\(\int \frac {x^3 \text {arctanh}(a x)^2}{(1-a^2 x^2)^2} \, dx\) [266]

Optimal result

Integrand size = 22, antiderivative size = 161 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {1}{4 a^4 \left (1-a^2 x^2\right )}-\frac {x \text {arctanh}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^2}{4 a^4}+\frac {\text {arctanh}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{3 a^4}-\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx=\frac {-\frac {1}{3} \text {arctanh}(a x)^3+\frac {1}{8} \left (1+2 \text {arctanh}(a x)^2\right ) \cosh (2 \text {arctanh}(a x))-\text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )-\frac {1}{4} \text {arctanh}(a x) \sinh (2 \text {arctanh}(a x))}{a^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6590, 6546, 6470, 6556, 6518, 241, 6620, 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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 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 = 24.04 (sec) , antiderivative size = 735, normalized size of antiderivative = 4.57

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{8 \left (a x +1\right )}-\frac {a x -1}{16 \left (a x +1\right )}+\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{8 a x -8}-\frac {a x +1}{16 \left (a x -1\right )}-\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\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 )^{2}}{4}}{a^{4}}\)	\(735\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a x +4}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{8 \left (a x +1\right )}-\frac {a x -1}{16 \left (a x +1\right )}+\frac {\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{8 a x -8}-\frac {a x +1}{16 \left (a x -1\right )}-\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\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 )^{2}}{4}}{a^{4}}\)	\(735\)
parts	\(\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2 a^{4}}-\frac {a \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5}}-\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3 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 {\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 ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{5}}-\frac {\operatorname {polylog}\left (3, -\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 )^{2}}{2 a^{5}}\right )}{2}\)	\(774\)

Input:

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

Output:

1/a^4*(-1/4*arctanh(a*x)^2/(a*x-1)+1/2*arctanh(a*x)^2*ln(a*x-1)+1/4*arctan 
h(a*x)^2/(a*x+1)+1/2*arctanh(a*x)^2*ln(a*x+1)-arctanh(a*x)^2*ln((a*x+1)/(- 
a^2*x^2+1)^(1/2))+1/3*arctanh(a*x)^3-1/8*arctanh(a*x)*(a*x-1)/(a*x+1)-1/16 
*(a*x-1)/(a*x+1)+1/8*arctanh(a*x)*(a*x+1)/(a*x-1)-1/16*(a*x+1)/(a*x-1)-arc 
tanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+1/2*polylog(3,-(a*x+1)^2/(-a^ 
2*x^2+1))-1/4*(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)^ 
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))*csgn(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)/(-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))^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*csgn(I*(a*x+1)^2/(a^2*x^2 
-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3-2*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)^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-3/4*a^3*integrate(x^3*log(a*x + 1)*log(-a*x + 1)/(a^7*x^4 - 2*a^5*x^2 + a 
^3), x) - 1/4*a^2*integrate(x^2*log(a*x + 1)*log(-a*x + 1)/(a^7*x^4 - 2*a^ 
5*x^2 + a^3), x) - 1/32*(a*(2/(a^7*x - a^6) - log(a*x + 1)/a^6 + log(a*x - 
 1)/a^6) + 4*log(-a*x + 1)/(a^7*x^2 - a^5))*a + 1/4*a*integrate(x*log(a*x 
+ 1)*log(-a*x + 1)/(a^7*x^4 - 2*a^5*x^2 + a^3), x) + 1/24*((a^2*x^2 - 1)*l 
og(-a*x + 1)^3 + 3*((a^2*x^2 - 1)*log(a*x + 1) - 1)*log(-a*x + 1)^2)/(a^6* 
x^2 - a^4) + 1/4*integrate(a^3*x^3*log(a*x + 1)^2/(a^7*x^4 - 2*a^5*x^2 + a 
^3), x) + 1/4*integrate(log(a*x + 1)*log(-a*x + 1)/(a^7*x^4 - 2*a^5*x^2 + 
a^3), x) + 1/4*integrate(log(-a*x + 1)/(a^7*x^4 - 2*a^5*x^2 + a^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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