\(\int \frac {\text {arctanh}(a x)}{x^2 (1-a^2 x^2)^2} \, dx\) [264]

Optimal result

Integrand size = 20, antiderivative size = 82 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=-\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{x}+\frac {a^2 x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {3}{4} a \text {arctanh}(a x)^2+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{4} \left (-\frac {2 \left (-2+3 a^2 x^2\right ) \text {arctanh}(a x)}{x \left (-1+a^2 x^2\right )}+3 a \text {arctanh}(a x)^2+a \left (\frac {1}{-1+a^2 x^2}+4 \log (a x)-2 \log \left (1-a^2 x^2\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6592, 6518, 241, 6544, 6452, 243, 47, 14, 16, 6510}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6510

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b 
, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && 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 6544

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

rule 6592

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

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.55

method	result	size
parallelrisch	\(\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+4 \ln \left (x \right ) a^{3} x^{3}-4 \ln \left (a x -1\right ) x^{3} a^{3}-4 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+a^{3} x^{3}-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-3 \operatorname {arctanh}\left (a x \right )^{2} a x -4 a \ln \left (x \right ) x +4 \ln \left (a x -1\right ) a x +4 a x \,\operatorname {arctanh}\left (a x \right )+4 \,\operatorname {arctanh}\left (a x \right )}{4 \left (a^{2} x^{2}-1\right ) x}\)	\(127\)
derivativedivides	\(a \left (-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {3 \ln \left (a x -1\right )^{2}}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {3 \ln \left (a x +1\right )^{2}}{16}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )}{2}-\frac {1}{8 \left (a x +1\right )}-\frac {\ln \left (a x -1\right )}{2}+\frac {1}{8 a x -8}+\ln \left (a x \right )\right )\)	\(164\)
default	\(a \left (-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {3 \ln \left (a x -1\right )^{2}}{16}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {3 \ln \left (a x +1\right )^{2}}{16}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )}{2}-\frac {1}{8 \left (a x +1\right )}-\frac {\ln \left (a x -1\right )}{2}+\frac {1}{8 a x -8}+\ln \left (a x \right )\right )\)	\(164\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{x}-\frac {\operatorname {arctanh}\left (a x \right ) a}{4 \left (a x +1\right )}+\frac {3 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right ) a}{4 \left (a x -1\right )}-\frac {3 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {a \left (\frac {3 \ln \left (a x -1\right )^{2}}{4}-\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {3 \ln \left (a x +1\right )^{2}}{4}-\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-4 \ln \left (x \right )+2 \ln \left (a x +1\right )+\frac {1}{2 a x +2}+2 \ln \left (a x -1\right )-\frac {1}{2 \left (a x -1\right )}\right )}{4}\)	\(167\)
risch	\(\frac {3 a \ln \left (a x +1\right )^{2}}{16}-\frac {\left (3 a^{3} x^{3} \ln \left (-a x +1\right )+6 a^{2} x^{2}-3 a x \ln \left (-a x +1\right )-4\right ) \ln \left (a x +1\right )}{8 \left (a^{2} x^{2}-1\right ) x}+\frac {3 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+16 \ln \left (x \right ) a^{3} x^{3}-8 \ln \left (a^{2} x^{2}-1\right ) a^{3} x^{3}+12 x^{2} \ln \left (-a x +1\right ) a^{2}-3 a \ln \left (-a x +1\right )^{2} x -16 a \ln \left (x \right ) x +8 a \ln \left (a^{2} x^{2}-1\right ) x +4 a x -8 \ln \left (-a x +1\right )}{16 \left (a^{2} x^{2}-1\right ) x}\)	\(194\)

Input:

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

Output:

1/4*(3*arctanh(a*x)^2*a^3*x^3+4*ln(x)*a^3*x^3-4*ln(a*x-1)*x^3*a^3-4*a^3*x^ 
3*arctanh(a*x)+a^3*x^3-6*a^2*x^2*arctanh(a*x)-3*arctanh(a*x)^2*a*x-4*a*ln( 
x)*x+4*ln(a*x-1)*a*x+4*a*x*arctanh(a*x)+4*arctanh(a*x))/(a^2*x^2-1)/x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.44 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {3 \, {\left (a^{3} x^{3} - a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, a x - 8 \, {\left (a^{3} x^{3} - a x\right )} \log \left (a^{2} x^{2} - 1\right ) + 16 \, {\left (a^{3} x^{3} - a x\right )} \log \left (x\right ) - 4 \, {\left (3 \, a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{16 \, {\left (a^{2} x^{3} - x\right )}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (68) = 136\).

Time = 0.89 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.09 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\begin {cases} \frac {4 a^{3} x^{3} \log {\left (x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a^{3} x^{3} \log {\left (x - \frac {1}{a} \right )}}{4 a^{2} x^{3} - 4 x} + \frac {3 a^{3} x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a^{3} x^{3} \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {6 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a x \log {\left (x \right )}}{4 a^{2} x^{3} - 4 x} + \frac {4 a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{2} x^{3} - 4 x} - \frac {3 a x \operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} + \frac {4 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} + \frac {a x}{4 a^{2} x^{3} - 4 x} + \frac {4 \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((4*a**3*x**3*log(x)/(4*a**2*x**3 - 4*x) - 4*a**3*x**3*log(x - 1/ 
a)/(4*a**2*x**3 - 4*x) + 3*a**3*x**3*atanh(a*x)**2/(4*a**2*x**3 - 4*x) - 4 
*a**3*x**3*atanh(a*x)/(4*a**2*x**3 - 4*x) - 6*a**2*x**2*atanh(a*x)/(4*a**2 
*x**3 - 4*x) - 4*a*x*log(x)/(4*a**2*x**3 - 4*x) + 4*a*x*log(x - 1/a)/(4*a* 
*2*x**3 - 4*x) - 3*a*x*atanh(a*x)**2/(4*a**2*x**3 - 4*x) + 4*a*x*atanh(a*x 
)/(4*a**2*x**3 - 4*x) + a*x/(4*a**2*x**3 - 4*x) + 4*atanh(a*x)/(4*a**2*x** 
3 - 4*x), Ne(a, 0)), (0, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (72) = 144\).

Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.83 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{16} \, a {\left (\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4}{a^{2} x^{2} - 1} + 8 \, \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right ) - 16 \, \log \left (x\right )\right )} + \frac {1}{4} \, {\left (3 \, a \log \left (a x + 1\right ) - 3 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 2\right )}}{a^{2} x^{3} - x}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

-1/16*a*((3*(a^2*x^2 - 1)*log(a*x + 1)^2 - 6*(a^2*x^2 - 1)*log(a*x + 1)*lo 
g(a*x - 1) + 3*(a^2*x^2 - 1)*log(a*x - 1)^2 - 4)/(a^2*x^2 - 1) + 8*log(a*x 
 + 1) + 8*log(a*x - 1) - 16*log(x)) + 1/4*(3*a*log(a*x + 1) - 3*a*log(a*x 
- 1) - 2*(3*a^2*x^2 - 2)/(a^2*x^3 - x))*arctanh(a*x)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.84 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.61 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {3\,a\,{\ln \left (a\,x+1\right )}^2}{16}+\frac {3\,a\,{\ln \left (1-a\,x\right )}^2}{16}+\frac {a}{2\,\left (2\,a^2\,x^2-2\right )}-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}+a\,\ln \left (x\right )-\ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,a^2\,x^2}{2}-1}{2\,x-2\,a^2\,x^3}+\frac {3\,a\,\ln \left (a\,x+1\right )}{8}\right )+\frac {\ln \left (a\,x+1\right )\,\left (\frac {3\,a\,x^2}{4}-\frac {1}{2\,a}\right )}{\frac {x}{a}-a\,x^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {3 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {atanh} \left (a x \right )^{2} a x -4 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-6 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+4 \mathit {atanh} \left (a x \right ) a x +4 \mathit {atanh} \left (a x \right )-4 \,\mathrm {log}\left (a^{2} x -a \right ) a^{3} x^{3}+4 \,\mathrm {log}\left (a^{2} x -a \right ) a x +4 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-4 \,\mathrm {log}\left (x \right ) a x +a^{3} x^{3}}{4 x \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

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