\(\int \frac {(1-a^2 x^2)^2 \text {arctanh}(a x)^2}{x} \, dx\) [208]

Optimal result

Integrand size = 22, antiderivative size = 186 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x} \, dx=\frac {a^2 x^2}{12}-\frac {3}{2} a x \text {arctanh}(a x)+\frac {1}{6} a^3 x^3 \text {arctanh}(a x)+\frac {3}{4} \text {arctanh}(a x)^2-a^2 x^2 \text {arctanh}(a x)^2+\frac {1}{4} a^4 x^4 \text {arctanh}(a x)^2+2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \] Output:

1/12*a^2*x^2-3/2*a*x*arctanh(a*x)+1/6*a^3*x^3*arctanh(a*x)+3/4*arctanh(a*x 
)^2-a^2*x^2*arctanh(a*x)^2+1/4*a^4*x^4*arctanh(a*x)^2-2*arctanh(a*x)^2*arc 
tanh(-1+2/(-a*x+1))-2/3*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*po 
lylog(3,-1+2/(-a*x+1))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x} \, dx=\frac {a^2 x^2}{12}-2 a x \text {arctanh}(a x)+\frac {1}{6} a x \left (3+a^2 x^2\right ) \text {arctanh}(a x)-\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {1}{4} \left (-1+a^4 x^4\right ) \text {arctanh}(a x)^2-\frac {2}{3} \text {arctanh}(a x)^3-\text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right ) \] Input:

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

Output:

(a^2*x^2)/12 - 2*a*x*ArcTanh[a*x] + (a*x*(3 + a^2*x^2)*ArcTanh[a*x])/6 - ( 
-1 + a^2*x^2)*ArcTanh[a*x]^2 + ((-1 + a^4*x^4)*ArcTanh[a*x]^2)/4 - (2*ArcT 
anh[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])] - (2*Log[1 - a^2*x^2])/3 + ArcTanh[a*x]*Pol 
yLog[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 = 0.77 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6574, 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.

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

\(\Big \downarrow \) 6574

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} a^4 x^4 \text {arctanh}(a x)^2+\frac {1}{6} a^3 x^3 \text {arctanh}(a x)-a^2 x^2 \text {arctanh}(a x)^2+\frac {a^2 x^2}{12}-\frac {2}{3} \log \left (1-a^2 x^2\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )-\frac {3}{2} a x \text {arctanh}(a x)+\frac {3}{4} \text {arctanh}(a x)^2+2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )\)

Input:

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

Output:

(a^2*x^2)/12 - (3*a*x*ArcTanh[a*x])/2 + (a^3*x^3*ArcTanh[a*x])/6 + (3*ArcT 
anh[a*x]^2)/4 - a^2*x^2*ArcTanh[a*x]^2 + (a^4*x^4*ArcTanh[a*x]^2)/4 + 2*Ar 
cTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (2*Log[1 - a^2*x^2])/3 - ArcTanh[a 
*x]*PolyLog[2, 1 - 2/(1 - a*x)] + ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x) 
] + PolyLog[3, 1 - 2/(1 - a*x)]/2 - PolyLog[3, -1 + 2/(1 - a*x)]/2
 

Defintions of rubi rules used

rule 2009

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

rule 6574

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

Maple [C] (warning: unable to verify)

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

Time = 7.38 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.94

method	result	size
derivativedivides	\(\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{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}+\frac {4 \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{3}-\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}-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+\frac {a x}{6}-\frac {1}{6}-\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 {3 \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {\left (a x -1\right )^{2}}{12}+\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 {\left (a^{2} x^{2}-4 a x +7\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{6}+\frac {\left (a x -3\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{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}{-\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}\)	\(733\)
default	\(\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}-a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{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}+\frac {4 \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{3}-\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}-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+\frac {a x}{6}-\frac {1}{6}-\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 {3 \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {\left (a x -1\right )^{2}}{12}+\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 {\left (a^{2} x^{2}-4 a x +7\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{6}+\frac {\left (a x -3\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{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}{-\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}\)	\(733\)
parts	\(\text {Expression too large to display}\)	\(1196\)

Input:

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

Output:

1/4*a^4*x^4*arctanh(a*x)^2-a^2*x^2*arctanh(a*x)^2+arctanh(a*x)^2*ln(a*x)-a 
rctanh(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*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*p 
olylog(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))+4/3*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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^2-(a*x+1)*arctanh(a*x)+1 
/6*a*x-1/6-1/2*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+3/4*arctanh(a* 
x)^2+1/12*(a*x-1)^2+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+1/6*(a^2*x^2-4*a*x+7)*(a*x+1)*arctanh(a* 
x)+1/2*(a*x-3)*(a*x+1)*arctanh(a*x)+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
 

Fricas [F]

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

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

Output:

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/16*(a^4*x^4 - 4*a^2*x^2)*log(-a*x + 1)^2 - 1/4*integrate(-1/2*(2*(a^5*x^ 
5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*x + 1)^2 - (a^5*x^5 - 
 4*a^3*x^3 + 4*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a 
*x + 1))*log(-a*x + 1))/(a*x^2 - x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x} \, dx=\frac {\mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}}{4}-\mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+\frac {3 \mathit {atanh} \left (a x \right )^{2}}{4}+\frac {\mathit {atanh} \left (a x \right ) a^{3} x^{3}}{6}-\frac {3 \mathit {atanh} \left (a x \right ) a x}{2}-\frac {4 \mathit {atanh} \left (a x \right )}{3}+\int \frac {\mathit {atanh} \left (a x \right )^{2}}{x}d x -\frac {4 \,\mathrm {log}\left (a^{2} x -a \right )}{3}+\frac {a^{2} x^{2}}{12} \] Input:

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

Output:

(3*atanh(a*x)**2*a**4*x**4 - 12*atanh(a*x)**2*a**2*x**2 + 9*atanh(a*x)**2 
+ 2*atanh(a*x)*a**3*x**3 - 18*atanh(a*x)*a*x - 16*atanh(a*x) + 12*int(atan 
h(a*x)**2/x,x) - 16*log(a**2*x - a) + a**2*x**2)/12