\(\int \frac {\text {arctanh}(a x)^3}{(1-a^2 x^2)^3} \, dx\) [317]

Optimal result

Integrand size = 19, antiderivative size = 203 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {3}{128 a \left (1-a^2 x^2\right )^2}-\frac {45}{128 a \left (1-a^2 x^2\right )}+\frac {3 x \text {arctanh}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {45 x \text {arctanh}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {45 \text {arctanh}(a x)^2}{128 a}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arctanh}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {3 \text {arctanh}(a x)^4}{32 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.55 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-48+45 a^2 x^2+\left (102 a x-90 a^3 x^3\right ) \text {arctanh}(a x)+3 \left (-17-6 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)^2+\left (80 a x-48 a^3 x^3\right ) \text {arctanh}(a x)^3+12 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^4}{128 a \left (-1+a^2 x^2\right )^2} \] Input:

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

Output:

(-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 + (80*a*x - 48*a^3*x^3)*ArcTanh[a*x]^3 + 1 
2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)/(128*a*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.39, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6526, 6518, 6522, 6518, 241, 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.

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

\(\Big \downarrow \) 6526

\(\displaystyle \frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {3}{8} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\)

Input:

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

Output:

(-3*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x]^3)/(4*(1 - a^ 
2*x^2)^2) + (3*(-1/16*1/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x])/(4*(1 - a^2 
*x^2)^2) + (3*(-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)))/4))/8 + (3*((x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2 
)) + ArcTanh[a*x]^4/(8*a) - (3*a*(ArcTanh[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))/4
 

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 6522

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo 
l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + 
 e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(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]), x], x]) /; Fre 
eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
 

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 [A] (verified)

Time = 12.40 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(-\frac {-12 \operatorname {arctanh}\left (a x \right )^{4} a^{4} x^{4}-45 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+48 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}-48 a^{4} x^{4}+24 \operatorname {arctanh}\left (a x \right )^{4} a^{2} x^{2}+90 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+18 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-80 \operatorname {arctanh}\left (a x \right )^{3} a x +51 a^{2} x^{2}-12 \operatorname {arctanh}\left (a x \right )^{4}-102 a x \,\operatorname {arctanh}\left (a x \right )+51 \operatorname {arctanh}\left (a x \right )^{2}}{128 \left (a^{2} x^{2}-1\right )^{2} a}\)	\(150\)
risch	\(\frac {3 \ln \left (a x +1\right )^{4}}{512 a}-\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-10 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{128 \left (a^{2} x^{2}-1\right )^{2} a}+\frac {3 \left (6 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+15 a^{4} x^{4}+24 a^{3} x^{3} \ln \left (-a x +1\right )-12 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-6 a^{2} x^{2}-40 a x \ln \left (-a x +1\right )+6 \ln \left (-a x +1\right )^{2}-17\right ) \ln \left (a x +1\right )^{2}}{512 a \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}-\frac {3 \left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+15 x^{4} \ln \left (-a x +1\right ) a^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+30 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-20 a \ln \left (-a x +1\right )^{2} x +2 \ln \left (-a x +1\right )^{3}-34 a x -17 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{256 a \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}+\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{4}+45 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+24 a^{3} x^{3} \ln \left (-a x +1\right )^{3}-6 a^{2} x^{2} \ln \left (-a x +1\right )^{4}+180 a^{3} x^{3} \ln \left (-a x +1\right )-18 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-40 a x \ln \left (-a x +1\right )^{3}+180 a^{2} x^{2}+3 \ln \left (-a x +1\right )^{4}-204 a x \ln \left (-a x +1\right )-51 \ln \left (-a x +1\right )^{2}-192}{512 a \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}\)	\(567\)
derivativedivides	\(\text {Expression too large to display}\)	\(913\)
default	\(\text {Expression too large to display}\)	\(913\)
parts	\(\text {Expression too large to display}\)	\(1002\)

Input:

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

Output:

-1/128*(-12*arctanh(a*x)^4*a^4*x^4-45*a^4*x^4*arctanh(a*x)^2+48*arctanh(a* 
x)^3*a^3*x^3-48*a^4*x^4+24*arctanh(a*x)^4*a^2*x^2+90*a^3*x^3*arctanh(a*x)+ 
18*a^2*x^2*arctanh(a*x)^2-80*arctanh(a*x)^3*a*x+51*a^2*x^2-12*arctanh(a*x) 
^4-102*a*x*arctanh(a*x)+51*arctanh(a*x)^2)/(a^2*x^2-1)^2/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 180 \, a^{2} x^{2} - 8 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 3 \, {\left (15 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 17\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (15 \, a^{3} x^{3} - 17 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 192}{512 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \] Input:

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

Output:

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)*log( 
-(a*x + 1)/(a*x - 1)) - 192)/(a^5*x^4 - 2*a^3*x^2 + a)
 

Sympy [F]

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

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

Output:

-Integral(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. 663 vs. \(2 (175) = 350\).

Time = 0.08 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.27 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-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 - 1 
0*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
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 5.37 (sec) , antiderivative size = 736, normalized size of antiderivative = 3.63 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\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} \] Input:

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

Output:

((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 - 2 
56*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)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.74 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {24 \mathit {atanh} \left (a x \right )^{4} a^{4} x^{4}-48 \mathit {atanh} \left (a x \right )^{4} a^{2} x^{2}+24 \mathit {atanh} \left (a x \right )^{4}-96 \mathit {atanh} \left (a x \right )^{3} a^{3} x^{3}+160 \mathit {atanh} \left (a x \right )^{3} a x +90 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-36 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-102 \mathit {atanh} \left (a x \right )^{2}-180 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+204 \mathit {atanh} \left (a x \right ) a x +45 a^{4} x^{4}-51}{256 a \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(24*atanh(a*x)**4*a**4*x**4 - 48*atanh(a*x)**4*a**2*x**2 + 24*atanh(a*x)** 
4 - 96*atanh(a*x)**3*a**3*x**3 + 160*atanh(a*x)**3*a*x + 90*atanh(a*x)**2* 
a**4*x**4 - 36*atanh(a*x)**2*a**2*x**2 - 102*atanh(a*x)**2 - 180*atanh(a*x 
)*a**3*x**3 + 204*atanh(a*x)*a*x + 45*a**4*x**4 - 51)/(256*a*(a**4*x**4 - 
2*a**2*x**2 + 1))