\(\int \frac {(1-a^2 x^2)^2 \text {arctanh}(a x)^2}{x^5} \, dx\) [212]

Optimal result

Integrand size = 22, antiderivative size = 214 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \text {arctanh}(a x)}{6 x^3}+\frac {3 a^3 \text {arctanh}(a x)}{2 x}-\frac {3}{4} a^4 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{4 x^4}+\frac {a^2 \text {arctanh}(a x)^2}{x^2}+2 a^4 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-\frac {4}{3} a^4 \log (x)+\frac {2}{3} a^4 \log \left (1-a^2 x^2\right )-a^4 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+a^4 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )+\frac {1}{2} a^4 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \] Output:

-1/12*a^2/x^2-1/6*a*arctanh(a*x)/x^3+3/2*a^3*arctanh(a*x)/x-3/4*a^4*arctan 
h(a*x)^2-1/4*arctanh(a*x)^2/x^4+a^2*arctanh(a*x)^2/x^2-2*a^4*arctanh(a*x)^ 
2*arctanh(-1+2/(-a*x+1))-4/3*a^4*ln(x)+2/3*a^4*ln(-a^2*x^2+1)-a^4*arctanh( 
a*x)*polylog(2,1-2/(-a*x+1))+a^4*arctanh(a*x)*polylog(2,-1+2/(-a*x+1))+1/2 
*a^4*polylog(3,1-2/(-a*x+1))-1/2*a^4*polylog(3,-1+2/(-a*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.11 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\frac {1}{24} \left (2 a^4+i a^4 \pi ^3-\frac {2 a^2}{x^2}-\frac {4 a \text {arctanh}(a x)}{x^3}+\frac {36 a^3 \text {arctanh}(a x)}{x}-18 a^4 \text {arctanh}(a x)^2-\frac {6 \text {arctanh}(a x)^2}{x^4}+\frac {24 a^2 \text {arctanh}(a x)^2}{x^2}-16 a^4 \text {arctanh}(a x)^3-24 a^4 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+24 a^4 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-32 a^4 \log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )+24 a^4 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+24 a^4 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+12 a^4 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )-12 a^4 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )\right ) \] Input:

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^5,x]
 

Output:

(2*a^4 + I*a^4*Pi^3 - (2*a^2)/x^2 - (4*a*ArcTanh[a*x])/x^3 + (36*a^3*ArcTa 
nh[a*x])/x - 18*a^4*ArcTanh[a*x]^2 - (6*ArcTanh[a*x]^2)/x^4 + (24*a^2*ArcT 
anh[a*x]^2)/x^2 - 16*a^4*ArcTanh[a*x]^3 - 24*a^4*ArcTanh[a*x]^2*Log[1 + E^ 
(-2*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3 
2*a^4*Log[(a*x)/Sqrt[1 - a^2*x^2]] + 24*a^4*ArcTanh[a*x]*PolyLog[2, -E^(-2 
*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + 12* 
a^4*PolyLog[3, -E^(-2*ArcTanh[a*x])] - 12*a^4*PolyLog[3, E^(2*ArcTanh[a*x] 
)])/24
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 214, 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.

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

\(\Big \downarrow \) 6574

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

\(\Big \downarrow \) 2009

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

Input:

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^5,x]
 

Output:

-1/12*a^2/x^2 - (a*ArcTanh[a*x])/(6*x^3) + (3*a^3*ArcTanh[a*x])/(2*x) - (3 
*a^4*ArcTanh[a*x]^2)/4 - ArcTanh[a*x]^2/(4*x^4) + (a^2*ArcTanh[a*x]^2)/x^2 
 + 2*a^4*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (4*a^4*Log[x])/3 + (2*a 
^4*Log[1 - a^2*x^2])/3 - a^4*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + a^ 
4*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] + (a^4*PolyLog[3, 1 - 2/(1 - a 
*x)])/2 - (a^4*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 = 9.86 (sec) , antiderivative size = 1124, normalized size of antiderivative = 5.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1124\)
default	\(\text {Expression too large to display}\)	\(1124\)
parts	\(\text {Expression too large to display}\)	\(1571\)

Input:

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x,method=_RETURNVERBOSE)
 

Output:

a^4*(arctanh(a*x)^2/a^2/x^2-1/4*arctanh(a*x)^2/a^4/x^4-1/24*(-(-a^2*x^2+1) 
^(1/2)*a^2*x^2+5*a^3*x^3+3*a*x*(-a^2*x^2+1)^(1/2)-2*(-a^2*x^2+1)^(1/2)-3*a 
*x+2)*arctanh(a*x)/a^3/x^3+1/8*(-a*x*(-a^2*x^2+1)^(1/2)+2*a^2*x^2+(-a^2*x^ 
2+1)^(1/2)+a*x-1)*arctanh(a*x)/a^2/x^2-1/24*((-a^2*x^2+1)^(1/2)*a^2*x^2+5* 
a^3*x^3-3*a*x*(-a^2*x^2+1)^(1/2)+2*(-a^2*x^2+1)^(1/2)-3*a*x+2)*arctanh(a*x 
)/a^3/x^3+1/8*(a*x*(-a^2*x^2+1)^(1/2)+2*a^2*x^2-(-a^2*x^2+1)^(1/2)+a*x-1)* 
arctanh(a*x)/a^2/x^2-4/3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+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))*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)^2-1/12/(a* 
x-(-a^2*x^2+1)^(1/2)+1)*(-a^2*x^2+1)^(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))-arctanh(a*x)*polylog(2,- 
(a*x+1)^2/(-a^2*x^2+1))-3/4*arctanh(a*x)^2+2*arctanh(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) 
)-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))+arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(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)^2-4/3*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+1/12/(a*x+(-a^2*x^2+1)^(1/2)+ 
1)*(-a^2*x^2+1)^(1/2)+1/24*(a*x-1)/((-a^2*x^2+1)^(1/2)+1)-1/24*(a*x-1)/((- 
a^2*x^2+1)^(1/2)-1)+arctanh(a*x)^2*ln(a*x)+1/2*polylog(3,-(a*x+1)^2/(-a^2* 
x^2+1))+5/8*(a*x-(-a^2*x^2+1)^(1/2)+1)/a/x*arctanh(a*x)+5/8*arctanh(a*x...
 

Fricas [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{5}} \,d x } \] Input:

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="fricas")
 

Output:

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^5, x)
 

Sympy [F]

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

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**5,x)
 

Output:

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**5, x)
 

Maxima [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{5}} \,d x } \] Input:

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="maxima")
 

Output:

1/16*(4*a^2*x^2 - 1)*log(-a*x + 1)^2/x^4 - 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 - (4*a^3*x^3 - 
 a*x + 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^6 - x^5), x)
 

Giac [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{5}} \,d x } \] Input:

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="giac")
 

Output:

integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2/x^5, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^5} \,d x \] Input:

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^5,x)
 

Output:

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^5, x)
 

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^5} \, dx=\frac {-9 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+12 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-3 \mathit {atanh} \left (a x \right )^{2}+16 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+18 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-2 \mathit {atanh} \left (a x \right ) a x +12 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{x}d x \right ) a^{4} x^{4}+16 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}-16 \,\mathrm {log}\left (x \right ) a^{4} x^{4}-a^{2} x^{2}}{12 x^{4}} \] Input:

int((-a^2*x^2+1)^2*atanh(a*x)^2/x^5,x)
 

Output:

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