x + 1)*a^4) + 3*(a*x - 
 1)*log(-(a*x + 1)/(a*x - 1))^2/((a*x + 1)*a^4) + 6*(a*x - 1)*log(-(a*x + 
1)/(a*x - 1))/((a*x + 1)*a^4) + 6*(a*x - 1)/((a*x + 1)*a^4))
 

Mupad [B] (verification not implemented)

Time = 4.27 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.29 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {3\,{\ln \left (a\,x+1\right )}^2}{32\,a}-\frac {3}{2\,\left (4\,a-4\,a^3\,x^2\right )}-\frac {3\,{\ln \left (1-a\,x\right )}^2}{16\,a-16\,a^3\,x^2}+\frac {3\,{\ln \left (1-a\,x\right )}^2}{32\,a}+\frac {{\ln \left (a\,x+1\right )}^4}{128\,a}+\frac {{\ln \left (1-a\,x\right )}^4}{128\,a}-\frac {3\,{\ln \left (a\,x+1\right )}^2}{16\,\left (a-a^3\,x^2\right )}-\frac {3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{16\,a}-\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^3}{32\,a}-\frac {{\ln \left (a\,x+1\right )}^3\,\ln \left (1-a\,x\right )}{32\,a}-\frac {3\,x\,\ln \left (a\,x+1\right )}{8\,\left (a^2\,x^2-1\right )}+\frac {6\,x\,\ln \left (1-a\,x\right )}{16\,a^2\,x^2-16}+\frac {3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{8\,a-8\,a^3\,x^2}+\frac {3\,{\ln \left (a\,x+1\right )}^2\,{\ln \left (1-a\,x\right )}^2}{64\,a}-\frac {x\,{\ln \left (a\,x+1\right )}^3}{16\,\left (a^2\,x^2-1\right )}+\frac {x\,{\ln \left (1-a\,x\right )}^3}{2\,\left (8\,a^2\,x^2-8\right )}-\frac {6\,x\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{32\,a^2\,x^2-32}+\frac {6\,x\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{32\,a^2\,x^2-32} \] Input:

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

Output:

(3*log(a*x + 1)^2)/(32*a) - 3/(2*(4*a - 4*a^3*x^2)) - (3*log(1 - a*x)^2)/( 
16*a - 16*a^3*x^2) + (3*log(1 - a*x)^2)/(32*a) + log(a*x + 1)^4/(128*a) + 
log(1 - a*x)^4/(128*a) - (3*log(a*x + 1)^2)/(16*(a - a^3*x^2)) - (3*log(a* 
x + 1)*log(1 - a*x))/(16*a) - (log(a*x + 1)*log(1 - a*x)^3)/(32*a) - (log( 
a*x + 1)^3*log(1 - a*x))/(32*a) - (3*x*log(a*x + 1))/(8*(a^2*x^2 - 1)) + ( 
6*x*log(1 - a*x))/(16*a^2*x^2 - 16) + (3*log(a*x + 1)*log(1 - a*x))/(8*a - 
 8*a^3*x^2) + (3*log(a*x + 1)^2*log(1 - a*x)^2)/(64*a) - (x*log(a*x + 1)^3 
)/(16*(a^2*x^2 - 1)) + (x*log(1 - a*x)^3)/(2*(8*a^2*x^2 - 8)) - (6*x*log(a 
*x + 1)*log(1 - a*x)^2)/(32*a^2*x^2 - 32) + (6*x*log(a*x + 1)^2*log(1 - a* 
x))/(32*a^2*x^2 - 32)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {\mathit {atanh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{4}-4 \mathit {atanh} \left (a x \right )^{3} a x +3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {atanh} \left (a x \right )^{2}-6 \mathit {atanh} \left (a x \right ) a x +3 a^{2} x^{2}}{8 a \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

(atanh(a*x)**4*a**2*x**2 - atanh(a*x)**4 - 4*atanh(a*x)**3*a*x + 3*atanh(a 
*x)**2*a**2*x**2 + 3*atanh(a*x)**2 - 6*atanh(a*x)*a*x + 3*a**2*x**2)/(8*a* 
(a**2*x**2 - 1))