3.117 \(\int x^8 (a+b \tanh ^{-1}(c x^3))^2 \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x^3}\right )}{9 c^3}+\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{9 c^3}-\frac{2 b \log \left (\frac{2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{9 c^3}+\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac{b x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{9 c}+\frac{b^2 x^3}{9 c^2}-\frac{b^2 \tanh ^{-1}\left (c x^3\right )}{9 c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.30934, antiderivative size = 536, normalized size of antiderivative = 3.67, number of steps used = 53, number of rules used = 19, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.187, Rules used = {6099, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ -\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )}{18 c^3}+\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )}{18 c^3}-\frac{a b x^3}{9 c^2}-\frac{1}{108} b \left (\frac{2 \left (1-c x^3\right )^3}{c^3}-\frac{9 \left (1-c x^3\right )^2}{c^3}+\frac{18 \left (1-c x^3\right )}{c^3}-\frac{6 \log \left (1-c x^3\right )}{c^3}\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{b \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 c^3}+\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{54} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{18} b x^9 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{36 c}+\frac{19 b^2 x^3}{108 c^2}-\frac{b^2 \left (1-c x^3\right )^3}{162 c^3}+\frac{b^2 \left (1-c x^3\right )^2}{24 c^3}+\frac{b^2 \log ^2\left (1-c x^3\right )}{36 c^3}+\frac{b^2 \log ^2\left (c x^3+1\right )}{36 c^3}-\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{18 c^3}+\frac{b^2 \log \left (1-c x^3\right )}{108 c^3}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{18 c^3}-\frac{b^2 \log \left (c x^3+1\right )}{18 c^3}-\frac{5 b^2 x^6}{216 c}+\frac{1}{36} b^2 x^9 \log ^2\left (c x^3+1\right )+\frac{b^2 x^6 \log \left (c x^3+1\right )}{18 c}-\frac{1}{162} b^2 x^9 \]

Warning: Unable to verify antiderivative.

[In]

Int[x^8*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

-(a*b*x^3)/(9*c^2) + (19*b^2*x^3)/(108*c^2) - (5*b^2*x^6)/(216*c) - (b^2*x^9)/162 + (b^2*(1 - c*x^3)^2)/(24*c^
3) - (b^2*(1 - c*x^3)^3)/(162*c^3) + (b^2*Log[1 - c*x^3])/(108*c^3) - (b^2*(1 - c*x^3)*Log[1 - c*x^3])/(18*c^3
) + (b^2*Log[1 - c*x^3]^2)/(36*c^3) + (b*x^6*(2*a - b*Log[1 - c*x^3]))/(36*c) - (b*x^9*(2*a - b*Log[1 - c*x^3]
))/54 + (x^9*(2*a - b*Log[1 - c*x^3])^2)/36 - (b*(2*a - b*Log[1 - c*x^3])*((18*(1 - c*x^3))/c^3 - (9*(1 - c*x^
3)^2)/c^3 + (2*(1 - c*x^3)^3)/c^3 - (6*Log[1 - c*x^3])/c^3))/108 + (b*(2*a - b*Log[1 - c*x^3])*Log[(1 + c*x^3)
/2])/(18*c^3) - (b^2*Log[1 + c*x^3])/(18*c^3) + (b^2*x^6*Log[1 + c*x^3])/(18*c) + (b^2*Log[(1 - c*x^3)/2]*Log[
1 + c*x^3])/(18*c^3) + (b*x^9*(2*a - b*Log[1 - c*x^3])*Log[1 + c*x^3])/18 + (b^2*Log[1 + c*x^3]^2)/(36*c^3) +
(b^2*x^9*Log[1 + c*x^3]^2)/36 - (b^2*PolyLog[2, (1 - c*x^3)/2])/(18*c^3) + (b^2*PolyLog[2, (1 + c*x^3)/2])/(18
*c^3)

Rule 6099

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] &&
IntegerQ[m] && IntegerQ[n]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

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

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]))/(r +
1), x] + (-Dist[(g*j*m)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[(b*e*n*
p)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /
; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

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

Rule 2391

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

Rule 2410

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin{align*} \int x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^8 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{2} b x^8 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{4} b^2 x^8 \log ^2\left (1+c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^8 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \, dx-\frac{1}{2} b \int x^8 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right ) \, dx+\frac{1}{4} b^2 \int x^8 \log ^2\left (1+c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int x^2 (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )+\frac{1}{12} b^2 \operatorname{Subst}\left (\int x^2 \log ^2(1+c x) \, dx,x,x^3\right )\\ &=\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{x^3 (2 a-b \log (1-c x))}{1-c x} \, dx,x,x^3\right )+\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{x^3 (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+c x)}{1-c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )+\frac{1}{18} b \operatorname{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 )+\frac{1}{18} (b c) \operatorname{Subst}\left (\int \left (\frac{-2 a+b \log (1-c x)}{c^3}-\frac{x (-2 a+b \log (1-c x))}{c^2}+\frac{x^2 (-2 a+b \log (1-c x))}{c}-\frac{-2 a+b \log (1-c x)}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c\right ) \operatorname{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{1}{18} \left (b^2 c\right ) \operatorname{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{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{108} b \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{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )+\frac{1}{18} b \operatorname{Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \, dx,x,x^3\right )+\frac{1}{18} b^2 \operatorname{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{b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{18 c^2}-\frac{b \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{18 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{18 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^3\right )}{18 c^2}-\frac{b \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{18 c}+2 \frac{b^2 \operatorname{Subst}\left (\int x \log (1+c x) \, dx,x,x^3\right )}{18 c}\\ &=-\frac{a b x^3}{9 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{36 c}-\frac{1}{54} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{108} b \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 ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 c^3}+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )-\frac{1}{36} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-c x} \, dx,x,x^3\right )+2 \left (\frac{b^2 x^6 \log \left (1+c x^3\right )}{36 c}-\frac{1}{36} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1+c x} \, dx,x,x^3\right )\right )+\frac{b^2 \operatorname{Subst}\left (\int \frac{x \left (-18+9 x-2 x^2\right )+6 \log (x)}{x} \, dx,x,1-c x^3\right )}{108 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^3\right )}{18 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{18 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{18 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{18 c^2}+\frac{1}{54} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3}{1-c x} \, dx,x,x^3\right )\\ &=-\frac{a b x^3}{9 c^2}+\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{36 c}-\frac{1}{54} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{108} b \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 ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 c^3}+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \log ^2\left (1+c x^3\right )}{36 c^3}+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )-\frac{1}{36} b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )+2 \left (\frac{b^2 x^6 \log \left (1+c x^3\right )}{36 c}-\frac{1}{36} b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x}{c}+\frac{1}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )\right )+\frac{b^2 \operatorname{Subst}\left (\int \left (-18+9 x-2 x^2+\frac{6 \log (x)}{x}\right ) \, dx,x,1-c x^3\right )}{108 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{18 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{18 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{18 c^3}+\frac{1}{54} \left (b^2 c\right ) \operatorname{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 x^3}{9 c^2}+\frac{13 b^2 x^3}{108 c^2}+\frac{b^2 x^6}{216 c}-\frac{b^2 x^9}{162}+\frac{b^2 \left (1-c x^3\right )^2}{24 c^3}-\frac{b^2 \left (1-c x^3\right )^3}{162 c^3}+\frac{b^2 \log \left (1-c x^3\right )}{108 c^3}-\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{18 c^3}+\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{36 c}-\frac{1}{54} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{108} b \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 ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 c^3}+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \log ^2\left (1+c x^3\right )}{36 c^3}+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )+2 \left (\frac{b^2 x^3}{36 c^2}-\frac{b^2 x^6}{72 c}-\frac{b^2 \log \left (1+c x^3\right )}{36 c^3}+\frac{b^2 x^6 \log \left (1+c x^3\right )}{36 c}\right )-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{18 c^3}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x^3\right )}{18 c^3}\\ &=-\frac{a b x^3}{9 c^2}+\frac{13 b^2 x^3}{108 c^2}+\frac{b^2 x^6}{216 c}-\frac{b^2 x^9}{162}+\frac{b^2 \left (1-c x^3\right )^2}{24 c^3}-\frac{b^2 \left (1-c x^3\right )^3}{162 c^3}+\frac{b^2 \log \left (1-c x^3\right )}{108 c^3}-\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{18 c^3}+\frac{b^2 \log ^2\left (1-c x^3\right )}{36 c^3}+\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{36 c}-\frac{1}{54} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{108} b \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 ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 c^3}+\frac{1}{18} b x^9 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \log ^2\left (1+c x^3\right )}{36 c^3}+\frac{1}{36} b^2 x^9 \log ^2\left (1+c x^3\right )+2 \left (\frac{b^2 x^3}{36 c^2}-\frac{b^2 x^6}{72 c}-\frac{b^2 \log \left (1+c x^3\right )}{36 c^3}+\frac{b^2 x^6 \log \left (1+c x^3\right )}{36 c}\right )-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{18 c^3}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{18 c^3}\\ \end{align*}

Mathematica [A]  time = 0.271671, size = 132, normalized size = 0.9 \[ \frac{b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+a^2 c^3 x^9+a b c^2 x^6+a b \log \left (c^2 x^6-1\right )+b \tanh ^{-1}\left (c x^3\right ) \left (2 a c^3 x^9+b c^2 x^6-2 b \log \left (e^{-2 \tanh ^{-1}\left (c x^3\right )}+1\right )-b\right )+b^2 \left (c^3 x^9-1\right ) \tanh ^{-1}\left (c x^3\right )^2+b^2 c x^3}{9 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^8*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{8} \left ( a+b{\it Artanh} \left ( c{x}^{3} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{9} \, a^{2} x^{9} + \frac{1}{9} \,{\left (2 \, x^{9} \operatorname{artanh}\left (c x^{3}\right ) +{\left (\frac{x^{6}}{c^{2}} + \frac{\log \left (c^{2} x^{6} - 1\right )}{c^{4}}\right )} c\right )} a b + \frac{1}{648} \,{\left (18 \, x^{9} \log \left (-c x^{3} + 1\right )^{2} - 2 \, c^{4}{\left (\frac{2 \,{\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{6}} - \frac{3 \, \log \left (c x^{3} + 1\right )}{c^{7}} + \frac{3 \, \log \left (c x^{3} - 1\right )}{c^{7}}\right )} + 3 \,{\left (\frac{x^{6}}{c^{4}} + \frac{\log \left (c^{2} x^{6} - 1\right )}{c^{6}}\right )} c^{3} + 1944 \, c^{3} \int \frac{x^{11} \log \left (c x^{3} + 1\right )}{9 \,{\left (c^{4} x^{6} - c^{2}\right )}}\,{d x} - 9 \, c^{2}{\left (\frac{2 \, x^{3}}{c^{4}} - \frac{\log \left (c x^{3} + 1\right )}{c^{5}} + \frac{\log \left (c x^{3} - 1\right )}{c^{5}}\right )} - 6 \, c{\left (\frac{2 \, c^{2} x^{9} + 3 \, c x^{6} + 6 \, x^{3}}{c^{3}} + \frac{6 \, \log \left (c x^{3} - 1\right )}{c^{4}}\right )} \log \left (-c x^{3} + 1\right ) + 972 \, c \int \frac{x^{5} \log \left (c x^{3} + 1\right )}{9 \,{\left (c^{4} x^{6} - c^{2}\right )}}\,{d x} + \frac{6 \,{\left (3 \, c^{3} x^{9} \log \left (c x^{3} + 1\right )^{2} +{\left (2 \, c^{3} x^{9} - 3 \, c^{2} x^{6} + 6 \, c x^{3} - 6 \,{\left (c^{3} x^{9} + 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )\right )}}{c^{3}} + \frac{4 \, c^{3} x^{9} + 15 \, c^{2} x^{6} + 66 \, c x^{3} + 18 \, \log \left (c x^{3} - 1\right )^{2} + 66 \, \log \left (c x^{3} - 1\right )}{c^{3}} - \frac{18 \, \log \left (9 \, c^{4} x^{6} - 9 \, c^{2}\right )}{c^{3}} + 972 \, \int \frac{x^{2} \log \left (c x^{3} + 1\right )}{9 \,{\left (c^{4} x^{6} - c^{2}\right )}}\,{d x}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3))^2,x, algorithm="maxima")

[Out]

1/9*a^2*x^9 + 1/9*(2*x^9*arctanh(c*x^3) + (x^6/c^2 + log(c^2*x^6 - 1)/c^4)*c)*a*b + 1/648*(18*x^9*log(-c*x^3 +
1)^2 - 2*c^4*(2*(c^2*x^9 + 3*x^3)/c^6 - 3*log(c*x^3 + 1)/c^7 + 3*log(c*x^3 - 1)/c^7) + 3*(x^6/c^4 + log(c^2*x
^6 - 1)/c^6)*c^3 + 1944*c^3*integrate(1/9*x^11*log(c*x^3 + 1)/(c^4*x^6 - c^2), x) - 9*c^2*(2*x^3/c^4 - log(c*x
^3 + 1)/c^5 + log(c*x^3 - 1)/c^5) - 6*c*((2*c^2*x^9 + 3*c*x^6 + 6*x^3)/c^3 + 6*log(c*x^3 - 1)/c^4)*log(-c*x^3
+ 1) + 972*c*integrate(1/9*x^5*log(c*x^3 + 1)/(c^4*x^6 - c^2), x) + 6*(3*c^3*x^9*log(c*x^3 + 1)^2 + (2*c^3*x^9
- 3*c^2*x^6 + 6*c*x^3 - 6*(c^3*x^9 + 1)*log(c*x^3 + 1))*log(-c*x^3 + 1))/c^3 + (4*c^3*x^9 + 15*c^2*x^6 + 66*c
*x^3 + 18*log(c*x^3 - 1)^2 + 66*log(c*x^3 - 1))/c^3 - 18*log(9*c^4*x^6 - 9*c^2)/c^3 + 972*integrate(1/9*x^2*lo
g(c*x^3 + 1)/(c^4*x^6 - c^2), x))*b^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{8} \operatorname{artanh}\left (c x^{3}\right )^{2} + 2 \, a b x^{8} \operatorname{artanh}\left (c x^{3}\right ) + a^{2} x^{8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(a+b*atanh(c*x**3))**2,x)

[Out]

Exception raised: KeyError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{3}\right ) + a\right )}^{2} x^{8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(a+b*arctanh(c*x^3))^2,x, algorithm="giac")

[Out]

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