\(\int \frac {(1-a^2 x^2) \text {arctanh}(a x)^2}{x} \, dx\) [177]

Optimal result

Integrand size = 20, antiderivative size = 146 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x} \, dx=-a x \text {arctanh}(a x)+\frac {1}{2} \text {arctanh}(a x)^2-\frac {1}{2} a^2 x^2 \text {arctanh}(a x)^2+2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \log \left (1-a^2 x^2\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x} \, dx=-a x \text {arctanh}(a x)-\frac {1}{2} \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2-\frac {2}{3} \text {arctanh}(a x)^3-\text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\frac {1}{2} \log \left (1-a^2 x^2\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6576, 6448, 6452, 6542, 6436, 240, 6510, 6614, 6620, 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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6448

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6614

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 6436

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

rule 6448

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

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 6542

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

rule 6614

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

rule 6620

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]
 

Maple [C] (warning: unable to verify)

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

Time = 5.26 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.54

method	result	size
derivativedivides	\(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{2}-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}\)	\(663\)
default	\(-\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{2}-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}\)	\(663\)
parts	\(\text {Expression too large to display}\)	\(1111\)

Input:

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

Output:

-1/2*a^2*x^2*arctanh(a*x)^2+arctanh(a*x)^2*ln(a*x)-arctanh(a*x)^2*ln((a*x+ 
1)^2/(-a^2*x^2+1)-1)+arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arc 
tanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1)/(-a^2* 
x^2+1)^(1/2))+arctanh(a*x)^2*ln(1+(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))-2*polylog(3,-(a*x+1)/(-a^2*x^2+1 
)^(1/2))-arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+1/2*polylog(3,-(a 
*x+1)^2/(-a^2*x^2+1))+ln((a*x+1)^2/(-a^2*x^2+1)+1)+1/2*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))*csgn(I*(-(a*x+1)^2 
/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))*arctanh(a*x)^2-(a*x+1)*arctanh 
(a*x)+1/2*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1 
))^3*arctanh(a*x)^2+1/2*arctanh(a*x)^2-1/2*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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*a 
rctanh(a*x)^2-1/2*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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x} \, dx=-\frac {\mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}}{2}+\frac {\mathit {atanh} \left (a x \right )^{2}}{2}-\mathit {atanh} \left (a x \right ) a x -\mathit {atanh} \left (a x \right )+\int \frac {\mathit {atanh} \left (a x \right )^{2}}{x}d x -\mathrm {log}\left (a^{2} x -a \right ) \] Input:

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

Output:

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