\(\int \frac {\text {arctanh}(a x)^3}{x^2 (1-a^2 x^2)} \, dx\) [246]

Optimal result

Integrand size = 22, antiderivative size = 90 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{x}+\frac {1}{4} a \text {arctanh}(a x)^4+3 a \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=-a \left (-\frac {i \pi ^3}{8}+\text {arctanh}(a x)^3+\frac {\text {arctanh}(a x)^3}{a x}-\frac {1}{4} \text {arctanh}(a x)^4-3 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

-(a*((-1/8*I)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - ArcTanh[a*x]^ 
4/4 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLo 
g[2, E^(2*ArcTanh[a*x])] + (3*PolyLog[3, E^(2*ArcTanh[a*x])])/2))
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6544, 6452, 6510, 6550, 6494, 6618, 7164}

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.

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

-(ArcTanh[a*x]^3/x) + (a*ArcTanh[a*x]^4)/4 + 3*a*(ArcTanh[a*x]^3/3 + ArcTa 
nh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + 
 a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a)))
 

Defintions of rubi rules used

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 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 6618

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

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

Time = 16.44 (sec) , antiderivative size = 810, normalized size of antiderivative = 9.00

method	result	size
derivativedivides	\(a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{x a}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\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}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}-\operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\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}}{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}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}+\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}}{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}}{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}}{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}}{4}\right )\)	\(810\)
default	\(a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{x a}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\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}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}-\operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\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}}{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}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}+\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}}{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}}{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}}{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}}{4}\right )\)	\(810\)
parts	\(\text {Expression too large to display}\)	\(811\)

Input:

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

Output:

a*(-1/2*arctanh(a*x)^3*ln(a*x-1)-arctanh(a*x)^3/x/a+1/2*arctanh(a*x)^3*ln( 
a*x+1)-arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(3,-(a*x+1)/ 
(-a^2*x^2+1)^(1/2))-6*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*Pi*csgn( 
I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^3-1/4*I*Pi*csgn(I*(a*x+1)^2/( 
a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^3+1/4*arctanh(a*x)^4 
-arctanh(a*x)^3+3*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctan 
h(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*ln(1-(a*x+1 
)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2)) 
-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^3-1/4*I*Pi*csgn(I*( 
a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^3- 
1/2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^3+1/2*I*Pi*arct 
anh(a*x)^3+1/4*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)^3-1/4*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)^3-1/2*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a* 
x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^3+1/4*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)^3)
 

Fricas [F]

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

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

Output:

integral(-arctanh(a*x)^3/(a^2*x^4 - x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/64*(a*x*log(-a*x + 1)^4 - 4*(a*x*log(a*x + 1) + 2*a*x - 2)*log(-a*x + 1) 
^3 + 6*(a*x*log(a*x + 1)^2 - 4*(a*x + 1)*log(a*x + 1))*log(-a*x + 1)^2)/x 
- 1/8*integrate(1/2*(2*log(a*x + 1)^3 + 3*((a^3*x^3 + a^2*x^2 - 2)*log(a*x 
 + 1)^2 - 4*(a^3*x^3 + 2*a^2*x^2 + a*x)*log(a*x + 1))*log(-a*x + 1))/(a^2* 
x^4 - x^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,\left (a^2\,x^2-1\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(atanh(a*x)**4*a*x - 4*atanh(a*x)**3 - 12*int(atanh(a*x)**2/(a**2*x**3 - x 
),x)*a*x)/(4*x)