\(\int \frac {\text {arctanh}(a x)^3}{x^3 (1-a^2 x^2)} \, dx\) [247]

Optimal result

Integrand size = 22, antiderivative size = 200 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx=\frac {3}{2} a^2 \text {arctanh}(a x)^2-\frac {3 a \text {arctanh}(a x)^2}{2 x}+\frac {1}{2} a^2 \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \text {arctanh}(a x)^4+3 a^2 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )+a^2 \text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx=-\frac {1}{64} a^2 \left (-\pi ^4-96 \text {arctanh}(a x)^2+\frac {96 \text {arctanh}(a x)^2}{a x}+\frac {32 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}{a^2 x^2}+16 \text {arctanh}(a x)^4-192 \text {arctanh}(a x) \log \left (1-e^{-2 \text {arctanh}(a x)}\right )-64 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+96 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )-96 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+96 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

-1/64*(a^2*(-Pi^4 - 96*ArcTanh[a*x]^2 + (96*ArcTanh[a*x]^2)/(a*x) + (32*(1 
 - a^2*x^2)*ArcTanh[a*x]^3)/(a^2*x^2) + 16*ArcTanh[a*x]^4 - 192*ArcTanh[a* 
x]*Log[1 - E^(-2*ArcTanh[a*x])] - 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a 
*x])] + 96*PolyLog[2, E^(-2*ArcTanh[a*x])] - 96*ArcTanh[a*x]^2*PolyLog[2, 
E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] - 48* 
PolyLog[4, E^(2*ArcTanh[a*x])]))
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6544, 6452, 6544, 6452, 6510, 6550, 6494, 2897, 6618, 6622, 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.

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 6622

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 6622

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

Leaf count of result is larger than twice the leaf count of optimal. \(368\) vs. \(2(182)=364\).

Time = 35.40 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}+\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2}+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(369\)
default	\(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}+\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2}+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(369\)

Input:

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

Output:

a^2*(-1/4*arctanh(a*x)^4+1/2*arctanh(a*x)^2*(a*x*arctanh(a*x)+arctanh(a*x) 
+3*a*x)*(a*x-1)/a^2/x^2+arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3* 
arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylo 
g(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))+ar 
ctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,- 
(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1) 
^(1/2))+6*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2+3*arctan 
h(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*polylog(2,(a*x+1)/(-a^2*x^2+1)^( 
1/2))+3*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*polylog(2,-(a*x+1) 
/(-a^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral(-arctanh(a*x)^3/(a^2*x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, dx \] Input:

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

Output:

-Integral(atanh(a*x)**3/(a**2*x**5 - x**3), x)
 

Maxima [F]

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

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

Output:

1/64*(a^2*x^2*log(-a*x + 1)^4 + 4*(a^2*x^2*log(a*x + 1) + 1)*log(-a*x + 1) 
^3)/x^2 - 1/8*integrate(1/2*(2*log(a*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x 
+ 1) + 3*(a^2*x^2 + a*x + (a^4*x^4 + a^3*x^3 + 2)*log(a*x + 1))*log(-a*x + 
 1)^2)/(a^2*x^5 - x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a^{2} x^{5}-x^{3}}d x \right ) \] Input:

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

Output:

 - int(atanh(a*x)**3/(a**2*x**5 - x**3),x)