\(\int \frac {(1-a^2 x^2) \text {arctanh}(a x)^2}{x^5} \, dx\) [181]

Optimal result

Integrand size = 20, antiderivative size = 89 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \text {arctanh}(a x)}{6 x^3}+\frac {a^3 \text {arctanh}(a x)}{2 x}-\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x)+\frac {1}{6} a^4 \log \left (1-a^2 x^2\right ) \] Output:

-1/12*a^2/x^2-1/6*a*arctanh(a*x)/x^3+1/2*a^3*arctanh(a*x)/x-1/4*(-a^2*x^2+ 
1)^2*arctanh(a*x)^2/x^4-1/3*a^4*ln(x)+1/6*a^4*ln(-a^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=\frac {-a^2 x^2+\left (-2 a x+6 a^3 x^3\right ) \text {arctanh}(a x)-3 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2-4 a^4 x^4 \log (x)+2 a^4 x^4 \log \left (1-a^2 x^2\right )}{12 x^4} \] Input:

Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^5,x]
 

Output:

(-(a^2*x^2) + (-2*a*x + 6*a^3*x^3)*ArcTanh[a*x] - 3*(-1 + a^2*x^2)^2*ArcTa 
nh[a*x]^2 - 4*a^4*x^4*Log[x] + 2*a^4*x^4*Log[1 - a^2*x^2])/(12*x^4)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6570, 6576, 6452, 243, 47, 14, 16, 54, 2009}

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.

\(\displaystyle \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx\)

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

Int[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^5,x]
 

Output:

-1/4*((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^4 + (a*(-1/3*ArcTanh[a*x]/x^3 - a^ 
2*(-(ArcTanh[a*x]/x) + (a*(Log[x^2] - Log[1 - a^2*x^2]))/2) + (a*(-x^(-2) 
+ a^2*Log[x^2] - a^2*Log[1 - a^2*x^2]))/6))/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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 6570

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

rule 6576

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.27

method	result	size
parallelrisch	\(-\frac {3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+4 \ln \left (x \right ) a^{4} x^{4}-4 \ln \left (a x -1\right ) x^{4} a^{4}-4 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+a^{4} x^{4}-6 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+a^{2} x^{2}+2 a x \,\operatorname {arctanh}\left (a x \right )+3 \operatorname {arctanh}\left (a x \right )^{2}}{12 x^{4}}\)	\(113\)
derivativedivides	\(a^{4} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right )}{6 a^{3} x^{3}}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{16}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {1}{12 a^{2} x^{2}}-\frac {\ln \left (a x \right )}{3}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}\right )\)	\(172\)
default	\(a^{4} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right )}{6 a^{3} x^{3}}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{16}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {1}{12 a^{2} x^{2}}-\frac {\ln \left (a x \right )}{3}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}\right )\)	\(172\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 x^{4}}+\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2 x^{2}}+\frac {a^{3} \operatorname {arctanh}\left (a x \right )}{2 x}-\frac {a \,\operatorname {arctanh}\left (a x \right )}{6 x^{3}}+\frac {a^{4} \operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {a^{4} \operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {a^{4} \left (-\frac {3 \ln \left (a x -1\right )^{2}}{4}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {3 \ln \left (a x +1\right )^{2}}{4}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {1}{a^{2} x^{2}}+4 \ln \left (a x \right )-2 \ln \left (a x -1\right )-2 \ln \left (a x +1\right )\right )}{12}\)	\(174\)
risch	\(-\frac {\left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) \ln \left (a x +1\right )^{2}}{16 x^{4}}+\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-2 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{24 x^{4}}-\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+16 \ln \left (x \right ) a^{4} x^{4}-8 \ln \left (-a^{2} x^{2}+1\right ) a^{4} x^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )-6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2}-4 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}}{48 x^{4}}\)	\(209\)

Input:

int((-a^2*x^2+1)*arctanh(a*x)^2/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/12*(3*a^4*x^4*arctanh(a*x)^2+4*ln(x)*a^4*x^4-4*ln(a*x-1)*x^4*a^4-4*a^4* 
x^4*arctanh(a*x)+a^4*x^4-6*a^3*x^3*arctanh(a*x)-6*a^2*x^2*arctanh(a*x)^2+a 
^2*x^2+2*a*x*arctanh(a*x)+3*arctanh(a*x)^2)/x^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.21 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=\frac {8 \, a^{4} x^{4} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{4} x^{4} \log \left (x\right ) - 4 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{3} x^{3} - a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, x^{4}} \] Input:

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^5,x, algorithm="fricas")
 

Output:

1/48*(8*a^4*x^4*log(a^2*x^2 - 1) - 16*a^4*x^4*log(x) - 4*a^2*x^2 - 3*(a^4* 
x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 + 4*(3*a^3*x^3 - a*x)*log 
(-(a*x + 1)/(a*x - 1)))/x^4
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=\begin {cases} - \frac {a^{4} \log {\left (x \right )}}{3} + \frac {a^{4} \log {\left (x - \frac {1}{a} \right )}}{3} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} + \frac {a^{4} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {a^{3} \operatorname {atanh}{\left (a x \right )}}{2 x} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2 x^{2}} - \frac {a^{2}}{12 x^{2}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{4 x^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate((-a**2*x**2+1)*atanh(a*x)**2/x**5,x)
 

Output:

Piecewise((-a**4*log(x)/3 + a**4*log(x - 1/a)/3 - a**4*atanh(a*x)**2/4 + a 
**4*atanh(a*x)/3 + a**3*atanh(a*x)/(2*x) + a**2*atanh(a*x)**2/(2*x**2) - a 
**2/(12*x**2) - a*atanh(a*x)/(6*x**3) - atanh(a*x)**2/(4*x**4), Ne(a, 0)), 
 (0, True))
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (76) = 152\).

Time = 0.03 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.84 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=-\frac {1}{48} \, {\left (16 \, a^{2} \log \left (x\right ) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 8 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 4 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) - 4}{x^{2}}\right )} a^{2} - \frac {1}{12} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{4 \, x^{4}} \] Input:

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^5,x, algorithm="maxima")
 

Output:

-1/48*(16*a^2*log(x) - (3*a^2*x^2*log(a*x + 1)^2 + 3*a^2*x^2*log(a*x - 1)^ 
2 + 8*a^2*x^2*log(a*x - 1) - 2*(3*a^2*x^2*log(a*x - 1) - 4*a^2*x^2)*log(a* 
x + 1) - 4)/x^2)*a^2 - 1/12*(3*a^3*log(a*x + 1) - 3*a^3*log(a*x - 1) - 2*( 
3*a^2*x^2 - 1)/x^3)*a*arctanh(a*x) + 1/4*(2*a^2*x^2 - 1)*arctanh(a*x)^2/x^ 
4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (76) = 152\).

Time = 0.12 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.17 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=-\frac {1}{3} \, {\left (a^{3} \log \left (-\frac {a x + 1}{a x - 1} - 1\right ) - a^{3} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x + 1\right )}^{2} a^{3} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x - 1\right )}^{2} {\left (\frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {4 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} - \frac {{\left (a x + 1\right )} a^{3}}{{\left (a x - 1\right )} {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \] Input:

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^5,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.09 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.76 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx={\ln \left (1-a\,x\right )}^2\,\left (\frac {\frac {a^2\,x^2}{2}-\frac {1}{4}}{4\,x^4}-\frac {a^4}{16}\right )-\ln \left (1-a\,x\right )\,\left (\ln \left (a\,x+1\right )\,\left (\frac {\frac {a^2\,x^2}{2}-\frac {1}{4}}{2\,x^4}-\frac {a^4}{8}\right )+\frac {3\,a^5\,x-2\,a^4}{24\,a^3\,x^3}-\frac {3\,x\,a^5+2\,a^4}{24\,a^3\,x^3}-\frac {a\,\left (22\,a^3\,x^3-12\,a^2\,x^2+6\,a\,x-4\right )}{96\,x^3}+\frac {a\,\left (44\,a^3\,x^3+24\,a^2\,x^2+12\,a\,x+8\right )}{192\,x^3}\right )-\frac {a^4\,\ln \left (x\right )}{3}+{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {a^2\,x^2}{8}-\frac {1}{16}}{x^4}-\frac {a^4}{16}\right )+\frac {a^4\,\ln \left (a^2\,x^2-1\right )}{6}-\frac {a^2}{12\,x^2}+\frac {a\,\ln \left (a\,x+1\right )\,\left (\frac {a^2\,x^2}{4}-\frac {1}{12}\right )}{x^3} \] Input:

int(-(atanh(a*x)^2*(a^2*x^2 - 1))/x^5,x)
 

Output:

log(1 - a*x)^2*(((a^2*x^2)/2 - 1/4)/(4*x^4) - a^4/16) - log(1 - a*x)*(log( 
a*x + 1)*(((a^2*x^2)/2 - 1/4)/(2*x^4) - a^4/8) + (3*a^5*x - 2*a^4)/(24*a^3 
*x^3) - (3*a^5*x + 2*a^4)/(24*a^3*x^3) - (a*(6*a*x - 12*a^2*x^2 + 22*a^3*x 
^3 - 4))/(96*x^3) + (a*(12*a*x + 24*a^2*x^2 + 44*a^3*x^3 + 8))/(192*x^3)) 
- (a^4*log(x))/3 + log(a*x + 1)^2*(((a^2*x^2)/8 - 1/16)/x^4 - a^4/16) + (a 
^4*log(a^2*x^2 - 1))/6 - a^2/(12*x^2) + (a*log(a*x + 1)*((a^2*x^2)/4 - 1/1 
2))/x^3
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^5} \, dx=\frac {-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+6 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-3 \mathit {atanh} \left (a x \right )^{2}+4 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+6 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-2 \mathit {atanh} \left (a x \right ) a x +4 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{4} x^{4}-a^{2} x^{2}}{12 x^{4}} \] Input:

int((-a^2*x^2+1)*atanh(a*x)^2/x^5,x)
 

Output:

( - 3*atanh(a*x)**2*a**4*x**4 + 6*atanh(a*x)**2*a**2*x**2 - 3*atanh(a*x)** 
2 + 4*atanh(a*x)*a**4*x**4 + 6*atanh(a*x)*a**3*x**3 - 2*atanh(a*x)*a*x + 4 
*log(a**2*x - a)*a**4*x**4 - 4*log(x)*a**4*x**4 - a**2*x**2)/(12*x**4)