\(\int (1-a^2 x^2)^2 \text {arctanh}(a x)^3 \, dx\) [217]

Optimal result

Integrand size = 19, antiderivative size = 248 \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3 \, dx=-\frac {1-a^2 x^2}{20 a}-x \text {arctanh}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}+\frac {8 \text {arctanh}(a x)^3}{15 a}+\frac {8}{15} x \text {arctanh}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3-\frac {8 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {8 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a}+\frac {4 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{5 a} \] Output:

-1/20*(-a^2*x^2+1)/a-x*arctanh(a*x)-1/10*x*(-a^2*x^2+1)*arctanh(a*x)+2/5*( 
-a^2*x^2+1)*arctanh(a*x)^2/a+3/20*(-a^2*x^2+1)^2*arctanh(a*x)^2/a+8/15*arc 
tanh(a*x)^3/a+8/15*x*arctanh(a*x)^3+4/15*x*(-a^2*x^2+1)*arctanh(a*x)^3+1/5 
*x*(-a^2*x^2+1)^2*arctanh(a*x)^3-8/5*arctanh(a*x)^2*ln(2/(-a*x+1))/a-1/2*l 
n(-a^2*x^2+1)/a-8/5*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a+4/5*polylog(3,1 
-2/(-a*x+1))/a
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.74 \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3 \, dx=\frac {-3+3 a^2 x^2-66 a x \text {arctanh}(a x)+6 a^3 x^3 \text {arctanh}(a x)+33 \text {arctanh}(a x)^2-42 a^2 x^2 \text {arctanh}(a x)^2+9 a^4 x^4 \text {arctanh}(a x)^2-32 \text {arctanh}(a x)^3+60 a x \text {arctanh}(a x)^3-40 a^3 x^3 \text {arctanh}(a x)^3+12 a^5 x^5 \text {arctanh}(a x)^3-96 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-30 \log \left (1-a^2 x^2\right )+96 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{60 a} \] Input:

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

Output:

(-3 + 3*a^2*x^2 - 66*a*x*ArcTanh[a*x] + 6*a^3*x^3*ArcTanh[a*x] + 33*ArcTan 
h[a*x]^2 - 42*a^2*x^2*ArcTanh[a*x]^2 + 9*a^4*x^4*ArcTanh[a*x]^2 - 32*ArcTa 
nh[a*x]^3 + 60*a*x*ArcTanh[a*x]^3 - 40*a^3*x^3*ArcTanh[a*x]^3 + 12*a^5*x^5 
*ArcTanh[a*x]^3 - 96*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 30*Log[ 
1 - a^2*x^2] + 96*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 48*PolyL 
og[3, -E^(-2*ArcTanh[a*x])])/(60*a)
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6506, 6504, 6436, 240, 6506, 6436, 240, 6546, 6470, 6620, 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.

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

\(\Big \downarrow \) 6506

\(\displaystyle -\frac {3}{10} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)dx+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\)

\(\Big \downarrow \) 6504

\(\displaystyle -\frac {3}{10} \left (\frac {2}{3} \int \text {arctanh}(a x)dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\)

\(\Big \downarrow \) 6436

\(\displaystyle -\frac {3}{10} \left (\frac {2}{3} \left (x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 6506

\(\displaystyle \frac {4}{5} \left (-\int \text {arctanh}(a x)dx+\frac {2}{3} \int \text {arctanh}(a x)^3dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+a \int \frac {x}{1-a^2 x^2}dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\)

Input:

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

Output:

(3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/(20*a) + (x*(1 - a^2*x^2)^2*ArcTanh[a*x 
]^3)/5 - (3*((1 - a^2*x^2)/(6*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x])/3 + (2*( 
x*ArcTanh[a*x] + Log[1 - a^2*x^2]/(2*a)))/3))/10 + (4*(-(x*ArcTanh[a*x]) + 
 ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/3 
 - Log[1 - a^2*x^2]/(2*a) + (2*(x*ArcTanh[a*x]^3 - 3*a*(-1/3*ArcTanh[a*x]^ 
3/a^2 + ((ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*(ArcTanh[a*x]*PolyL 
og[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a)))/3))/5
 

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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 6504

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

rule 6506

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

rule 6546

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

rule 6620

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 = 6.51 (sec) , antiderivative size = 858, normalized size of antiderivative = 3.46

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(858\)
default	\(\text {Expression too large to display}\)	\(858\)
parts	\(\text {Expression too large to display}\)	\(865\)

Input:

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

Output:

1/a*(2/5*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+1/5*arctanh(a*x)^3*a^5*x^5+4/5*arctanh(a*x)^2*ln(a*x+1)-8/5*arctanh(a*x 
)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/10-2/3*arctanh(a*x)^3*a^3*x^3-4/5*I*P 
i*arctanh(a*x)^2+1/10*a*x+arctanh(a*x)^3*a*x+4/5*arctanh(a*x)^2*ln(a*x-1)- 
8/5*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+8/15*arctanh(a*x)^3+11 
/20*arctanh(a*x)^2+2/5*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)^2-2/5*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))*arctanh(a*x)^2 
+3/20*a^4*x^4*arctanh(a*x)^2-7/10*a^2*x^2*arctanh(a*x)^2-8/5*arctanh(a*x)^ 
2*ln(2)-4/5*(a*x+1)*arctanh(a*x)+1/20*(a*x-1)^2-2/5*I*Pi*csgn(I*(a*x+1)^2/ 
(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^2+4/5*I*Pi*csgn(I/( 
-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^2-2/5*I*Pi*csgn(I*(a*x+1)^2/(a^2 
*x^2-1))^3*arctanh(a*x)^2-4/5*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3*ar 
ctanh(a*x)^2+1/10*(a^2*x^2-4*a*x+7)*(a*x+1)*arctanh(a*x)+3/10*(a*x-3)*(a*x 
+1)*arctanh(a*x)+ln((a*x+1)^2/(-a^2*x^2+1)+1)-4/5*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)^2-2/5*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)/(-(a*x+1)^2/ 
(a^2*x^2-1)+1))^2*arctanh(a*x)^2+4/5*polylog(3,-(a*x+1)^2/(-a^2*x^2+1)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2400*(36*a^5*x^5 - 45*a^4*x^4 - 140*a^3*x^3 + 210*a^2*x^2 + 480*a*x - 6 
0*(3*a^5*x^5 - 10*a^3*x^3 + 15*a*x + 8)*log(a*x + 1))*log(-a*x + 1)^2/a - 
1/8*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1)/ 
a - 1/1440000*(288*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x 
 + 1) - 6)*(a*x - 1)^5 + 5625*(32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 1 
2*log(-a*x + 1) - 3)*(a*x - 1)^4 + 40000*(9*log(-a*x + 1)^3 - 9*log(-a*x + 
 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 90000*(4*log(-a*x + 1)^3 - 6*lo 
g(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 180000*(log(-a*x + 1)^3 
 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/432*(4*(9*log 
(-a*x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*( 
4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 
 108*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1) 
)/a - 1/8*integrate(-1/150*(150*(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)^3 + (36*a^5*x^5 - 45*a^4*x^4 - 140*a^3*x^3 + 210* 
a^2*x^2 - 450*(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 + 480*a*x - 60*(3*a^5*x^5 - 10*a^3*x^3 + 15*a*x + 8)*log(a*x + 1) 
)*log(-a*x + 1))/(a*x - 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3 \, dx=\frac {12 \mathit {atanh} \left (a x \right )^{3} a^{5} x^{5}-40 \mathit {atanh} \left (a x \right )^{3} a^{3} x^{3}+60 \mathit {atanh} \left (a x \right )^{3} a x +9 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-42 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+33 \mathit {atanh} \left (a x \right )^{2}+6 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-66 \mathit {atanh} \left (a x \right ) a x -60 \mathit {atanh} \left (a x \right )+96 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2} x}{a^{2} x^{2}-1}d x \right ) a^{2}-60 \,\mathrm {log}\left (a^{2} x -a \right )+3 a^{2} x^{2}}{60 a} \] Input:

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

Output:

(12*atanh(a*x)**3*a**5*x**5 - 40*atanh(a*x)**3*a**3*x**3 + 60*atanh(a*x)** 
3*a*x + 9*atanh(a*x)**2*a**4*x**4 - 42*atanh(a*x)**2*a**2*x**2 + 33*atanh( 
a*x)**2 + 6*atanh(a*x)*a**3*x**3 - 66*atanh(a*x)*a*x - 60*atanh(a*x) + 96* 
int((atanh(a*x)**2*x)/(a**2*x**2 - 1),x)*a**2 - 60*log(a**2*x - a) + 3*a** 
2*x**2)/(60*a)