3.2.24 \(\int x^8 (a+b \tanh ^{-1}(c x^3))^3 \, dx\) [124]

Optimal. Leaf size=231 \[ \frac {a b^2 x^3}{3 c^2}+\frac {b^3 x^3 \tanh ^{-1}\left (c x^3\right )}{3 c^2}-\frac {b \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c^3}+\frac {b x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c}+\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{9 c^3}+\frac {1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3-\frac {b \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \log \left (\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \log \left (1-c^2 x^6\right )}{6 c^3}-\frac {b^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{3 c^3}+\frac {b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{6 c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6039, 6037, 6127, 6021, 266, 6095, 6131, 6055, 6205, 6745} \begin {gather*} -\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 c^3}+\frac {a b^2 x^3}{3 c^2}+\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{9 c^3}-\frac {b \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c^3}-\frac {b \log \left (\frac {2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{3 c^3}+\frac {1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3+\frac {b x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 c}+\frac {b^3 \text {Li}_3\left (1-\frac {2}{1-c x^3}\right )}{6 c^3}+\frac {b^3 x^3 \tanh ^{-1}\left (c x^3\right )}{3 c^2}+\frac {b^3 \log \left (1-c^2 x^6\right )}{6 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*ArcTanh[c*x^3])^3,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[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 6037

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] - Dist[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 6039

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[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[S
implify[(m + 1)/n]]

Rule 6055

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] + Dist[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 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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 6127

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[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 6131

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] + Dist[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 6205

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcT
anh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[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 6745

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]

Rubi steps

\begin {align*} \int x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^8 \left (2 a-b \log \left (1-c x^3\right )\right )^3+\frac {3}{8} b x^8 \left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )-\frac {3}{8} b^2 x^8 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {1}{8} b^3 x^8 \log ^3\left (1+c x^3\right )\right ) \, dx\\ &=\frac {1}{8} \int x^8 \left (2 a-b \log \left (1-c x^3\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^8 \left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right ) \, dx-\frac {1}{8} \left (3 b^2\right ) \int x^8 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right ) \, dx+\frac {1}{8} b^3 \int x^8 \log ^3\left (1+c x^3\right ) \, dx\\ &=\frac {1}{24} \text {Subst}\left (\int x^2 (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )+\frac {1}{8} b \text {Subst}\left (\int x^2 (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^3\right )-\frac {1}{8} b^2 \text {Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^3\right )+\frac {1}{24} b^3 \text {Subst}\left (\int x^2 \log ^3(1+c x) \, dx,x,x^3\right )\\ &=\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {1}{24} \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^3}{c^2}-\frac {2 (1-c x) (2 a-b \log (1-c x))^3}{c^2}+\frac {(1-c x)^2 (2 a-b \log (1-c x))^3}{c^2}\right ) \, dx,x,x^3\right )+\frac {1}{24} b^3 \text {Subst}\left (\int \left (\frac {\log ^3(1+c x)}{c^2}-\frac {2 (1+c x) \log ^3(1+c x)}{c^2}+\frac {(1+c x)^2 \log ^3(1+c x)}{c^2}\right ) \, dx,x,x^3\right )-\frac {1}{24} (b c) \text {Subst}\left (\int \frac {x^3 (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^3 (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^3 (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )-\frac {1}{24} \left (b^3 c\right ) \text {Subst}\left (\int \frac {x^3 \log ^2(1+c x)}{1-c x} \, dx,x,x^3\right )\\ &=\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {\text {Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c^2}+\frac {\text {Subst}\left (\int (1-c x)^2 (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c^2}-\frac {\text {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{12 c^2}+\frac {b^3 \text {Subst}\left (\int \log ^3(1+c x) \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int (1+c x)^2 \log ^3(1+c x) \, dx,x,x^3\right )}{24 c^2}-\frac {b^3 \text {Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,x^3\right )}{12 c^2}-\frac {1}{24} (b c) \text {Subst}\left (\int \left (\frac {(-2 a+b \log (1-c x))^2}{c^3}-\frac {x (-2 a+b \log (1-c x))^2}{c^2}+\frac {x^2 (-2 a+b \log (1-c x))^2}{c}-\frac {(-2 a+b \log (1-c x))^2}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^3}+\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c^2}+\frac {x^2 (2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^3 (-1+c x)}\right ) \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^3}+\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c^2}-\frac {x^2 (2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} \left (b^3 c\right ) \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c^3}-\frac {x \log ^2(1+c x)}{c^2}-\frac {x^2 \log ^2(1+c x)}{c}-\frac {\log ^2(1+c x)}{c^3 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {1}{24} b \text {Subst}\left (\int x^2 (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )+\frac {1}{24} b^3 \text {Subst}\left (\int x^2 \log ^2(1+c x) \, dx,x,x^3\right )-\frac {\text {Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^3}-\frac {\text {Subst}\left (\int x^2 (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^3}+\frac {\text {Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{12 c^3}+\frac {b^3 \text {Subst}\left (\int \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^3}+\frac {b^3 \text {Subst}\left (\int x^2 \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int x \log ^3(x) \, dx,x,1+c x^3\right )}{12 c^3}-\frac {b \text {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{24 c^2}+\frac {b \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )}{24 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )}{12 c^2}+\frac {b^3 \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int \frac {\log ^2(1+c x)}{-1+c x} \, dx,x,x^3\right )}{24 c^2}+\frac {b \text {Subst}\left (\int x (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{24 c}+2 \frac {b^2 \text {Subst}\left (\int x (2 a-b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )}{12 c}+\frac {b^3 \text {Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^3\right )}{24 c}\\ &=-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}+2 \left (\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}-\frac {1}{24} b^2 \text {Subst}\left (\int \frac {x^2 (2 a-b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac {1}{24} b^3 \text {Subst}\left (\int \frac {x^2 \log (1+c x)}{1-c x} \, dx,x,x^3\right )\right )-\frac {b \text {Subst}\left (\int x^2 (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{24 c^3}+\frac {b \text {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{24 c^3}-\frac {b \text {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{8 c^3}+\frac {b \text {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{8 c^3}+\frac {b^2 \text {Subst}\left (\int \frac {(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^3\right )}{12 c^3}+\frac {b^2 \text {Subst}\left (\int \frac {\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{12 c^3}+\frac {b^3 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int x^2 \log ^2(x) \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{8 c^3}+\frac {b^3 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{8 c^3}+\frac {b^2 \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{12 c^2}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right ) \log (1+c x)}{1+c x} \, dx,x,x^3\right )}{12 c^2}+\frac {b \text {Subst}\left (\int \left (\frac {(-2 a+b \log (1-c x))^2}{c}-\frac {(1-c x) (-2 a+b \log (1-c x))^2}{c}\right ) \, dx,x,x^3\right )}{24 c}+\frac {b^3 \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,x^3\right )}{24 c}-\frac {1}{36} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^3 (-2 a+b \log (1-c x))}{1-c x} \, dx,x,x^3\right )-\frac {1}{36} \left (b^3 c\right ) \text {Subst}\left (\int \frac {x^3 \log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{16 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}+\frac {1}{36} b^2 \text {Subst}\left (\int \frac {\left (\frac {1}{c}-\frac {x}{c}\right )^3 (-2 a+b \log (x))}{x} \, dx,x,1-c x^3\right )+2 \left (\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}-\frac {1}{24} b^2 \text {Subst}\left (\int \left (-\frac {2 a-b \log (1-c x)}{c^2}+\frac {x (2 a-b \log (1-c x))}{c}+\frac {2 a-b \log (1-c x)}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} b^3 \text {Subst}\left (\int \left (-\frac {\log (1+c x)}{c^2}-\frac {x \log (1+c x)}{c}-\frac {\log (1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )\right )-\frac {b \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^3\right )}{24 c^3}-\frac {b^2 \text {Subst}\left (\int x^2 (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{36 c^3}-\frac {b^2 \text {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) (-2 a+b \log (x))}{x} \, dx,x,1-c x^3\right )}{12 c^3}+\frac {b^2 \text {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{8 c^3}-\frac {b^2 \text {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{4 c^3}+\frac {b^3 \text {Subst}\left (\int x^2 \log (x) \, dx,x,1+c x^3\right )}{36 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log ^2(x)}{2-x} \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{8 c^3}+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{4 c^3}+\frac {b \text {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{24 c^2}-\frac {b \text {Subst}\left (\int (1-c x) (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{24 c^2}-\frac {b^3 \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^3\right )}{24 c^2}-\frac {1}{36} \left (b^3 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+c x)}{c^3}-\frac {x \log (1+c x)}{c^2}+\frac {x^2 \log (1+c x)}{c}-\frac {\log (1+c x)}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a b^2 x^3}{3 c^2}-\frac {b^3 x^3}{6 c^2}+\frac {b^3 \left (1-c x^3\right )^2}{32 c^3}-\frac {b^3 \left (1-c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1+c x^3\right )^2}{32 c^3}-\frac {b^3 \left (1+c x^3\right )^3}{324 c^3}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{16 c^3}-\frac {b^2 \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )}{108 c^3}-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {1}{216} b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {18 \left (1-c x^3\right )}{c^3}-\frac {9 \left (1-c x^3\right )^2}{c^3}+\frac {2 \left (1-c x^3\right )^3}{c^3}-\frac {6 \log \left (1-c x^3\right )}{c^3}\right )+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{6 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{16 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log \left (1+c x^3\right )}{108 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{16 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{12 c^3}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}-\frac {1}{36} b^3 \text {Subst}\left (\int \frac {x \left (-18+9 x-2 x^2\right )+6 \log (x)}{6 c^3 x} \, dx,x,1-c x^3\right )-\frac {1}{36} b^3 \text {Subst}\left (\int x^2 \log (1+c x) \, dx,x,x^3\right )-\frac {b \text {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{24 c^3}+\frac {b \text {Subst}\left (\int x (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{24 c^3}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{24 c^3}+\frac {b^3 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{12 c^3}+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{4 c^3}-\frac {b^3 \text {Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{36 c^2}+\frac {b^3 \text {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,x^3\right )}{36 c^2}+\frac {b^3 \text {Subst}\left (\int x \log (1+c x) \, dx,x,x^3\right )}{36 c}+2 \left (\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}+\frac {b^2 \text {Subst}\left (\int (2 a-b \log (1-c x)) \, dx,x,x^3\right )}{24 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {2 a-b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{24 c^2}-\frac {b^2 \text {Subst}\left (\int x (2 a-b \log (1-c x)) \, dx,x,x^3\right )}{24 c}+\frac {b^3 \text {Subst}\left (\int x \log (1+c x) \, dx,x,x^3\right )}{24 c}\right )\\ &=\frac {a b^2 x^3}{3 c^2}+\frac {b^3 \left (1-c x^3\right )^2}{32 c^3}-\frac {b^3 \left (1-c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1+c x^3\right )^2}{32 c^3}-\frac {b^3 \left (1+c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{6 c^3}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{16 c^3}-\frac {b^2 \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )}{108 c^3}-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {1}{216} b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {18 \left (1-c x^3\right )}{c^3}-\frac {9 \left (1-c x^3\right )^2}{c^3}+\frac {2 \left (1-c x^3\right )^3}{c^3}-\frac {6 \log \left (1-c x^3\right )}{c^3}\right )+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{72 c}-\frac {1}{108} b^3 x^9 \log \left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{6 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{16 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log \left (1+c x^3\right )}{108 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{12 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^3\right )\right )}{12 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}-\frac {1}{72} b^3 \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,x^3\right )-\frac {b^2 \text {Subst}\left (\int x (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{24 c^3}+\frac {b^2 \text {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {x \left (-18+9 x-2 x^2\right )+6 \log (x)}{x} \, dx,x,1-c x^3\right )}{216 c^3}-\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{36 c^3}+\frac {b^3 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+c x^3\right )}{36 c^3}-\frac {b^3 \text {Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{24 c^3}+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{12 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{12 c^3}+2 \left (\frac {a b^2 x^3}{12 c^2}-\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{48 c}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}+\frac {1}{48} b^3 \text {Subst}\left (\int \frac {x^2}{1-c x} \, dx,x,x^3\right )-\frac {1}{48} b^3 \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,x^3\right )+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{24 c^2}-\frac {b^3 \text {Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{24 c^2}+\frac {b^3 \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{24 c^2}\right )+\frac {1}{108} \left (b^3 c\right ) \text {Subst}\left (\int \frac {x^3}{1+c x} \, dx,x,x^3\right )\\ &=\frac {a b^2 x^3}{2 c^2}-\frac {b^3 x^3}{18 c^2}+\frac {b^3 \left (1-c x^3\right )^2}{24 c^3}-\frac {b^3 \left (1-c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1+c x^3\right )^2}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{6 c^3}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{12 c^3}-\frac {b^2 \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )}{108 c^3}-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {1}{216} b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {18 \left (1-c x^3\right )}{c^3}-\frac {9 \left (1-c x^3\right )^2}{c^3}+\frac {2 \left (1-c x^3\right )^3}{c^3}-\frac {6 \log \left (1-c x^3\right )}{c^3}\right )+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{72 c}-\frac {1}{108} b^3 x^9 \log \left (1+c x^3\right )+\frac {2 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{9 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log \left (1+c x^3\right )}{108 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{12 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}-\frac {1}{72} b^3 \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {b^3 \text {Subst}\left (\int \left (-18+9 x-2 x^2+\frac {6 \log (x)}{x}\right ) \, dx,x,1-c x^3\right )}{216 c^3}+2 \left (\frac {a b^2 x^3}{12 c^2}-\frac {b^3 x^3}{24 c^2}-\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{48 c}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}+\frac {1}{48} b^3 \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {x}{c}-\frac {1}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{48} b^3 \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{24 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{24 c^3}+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{24 c^3}\right )+\frac {b^3 \text {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{12 c^3}+\frac {1}{108} \left (b^3 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^3}-\frac {x}{c^2}+\frac {x^2}{c}-\frac {1}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a b^2 x^3}{2 c^2}-\frac {7 b^3 x^3}{216 c^2}-\frac {5 b^3 x^6}{432 c}+\frac {b^3 x^9}{324}+\frac {b^3 \left (1-c x^3\right )^2}{48 c^3}+\frac {b^3 \left (1+c x^3\right )^2}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{4 c^3}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{12 c^3}-\frac {b^2 \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )}{108 c^3}-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {1}{216} b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {18 \left (1-c x^3\right )}{c^3}-\frac {9 \left (1-c x^3\right )^2}{c^3}+\frac {2 \left (1-c x^3\right )^3}{c^3}-\frac {6 \log \left (1-c x^3\right )}{c^3}\right )+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}-\frac {5 b^3 \log \left (1+c x^3\right )}{216 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{72 c}-\frac {1}{108} b^3 x^9 \log \left (1+c x^3\right )+\frac {2 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{9 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log \left (1+c x^3\right )}{108 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{12 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}+2 \left (\frac {a b^2 x^3}{12 c^2}-\frac {b^3 x^6}{48 c}-\frac {b^3 \log \left (1-c x^3\right )}{48 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^3}-\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}-\frac {b^3 \log \left (1+c x^3\right )}{48 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{48 c}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}+\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{24 c^3}+\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}\right )-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x^3\right )}{36 c^3}\\ &=\frac {a b^2 x^3}{2 c^2}-\frac {7 b^3 x^3}{216 c^2}-\frac {5 b^3 x^6}{432 c}+\frac {b^3 x^9}{324}+\frac {b^3 \left (1-c x^3\right )^2}{48 c^3}+\frac {b^3 \left (1+c x^3\right )^2}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^3}{324 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{4 c^3}-\frac {b^3 \log ^2\left (1-c x^3\right )}{72 c^3}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{12 c^3}-\frac {b^2 \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )}{108 c^3}-\frac {1}{72} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^3}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c^3}-\frac {b \left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{72 c^3}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^3}-\frac {\left (1-c x^3\right )^3 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{72 c^3}+\frac {1}{216} b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {18 \left (1-c x^3\right )}{c^3}-\frac {9 \left (1-c x^3\right )^2}{c^3}+\frac {2 \left (1-c x^3\right )^3}{c^3}-\frac {6 \log \left (1-c x^3\right )}{c^3}\right )+\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{12 c^3}-\frac {5 b^3 \log \left (1+c x^3\right )}{216 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{72 c}-\frac {1}{108} b^3 x^9 \log \left (1+c x^3\right )+\frac {2 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{9 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{12 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log \left (1+c x^3\right )}{108 c^3}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {1}{72} b^3 x^9 \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^3}+\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{12 c^3}-\frac {b^3 \left (1+c x^3\right )^3 \log ^2\left (1+c x^3\right )}{72 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{12 c^3}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{24 c^3}+\frac {1}{24} b^2 x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^3}-\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{24 c^3}+\frac {b^3 \left (1+c x^3\right )^3 \log ^3\left (1+c x^3\right )}{72 c^3}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}+2 \left (\frac {a b^2 x^3}{12 c^2}-\frac {b^3 x^6}{48 c}-\frac {b^3 \log \left (1-c x^3\right )}{48 c^3}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^3}-\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}-\frac {b^3 \log \left (1+c x^3\right )}{48 c^3}+\frac {b^3 x^6 \log \left (1+c x^3\right )}{48 c}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^3}+\frac {b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c}+\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{24 c^3}+\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^3}\right )-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^3\right )\right )}{6 c^3}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^3\right )\right )}{6 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 334, normalized size = 1.45 \begin {gather*} \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 \tanh ^{-1}\left (c x^3\right )+6 b^3 c x^3 \tanh ^{-1}\left (c x^3\right )+6 a b^2 c^2 x^6 \tanh ^{-1}\left (c x^3\right )+6 a^2 b c^3 x^9 \tanh ^{-1}\left (c x^3\right )-6 a b^2 \tanh ^{-1}\left (c x^3\right )^2-3 b^3 \tanh ^{-1}\left (c x^3\right )^2+3 b^3 c^2 x^6 \tanh ^{-1}\left (c x^3\right )^2+6 a b^2 c^3 x^9 \tanh ^{-1}\left (c x^3\right )^2-2 b^3 \tanh ^{-1}\left (c x^3\right )^3+2 b^3 c^3 x^9 \tanh ^{-1}\left (c x^3\right )^3-12 a b^2 \tanh ^{-1}\left (c x^3\right ) \log \left (1+e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )-6 b^3 \tanh ^{-1}\left (c x^3\right )^2 \log \left (1+e^{-2 \tanh ^{-1}\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 \tanh ^{-1}\left (c x^3\right )\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+3 b^3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )}{18 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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*ArcTanh[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*PolyLog[3, -E^(-2*ArcTanh[c*x^3])])/(18*c^3)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{8} \left (a +b \arctanh \left (c \,x^{3}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^8\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*atanh(c*x^3))^3,x)

[Out]

int(x^8*(a + b*atanh(c*x^3))^3, x)

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