\(\int \frac {\text {arctanh}(a x)^2}{x^2 (1-a^2 x^2)} \, dx\) [239]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{x}+\frac {1}{3} a \text {arctanh}(a x)^3+2 a \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=-a \left (-\frac {1}{3} \text {arctanh}(a x) \left (-\frac {3 \text {arctanh}(a x)}{a x}+\text {arctanh}(a x) (3+\text {arctanh}(a x))+6 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

-(a*(-1/3*(ArcTanh[a*x]*((-3*ArcTanh[a*x])/(a*x) + ArcTanh[a*x]*(3 + ArcTa 
nh[a*x]) + 6*Log[1 - E^(-2*ArcTanh[a*x])])) + PolyLog[2, E^(-2*ArcTanh[a*x 
])]))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6544, 6452, 6510, 6550, 6494, 2897}

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.

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 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 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 6544

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x 
], x] - Simp[e/(d*f^2)   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] && LtQ[m, -1]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 13.47 (sec) , antiderivative size = 4380, normalized size of antiderivative = 66.36

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4380\)
default	\(\text {Expression too large to display}\)	\(4380\)
parts	\(\text {Expression too large to display}\)	\(4383\)

Input:

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

Output:

a*(1/4*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 
)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/ 
2))+1/2*arctanh(a*x)^2*ln(a*x+1)-arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1 
/2))-arctanh(a*x)^2/a/x+1/4*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2- 
1)+1))*arctanh(a*x)^2-1/2*arctanh(a*x)^2*ln(a*x-1)+1/2*I*Pi*csgn(I/(-(a*x+ 
1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^2-1/4*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+ 
1)^2/(a^2*x^2-1)+1))*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog 
(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/3* 
arctanh(a*x)^3-arctanh(a*x)^2-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn( 
I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)*ln(1-(a 
*x+1)/(-a^2*x^2+1)^(1/2))-1/4*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^ 
2-1)+1))*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*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))*csgn(I*(a*x+1)^2/(a^2*x^2- 
1)/(-(a*x+1)^2/(a^2*x^2-1)+1))*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*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))*csgn( 
I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))*dilog(1+(a*x+1)/(-a^2* 
x^2+1)^(1/2))-1/4*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I*(a*x+1...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{4} - x^{2}}\, dx \] Input:

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

Output:

-Integral(atanh(a*x)**2/(a**2*x**4 - x**2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (63) = 126\).

Time = 0.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.59 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx=-\frac {1}{24} \, a^{2} {\left (\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {24 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {24 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} + \frac {1}{4} \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \left (x\right )\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:

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

Output:

-1/24*a^2*((3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x 
 - 1)^3 - 3*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1 
)^2)/a - 24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a + 
24*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 24*(log(-a*x + 1)*log(x) + dilo 
g(a*x))/a) + 1/4*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log 
(a*x - 1)^2 - 4*log(a*x - 1) + 8*log(x))*a*arctanh(a*x) + 1/2*(a*log(a*x + 
 1) - a*log(a*x - 1) - 2/x)*arctanh(a*x)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(atanh(a*x)**3*a*x - 3*atanh(a*x)**2 - 6*int(atanh(a*x)/(a**2*x**3 - x),x) 
*a*x)/(3*x)