3.123 \(\int \frac{(a+b \tanh ^{-1}(c x^3))^2}{x^{10}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,\frac{2}{c x^3+1}-1\right )+\frac{1}{9} c^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac{2}{9} b c^3 \log \left (2-\frac{2}{c x^3+1}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{9 x^6}-\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{9 x^9}-\frac{b^2 c^2}{9 x^3}+\frac{1}{9} b^2 c^3 \tanh ^{-1}\left (c x^3\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.31132, antiderivative size = 420, normalized size of antiderivative = 2.92, number of steps used = 59, number of rules used = 24, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.5, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2395, 2439, 2416, 36, 29, 2392, 2391, 2394, 2393, 2410, 2390} \[ -\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,-c x^3\right )+\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,c x^3\right )+\frac{1}{18} b^2 c^3 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{18} b c^3 \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{2}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^9}-\frac{b^2 c^2}{9 x^3}-\frac{1}{36} b^2 c^3 \log ^2\left (c x^3+1\right )+\frac{1}{18} b^2 c^3 \log \left (c x^3+1\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )-\frac{b^2 \log ^2\left (c x^3+1\right )}{36 x^9}-\frac{b^2 c \log \left (c x^3+1\right )}{18 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcTanh[c*x^3])^2/x^10,x]

[Out]

-(b^2*c^2)/(9*x^3) + (2*a*b*c^3*Log[x])/3 - (b*c*(2*a - b*Log[1 - c*x^3]))/(18*x^6) + (b*c^2*(2*a - b*Log[1 -
c*x^3]))/(18*x^3) - (b*c^2*(1 - c*x^3)*(2*a - b*Log[1 - c*x^3]))/(18*x^3) + (c^3*(2*a - b*Log[1 - c*x^3])^2)/3
6 - (2*a - b*Log[1 - c*x^3])^2/(36*x^9) - (b*c^3*(2*a - b*Log[1 - c*x^3])*Log[(1 + c*x^3)/2])/18 + (b^2*c^3*Lo
g[1 + c*x^3])/18 - (b^2*c*Log[1 + c*x^3])/(18*x^6) - (b^2*c^3*Log[(1 - c*x^3)/2]*Log[1 + c*x^3])/18 - (b*(2*a
- b*Log[1 - c*x^3])*Log[1 + c*x^3])/(18*x^9) - (b^2*c^3*Log[1 + c*x^3]^2)/36 - (b^2*Log[1 + c*x^3]^2)/(36*x^9)
- (b^2*c^3*PolyLog[2, -(c*x^3)])/9 + (b^2*c^3*PolyLog[2, c*x^3])/9 + (b^2*c^3*PolyLog[2, (1 - c*x^3)/2])/18 -
(b^2*c^3*PolyLog[2, (1 + c*x^3)/2])/18

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 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

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 2316

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

Rule 2315

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

Rule 2314

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

Rule 31

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

Rule 2319

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 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 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 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 \frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{x^{10}} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{4 x^{10}}-\frac{b \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{2 x^{10}}+\frac{b^2 \log ^2\left (1+c x^3\right )}{4 x^{10}}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{x^{10}} \, dx-\frac{1}{2} b \int \frac{\left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{x^{10}} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+c x^3\right )}{x^{10}} \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^4} \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x)) \log (1+c x)}{x^4} \, dx,x,x^3\right )+\frac{1}{12} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^4} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^3 (1-c x)} \, dx,x,x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^3 (1+c x)} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1-c x)} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{18} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, 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)}{x^3}-\frac{c (-2 a+b \log (1-c x))}{x^2}+\frac{c^2 (-2 a+b \log (1-c x))}{x}-\frac{c^3 (-2 a+b \log (1-c x))}{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)}{x^3}+\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \log (1+c x)}{-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)}{x^3}-\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \log (1+c x)}{1+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{18} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^3} \, dx,x,x^3\right )+2 \left (\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3} \, dx,x,x^3\right )\right )+\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{18} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x} \, dx,x,x^3\right )+2 \left (\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,x^3\right )\right )+\frac{1}{18} \left (b c^4\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=\frac{1}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{36} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-c x^3\right )+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1-c x)} \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+c x)} \, dx,x,x^3\right )\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1-c x)} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )\\ &=\frac{1}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (c x^3\right )-\frac{1}{36} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{c^2}{(-1+x)^2}-\frac{c^2}{-1+x}+\frac{c^2}{x}\right ) \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c}{x}+\frac{c^2}{1+c x}\right ) \, dx,x,x^3\right )\right )-\frac{1}{18} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )+\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x} \, dx,x,x^3\right )\\ &=-\frac{b^2 c^2}{18 x^3}+\frac{2}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+2 \left (-\frac{b^2 c^2}{36 x^3}-\frac{1}{12} b^2 c^3 \log (x)+\frac{1}{36} b^2 c^3 \log \left (1+c x^3\right )-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}\right )-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )+\frac{1}{18} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )\\ &=-\frac{b^2 c^2}{18 x^3}+\frac{2}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+2 \left (-\frac{b^2 c^2}{36 x^3}-\frac{1}{12} b^2 c^3 \log (x)+\frac{1}{36} b^2 c^3 \log \left (1+c x^3\right )-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}\right )-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{9} b^2 c^3 \text{Li}_2\left (c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.362807, size = 159, normalized size = 1.1 \[ -\frac{b^2 c^3 x^9 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+a^2-2 a b c^3 x^9 \log \left (c x^3\right )+a b c^3 x^9 \log \left (1-c^2 x^6\right )+b \tanh ^{-1}\left (c x^3\right ) \left (2 a-b c^3 x^9-2 b c^3 x^9 \log \left (1-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+b c x^3\right )+a b c x^3+b^2 c^2 x^6+b^2 \left (1-c^3 x^9\right ) \tanh ^{-1}\left (c x^3\right )^2}{9 x^9} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c*x^3])^2/x^10,x]

[Out]

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

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{3} \right ) \right ) ^{2}}{{x}^{10}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^3))^2/x^10,x)

[Out]

int((a+b*arctanh(c*x^3))^2/x^10,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{9} \,{\left ({\left (c^{2} \log \left (c^{2} x^{6} - 1\right ) - c^{2} \log \left (x^{6}\right ) + \frac{1}{x^{6}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{9}}\right )} a b - \frac{1}{36} \, b^{2}{\left (\frac{\log \left (-c x^{3} + 1\right )^{2}}{x^{9}} + 9 \, \int -\frac{3 \,{\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )^{2} + 2 \,{\left (c x^{3} - 3 \,{\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )}{3 \,{\left (c x^{13} - x^{10}\right )}}\,{d x}\right )} - \frac{a^{2}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^10,x, algorithm="maxima")

[Out]

-1/9*((c^2*log(c^2*x^6 - 1) - c^2*log(x^6) + 1/x^6)*c + 2*arctanh(c*x^3)/x^9)*a*b - 1/36*b^2*(log(-c*x^3 + 1)^
2/x^9 + 9*integrate(-1/3*(3*(c*x^3 - 1)*log(c*x^3 + 1)^2 + 2*(c*x^3 - 3*(c*x^3 - 1)*log(c*x^3 + 1))*log(-c*x^3
+ 1))/(c*x^13 - x^10), x)) - 1/9*a^2/x^9

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x^{3}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{3}\right ) + a^{2}}{x^{10}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^10,x, algorithm="fricas")

[Out]

integral((b^2*arctanh(c*x^3)^2 + 2*a*b*arctanh(c*x^3) + a^2)/x^10, 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((a+b*atanh(c*x**3))**2/x**10,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x^{3}\right ) + a\right )}^{2}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))^2/x^10,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^3) + a)^2/x^10, x)