\(\int \frac {x^2 \text {arctanh}(a x)^2}{(1-a^2 x^2)^3} \, dx\) [309]

Optimal result

Integrand size = 22, antiderivative size = 163 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{64 a^3}-\frac {\text {arctanh}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\text {arctanh}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \text {arctanh}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^3}{24 a^3} \] Output:

1/32*x/a^2/(-a^2*x^2+1)^2-1/64*x/a^2/(-a^2*x^2+1)-1/64*arctanh(a*x)/a^3-1/ 
8*arctanh(a*x)/a^3/(-a^2*x^2+1)^2+1/8*arctanh(a*x)/a^3/(-a^2*x^2+1)+1/4*x* 
arctanh(a*x)^2/a^2/(-a^2*x^2+1)^2-1/8*x*arctanh(a*x)^2/a^2/(-a^2*x^2+1)-1/ 
24*arctanh(a*x)^3/a^3
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {6 a x \left (1+a^2 x^2\right )-48 a^2 x^2 \text {arctanh}(a x)+48 \left (a x+a^3 x^3\right ) \text {arctanh}(a x)^2-16 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^3+3 \left (-1+a^2 x^2\right )^2 \log (1-a x)-3 \left (-1+a^2 x^2\right )^2 \log (1+a x)}{384 a^3 \left (-1+a^2 x^2\right )^2} \] Input:

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

Output:

(6*a*x*(1 + a^2*x^2) - 48*a^2*x^2*ArcTanh[a*x] + 48*(a*x + a^3*x^3)*ArcTan 
h[a*x]^2 - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x]^3 + 3*(-1 + a^2*x^2)^2*Log[1 - 
 a*x] - 3*(-1 + a^2*x^2)^2*Log[1 + a*x])/(384*a^3*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6590, 6518, 6526, 215, 215, 219, 6518, 6556, 215, 219}

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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 6526

\(\displaystyle \frac {\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {\frac {3}{4} \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{a^2}-\frac {\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}}{a^2}\)

Input:

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

Output:

-(((x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTan 
h[a*x]/(2*a^2*(1 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/ 
(2*a)))/a^2) + (-1/8*ArcTanh[a*x]/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x]^2) 
/(4*(1 - a^2*x^2)^2) + (x/(4*(1 - a^2*x^2)^2) + (3*(x/(2*(1 - a^2*x^2)) + 
ArcTanh[a*x]/(2*a)))/4)/8 + (3*((x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + Arc 
Tanh[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))))/4)/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 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 6526

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 
*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p 
/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1 
)*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2))   Int 
[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
 

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]
 

Maple [A] (verified)

Time = 50.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(-\frac {8 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{4}+3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-24 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}-16 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}-3 a^{3} x^{3}+18 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-24 \operatorname {arctanh}\left (a x \right )^{2} a x +8 \operatorname {arctanh}\left (a x \right )^{3}-3 a x +3 \,\operatorname {arctanh}\left (a x \right )}{192 \left (a^{2} x^{2}-1\right )^{2} a^{3}}\)	\(120\)
risch	\(-\frac {\ln \left (a x +1\right )^{3}}{192 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{64 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (a^{4} x^{4} \ln \left (-a x +1\right )^{2}+4 a^{3} x^{3} \ln \left (-a x +1\right )-2 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}\right ) \ln \left (a x +1\right )}{64 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 x^{4} \ln \left (-a x +1\right ) a^{4}-3 \ln \left (a x +1\right ) a^{4} x^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 a^{3} x^{3}+18 x^{2} \ln \left (-a x +1\right ) a^{2}+6 \ln \left (a x +1\right ) a^{2} x^{2}+12 a \ln \left (-a x +1\right )^{2} x +2 \ln \left (-a x +1\right )^{3}+6 a x +3 \ln \left (-a x +1\right )-3 \ln \left (a x +1\right )}{384 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(394\)
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a x +16}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{16}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a x -16}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{16}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-\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 )^{2}}{16}+\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 )^{2}}{32}+\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 )^{2}}{32}+\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 )^{2}}{16}+\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 )^{2}}{16}-\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 )^{2}}{32}+\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 )^{2}}{32}-\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 )^{2}}{32}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{16}+\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 )^{2}}{32}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{24}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )^{2}}{128 \left (a x -1\right )^{2}}+\frac {\left (a x +1\right )^{2}}{512 \left (a x -1\right )^{2}}-\frac {\left (a x -1\right )^{2} \operatorname {arctanh}\left (a x \right )}{128 \left (a x +1\right )^{2}}-\frac {\left (a x -1\right )^{2}}{512 \left (a x +1\right )^{2}}}{a^{3}}\)	\(765\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a x +16}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{16}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a x -1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{16 a x -16}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{16}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-\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 )^{2}}{16}+\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 )^{2}}{32}+\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 )^{2}}{32}+\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 )^{2}}{16}+\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 )^{2}}{16}-\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 )^{2}}{32}+\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 )^{2}}{32}-\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 )^{2}}{32}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{2}}{16}+\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 )^{2}}{32}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{24}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )^{2}}{128 \left (a x -1\right )^{2}}+\frac {\left (a x +1\right )^{2}}{512 \left (a x -1\right )^{2}}-\frac {\left (a x -1\right )^{2} \operatorname {arctanh}\left (a x \right )}{128 \left (a x +1\right )^{2}}-\frac {\left (a x -1\right )^{2}}{512 \left (a x +1\right )^{2}}}{a^{3}}\)	\(765\)
parts	\(\text {Expression too large to display}\)	\(831\)

Input:

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

Output:

-1/192*(8*arctanh(a*x)^3*a^4*x^4+3*a^4*x^4*arctanh(a*x)-24*arctanh(a*x)^2* 
a^3*x^3-16*arctanh(a*x)^3*a^2*x^2-3*a^3*x^3+18*a^2*x^2*arctanh(a*x)-24*arc 
tanh(a*x)^2*a*x+8*arctanh(a*x)^3-3*a*x+3*arctanh(a*x))/(a^2*x^2-1)^2/a^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, a x - 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{384 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \] Input:

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

Output:

1/384*(6*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^3 
 + 12*(a^3*x^3 + a*x)*log(-(a*x + 1)/(a*x - 1))^2 + 6*a*x - 3*(a^4*x^4 + 6 
*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^4 - 2*a^5*x^2 + a^3)
 

Sympy [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \] Input:

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

Output:

-Integral(x**2*atanh(a*x)**2/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), 
x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (141) = 282\).

Time = 0.04 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.38 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 6 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{384 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )}} - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a \operatorname {artanh}\left (a x\right )}{32 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \] Input:

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

Output:

1/16*(2*(a^2*x^3 + x)/(a^6*x^4 - 2*a^4*x^2 + a^2) - log(a*x + 1)/a^3 + log 
(a*x - 1)/a^3)*arctanh(a*x)^2 + 1/384*(6*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 
+ 1)*log(a*x + 1)^3 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*log(a*x - 
 1) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 + 6*a*x - 3*(a^4*x^4 - 2* 
a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 + 1)*log(a*x + 1) + 3 
*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a^2/(a^9*x^4 - 2*a^7*x^2 + a^5) - 
 1/32*(4*a^2*x^2 - (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 2*(a^4*x^4 - 
 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - (a^4*x^4 - 2*a^2*x^2 + 1)*log( 
a*x - 1)^2)*a*arctanh(a*x)/(a^8*x^4 - 2*a^6*x^2 + a^4)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.81 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.15 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,a\,x^3}{2}-\frac {x}{2\,a}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {\frac {x}{2\,a}-\frac {3\,a\,x^3}{2}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {{\ln \left (a\,x+1\right )}^2}{64\,a^3}-\frac {\ln \left (a\,x+1\right )\,\left (2\,a^2\,x^3+2\,x\right )}{32\,a^6\,x^4-64\,a^4\,x^2+32\,a^2}\right )+\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\ln \left (a\,x+1\right )}{64\,a^3}-\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{4\,a^4\,x^4-8\,a^2\,x^2+4}\right )-\frac {{\ln \left (a\,x+1\right )}^3}{192\,a^3}+\frac {{\ln \left (1-a\,x\right )}^3}{192\,a^3}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (\frac {x}{32\,a^3}+\frac {x^3}{32\,a}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}-\frac {x^2\,\ln \left (a\,x+1\right )}{16\,a^2\,\left (\frac {1}{a}-2\,a\,x^2+a^3\,x^4\right )}+\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3} \] Input:

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

Output:

log(1 - a*x)*(((3*a*x^3)/2 - x/(2*a) + x^2)/(32*a - 64*a^3*x^2 + 32*a^5*x^ 
4) + (x/(2*a) - (3*a*x^3)/2 + x^2)/(32*a - 64*a^3*x^2 + 32*a^5*x^4) + log( 
a*x + 1)^2/(64*a^3) - (log(a*x + 1)*(2*x + 2*a^2*x^3))/(32*a^2 - 64*a^4*x^ 
2 + 32*a^6*x^4)) + (x/(8*a^2) + x^3/8)/(8*a^4*x^4 - 16*a^2*x^2 + 8) - log( 
1 - a*x)^2*(log(a*x + 1)/(64*a^3) - (x/(8*a^2) + x^3/8)/(4*a^4*x^4 - 8*a^2 
*x^2 + 4)) - log(a*x + 1)^3/(192*a^3) + log(1 - a*x)^3/(192*a^3) + (atan(a 
*x*1i)*1i)/(64*a^3) + (log(a*x + 1)^2*(x/(32*a^3) + x^3/(32*a)))/(1/a - 2* 
a*x^2 + a^3*x^4) - (x^2*log(a*x + 1))/(16*a^2*(1/a - 2*a*x^2 + a^3*x^4))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.26 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-16 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+32 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-16 \mathit {atanh} \left (a x \right )^{3}+48 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}+48 \mathit {atanh} \left (a x \right )^{2} a x -24 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-24 \mathit {atanh} \left (a x \right )-9 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}+18 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}-9 \,\mathrm {log}\left (a^{2} x -a \right )+9 \,\mathrm {log}\left (a^{2} x +a \right ) a^{4} x^{4}-18 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}+9 \,\mathrm {log}\left (a^{2} x +a \right )+6 a^{3} x^{3}+6 a x}{384 a^{3} \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 16*atanh(a*x)**3*a**4*x**4 + 32*atanh(a*x)**3*a**2*x**2 - 16*atanh(a*x 
)**3 + 48*atanh(a*x)**2*a**3*x**3 + 48*atanh(a*x)**2*a*x - 24*atanh(a*x)*a 
**4*x**4 - 24*atanh(a*x) - 9*log(a**2*x - a)*a**4*x**4 + 18*log(a**2*x - a 
)*a**2*x**2 - 9*log(a**2*x - a) + 9*log(a**2*x + a)*a**4*x**4 - 18*log(a** 
2*x + a)*a**2*x**2 + 9*log(a**2*x + a) + 6*a**3*x**3 + 6*a*x)/(384*a**3*(a 
**4*x**4 - 2*a**2*x**2 + 1))