\(\int \frac {(1-a^2 x^2) \text {arctanh}(a x)^2}{x^6} \, dx\) [182]

Optimal result

Integrand size = 20, antiderivative size = 143 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^6} \, dx=-\frac {a^2}{30 x^3}+\frac {a^4}{30 x}-\frac {1}{30} a^5 \text {arctanh}(a x)-\frac {a \text {arctanh}(a x)}{10 x^4}+\frac {2 a^3 \text {arctanh}(a x)}{15 x^2}-\frac {2}{15} a^5 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{5 x^5}+\frac {a^2 \text {arctanh}(a x)^2}{3 x^3}-\frac {4}{15} a^5 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {2}{15} a^5 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

-1/30*a^2/x^3+1/30*a^4/x-1/30*a^5*arctanh(a*x)-1/10*a*arctanh(a*x)/x^4+2/1 
5*a^3*arctanh(a*x)/x^2-2/15*a^5*arctanh(a*x)^2-1/5*arctanh(a*x)^2/x^5+1/3* 
a^2*arctanh(a*x)^2/x^3-4/15*a^5*arctanh(a*x)*ln(2-2/(a*x+1))+2/15*a^5*poly 
log(2,-1+2/(a*x+1))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^6} \, dx=\frac {a^2 x^2 \left (-1+a^2 x^2\right )-2 \left (3-5 a^2 x^2+2 a^5 x^5\right ) \text {arctanh}(a x)^2-a x \text {arctanh}(a x) \left (3-4 a^2 x^2+a^4 x^4+8 a^4 x^4 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )+4 a^5 x^5 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )}{30 x^5} \] Input:

Integrate[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^6,x]
 

Output:

(a^2*x^2*(-1 + a^2*x^2) - 2*(3 - 5*a^2*x^2 + 2*a^5*x^5)*ArcTanh[a*x]^2 - a 
*x*ArcTanh[a*x]*(3 - 4*a^2*x^2 + a^4*x^4 + 8*a^4*x^4*Log[1 - E^(-2*ArcTanh 
[a*x])]) + 4*a^5*x^5*PolyLog[2, E^(-2*ArcTanh[a*x])])/(30*x^5)
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.71, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6576, 6452, 6544, 6452, 264, 219, 264, 219, 6544, 6452, 264, 219, 6550, 6494, 2897}

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.

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

\(\Big \downarrow \) 6576

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^6}dx-a^2 \int \frac {\text {arctanh}(a x)^2}{x^4}dx\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {2}{5} a \int \frac {\text {arctanh}(a x)}{x^5 \left (1-a^2 x^2\right )}dx-\left (a^2 \left (\frac {2}{3} a \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6544

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\int \frac {\text {arctanh}(a x)}{x^3}dx\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\int \frac {\text {arctanh}(a x)}{x^5}dx\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6452

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 264

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx-\frac {1}{x}\right )-\frac {\text {arctanh}(a x)}{2 x^2}\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \left (a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \left (a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 264

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \left (a^2 \left (a^2 \int \frac {1}{1-a^2 x^2}dx-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6544

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\int \frac {\text {arctanh}(a x)}{x^3}dx\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6452

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 264

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx-\frac {1}{x}\right )-\frac {\text {arctanh}(a x)}{2 x^2}\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6550

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 6494

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

\(\Big \downarrow \) 2897

\(\displaystyle -\left (a^2 \left (\frac {2}{3} a \left (a^2 \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )-\frac {\text {arctanh}(a x)^2}{3 x^3}\right )\right )+\frac {2}{5} a \left (a^2 \left (a^2 \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a \left (a \text {arctanh}(a x)-\frac {1}{x}\right )\right )+\frac {1}{4} a \left (a^2 \left (a \text {arctanh}(a x)-\frac {1}{x}\right )-\frac {1}{3 x^3}\right )-\frac {\text {arctanh}(a x)}{4 x^4}\right )-\frac {\text {arctanh}(a x)^2}{5 x^5}\)

Input:

Int[((1 - a^2*x^2)*ArcTanh[a*x]^2)/x^6,x]
 

Output:

-1/5*ArcTanh[a*x]^2/x^5 - a^2*(-1/3*ArcTanh[a*x]^2/x^3 + (2*a*(-1/2*ArcTan 
h[a*x]/x^2 + (a*(-x^(-1) + a*ArcTanh[a*x]))/2 + a^2*(ArcTanh[a*x]^2/2 + Ar 
cTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)))/3) + ( 
2*a*(-1/4*ArcTanh[a*x]/x^4 + (a*(-1/3*1/x^3 + a^2*(-x^(-1) + a*ArcTanh[a*x 
])))/4 + a^2*(-1/2*ArcTanh[a*x]/x^2 + (a*(-x^(-1) + a*ArcTanh[a*x]))/2 + a 
^2*(ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 
 2/(1 + a*x)]/2))))/5
 

Defintions of rubi rules used

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 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 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 6550

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

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 = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(a^{5} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 a^{5} x^{5}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{10 a^{4} x^{4}}+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{15 a^{2} x^{2}}-\frac {4 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {1}{30 a^{3} x^{3}}+\frac {1}{30 a x}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}+\frac {2 \operatorname {dilog}\left (a x \right )}{15}+\frac {2 \operatorname {dilog}\left (a x +1\right )}{15}+\frac {2 \ln \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}\right )\)	\(219\)
default	\(a^{5} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 a^{5} x^{5}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{10 a^{4} x^{4}}+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{15 a^{2} x^{2}}-\frac {4 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {1}{30 a^{3} x^{3}}+\frac {1}{30 a x}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}+\frac {2 \operatorname {dilog}\left (a x \right )}{15}+\frac {2 \operatorname {dilog}\left (a x +1\right )}{15}+\frac {2 \ln \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}\right )\)	\(219\)
parts	\(\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{3 x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 x^{5}}-\frac {a \,\operatorname {arctanh}\left (a x \right )}{10 x^{4}}+\frac {2 a^{3} \operatorname {arctanh}\left (a x \right )}{15 x^{2}}-\frac {4 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}+\frac {2 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {a^{5} \left (-4 \operatorname {dilog}\left (a x \right )-4 \operatorname {dilog}\left (a x +1\right )-4 \ln \left (a x \right ) \ln \left (a x +1\right )-\ln \left (a x -1\right )^{2}+4 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+2 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-2 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )+\ln \left (a x +1\right )^{2}+\frac {1}{a^{3} x^{3}}-\frac {1}{a x}-\frac {\ln \left (a x -1\right )}{2}+\frac {\ln \left (a x +1\right )}{2}\right )}{30}\)	\(222\)

Input:

int((-a^2*x^2+1)*arctanh(a*x)^2/x^6,x,method=_RETURNVERBOSE)
 

Output:

a^5*(-1/5*arctanh(a*x)^2/a^5/x^5+1/3*arctanh(a*x)^2/a^3/x^3-1/10*arctanh(a 
*x)/a^4/x^4+2/15*arctanh(a*x)/a^2/x^2-4/15*arctanh(a*x)*ln(a*x)+2/15*arcta 
nh(a*x)*ln(a*x-1)+2/15*arctanh(a*x)*ln(a*x+1)-1/30/a^3/x^3+1/30/a/x+1/60*l 
n(a*x-1)-1/60*ln(a*x+1)+2/15*dilog(a*x)+2/15*dilog(a*x+1)+2/15*ln(a*x)*ln( 
a*x+1)+1/30*ln(a*x-1)^2-2/15*dilog(1/2*a*x+1/2)-1/15*ln(a*x-1)*ln(1/2*a*x+ 
1/2)+1/15*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/30*ln(a*x+1)^2)
 

Fricas [F]

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

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^6,x, algorithm="fricas")
 

Output:

integral(-(a^2*x^2 - 1)*arctanh(a*x)^2/x^6, x)
 

Sympy [F]

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

integrate((-a**2*x**2+1)*atanh(a*x)**2/x**6,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.59 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^6} \, dx=-\frac {1}{60} \, {\left (8 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a^{3} - 8 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a^{3} + 8 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a^{3} + a^{3} \log \left (a x + 1\right ) - a^{3} \log \left (a x - 1\right ) + \frac {2 \, {\left (a^{3} x^{3} \log \left (a x + 1\right )^{2} - 2 \, a^{3} x^{3} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a^{3} x^{3} \log \left (a x - 1\right )^{2} - a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a^{2} + \frac {1}{30} \, {\left (4 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{4} \log \left (x^{2}\right ) + \frac {4 \, a^{2} x^{2} - 3}{x^{4}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (5 \, a^{2} x^{2} - 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{15 \, x^{5}} \] Input:

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^6,x, algorithm="maxima")
 

Output:

-1/60*(8*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a^3 - 8 
*(log(a*x + 1)*log(x) + dilog(-a*x))*a^3 + 8*(log(-a*x + 1)*log(x) + dilog 
(a*x))*a^3 + a^3*log(a*x + 1) - a^3*log(a*x - 1) + 2*(a^3*x^3*log(a*x + 1) 
^2 - 2*a^3*x^3*log(a*x + 1)*log(a*x - 1) - a^3*x^3*log(a*x - 1)^2 - a^2*x^ 
2 + 1)/x^3)*a^2 + 1/30*(4*a^4*log(a^2*x^2 - 1) - 4*a^4*log(x^2) + (4*a^2*x 
^2 - 3)/x^4)*a*arctanh(a*x) + 1/15*(5*a^2*x^2 - 3)*arctanh(a*x)^2/x^5
 

Giac [F]

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

integrate((-a^2*x^2+1)*arctanh(a*x)^2/x^6,x, algorithm="giac")
 

Output:

integrate(-(a^2*x^2 - 1)*arctanh(a*x)^2/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^6} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^6} \,d x \] Input:

int(-(atanh(a*x)^2*(a^2*x^2 - 1))/x^6,x)
 

Output:

-int((atanh(a*x)^2*(a^2*x^2 - 1))/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^6} \, dx=\frac {10 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-6 \mathit {atanh} \left (a x \right )^{2}-\mathit {atanh} \left (a x \right ) a^{5} x^{5}+4 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-3 \mathit {atanh} \left (a x \right ) a x +8 \left (\int \frac {\mathit {atanh} \left (a x \right )}{a^{2} x^{3}-x}d x \right ) a^{5} x^{5}+a^{4} x^{4}-a^{2} x^{2}}{30 x^{5}} \] Input:

int((-a^2*x^2+1)*atanh(a*x)^2/x^6,x)
 

Output:

(10*atanh(a*x)**2*a**2*x**2 - 6*atanh(a*x)**2 - atanh(a*x)*a**5*x**5 + 4*a 
tanh(a*x)*a**3*x**3 - 3*atanh(a*x)*a*x + 8*int(atanh(a*x)/(a**2*x**3 - x), 
x)*a**5*x**5 + a**4*x**4 - a**2*x**2)/(30*x**5)