\(\int \frac {\text {arctanh}(a x)^3}{x (1-a^2 x^2)^2} \, dx\) [277]

Optimal result

Integrand size = 22, antiderivative size = 193 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx=-\frac {3 a x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} \text {arctanh}(a x)+\frac {3 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \text {arctanh}(a x)^3+\frac {\text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \] Output:

-3*a*x/(-8*a^2*x^2+8)-3/8*arctanh(a*x)+3*arctanh(a*x)/(-4*a^2*x^2+4)-3*a*x 
*arctanh(a*x)^2/(-4*a^2*x^2+4)-1/4*arctanh(a*x)^3+arctanh(a*x)^3/(-2*a^2*x 
^2+2)+1/4*arctanh(a*x)^4+arctanh(a*x)^3*ln(2-2/(a*x+1))-3/2*arctanh(a*x)^2 
*polylog(2,-1+2/(a*x+1))-3/2*arctanh(a*x)*polylog(3,-1+2/(a*x+1))-3/4*poly 
log(4,-1+2/(a*x+1))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.70 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{64} \left (\pi ^4-16 \text {arctanh}(a x)^4+24 \text {arctanh}(a x) \cosh (2 \text {arctanh}(a x))+16 \text {arctanh}(a x)^3 \cosh (2 \text {arctanh}(a x))+64 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+96 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-96 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )-12 \sinh (2 \text {arctanh}(a x))-24 \text {arctanh}(a x)^2 \sinh (2 \text {arctanh}(a x))\right ) \] Input:

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

Output:

(Pi^4 - 16*ArcTanh[a*x]^4 + 24*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 16*ArcT 
anh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 64*ArcTanh[a*x]^3*Log[1 - 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])] - 12*S 
inh[2*ArcTanh[a*x]] - 24*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]])/64
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6592, 6550, 6494, 6556, 6518, 6556, 215, 219, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6518

\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 6556

\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 215

\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 6618

\(\displaystyle -3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 6622

\(\displaystyle -3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\)

Input:

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

Output:

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

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 [C] (warning: unable to verify)

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

Time = 65.90 (sec) , antiderivative size = 1300, normalized size of antiderivative = 6.74

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1300\)
default	\(\text {Expression too large to display}\)	\(1300\)
parts	\(\text {Expression too large to display}\)	\(1711\)

Input:

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

Output:

1/4*arctanh(a*x)^3/(a*x+1)-1/2*arctanh(a*x)^3*ln(a*x+1)-1/4*arctanh(a*x)^3 
/(a*x-1)-1/2*arctanh(a*x)^3*ln(a*x-1)+arctanh(a*x)^3*ln(a*x)+arctanh(a*x)^ 
3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*arctanh(a*x)^4-3/16*arctanh(a*x)^2*(a 
*x-1)/(a*x+1)-3/16*arctanh(a*x)*(a*x-1)/(a*x+1)-3/32*(a*x-1)/(a*x+1)+3/16* 
(a*x+1)*arctanh(a*x)^2/(a*x-1)-3/16*arctanh(a*x)*(a*x+1)/(a*x-1)+3/32*(a*x 
+1)/(a*x-1)-arctanh(a*x)^3*ln((a*x+1)^2/(-a^2*x^2+1)-1)+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)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylo 
g(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+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)*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))+1/4*(-2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2-2*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)-1)/(-(a*x+1)^2/(a^ 
2*x^2-1)+1))^2+2*I*Pi-2*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)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+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))-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))+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+2*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)+1))*csg 
n(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))+2*I*Pi*csgn(...
 

Fricas [F]

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

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

Output:

integral(arctanh(a*x)^3/(a^4*x^5 - 2*a^2*x^3 + x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(atanh(a*x)**3/(a**4*x**5 - 2*a**2*x**3 + x),x)