\(\int \frac {\text {arctanh}(a x)^3}{x^2 (1-a^2 x^2)^2} \, dx\) [278]

Optimal result

Integrand size = 22, antiderivative size = 191 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \text {arctanh}(a x)^2-\frac {3 a \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{x}+\frac {a^2 x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \text {arctanh}(a x)^4+3 a \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:

-3*a/(-8*a^2*x^2+8)+3*a^2*x*arctanh(a*x)/(-4*a^2*x^2+4)+3/8*a*arctanh(a*x) 
^2-3*a*arctanh(a*x)^2/(-4*a^2*x^2+4)+a*arctanh(a*x)^3-arctanh(a*x)^3/x+a^2 
*x*arctanh(a*x)^3/(-2*a^2*x^2+2)+3/8*a*arctanh(a*x)^4+3*a*arctanh(a*x)^2*l 
n(2-2/(a*x+1))-3*a*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-3/2*a*polylog(3,-1 
+2/(a*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{16} a \left (2 i \pi ^3-16 \text {arctanh}(a x)^3-\frac {16 \text {arctanh}(a x)^3}{a x}+6 \text {arctanh}(a x)^4-3 \cosh (2 \text {arctanh}(a x))-6 \text {arctanh}(a x)^2 \cosh (2 \text {arctanh}(a x))+48 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+48 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-24 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \sinh (2 \text {arctanh}(a x))+4 \text {arctanh}(a x)^3 \sinh (2 \text {arctanh}(a x))\right ) \] Input:

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

Output:

(a*((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 - (16*ArcTanh[a*x]^3)/(a*x) + 6*ArcTanh 
[a*x]^4 - 3*Cosh[2*ArcTanh[a*x]] - 6*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]] + 
 48*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 48*ArcTanh[a*x]*PolyLog[2 
, E^(2*ArcTanh[a*x])] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] + 6*ArcTanh[a*x] 
*Sinh[2*ArcTanh[a*x]] + 4*ArcTanh[a*x]^3*Sinh[2*ArcTanh[a*x]]))/16
 

Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6592, 6518, 6544, 6452, 6510, 6550, 6494, 6556, 6518, 241, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6518

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )}dx\)

\(\Big \downarrow \) 6544

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^3}{x^2}dx\)

\(\Big \downarrow \) 6452

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6510

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6550

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (\int \frac {\text {arctanh}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \text {arctanh}(a x)^3\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6494

\(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6556

\(\displaystyle a^2 \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6518

\(\displaystyle a^2 \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 241

\(\displaystyle 3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 6618

\(\displaystyle 3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

\(\Big \downarrow \) 7164

\(\displaystyle a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\)

Input:

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

Output:

-(ArcTanh[a*x]^3/x) + (a*ArcTanh[a*x]^4)/4 + a^2*((x*ArcTanh[a*x]^3)/(2*(1 
 - a^2*x^2)) + ArcTanh[a*x]^4/(8*a) - (3*a*(ArcTanh[a*x]^2/(2*a^2*(1 - a^2 
*x^2)) - (-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + 
ArcTanh[a*x]^2/(4*a))/a))/2) + 3*a*(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 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 6518

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy 
mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + 
 b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2)   Int[x*( 
(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
 

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 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -1]
 

rule 6592

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh 
[c*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers 
Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

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 [A] (verified)

Time = 36.93 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(a \left (\frac {3 \operatorname {arctanh}\left (a x \right )^{4}}{8}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{32 \left (a x -1\right )}+\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{32 a x +32}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(271\)
default	\(a \left (\frac {3 \operatorname {arctanh}\left (a x \right )^{4}}{8}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{32 \left (a x -1\right )}+\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{32 a x +32}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(271\)

Input:

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

Output:

a*(3/8*arctanh(a*x)^4-1/32*(a*x+1)*(4*arctanh(a*x)^3-6*arctanh(a*x)^2+6*ar 
ctanh(a*x)-3)/(a*x-1)+1/32*(4*arctanh(a*x)^3+6*arctanh(a*x)^2+6*arctanh(a* 
x)+3)*(a*x-1)/(a*x+1)+arctanh(a*x)^3/a/x*(a*x-1)-2*arctanh(a*x)^3+3*arctan 
h(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(2,-(a*x+1 
)/(-a^2*x^2+1)^(1/2))-6*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a 
*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(2,(a*x+1)/(- 
a^2*x^2+1)^(1/2))-6*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral(arctanh(a*x)^3/(a^4*x^6 - 2*a^2*x^4 + x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx=\frac {3 \mathit {atanh} \left (a x \right )^{4} a^{3} x^{3}-3 \mathit {atanh} \left (a x \right )^{4} a x -12 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}+8 \mathit {atanh} \left (a x \right )^{3}+9 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}+9 \mathit {atanh} \left (a x \right )^{2} a x -18 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+24 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a^{3} x^{3}-24 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a x +9 a^{3} x^{3}}{8 x \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

(3*atanh(a*x)**4*a**3*x**3 - 3*atanh(a*x)**4*a*x - 12*atanh(a*x)**3*a**2*x 
**2 + 8*atanh(a*x)**3 + 9*atanh(a*x)**2*a**3*x**3 + 9*atanh(a*x)**2*a*x - 
18*atanh(a*x)*a**2*x**2 + 24*int(atanh(a*x)**2/(a**4*x**5 - 2*a**2*x**3 + 
x),x)*a**3*x**3 - 24*int(atanh(a*x)**2/(a**4*x**5 - 2*a**2*x**3 + x),x)*a* 
x + 9*a**3*x**3)/(8*x*(a**2*x**2 - 1))