\(\int x^8 (a+b \text {arctanh}(c x^3))^3 \, dx\) [124]

Optimal result

Integrand size = 16, antiderivative size = 231 \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {a b^2 x^3}{3 c^2}+\frac {b^3 x^3 \text {arctanh}\left (c x^3\right )}{3 c^2}-\frac {b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 c^3}+\frac {b x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 c}+\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{9 c^3}+\frac {1}{9} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-\frac {b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \log \left (1-c^2 x^6\right )}{6 c^3}-\frac {b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{6 c^3} \] Output:

1/3*a*b^2*x^3/c^2+1/3*b^3*x^3*arctanh(c*x^3)/c^2-1/6*b*(a+b*arctanh(c*x^3) 
)^2/c^3+1/6*b*x^6*(a+b*arctanh(c*x^3))^2/c+1/9*(a+b*arctanh(c*x^3))^3/c^3+ 
1/9*x^9*(a+b*arctanh(c*x^3))^3-1/3*b*(a+b*arctanh(c*x^3))^2*ln(2/(-c*x^3+1 
))/c^3+1/6*b^3*ln(-c^2*x^6+1)/c^3-1/3*b^2*(a+b*arctanh(c*x^3))*polylog(2,1 
-2/(-c*x^3+1))/c^3+1/6*b^3*polylog(3,1-2/(-c*x^3+1))/c^3
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.45 \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {6 a b^2 c x^3+3 a^2 b c^2 x^6+2 a^3 c^3 x^9-6 a b^2 \text {arctanh}\left (c x^3\right )+6 b^3 c x^3 \text {arctanh}\left (c x^3\right )+6 a b^2 c^2 x^6 \text {arctanh}\left (c x^3\right )+6 a^2 b c^3 x^9 \text {arctanh}\left (c x^3\right )-6 a b^2 \text {arctanh}\left (c x^3\right )^2-3 b^3 \text {arctanh}\left (c x^3\right )^2+3 b^3 c^2 x^6 \text {arctanh}\left (c x^3\right )^2+6 a b^2 c^3 x^9 \text {arctanh}\left (c x^3\right )^2-2 b^3 \text {arctanh}\left (c x^3\right )^3+2 b^3 c^3 x^9 \text {arctanh}\left (c x^3\right )^3-12 a b^2 \text {arctanh}\left (c x^3\right ) \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )-6 b^3 \text {arctanh}\left (c x^3\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )+3 a^2 b \log \left (1-c^2 x^6\right )+3 b^3 \log \left (1-c^2 x^6\right )+6 b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+3 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )}{18 c^3} \] Input:

Integrate[x^8*(a + b*ArcTanh[c*x^3])^3,x]
 

Output:

(6*a*b^2*c*x^3 + 3*a^2*b*c^2*x^6 + 2*a^3*c^3*x^9 - 6*a*b^2*ArcTanh[c*x^3] 
+ 6*b^3*c*x^3*ArcTanh[c*x^3] + 6*a*b^2*c^2*x^6*ArcTanh[c*x^3] + 6*a^2*b*c^ 
3*x^9*ArcTanh[c*x^3] - 6*a*b^2*ArcTanh[c*x^3]^2 - 3*b^3*ArcTanh[c*x^3]^2 + 
 3*b^3*c^2*x^6*ArcTanh[c*x^3]^2 + 6*a*b^2*c^3*x^9*ArcTanh[c*x^3]^2 - 2*b^3 
*ArcTanh[c*x^3]^3 + 2*b^3*c^3*x^9*ArcTanh[c*x^3]^3 - 12*a*b^2*ArcTanh[c*x^ 
3]*Log[1 + E^(-2*ArcTanh[c*x^3])] - 6*b^3*ArcTanh[c*x^3]^2*Log[1 + E^(-2*A 
rcTanh[c*x^3])] + 3*a^2*b*Log[1 - c^2*x^6] + 3*b^3*Log[1 - c^2*x^6] + 6*b^ 
2*(a + b*ArcTanh[c*x^3])*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + 3*b^3*PolyLo 
g[3, -E^(-2*ArcTanh[c*x^3])])/(18*c^3)
 

Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6454, 6452, 6542, 6452, 6542, 2009, 6510, 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.

\(\displaystyle \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx\)

\(\Big \downarrow \) 6454

\(\displaystyle \frac {1}{3} \int x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3dx^3\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \int \frac {x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3\right )\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\int x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \int \frac {x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{1-c^2 x^6}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c x^3\right )\right )dx^3}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 6510

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c^2 x^6}dx^3}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{1-c x^3}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \log \left (\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} x^9 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{c}-2 b \left (\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}\right )}{c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )}{c^2}\right )\right )\)

Input:

Int[x^8*(a + b*ArcTanh[c*x^3])^3,x]
 

Output:

((x^9*(a + b*ArcTanh[c*x^3])^3)/3 - b*c*(-(((x^6*(a + b*ArcTanh[c*x^3])^2) 
/2 - b*c*((a + b*ArcTanh[c*x^3])^2/(2*b*c^3) - (a*x^3 + b*x^3*ArcTanh[c*x^ 
3] + (b*Log[1 - c^2*x^6])/(2*c))/c^2))/c^2) + (-1/3*(a + b*ArcTanh[c*x^3]) 
^3/(b*c^2) + (((a + b*ArcTanh[c*x^3])^2*Log[2/(1 - c*x^3)])/c - 2*b*(-1/2* 
((a + b*ArcTanh[c*x^3])*PolyLog[2, 1 - 2/(1 - c*x^3)])/c + (b*PolyLog[3, 1 
 - 2/(1 - c*x^3)])/(4*c)))/c)/c^2))/3
 

Defintions of rubi rules used

rule 2009

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

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 6454

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x 
], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl 
ify[(m + 1)/n]]
 

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 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 6542

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

\[\int x^{8} {\left (a +b \,\operatorname {arctanh}\left (c \,x^{3}\right )\right )}^{3}d x\]

Input:

int(x^8*(a+b*arctanh(c*x^3))^3,x)
 

Output:

int(x^8*(a+b*arctanh(c*x^3))^3,x)
 

Fricas [F]

\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \] Input:

integrate(x^8*(a+b*arctanh(c*x^3))^3,x, algorithm="fricas")
 

Output:

integral(b^3*x^8*arctanh(c*x^3)^3 + 3*a*b^2*x^8*arctanh(c*x^3)^2 + 3*a^2*b 
*x^8*arctanh(c*x^3) + a^3*x^8, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\text {Timed out} \] Input:

integrate(x**8*(a+b*atanh(c*x**3))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \] Input:

integrate(x^8*(a+b*arctanh(c*x^3))^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/9*a^3*x^9 + 1/6*(2*x^9*arctanh(c*x^3) + (x^6/c^2 + log(c^2*x^6 - 1)/c^4) 
*c)*a^2*b - 1/72*((b^3*c^3*x^9 - b^3)*log(-c*x^3 + 1)^3 - 3*(2*a*b^2*c^3*x 
^9 + b^3*c^2*x^6 + (b^3*c^3*x^9 + b^3)*log(c*x^3 + 1))*log(-c*x^3 + 1)^2)/ 
c^3 - integrate(-1/8*((b^3*c^3*x^11 - b^3*c^2*x^8)*log(c*x^3 + 1)^3 + 6*(a 
*b^2*c^3*x^11 - a*b^2*c^2*x^8)*log(c*x^3 + 1)^2 - (4*a*b^2*c^3*x^11 + 2*b^ 
3*c^2*x^8 + 3*(b^3*c^3*x^11 - b^3*c^2*x^8)*log(c*x^3 + 1)^2 - 2*(6*a*b^2*c 
^2*x^8 - (6*a*b^2*c^3 + b^3*c^3)*x^11 - b^3*x^2)*log(c*x^3 + 1))*log(-c*x^ 
3 + 1))/(c^3*x^3 - c^2), x)
 

Giac [F]

\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{8} \,d x } \] Input:

integrate(x^8*(a+b*arctanh(c*x^3))^3,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^3) + a)^3*x^8, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\int x^8\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3 \,d x \] Input:

int(x^8*(a + b*atanh(c*x^3))^3,x)
 

Output:

int(x^8*(a + b*atanh(c*x^3))^3, x)
 

Reduce [F]

\[ \int x^8 \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \, dx=\frac {2 \mathit {atanh} \left (c \,x^{3}\right )^{3} b^{3} c^{3} x^{9}-2 \mathit {atanh} \left (c \,x^{3}\right )^{3} b^{3} c \,x^{3}+6 \mathit {atanh} \left (c \,x^{3}\right )^{2} a \,b^{2} c^{3} x^{9}-6 \mathit {atanh} \left (c \,x^{3}\right )^{2} a \,b^{2} c \,x^{3}+3 \mathit {atanh} \left (c \,x^{3}\right )^{2} b^{3} c^{2} x^{6}-3 \mathit {atanh} \left (c \,x^{3}\right )^{2} b^{3}+6 \mathit {atanh} \left (c \,x^{3}\right ) a^{2} b \,c^{3} x^{9}-6 \mathit {atanh} \left (c \,x^{3}\right ) a^{2} b +6 \mathit {atanh} \left (c \,x^{3}\right ) a \,b^{2} c^{2} x^{6}-6 \mathit {atanh} \left (c \,x^{3}\right ) a \,b^{2}+6 \mathit {atanh} \left (c \,x^{3}\right ) b^{3} c \,x^{3}-6 \mathit {atanh} \left (c \,x^{3}\right ) b^{3}+6 \left (\int \mathit {atanh} \left (c \,x^{3}\right )^{3} x^{2}d x \right ) b^{3} c +18 \left (\int \mathit {atanh} \left (c \,x^{3}\right )^{2} x^{2}d x \right ) a \,b^{2} c +6 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) a^{2} b +6 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b^{3}+6 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) a^{2} b +6 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b^{3}+2 a^{3} c^{3} x^{9}+3 a^{2} b \,c^{2} x^{6}+6 a \,b^{2} c \,x^{3}}{18 c^{3}} \] Input:

int(x^8*(a+b*atanh(c*x^3))^3,x)
 

Output:

(2*atanh(c*x**3)**3*b**3*c**3*x**9 - 2*atanh(c*x**3)**3*b**3*c*x**3 + 6*at 
anh(c*x**3)**2*a*b**2*c**3*x**9 - 6*atanh(c*x**3)**2*a*b**2*c*x**3 + 3*ata 
nh(c*x**3)**2*b**3*c**2*x**6 - 3*atanh(c*x**3)**2*b**3 + 6*atanh(c*x**3)*a 
**2*b*c**3*x**9 - 6*atanh(c*x**3)*a**2*b + 6*atanh(c*x**3)*a*b**2*c**2*x** 
6 - 6*atanh(c*x**3)*a*b**2 + 6*atanh(c*x**3)*b**3*c*x**3 - 6*atanh(c*x**3) 
*b**3 + 6*int(atanh(c*x**3)**3*x**2,x)*b**3*c + 18*int(atanh(c*x**3)**2*x* 
*2,x)*a*b**2*c + 6*log(c**(2/3)*x**2 - c**(1/3)*x + 1)*a**2*b + 6*log(c**( 
2/3)*x**2 - c**(1/3)*x + 1)*b**3 + 6*log(c**(2/3)*x + c**(1/3))*a**2*b + 6 
*log(c**(2/3)*x + c**(1/3))*b**3 + 2*a**3*c**3*x**9 + 3*a**2*b*c**2*x**6 + 
 6*a*b**2*c*x**3)/(18*c**3)