3.4.12 \(\int \frac {\text {arctanh}(a x)^2}{x (1-a^2 x^2)^3} \, dx\) [312]

3.4.12.1 Optimal result

Integrand size = 22, antiderivative size = 196 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^3} \, dx=\frac {1}{32 \left (1-a^2 x^2\right )^2}+\frac {11}{32 \left (1-a^2 x^2\right )}-\frac {a x \text {arctanh}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {11 a x \text {arctanh}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {11}{32} \text {arctanh}(a x)^2+\frac {\text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {\text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]

output

1/32/(-a^2*x^2+1)^2+11/32/(-a^2*x^2+1)-1/8*a*x*arctanh(a*x)/(-a^2*x^2+1)^2 
-11/16*a*x*arctanh(a*x)/(-a^2*x^2+1)-11/32*arctanh(a*x)^2+1/4*arctanh(a*x) 
^2/(-a^2*x^2+1)^2+1/2*arctanh(a*x)^2/(-a^2*x^2+1)+1/3*arctanh(a*x)^3+arcta 
nh(a*x)^2*ln(2-2/(a*x+1))-arctanh(a*x)*polylog(2,-1+2/(a*x+1))-1/2*polylog 
(3,-1+2/(a*x+1))
 

3.4.12.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.66 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^3} \, dx=\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {1}{768} \left (32 i \pi ^3-256 \text {arctanh}(a x)^3+144 \cosh (2 \text {arctanh}(a x))+3 \cosh (4 \text {arctanh}(a x))+24 \text {arctanh}(a x)^2 \left (12 \cosh (2 \text {arctanh}(a x))+\cosh (4 \text {arctanh}(a x))+32 \log \left (1-e^{2 \text {arctanh}(a x)}\right )\right )-384 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-12 \text {arctanh}(a x) (24 \sinh (2 \text {arctanh}(a x))+\sinh (4 \text {arctanh}(a x)))\right ) \]

input

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

output

ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((32*I)*Pi^3 - 256*ArcTanh[a 
*x]^3 + 144*Cosh[2*ArcTanh[a*x]] + 3*Cosh[4*ArcTanh[a*x]] + 24*ArcTanh[a*x 
]^2*(12*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] + 32*Log[1 - E^(2*ArcT 
anh[a*x])]) - 384*PolyLog[3, E^(2*ArcTanh[a*x])] - 12*ArcTanh[a*x]*(24*Sin 
h[2*ArcTanh[a*x]] + Sinh[4*ArcTanh[a*x]]))/768
 

3.4.12.3 Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6592, 6556, 6522, 6518, 241, 6592, 6550, 6494, 6556, 6518, 241, 6618, 7164}

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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6522

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.4.12.3.1 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 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 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 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 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 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]
 

rule 6618

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

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

3.4.12.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.94 (sec) , antiderivative size = 1305, normalized size of antiderivative = 6.66

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1305\)
default	\(\text {Expression too large to display}\)	\(1305\)
parts	\(\text {Expression too large to display}\)	\(1716\)

input

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

output

-3/16*arctanh(a*x)*(a*x-1)/(a*x+1)+3/16*(a*x+1)*arctanh(a*x)/(a*x-1)+2*arc 
tanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2, 
(a*x+1)/(-a^2*x^2+1)^(1/2))+1/16*arctanh(a*x)^2/(a*x-1)^2+1/16*arctanh(a*x 
)^2/(a*x+1)^2-2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1) 
/(-a^2*x^2+1)^(1/2))-1/3*arctanh(a*x)^3-1/128*arctanh(a*x)*(a*x+1)^2/(a*x- 
1)^2+1/128*(a*x-1)^2*arctanh(a*x)/(a*x+1)^2+1/512*(a*x+1)^2/(a*x-1)^2-3/32 
/(a*x+1)*(a*x-1)+1/32*(16*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1 
)^2/(a^2*x^2-1)))^3+16*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x 
+1)^2/(a^2*x^2-1))^2-8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2 
*x^2-1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+16*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2 
*x^2-1)))^3+8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1))) 
^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))+8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^ 
3-16*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2+8*I*Pi*csgn(I*(a*x+1)/(-a^2* 
x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+16*I*Pi-16*I*Pi*csgn(I*(-(a* 
x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^ 
2*x^2-1)))+8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^ 
3+16*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1 
)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))-16*I*Pi* 
csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a 
*x+1)^2/(a^2*x^2-1)))^2-8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^...
 

3.4.12.5 Fricas [F]

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

input

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

output

integral(-arctanh(a*x)^2/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)
 

3.4.12.6 Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^3} \, dx=- \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \]

input

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

output

-Integral(atanh(a*x)**2/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x**3 - x), x)
 

3.4.12.7 Maxima [F]

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

input

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

output

1/2*a^6*integrate(1/2*x^6*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 
+ 3*a^2*x^3 - x), x) + 1/2*a^5*integrate(1/2*x^5*log(a*x + 1)*log(-a*x + 1 
)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 1/256*(a*(2*(5*a^2*x^2 + 3*a 
*x - 6)/(a^8*x^3 - a^7*x^2 - a^6*x + a^5) - 5*log(a*x + 1)/a^5 + 5*log(a*x 
 - 1)/a^5) + 16*(2*a^2*x^2 - 1)*log(-a*x + 1)/(a^8*x^4 - 2*a^6*x^2 + a^4)) 
*a^4 - a^4*integrate(1/2*x^4*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x 
^5 + 3*a^2*x^3 - x), x) - a^3*integrate(1/2*x^3*log(a*x + 1)*log(-a*x + 1) 
/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) + 1/2*a^3*integrate(1/2*x^3*log 
(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 3/512*(a*(2*(3*a^2* 
x^2 - 3*a*x - 2)/(a^6*x^3 - a^5*x^2 - a^4*x + a^3) - 3*log(a*x + 1)/a^3 + 
3*log(a*x - 1)/a^3) - 16*log(-a*x + 1)/(a^6*x^4 - 2*a^4*x^2 + a^2))*a^2 + 
1/2*a^2*integrate(1/2*x^2*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 
+ 3*a^2*x^3 - x), x) + 1/2*a*integrate(1/2*x*log(a*x + 1)*log(-a*x + 1)/(a 
^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 3/4*a*integrate(1/2*x*log(-a*x + 
 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 1/48*(2*(a^4*x^4 - 2*a^2*x 
^2 + 1)*log(-a*x + 1)^3 + 3*(2*a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a 
*x + 1) - 3)*log(-a*x + 1)^2)/(a^4*x^4 - 2*a^2*x^2 + 1) - 1/2*integrate(1/ 
2*log(a*x + 1)^2/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) + integrate(1/2 
*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)
 

3.4.12.8 Giac [F]

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

input

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

output

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

3.4.12.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^3} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x\,{\left (a^2\,x^2-1\right )}^3} \,d x \]

input

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

output

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