\(\int \frac {(1-a^2 x^2)^2 \text {arctanh}(a x)^2}{x^3} \, dx\) [210]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+a^3 x \text {arctanh}(a x)-\frac {\text {arctanh}(a x)^2}{2 x^2}+\frac {1}{2} a^4 x^2 \text {arctanh}(a x)^2-4 a^2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )+a^2 \log (x)+2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )-a^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )+a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \] Output:

-a*arctanh(a*x)/x+a^3*x*arctanh(a*x)-1/2*arctanh(a*x)^2/x^2+1/2*a^4*x^2*ar 
ctanh(a*x)^2+4*a^2*arctanh(a*x)^2*arctanh(-1+2/(-a*x+1))+a^2*ln(x)+2*a^2*a 
rctanh(a*x)*polylog(2,1-2/(-a*x+1))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(-a* 
x+1))-a^2*polylog(3,1-2/(-a*x+1))+a^2*polylog(3,-1+2/(-a*x+1))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.23 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+a^3 x \text {arctanh}(a x)+\frac {1}{2} a^2 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2}{2 x^2}+\frac {4}{3} a^2 \text {arctanh}(a x)^3+2 a^2 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-2 a^2 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+a^2 \log (x)-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 162, 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.

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

\(\Big \downarrow \) 6574

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 6.24 (sec) , antiderivative size = 779, normalized size of antiderivative = 4.81

method	result	size
derivativedivides	\(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-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}-\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}+\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+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}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+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}-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}\right )\)	\(779\)
default	\(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-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}-\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}+\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )+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}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+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}-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}\right )\)	\(779\)
parts	\(\text {Expression too large to display}\)	\(1178\)

Input:

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

Output:

a^2*(1/2*a^2*x^2*arctanh(a*x)^2-2*arctanh(a*x)^2*ln(a*x)-1/2*arctanh(a*x)^ 
2/a^2/x^2+2*arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)-2*arctanh(a*x)^2*l 
n(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-4*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^ 
2+1)^(1/2))+4*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)^2*ln(1 
-(a*x+1)/(-a^2*x^2+1)^(1/2))-4*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1) 
^(1/2))+4*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,- 
(a*x+1)^2/(-a^2*x^2+1))-polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-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-ln((a 
*x+1)^2/(-a^2*x^2+1)+1)-1/2*(a*x-(-a^2*x^2+1)^(1/2)+1)/a/x*arctanh(a*x)-1/ 
2*arctanh(a*x)*(a*x+(-a^2*x^2+1)^(1/2)+1)/a/x+(a*x+1)*arctanh(a*x)+I*Pi*cs 
gn(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+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+ln( 
1+(a*x+1)/(-a^2*x^2+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 
-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]

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

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

Output:

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^3, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/16*(2*x^2*log(-a*x + 1) - a*((a*x^2 + 2*x)/a^2 + 2*log(a*x - 1)/a^3))*a 
^4 - 1/2*a^4*integrate(x*log(a*x + 1)*log(-a*x + 1), x) + 1/4*a^3*integrat 
e(a*x*log(a*x + 1)^2, x) + 1/4*a^3*integrate(log(a*x + 1)^2/(a^3*x^3), x) 
+ 1/4*(a*x - (a*x - 1)*log(-a*x + 1) - 1)*a^2 - 1/2*a^2*integrate(log(a*x 
+ 1)^2/x, x) + a^2*integrate(log(a*x + 1)*log(-a*x + 1)/x, x) - 1/4*a^2*in 
tegrate(log(-a*x + 1)/x, x) - 1/4*(a*(log(a*x - 1) - log(x)) - log(-a*x + 
1)/x)*a + 1/8*(a^4*x^4 - 1)*log(-a*x + 1)^2/x^2 - 1/2*integrate(log(a*x + 
1)*log(-a*x + 1)/x^3, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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