\(\int \frac {\text {arctanh}(a x)^2}{x^3 (1-a^2 x^2)} \, dx\) [240]

Optimal result

Integrand size = 22, antiderivative size = 138 \[ \int \frac {\text {arctanh}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx=-\frac {a \text {arctanh}(a x)}{x}+\frac {1}{2} a^2 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \text {arctanh}(a x)^3+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {\text {arctanh}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx=-a^2 \left (-\frac {i \pi ^3}{24}+\frac {\text {arctanh}(a x)}{a x}+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a^2 x^2}+\frac {1}{3} \text {arctanh}(a x)^3-\text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

-(a^2*((-1/24*I)*Pi^3 + ArcTanh[a*x]/(a*x) + ((1 - a^2*x^2)*ArcTanh[a*x]^2 
)/(2*a^2*x^2) + ArcTanh[a*x]^3/3 - ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x 
])] - Log[(a*x)/Sqrt[1 - a^2*x^2]] - ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[ 
a*x])] + PolyLog[3, E^(2*ArcTanh[a*x])]/2))
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6544, 6452, 6544, 6452, 243, 47, 14, 16, 6510, 6550, 6494, 6618, 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.

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

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

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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]
 

rule 6618

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 = 23.34 (sec) , antiderivative size = 1316, normalized size of antiderivative = 9.54

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1316\)
default	\(\text {Expression too large to display}\)	\(1316\)
parts	\(\text {Expression too large to display}\)	\(1735\)

Input:

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

Output:

a^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))-1/2*arctanh(a*x)^2/a^2/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)^2+ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*arcta 
nh(a*x)^2*ln(a*x-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+1/4*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*cs 
gn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^2+1/ 
2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^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))-1/3*arct 
anh(a*x)^3+1/2*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(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)^2+1/2*I*Pi*csgn(I*(a*x+1)/( 
-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^2+1/4*I*Pi 
*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctan 
h(a*x)^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*arctanh(a*x)^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+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)...
 

Fricas [F]

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

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

Output:

integral(-arctanh(a*x)^2/(a^2*x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, dx \] Input:

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

Output:

-Integral(atanh(a*x)**2/(a**2*x**5 - x**3), x)
 

Maxima [F]

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

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

Output:

-1/24*(a^2*x^2*log(-a*x + 1)^3 + 3*(a^2*x^2*log(a*x + 1) + 1)*log(-a*x + 1 
)^2)/x^2 + 1/4*integrate(-(log(a*x + 1)^2 - (a^2*x^2 + a*x + (a^4*x^4 + a^ 
3*x^3 + 2)*log(a*x + 1))*log(-a*x + 1))/(a^2*x^5 - x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{2} x^{5}-x^{3}}d x \right ) \] Input:

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

Output:

 - int(atanh(a*x)**2/(a**2*x**5 - x**3),x)