\(\int \frac {x \text {arctanh}(a x)^2}{(1-a^2 x^2)^3} \, dx\) [310]

Optimal result

Integrand size = 20, antiderivative size = 125 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {3}{32 a^2 \left (1-a^2 x^2\right )}-\frac {x \text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3 x \text {arctanh}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)^2}{32 a^2}+\frac {\text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2} \] Output:

1/32/a^2/(-a^2*x^2+1)^2+3/32/a^2/(-a^2*x^2+1)-1/8*x*arctanh(a*x)/a/(-a^2*x 
^2+1)^2-3/16*x*arctanh(a*x)/a/(-a^2*x^2+1)-3/32*arctanh(a*x)^2/a^2+1/4*arc 
tanh(a*x)^2/a^2/(-a^2*x^2+1)^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {4-3 a^2 x^2+2 a x \left (-5+3 a^2 x^2\right ) \text {arctanh}(a x)+\left (5+6 a^2 x^2-3 a^4 x^4\right ) \text {arctanh}(a x)^2}{32 a^2 \left (-1+a^2 x^2\right )^2} \] Input:

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

Output:

(4 - 3*a^2*x^2 + 2*a*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x] + (5 + 6*a^2*x^2 - 3* 
a^4*x^4)*ArcTanh[a*x]^2)/(32*a^2*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6556, 6522, 6518, 241}

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.

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6522

\(\displaystyle \frac {\text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}}{2 a}\)

\(\Big \downarrow \) 6518

\(\displaystyle \frac {\text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}}{2 a}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {\text {arctanh}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}}{2 a}\)

Input:

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

Output:

ArcTanh[a*x]^2/(4*a^2*(1 - a^2*x^2)^2) - (-1/16*1/(a*(1 - a^2*x^2)^2) + (x 
*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*(-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)))/4)/(2*a)
 

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 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 6522

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo 
l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + 
 e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(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]), x], x]) /; Fre 
eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -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]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(-\frac {3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+4 a^{4} x^{4}-6 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-5 a^{2} x^{2}+10 a x \,\operatorname {arctanh}\left (a x \right )-5 \operatorname {arctanh}\left (a x \right )^{2}}{32 \left (a^{2} x^{2}-1\right )^{2} a^{2}}\)	\(90\)
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a^{2} x^{2}-1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}+\frac {3 \ln \left (a x -1\right )^{2}}{128}-\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {3 \ln \left (a x +1\right )^{2}}{128}-\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {1}{128 \left (a x -1\right )^{2}}-\frac {7}{128 \left (a x -1\right )}+\frac {1}{128 \left (a x +1\right )^{2}}+\frac {7}{128 \left (a x +1\right )}}{a^{2}}\)	\(197\)
default	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 \left (a^{2} x^{2}-1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}+\frac {3 \ln \left (a x -1\right )^{2}}{128}-\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {3 \ln \left (a x +1\right )^{2}}{128}-\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {1}{128 \left (a x -1\right )^{2}}-\frac {7}{128 \left (a x -1\right )}+\frac {1}{128 \left (a x +1\right )^{2}}+\frac {7}{128 \left (a x +1\right )}}{a^{2}}\)	\(197\)
parts	\(\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{2} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{2 a^{2}}\)	\(202\)
risch	\(-\frac {\left (3 a^{4} x^{4}-6 a^{2} x^{2}-5\right ) \ln \left (a x +1\right )^{2}}{128 a^{2} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}+\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-10 a x -5 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{64 a^{2} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}-\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+12 a^{3} x^{3} \ln \left (-a x +1\right )-6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+12 a^{2} x^{2}-20 a x \ln \left (-a x +1\right )-5 \ln \left (-a x +1\right )^{2}-16}{128 a^{2} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}\)	\(257\)
orering	\(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (60 a^{6} x^{6}-49 a^{4} x^{4}-34 a^{2} x^{2}-15\right ) \operatorname {arctanh}\left (a x \right )^{2}}{32 a^{2} \left (-a^{2} x^{2}+1\right )^{3}}+\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (24 a^{4} x^{4}-23 a^{2} x^{2}-10\right ) \left (\frac {\operatorname {arctanh}\left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {2 x \,\operatorname {arctanh}\left (a x \right ) a}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {6 x^{2} \operatorname {arctanh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{4}}\right )}{32 a^{2}}+\frac {x \left (4 a^{2} x^{2}-5\right ) \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (\frac {4 \,\operatorname {arctanh}\left (a x \right ) a}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {18 \operatorname {arctanh}\left (a x \right )^{2} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {2 x \,a^{2}}{\left (-a^{2} x^{2}+1\right )^{5}}+\frac {28 x^{2} \operatorname {arctanh}\left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{5}}+\frac {48 x^{3} \operatorname {arctanh}\left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{5}}\right )}{64 a^{2}}\)	\(306\)

Input:

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

Output:

-1/32*(3*a^4*x^4*arctanh(a*x)^2+4*a^4*x^4-6*a^3*x^3*arctanh(a*x)-6*a^2*x^2 
*arctanh(a*x)^2-5*a^2*x^2+10*a*x*arctanh(a*x)-5*arctanh(a*x)^2)/(a^2*x^2-1 
)^2/a^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {12 \, a^{2} x^{2} + {\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{128 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \] Input:

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

Output:

-1/128*(12*a^2*x^2 + (3*a^4*x^4 - 6*a^2*x^2 - 5)*log(-(a*x + 1)/(a*x - 1)) 
^2 - 4*(3*a^3*x^3 - 5*a*x)*log(-(a*x + 1)/(a*x - 1)) - 16)/(a^6*x^4 - 2*a^ 
4*x^2 + a^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.65 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {{\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )}{32 \, a} - \frac {12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16}{128 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} + \frac {\operatorname {artanh}\left (a x\right )^{2}}{4 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \] Input:

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

Output:

1/32*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 3 
*log(a*x - 1)/a)*arctanh(a*x)/a - 1/128*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x 
^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x 
- 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)/(a^6*x^4 - 2*a^4*x 
^2 + a^2) + 1/4*arctanh(a*x)^2/((a^2*x^2 - 1)^2*a^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (108) = 216\).

Time = 0.12 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.01 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{512} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} a \] Input:

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

Output:

-1/512*(2*((a*x - 1)^2*(4*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) - (a* 
x + 1)^2/((a*x - 1)^2*a^3) + 4*(a*x + 1)/((a*x - 1)*a^3))*log(-(a*x + 1)/( 
a*x - 1))^2 + 2*((a*x - 1)^2*(8*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) 
 + (a*x + 1)^2/((a*x - 1)^2*a^3) - 8*(a*x + 1)/((a*x - 1)*a^3))*log(-(a*x 
+ 1)/(a*x - 1)) + (a*x - 1)^2*(16*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^ 
3) - (a*x + 1)^2/((a*x - 1)^2*a^3) + 16*(a*x + 1)/((a*x - 1)*a^3))*a
 

Mupad [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.55 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx={\ln \left (a\,x+1\right )}^2\,\left (\frac {1}{16\,a^3\,\left (\frac {1}{a}-2\,a\,x^2+a^3\,x^4\right )}-\frac {3}{128\,a^2}\right )-{\ln \left (1-a\,x\right )}^2\,\left (\frac {3}{128\,a^2}-\frac {1}{4\,a^2\,\left (4\,a^4\,x^4-8\,a^2\,x^2+4\right )}\right )-\ln \left (1-a\,x\right )\,\left (\frac {\frac {1}{4\,a}-\frac {5\,x}{8}+\frac {3\,a^2\,x^3}{8}}{8\,a^5\,x^4-16\,a^3\,x^2+8\,a}-\frac {\frac {5\,x}{8}+\frac {1}{4\,a}-\frac {3\,a^2\,x^3}{8}}{8\,a^5\,x^4-16\,a^3\,x^2+8\,a}+\ln \left (a\,x+1\right )\,\left (\frac {1}{4\,a^2\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}-\frac {3\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}{32\,a^2\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}\right )\right )+\frac {\frac {2}{a^2}-\frac {3\,x^2}{2}}{16\,a^4\,x^4-32\,a^2\,x^2+16}-\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{32\,a^2}-\frac {3\,x^3}{32}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \] Input:

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

Output:

log(a*x + 1)^2*(1/(16*a^3*(1/a - 2*a*x^2 + a^3*x^4)) - 3/(128*a^2)) - log( 
1 - a*x)^2*(3/(128*a^2) - 1/(4*a^2*(4*a^4*x^4 - 8*a^2*x^2 + 4))) - log(1 - 
 a*x)*((1/(4*a) - (5*x)/8 + (3*a^2*x^3)/8)/(8*a - 16*a^3*x^2 + 8*a^5*x^4) 
- ((5*x)/8 + 1/(4*a) - (3*a^2*x^3)/8)/(8*a - 16*a^3*x^2 + 8*a^5*x^4) + log 
(a*x + 1)*(1/(4*a^2*(2*a^4*x^4 - 4*a^2*x^2 + 2)) - (3*(a^4*x^4 - 2*a^2*x^2 
 + 1))/(32*a^2*(2*a^4*x^4 - 4*a^2*x^2 + 2)))) + (2/a^2 - (3*x^2)/2)/(16*a^ 
4*x^4 - 32*a^2*x^2 + 16) - (log(a*x + 1)*((5*x)/(32*a^2) - (3*x^3)/32))/(1 
/a - 2*a*x^2 + a^3*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.72 \[ \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-6 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+12 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+10 \mathit {atanh} \left (a x \right )^{2}+12 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-20 \mathit {atanh} \left (a x \right ) a x -3 a^{4} x^{4}+5}{64 a^{2} \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 6*atanh(a*x)**2*a**4*x**4 + 12*atanh(a*x)**2*a**2*x**2 + 10*atanh(a*x) 
**2 + 12*atanh(a*x)*a**3*x**3 - 20*atanh(a*x)*a*x - 3*a**4*x**4 + 5)/(64*a 
**2*(a**4*x**4 - 2*a**2*x**2 + 1))