3.125 \(\int x^5 (a+b \tanh ^{-1}(c x^3))^3 \, dx\)

Optimal. Leaf size=139 \[ -\frac{b^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x^3}\right )}{2 c^2}-\frac{b^2 \log \left (\frac{2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{c^2}-\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{6 c^2}+\frac{b \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{2 c^2}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3+\frac{b x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{2 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 4.20449, antiderivative size = 479, normalized size of antiderivative = 3.45, number of steps used = 155, number of rules used = 30, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.875, Rules used = {6099, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2439, 2416, 2396, 2433, 2374, 6589, 2411, 43, 2334, 12, 14, 2301, 6742, 2430, 2394, 2393, 2391, 2395, 2375, 2317, 2425} \[ -\frac{b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )}{4 c^2}+\frac{b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )}{4 c^2}-\frac{b^2 \log ^2\left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{16 c^2}+\frac{b^2 \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{4 c^2}+\frac{1}{16} b^2 x^6 \log ^2\left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{b^2 x^3 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{4 c}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}-\frac{b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac{b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac{1}{16} b x^6 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac{b^3 \left (c x^3+1\right )^2 \log ^3\left (c x^3+1\right )}{48 c^2}-\frac{b^3 \left (c x^3+1\right ) \log ^3\left (c x^3+1\right )}{24 c^2}+\frac{b^3 \left (c x^3+1\right ) \log ^2\left (c x^3+1\right )}{8 c^2}+\frac{b^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{4 c^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^5*(a + b*ArcTanh[c*x^3])^3,x]

[Out]

-(b*(1 - c*x^3)*(2*a - b*Log[1 - c*x^3])^2)/(8*c^2) - ((1 - c*x^3)*(2*a - b*Log[1 - c*x^3])^3)/(24*c^2) + ((1
- c*x^3)^2*(2*a - b*Log[1 - c*x^3])^3)/(48*c^2) + (b^2*(2*a - b*Log[1 - c*x^3])*Log[(1 + c*x^3)/2])/(4*c^2) +
(b^3*Log[(1 - c*x^3)/2]*Log[1 + c*x^3])/(4*c^2) + (b^2*x^3*(2*a - b*Log[1 - c*x^3])*Log[1 + c*x^3])/(4*c) - (b
*(2*a - b*Log[1 - c*x^3])^2*Log[1 + c*x^3])/(16*c^2) + (b*x^6*(2*a - b*Log[1 - c*x^3])^2*Log[1 + c*x^3])/16 +
(b^3*(1 + c*x^3)*Log[1 + c*x^3]^2)/(8*c^2) - (b^2*(2*a - b*Log[1 - c*x^3])*Log[1 + c*x^3]^2)/(16*c^2) + (b^2*x
^6*(2*a - b*Log[1 - c*x^3])*Log[1 + c*x^3]^2)/16 - (b^3*(1 + c*x^3)*Log[1 + c*x^3]^3)/(24*c^2) + (b^3*(1 + c*x
^3)^2*Log[1 + c*x^3]^3)/(48*c^2) - (b^3*PolyLog[2, (1 - c*x^3)/2])/(4*c^2) + (b^3*PolyLog[2, (1 + c*x^3)/2])/(
4*c^2)

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 2401

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

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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 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]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -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 2396

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

Rule 2433

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(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]

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 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 2375

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 2317

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

Rule 2425

Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)))/(x_), x_Symbol] :> Simp[(Log[
f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m), x] - Dist[(b*e*n)/(2*m), Int[Log[f*x^m]^2/(d + e*x), x], x] /; Fre
eQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin{align*} \int x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^3+\frac{3}{8} b x^5 \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^5 \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^5 \log ^3\left (1+c x^3\right )\right ) \, dx\\ &=\frac{1}{8} \int x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^3 \, dx+\frac{1}{8} (3 b) \int x^5 \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^5 \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^5 \log ^3\left (1+c x^3\right ) \, dx\\ &=\frac{1}{24} \operatorname{Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )+\frac{1}{8} b \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^3\right )-\frac{1}{8} b^2 \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^3\right )+\frac{1}{24} b^3 \operatorname{Subst}\left (\int x \log ^3(1+c x) \, dx,x,x^3\right )\\ &=\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac{1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac{1}{24} \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x))^3}{c}-\frac{(1-c x) (2 a-b \log (1-c x))^3}{c}\right ) \, dx,x,x^3\right )+\frac{1}{24} b^3 \operatorname{Subst}\left (\int \left (-\frac{\log ^3(1+c x)}{c}+\frac{(1+c x) \log ^3(1+c x)}{c}\right ) \, dx,x,x^3\right )-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^3\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )-\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2 \log ^2(1+c x)}{1-c x} \, dx,x,x^3\right )\\ &=\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac{1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac{\operatorname{Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c}-\frac{\operatorname{Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(1+c x) \, dx,x,x^3\right )}{24 c}+\frac{b^3 \operatorname{Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,x^3\right )}{24 c}-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \left (-\frac{(-2 a+b \log (1-c x))^2}{c^2}+\frac{x (-2 a+b \log (1-c x))^2}{c}+\frac{(-2 a+b \log (1-c x))^2}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c^2}+\frac{x (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^2 (-1+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c^2}-\frac{x (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^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c^2}-\frac{x \log ^2(1+c x)}{c}-\frac{\log ^2(1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac{1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac{1}{16} b \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )+\frac{1}{16} b^3 \operatorname{Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^3\right )-\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^2}+\frac{\operatorname{Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^2}+\frac{b^3 \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^2}+\frac{b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}-\frac{b \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )}{16 c}+2 \frac{b^2 \operatorname{Subst}\left (\int (2 a-b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )}{8 c}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{8 c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{-1+c x} \, dx,x,x^3\right )}{16 c}\\ &=-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\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 )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \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 )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}-\frac{1}{16} b \operatorname{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 )+\frac{1}{16} b^3 \operatorname{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 )+2 \left (\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac{1}{8} b^2 \operatorname{Subst}\left (\int \frac{x (2 a-b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \frac{x \log (1+c x)}{1-c x} \, dx,x,x^3\right )\right )+\frac{b \operatorname{Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac{b \operatorname{Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac{b \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}-\frac{b^3 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^2 \operatorname{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 )}{8 c}-\frac{b^3 \operatorname{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 )}{8 c}\\ &=-\frac{3 b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac{b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{32 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\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 )}{16 c^2}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac{3 b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac{b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{32 c^2}+\frac{b^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac{b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac{1}{8} b^2 \operatorname{Subst}\left (\int \left (\frac{2 a-b \log (1-c x)}{c}-\frac{2 a-b \log (1-c x)}{c (1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c}-\frac{\log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^3\right )\right )-\frac{b \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^3\right )}{16 c^2}+\frac{b^2 \operatorname{Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{16 c^2}+\frac{b^2 \operatorname{Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{8 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) (-2 a+b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^2 \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{4 c^2}+\frac{b^3 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+c x^3\right )}{16 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{4 c^2}-\frac{b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}+\frac{b \operatorname{Subst}\left (\int (1-c x) (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}-\frac{b^3 \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}\\ &=\frac{3 a b^2 x^3}{4 c}+\frac{3 b^3 x^3}{8 c}+\frac{b^3 \left (1-c x^3\right )^2}{64 c^2}-\frac{b^3 \left (1+c x^3\right )^2}{64 c^2}+\frac{b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{32 c^2}-\frac{3 b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac{b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{32 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac{3 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{32 c^2}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac{3 b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac{b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{32 c^2}-\frac{b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+\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 )}{8 c^2}+\frac{b^3 \log \left (1+c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac{b \operatorname{Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac{b \operatorname{Subst}\left (\int x (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{4 c^2}+2 \left (\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac{b^2 \operatorname{Subst}\left (\int (2 a-b \log (1-c x)) \, dx,x,x^3\right )}{8 c}+\frac{b^2 \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{8 c}\right )\\ &=\frac{3 a b^2 x^3}{4 c}+\frac{3 b^3 x^3}{4 c}+\frac{b^3 \left (1-c x^3\right )^2}{64 c^2}-\frac{b^3 \left (1+c x^3\right )^2}{64 c^2}+\frac{3 b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}+\frac{b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{32 c^2}-\frac{b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac{3 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{32 c^2}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \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^2}-\frac{b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1-c x^3\right )\right )}{8 c^2}-\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac{b^2 \operatorname{Subst}\left (\int x (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{16 c^2}-\frac{b^2 \operatorname{Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}+2 \left (-\frac{a b^2 x^3}{4 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 )}{8 c^2}+\frac{b^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{8 c}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{8 c}\right )\\ &=\frac{a b^2 x^3}{2 c}+\frac{5 b^3 x^3}{8 c}+\frac{3 b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}-\frac{b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac{b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \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^2}-\frac{b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (-\frac{a b^2 x^3}{4 c}-\frac{b^3 x^3}{8 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 )}{8 c^2}+\frac{b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}\right )-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}\\ &=\frac{a b^2 x^3}{2 c}+\frac{b^3 x^3}{2 c}+\frac{b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{4 c^2}-\frac{b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac{\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac{b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \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^2}-\frac{b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac{1}{16} b^2 x^6 \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^2}+\frac{b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (-\frac{a b^2 x^3}{4 c}-\frac{b^3 x^3}{4 c}-\frac{b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}+\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 )}{8 c^2}+\frac{b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac{b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{8 c^2}+\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{8 c^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.298934, size = 185, normalized size = 1.33 \[ \frac{6 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+a \left (2 a^2 c^2 x^6+6 a b c x^3+3 a b \log \left (1-c x^3\right )-3 a b \log \left (c x^3+1\right )+6 b^2 \log \left (1-c^2 x^6\right )\right )+6 b^2 \left (c x^3-1\right ) \tanh ^{-1}\left (c x^3\right )^2 \left (a c x^3+a+b\right )+6 b \tanh ^{-1}\left (c x^3\right ) \left (a c x^3 \left (a c x^3+2 b\right )-2 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c x^3\right )}+1\right )\right )+2 b^3 \left (c^2 x^6-1\right ) \tanh ^{-1}\left (c x^3\right )^3}{12 c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5*(a + b*ArcTanh[c*x^3])^3,x]

[Out]

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

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Maple [C]  time = 0.316, size = 750, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctanh(c*x^3))^3,x)

[Out]

1/48*b^3*(c^2*x^6-1)/c^2*ln(c*x^3+1)^3+1/16*b^2*(-x^6*b*ln(-c*x^3+1)*c^2+2*a*c^2*x^6+2*b*c*x^3+b*ln(-c*x^3+1)-
2*a+2*b)/c^2*ln(c*x^3+1)^2+(1/16*b^3*(c^2*x^6-1)/c^2*ln(-c*x^3+1)^2-1/4*x^3*b^2*(a*c*x^3+b)/c*ln(-c*x^3+1)+1/4
*b*(a^2*c^2*x^6+2*a*b*c*x^3+b*ln(-c*x^3+1)*a+b^2*ln(-c*x^3+1))/c^2)*ln(c*x^3+1)-1/2*a*b^2/c*x^3*ln(-c*x^3+1)+1
/4/c^2*b^3*ln(-c*x^3+1)-1/4/c^2*b^3*ln(c*x^3-1)+3/4*b^2/c*Sum(-2/3*(ln(x-_alpha)*ln(-c*x^3+1)+3*c*(-1/3*ln(x-_
alpha)*(ln((RootOf(_Z^2+_Z*_alpha+_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+_Z*_alpha+_alpha^2,index=1))+ln((Roo
tOf(_Z^2+_Z*_alpha+_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+_Z*_alpha+_alpha^2,index=2))+ln(1/2*(x+_alpha)/_alp
ha))/c-1/3*(dilog((RootOf(_Z^2+_Z*_alpha+_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+_Z*_alpha+_alpha^2,index=1))+
dilog((RootOf(_Z^2+_Z*_alpha+_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+_Z*_alpha+_alpha^2,index=2))+dilog(1/2*(x
+_alpha)/_alpha))/c))*b/c,_alpha=RootOf(_Z^3*c+1))-1/8*b^3/c^2+1/8*b^3/c*x^3*ln(-c*x^3+1)^2-1/4*a^2*b*x^6*ln(-
c*x^3+1)+1/4*a^2*b/c^2*ln(c*x^3-1)+1/8*a*b^2*x^6*ln(-c*x^3+1)^2+3/8*a*b^2/c^2*ln(-c*x^3+1)-1/8*a*b^2/c^2*ln(-c
*x^3+1)^2+1/6*x^6*a^3-1/48*b^3*x^6*ln(-c*x^3+1)^3-1/8*b^3/c^2*ln(-c*x^3+1)^2+1/48*b^3/c^2*ln(-c*x^3+1)^3+1/2*a
*b^2/c^2*ln(c*x^3+1)+1/8*b^2/c^2*a*ln(c*x^3-1)-1/4*a^2*b/c^2*ln(c*x^3+1)+1/2/c*a^2*b*x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^3,x, algorithm="maxima")

[Out]

1/2*a*b^2*x^6*arctanh(c*x^3)^2 + 1/6*a^3*x^6 + 1/4*(2*x^6*arctanh(c*x^3) + c*(2*x^3/c^2 - log(c*x^3 + 1)/c^3 +
log(c*x^3 - 1)/c^3))*a^2*b + 1/8*(4*c*(2*x^3/c^2 - log(c*x^3 + 1)/c^3 + log(c*x^3 - 1)/c^3)*arctanh(c*x^3) -
(2*(log(c*x^3 - 1) - 2)*log(c*x^3 + 1) - log(c*x^3 + 1)^2 - log(c*x^3 - 1)^2 - 4*log(c*x^3 - 1))/c^2)*a*b^2 -
1/192*(4*x^6*log(-c*x^3 + 1)^3 + 3*(x^6/c^3 + log(c^2*x^6 - 1)/c^5)*c^3 - 6*c*((c*x^6 + 2*x^3)/c^2 + 2*log(c*x
^3 - 1)/c^3)*log(-c*x^3 + 1)^2 + 21*c^2*(2*x^3/c^3 - log(c*x^3 + 1)/c^4 + log(c*x^3 - 1)/c^4) + c*(6*(c^2*x^6
+ 6*c*x^3 + 2*log(c*x^3 - 1)^2 + 6*log(c*x^3 - 1))*log(-c*x^3 + 1)/c^3 - (3*c^2*x^6 + 42*c*x^3 + 4*log(c*x^3 -
1)^3 + 18*log(c*x^3 - 1)^2 + 42*log(c*x^3 - 1))/c^3) - 1728*c*integrate(1/4*x^5*log(c*x^3 + 1)/(c^3*x^6 - c),
x) - 2*(12*c*x^3*log(c*x^3 + 1)^2 + 2*(c^2*x^6 - 1)*log(c*x^3 + 1)^3 - 3*(c^2*x^6 - 2*c*x^3 - 2*(c^2*x^6 - 1)
*log(c*x^3 + 1) + 1)*log(-c*x^3 + 1)^2 + 3*(c^2*x^6 + 6*c*x^3 - 2*(c^2*x^6 - 1)*log(c*x^3 + 1)^2 - 8*(c*x^3 +
1)*log(c*x^3 + 1))*log(-c*x^3 + 1))/c^2 + 18*log(4*c^3*x^6 - 4*c)/c^2 - 576*integrate(1/4*x^2*log(c*x^3 + 1)/(
c^3*x^6 - c), x))*b^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{5} \operatorname{artanh}\left (c x^{3}\right )^{3} + 3 \, a b^{2} x^{5} \operatorname{artanh}\left (c x^{3}\right )^{2} + 3 \, a^{2} b x^{5} \operatorname{artanh}\left (c x^{3}\right ) + a^{3} x^{5}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^3,x, algorithm="fricas")

[Out]

integral(b^3*x^5*arctanh(c*x^3)^3 + 3*a*b^2*x^5*arctanh(c*x^3)^2 + 3*a^2*b*x^5*arctanh(c*x^3) + a^3*x^5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atanh(c*x**3))**3,x)

[Out]

Timed out

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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 )}^{3} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctanh(c*x^3))^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^3) + a)^3*x^5, x)