\(\int \frac {x^3 \text {arctanh}(a x)^3}{(1-a^2 x^2)^2} \, dx\) [273]

Optimal result

Integrand size = 22, antiderivative size = 227 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {3 x}{8 a^3 \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)}{8 a^4}+\frac {3 \text {arctanh}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x \text {arctanh}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^3}{4 a^4}+\frac {\text {arctanh}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{4 a^4}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{4 a^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {-4 \text {arctanh}(a x)^4+6 \text {arctanh}(a x) \cosh (2 \text {arctanh}(a x))+4 \text {arctanh}(a x)^3 \cosh (2 \text {arctanh}(a x))-16 \text {arctanh}(a x)^3 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+24 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+12 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )-3 \sinh (2 \text {arctanh}(a x))-6 \text {arctanh}(a x)^2 \sinh (2 \text {arctanh}(a x))}{16 a^4} \] Input:

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

Output:

(-4*ArcTanh[a*x]^4 + 6*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 4*ArcTanh[a*x]^ 
3*Cosh[2*ArcTanh[a*x]] - 16*ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 
24*ArcTanh[a*x]^2*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 24*ArcTanh[a*x]*PolyL 
og[3, -E^(-2*ArcTanh[a*x])] + 12*PolyLog[4, -E^(-2*ArcTanh[a*x])] - 3*Sinh 
[2*ArcTanh[a*x]] - 6*ArcTanh[a*x]^2*Sinh[2*ArcTanh[a*x]])/(16*a^4)
 

Rubi [A] (verified)

Time = 1.92 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6590, 6546, 6470, 6556, 6518, 6556, 215, 219, 6620, 6624, 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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 6624

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

(ArcTanh[a*x]^3/(2*a^2*(1 - a^2*x^2)) - (3*((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))/a^2 - (-1/4*ArcTa 
nh[a*x]^4/a^2 + ((ArcTanh[a*x]^3*Log[2/(1 - a*x)])/a - 3*(-1/2*(ArcTanh[a* 
x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/a + (ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - 
a*x)])/(2*a) - PolyLog[4, 1 - 2/(1 - a*x)]/(4*a)))/a)/a^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 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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 6590

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

rule 6620

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 6624

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] && E 
qQ[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 = 27.72 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.55

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x -1\right )}+\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 )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {3 \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}-\frac {\left (i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 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}-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 )-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}+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 )+2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+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}+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}-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+2 i \pi +4 \ln \left (2\right )+1\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}}{a^{4}}\)	\(806\)
default	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 \left (a x -1\right )}+\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 )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {3 \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}-\frac {\left (i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 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}-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 )-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}+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 )+2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+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}+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}-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+2 i \pi +4 \ln \left (2\right )+1\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}}{a^{4}}\)	\(806\)
parts	\(\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 a^{4} \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4 a^{4} \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2 a^{4}}-\frac {3 a \left (\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{5}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{3 a^{5}}+\frac {\left (a x -1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x +1\right ) a^{5}}+\frac {\left (a x -1\right ) \operatorname {arctanh}\left (a x \right )}{4 \left (a x +1\right ) a^{5}}+\frac {a x -1}{8 \left (a x +1\right ) a^{5}}-\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x -1\right ) a^{5}}+\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{4 \left (a x -1\right ) a^{5}}-\frac {a x +1}{8 \left (a x -1\right ) a^{5}}+\frac {2 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{5}}-\frac {2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{5}}+\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{5}}+\frac {\left (i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 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}-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 )-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}+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 )+2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+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}+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}-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+2 i \pi +4 \ln \left (2\right )+1\right ) \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{5}}\right )}{4}\)	\(853\)

Input:

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

Output:

1/a^4*(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+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)-3/2* 
arctanh(a*x)^2*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+3/2*arctanh(a*x)*polylog 
(3,-(a*x+1)^2/(-a^2*x^2+1))-3/4*polylog(4,-(a*x+1)^2/(-a^2*x^2+1))-1/4*(2* 
I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3+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-I*Pi*cs 
gn(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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csg 
n(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+I*Pi*csgn(I*(a*x+1 
)^2/(a^2*x^2-1))^3+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+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))-2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+2*I*Pi+4*ln(2 
)+1)*arctanh(a*x)^3)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^3*arctanh(a*x)^3/(-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)* 
log(-a*x + 1)^3)/(a^6*x^2 - a^4) + 1/8*integrate(1/2*(2*a^3*x^3*log(a*x + 
1)^3 - 6*a^3*x^3*log(a*x + 1)^2*log(-a*x + 1) - 3*(a*x - (3*a^3*x^3 + a^2* 
x^2 - a*x - 1)*log(a*x + 1) + 1)*log(-a*x + 1)^2)/(a^7*x^4 - 2*a^5*x^2 + a 
^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((atanh(a*x)**3*x**3)/(a**4*x**4 - 2*a**2*x**2 + 1),x)