\(\int (1-a^2 x^2)^3 \text {arctanh}(a x)^2 \, dx\) [225]

Optimal result

Integrand size = 19, antiderivative size = 227 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {16 \text {arctanh}(a x)^2}{35 a}+\frac {16}{35} x \text {arctanh}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2-\frac {32 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {16 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a} \] Output:

-38/105*x+19/315*a^2*x^3-1/105*a^4*x^5+8/35*(-a^2*x^2+1)*arctanh(a*x)/a+3/ 
35*(-a^2*x^2+1)^2*arctanh(a*x)/a+1/21*(-a^2*x^2+1)^3*arctanh(a*x)/a+16/35* 
arctanh(a*x)^2/a+16/35*x*arctanh(a*x)^2+8/35*x*(-a^2*x^2+1)*arctanh(a*x)^2 
+6/35*x*(-a^2*x^2+1)^2*arctanh(a*x)^2+1/7*x*(-a^2*x^2+1)^3*arctanh(a*x)^2- 
32/35*arctanh(a*x)*ln(2/(-a*x+1))/a-16/35*polylog(2,1-2/(-a*x+1))/a
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.55 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {114 a x-19 a^3 x^3+3 a^5 x^5+9 (-1+a x)^4 \left (16+29 a x+20 a^2 x^2+5 a^3 x^3\right ) \text {arctanh}(a x)^2+3 \text {arctanh}(a x) \left (-38+57 a^2 x^2-24 a^4 x^4+5 a^6 x^6+96 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-144 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{315 a} \] Input:

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

Output:

-1/315*(114*a*x - 19*a^3*x^3 + 3*a^5*x^5 + 9*(-1 + a*x)^4*(16 + 29*a*x + 2 
0*a^2*x^2 + 5*a^3*x^3)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(-38 + 57*a^2*x^2 - 
 24*a^4*x^4 + 5*a^6*x^6 + 96*Log[1 + E^(-2*ArcTanh[a*x])]) - 144*PolyLog[2 
, -E^(-2*ArcTanh[a*x])])/a
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6506, 210, 2009, 6506, 2009, 6506, 24, 6436, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6506

\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx-\frac {1}{21} \int \left (1-a^2 x^2\right )^2dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}\)

\(\Big \downarrow \) 210

\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx-\frac {1}{21} \int \left (a^4 x^4-2 a^2 x^2+1\right )dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 6506

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx-\frac {1}{10} \int \left (1-a^2 x^2\right )dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 6506

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx-\frac {\int 1dx}{3}+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\)

Input:

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

Output:

(-x + (2*a^2*x^3)/3 - (a^4*x^5)/5)/21 + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21 
*a) + (x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2)/7 + (6*((-x + (a^2*x^3)/3)/10 + ( 
(1 - a^2*x^2)^2*ArcTanh[a*x])/(10*a) + (x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/ 
5 + (4*(-1/3*x + ((1 - a^2*x^2)*ArcTanh[a*x])/(3*a) + (x*(1 - a^2*x^2)*Arc 
Tanh[a*x]^2)/3 + (2*(x*ArcTanh[a*x]^2 - 2*a*(-1/2*ArcTanh[a*x]^2/a^2 + ((A 
rcTanh[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)))/ 
3))/5))/7
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

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 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]
 

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{35}-\frac {19 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\)	\(222\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{35}-\frac {19 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\)	\(222\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{5}}{5}-x^{3} a^{2} \operatorname {arctanh}\left (a x \right )^{2}+x \operatorname {arctanh}\left (a x \right )^{2}-\frac {a^{5} \operatorname {arctanh}\left (a x \right ) x^{6}}{21}+\frac {8 a^{3} \operatorname {arctanh}\left (a x \right ) x^{4}}{35}-\frac {19 x^{2} \operatorname {arctanh}\left (a x \right ) a}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35 a}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35 a}+\frac {-48 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )-24 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )+12 \ln \left (a x -1\right )^{2}+24 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )-12 \ln \left (a x +1\right )^{2}-a^{5} x^{5}+\frac {19 a^{3} x^{3}}{3}-38 a x -19 \ln \left (a x -1\right )+19 \ln \left (a x +1\right )}{105 a}\)	\(227\)
risch	\(-\frac {38 x}{105}+\frac {4 a^{3} \ln \left (a x +1\right ) x^{4}}{35}-\frac {4 a^{3} \ln \left (-a x +1\right ) x^{4}}{35}+\frac {a^{5} \ln \left (-a x +1\right ) x^{6}}{42}-\frac {a^{5} \ln \left (a x +1\right ) x^{6}}{42}+\frac {19 a^{2} x^{3}}{315}-\frac {a^{4} x^{5}}{105}+\frac {19 a \ln \left (-a x +1\right ) x^{2}}{70}+\frac {19 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{70 a}-\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a}+\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}-\frac {19 a \ln \left (a x +1\right ) x^{2}}{70}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {x \ln \left (-a x +1\right )}{2}+\frac {3 a^{4} \ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {67 \ln \left (a x +1\right )}{210 a}-\frac {1276 \ln \left (a x -1\right )}{3675 a}-\frac {2453 \ln \left (-a x +1\right )}{7350 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}+\frac {3 a^{4} \ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {\ln \left (-a x +1\right )^{2} x}{4}-\frac {4 \ln \left (-a x +1\right )^{2}}{35 a}+\frac {\ln \left (a x +1\right )^{2} x}{4}+\frac {4 \ln \left (a x +1\right )^{2}}{35 a}-\frac {3 a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}-\frac {a^{6} \ln \left (-a x +1\right )^{2} x^{7}}{28}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{4}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{4}-\frac {20469}{42875 a}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{2}-\frac {a^{6} \ln \left (a x +1\right )^{2} x^{7}}{28}+\frac {a^{6} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{14}\)	\(509\)

Input:

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

Output:

1/a*(-1/7*arctanh(a*x)^2*a^7*x^7+3/5*arctanh(a*x)^2*a^5*x^5-arctanh(a*x)^2 
*a^3*x^3+arctanh(a*x)^2*a*x-1/21*arctanh(a*x)*a^6*x^6+8/35*a^4*x^4*arctanh 
(a*x)-19/35*a^2*x^2*arctanh(a*x)+16/35*arctanh(a*x)*ln(a*x-1)+16/35*arctan 
h(a*x)*ln(a*x+1)-16/35*dilog(1/2*a*x+1/2)-8/35*ln(a*x-1)*ln(1/2*a*x+1/2)+4 
/35*ln(a*x-1)^2+8/35*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-4/35*ln( 
a*x+1)^2-1/105*a^5*x^5+19/315*a^3*x^3-38/105*a*x-19/105*ln(a*x-1)+19/105*l 
n(a*x+1))
 

Fricas [F]

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

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

Output:

integral(-(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^2, x)
 

Sympy [F]

\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=- \int 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx \] Input:

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

Output:

-Integral(3*a**2*x**2*atanh(a*x)**2, x) - Integral(-3*a**4*x**4*atanh(a*x) 
**2, x) - Integral(a**6*x**6*atanh(a*x)**2, x) - Integral(-atanh(a*x)**2, 
x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {1}{315} \, a^{2} {\left (\frac {3 \, a^{5} x^{5} - 19 \, a^{3} x^{3} + 114 \, a x + 36 \, \log \left (a x + 1\right )^{2} - 72 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 36 \, \log \left (a x - 1\right )^{2} + 57 \, \log \left (a x - 1\right )}{a^{3}} + \frac {144 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} - \frac {57 \, \log \left (a x + 1\right )}{a^{3}}\right )} - \frac {1}{105} \, {\left (5 \, a^{4} x^{6} - 24 \, a^{2} x^{4} + 57 \, x^{2} - \frac {48 \, \log \left (a x + 1\right )}{a^{2}} - \frac {48 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 35 \, a^{2} x^{3} - 35 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:

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

Output:

-1/315*a^2*((3*a^5*x^5 - 19*a^3*x^3 + 114*a*x + 36*log(a*x + 1)^2 - 72*log 
(a*x + 1)*log(a*x - 1) - 36*log(a*x - 1)^2 + 57*log(a*x - 1))/a^3 + 144*(l 
og(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3 - 57*log(a*x + 
 1)/a^3) - 1/105*(5*a^4*x^6 - 24*a^2*x^4 + 57*x^2 - 48*log(a*x + 1)/a^2 - 
48*log(a*x - 1)/a^2)*a*arctanh(a*x) - 1/35*(5*a^6*x^7 - 21*a^4*x^5 + 35*a^ 
2*x^3 - 35*x)*arctanh(a*x)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=\frac {-45 \mathit {atanh} \left (a x \right )^{2} a^{7} x^{7}+189 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}-315 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}+315 \mathit {atanh} \left (a x \right )^{2} a x -15 \mathit {atanh} \left (a x \right ) a^{6} x^{6}+72 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-171 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+114 \mathit {atanh} \left (a x \right )+288 \left (\int \frac {\mathit {atanh} \left (a x \right ) x}{a^{2} x^{2}-1}d x \right ) a^{2}-3 a^{5} x^{5}+19 a^{3} x^{3}-114 a x}{315 a} \] Input:

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

Output:

( - 45*atanh(a*x)**2*a**7*x**7 + 189*atanh(a*x)**2*a**5*x**5 - 315*atanh(a 
*x)**2*a**3*x**3 + 315*atanh(a*x)**2*a*x - 15*atanh(a*x)*a**6*x**6 + 72*at 
anh(a*x)*a**4*x**4 - 171*atanh(a*x)*a**2*x**2 + 114*atanh(a*x) + 288*int(( 
atanh(a*x)*x)/(a**2*x**2 - 1),x)*a**2 - 3*a**5*x**5 + 19*a**3*x**3 - 114*a 
*x)/(315*a)