\(\int \frac {\text {arctanh}(a x)^2}{x^2 (1-a^2 x^2)^2} \, dx\) [271]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {a^2 x}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a \text {arctanh}(a x)-\frac {a \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{x}+\frac {a^2 x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} a \text {arctanh}(a x)^3+2 a \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

a^2*x/(-4*a^2*x^2+4)+1/4*a*arctanh(a*x)-a*arctanh(a*x)/(-2*a^2*x^2+2)+a*ar 
ctanh(a*x)^2-arctanh(a*x)^2/x+a^2*x*arctanh(a*x)^2/(-2*a^2*x^2+2)+1/2*a*ar 
ctanh(a*x)^3+2*a*arctanh(a*x)*ln(2-2/(a*x+1))-a*polylog(2,-1+2/(a*x+1))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.68 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {4 a x \text {arctanh}(a x)^3-2 a x \text {arctanh}(a x) \left (\cosh (2 \text {arctanh}(a x))-8 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )-8 a x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )+a x \sinh (2 \text {arctanh}(a x))+2 \text {arctanh}(a x)^2 (-4+4 a x+a x \sinh (2 \text {arctanh}(a x)))}{8 x} \] Input:

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

Output:

(4*a*x*ArcTanh[a*x]^3 - 2*a*x*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] - 8*Log[1 
 - E^(-2*ArcTanh[a*x])]) - 8*a*x*PolyLog[2, E^(-2*ArcTanh[a*x])] + a*x*Sin 
h[2*ArcTanh[a*x]] + 2*ArcTanh[a*x]^2*(-4 + 4*a*x + a*x*Sinh[2*ArcTanh[a*x] 
]))/(8*x)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6592, 6518, 6544, 6452, 6510, 6550, 6494, 2897, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

\(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\)

\(\Big \downarrow \) 6556

\(\displaystyle a^2 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\)

\(\Big \downarrow \) 215

\(\displaystyle a^2 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\)

\(\Big \downarrow \) 219

\(\displaystyle a^2 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\)

Input:

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

Output:

-(ArcTanh[a*x]^2/x) + (a*ArcTanh[a*x]^3)/3 + a^2*((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))) + 2*a*(ArcTanh[a*x] 
^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)
 

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 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6494

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

rule 6510

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b 
, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && 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 6544

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

rule 6550

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ 
d   Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), 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]
 

rule 6592

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 20.66 (sec) , antiderivative size = 3005, normalized size of antiderivative = 21.16

method	result	size
default	\(\text {Expression too large to display}\)	\(3005\)
parts	\(\text {Expression too large to display}\)	\(3015\)
derivativedivides	\(\text {Expression too large to display}\)	\(3050\)

Input:

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

Output:

a*(3/4*arctanh(a*x)^2*ln(a*x+1)-3/2*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1) 
^(1/2))-arctanh(a*x)^2/a/x+1/8*arctanh(a*x)*(a*x+1)/(a*x-1)+1/8*arctanh(a* 
x)*(a*x-1)/(a*x+1)-3/4*arctanh(a*x)^2*ln(a*x-1)+polylog(2,-(a*x+1)/(-a^2*x 
^2+1)^(1/2))+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*arctanh(a*x)^3-arct 
anh(a*x)^2+3/8*I*Pi*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))*(arcta 
nh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-dilog((a*x+1)/(-a^2*x^2+1)^(1/2)) 
+dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2)))+3/4*I*Pi*arctanh(a*x)^2+1/16*(a*x-1) 
/(a*x+1)-1/16*(a*x+1)/(a*x-1)+3/8*I*Pi*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*(arctanh(a*x)*ln(1+( 
a*x+1)/(-a^2*x^2+1)^(1/2))-dilog((a*x+1)/(-a^2*x^2+1)^(1/2))+dilog(1+(a*x+ 
1)/(-a^2*x^2+1)^(1/2)))-3/8*I*Pi*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))*(-arctanh(a*x)^2+arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arct 
anh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,-(a*x+1)/(-a^2*x^2+1)^ 
(1/2))+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2)))+3/8*I*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 
*(-arctanh(a*x)^2+arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a* 
x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2)) 
+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2)))-3/4*I*Pi*csgn(I*(a*x+1)/(-a^2*x...
 

Fricas [F]

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

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

Output:

integral(arctanh(a*x)^2/(a^4*x^6 - 2*a^2*x^4 + x^2), x)
 

Sympy [F]

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

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

Output:

Integral(atanh(a*x)**2/(x**2*(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. 406 vs. \(2 (128) = 256\).

Time = 0.04 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.86 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{16} \, a^{2} {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + {\left (4 \, a^{2} x^{2} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right ) - 4\right )} \log \left (a x + 1\right )^{2} - 4 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4 \, a x + {\left (3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 8 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a^{3} x^{2} - a} + \frac {16 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {16 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {16 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} + \frac {2 \, \log \left (a x + 1\right )}{a} - \frac {2 \, \log \left (a x - 1\right )}{a}\right )} - \frac {1}{8} \, a {\left (\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4}{a^{2} x^{2} - 1} + 8 \, \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right ) - 16 \, \log \left (x\right )\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{4} \, {\left (3 \, a \log \left (a x + 1\right ) - 3 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 2\right )}}{a^{2} x^{3} - x}\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:

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

Output:

1/16*a^2*(((a^2*x^2 - 1)*log(a*x + 1)^3 - (a^2*x^2 - 1)*log(a*x - 1)^3 + ( 
4*a^2*x^2 - 3*(a^2*x^2 - 1)*log(a*x - 1) - 4)*log(a*x + 1)^2 - 4*(a^2*x^2 
- 1)*log(a*x - 1)^2 - 4*a*x + (3*(a^2*x^2 - 1)*log(a*x - 1)^2 - 8*(a^2*x^2 
 - 1)*log(a*x - 1))*log(a*x + 1))/(a^3*x^2 - a) + 16*(log(a*x - 1)*log(1/2 
*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 16*(log(a*x + 1)*log(x) + dilog(- 
a*x))/a + 16*(log(-a*x + 1)*log(x) + dilog(a*x))/a + 2*log(a*x + 1)/a - 2* 
log(a*x - 1)/a) - 1/8*a*((3*(a^2*x^2 - 1)*log(a*x + 1)^2 - 6*(a^2*x^2 - 1) 
*log(a*x + 1)*log(a*x - 1) + 3*(a^2*x^2 - 1)*log(a*x - 1)^2 - 4)/(a^2*x^2 
- 1) + 8*log(a*x + 1) + 8*log(a*x - 1) - 16*log(x))*arctanh(a*x) + 1/4*(3* 
a*log(a*x + 1) - 3*a*log(a*x - 1) - 2*(3*a^2*x^2 - 2)/(a^2*x^3 - x))*arcta 
nh(a*x)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {4 \mathit {atanh} \left (a x \right )^{3} a^{3} x^{3}-4 \mathit {atanh} \left (a x \right )^{3} a x -12 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+8 \mathit {atanh} \left (a x \right )^{2}+12 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+16 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a^{3} x^{3}-16 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a x +3 \,\mathrm {log}\left (a^{2} x -a \right ) a^{3} x^{3}-3 \,\mathrm {log}\left (a^{2} x -a \right ) a x -3 \,\mathrm {log}\left (a^{2} x +a \right ) a^{3} x^{3}+3 \,\mathrm {log}\left (a^{2} x +a \right ) a x -6 a^{2} x^{2}}{8 x \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

(4*atanh(a*x)**3*a**3*x**3 - 4*atanh(a*x)**3*a*x - 12*atanh(a*x)**2*a**2*x 
**2 + 8*atanh(a*x)**2 + 12*atanh(a*x)*a**3*x**3 + 16*int(atanh(a*x)/(a**4* 
x**5 - 2*a**2*x**3 + x),x)*a**3*x**3 - 16*int(atanh(a*x)/(a**4*x**5 - 2*a* 
*2*x**3 + x),x)*a*x + 3*log(a**2*x - a)*a**3*x**3 - 3*log(a**2*x - a)*a*x 
- 3*log(a**2*x + a)*a**3*x**3 + 3*log(a**2*x + a)*a*x - 6*a**2*x**2)/(8*x* 
(a**2*x**2 - 1))