3.116 \(\int x^{11} (a+b \tanh ^{-1}(c x^3))^2 \, dx\)

Optimal. Leaf size=125 \[ \frac{a b x^3}{6 c^3}-\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{12 c^4}+\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac{b x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{18 c}+\frac{b^2 x^6}{36 c^2}+\frac{b^2 \log \left (1-c^2 x^6\right )}{9 c^4}+\frac{b^2 x^3 \tanh ^{-1}\left (c x^3\right )}{6 c^3} \]

[Out]

(a*b*x^3)/(6*c^3) + (b^2*x^6)/(36*c^2) + (b^2*x^3*ArcTanh[c*x^3])/(6*c^3) + (b*x^9*(a + b*ArcTanh[c*x^3]))/(18
*c) - (a + b*ArcTanh[c*x^3])^2/(12*c^4) + (x^12*(a + b*ArcTanh[c*x^3])^2)/12 + (b^2*Log[1 - c^2*x^6])/(9*c^4)

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Rubi [C]  time = 1.54585, antiderivative size = 636, normalized size of antiderivative = 5.09, number of steps used = 62, 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} \[ \frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )}{24 c^4}+\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}-\frac{1}{288} b \left (-\frac{3 \left (1-c x^3\right )^4}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{48 \left (1-c x^3\right )}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\right ) \left (2 a-b \log \left (1-c x^3\right )\right )-\frac{b \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{24 c^4}+\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{24} b x^{12} \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}+\frac{b^2 x^6}{192 c^2}+\frac{23 b^2 x^3}{288 c^3}+\frac{b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac{b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac{b^2 \left (1-c x^3\right )^2}{16 c^4}+\frac{b^2 \log ^2\left (1-c x^3\right )}{48 c^4}-\frac{b^2 \log ^2\left (c x^3+1\right )}{48 c^4}+\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}-\frac{5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac{b^2 \left (c x^3+1\right ) \log \left (c x^3+1\right )}{12 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{24 c^4}+\frac{b^2 \log \left (c x^3+1\right )}{36 c^4}-\frac{7 b^2 x^9}{864 c}+\frac{1}{48} b^2 x^{12} \log ^2\left (c x^3+1\right )+\frac{b^2 x^9 \log \left (c x^3+1\right )}{36 c}-\frac{1}{384} b^2 x^{12} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^11*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

(a*b*x^3)/(12*c^3) + (23*b^2*x^3)/(288*c^3) + (b^2*x^6)/(192*c^2) - (7*b^2*x^9)/(864*c) - (b^2*x^12)/384 + (b^
2*(1 - c*x^3)^2)/(16*c^4) - (b^2*(1 - c*x^3)^3)/(54*c^4) + (b^2*(1 - c*x^3)^4)/(384*c^4) - (5*b^2*Log[1 - c*x^
3])/(288*c^4) + (b^2*(1 - c*x^3)*Log[1 - c*x^3])/(24*c^4) + (b^2*Log[1 - c*x^3]^2)/(48*c^4) - (b*x^6*(2*a - b*
Log[1 - c*x^3]))/(48*c^2) + (b*x^9*(2*a - b*Log[1 - c*x^3]))/(72*c) - (b*x^12*(2*a - b*Log[1 - c*x^3]))/96 + (
x^12*(2*a - b*Log[1 - c*x^3])^2)/48 - (b*(2*a - b*Log[1 - c*x^3])*((48*(1 - c*x^3))/c^4 - (36*(1 - c*x^3)^2)/c
^4 + (16*(1 - c*x^3)^3)/c^4 - (3*(1 - c*x^3)^4)/c^4 - (12*Log[1 - c*x^3])/c^4))/288 - (b*(2*a - b*Log[1 - c*x^
3])*Log[(1 + c*x^3)/2])/(24*c^4) + (b^2*Log[1 + c*x^3])/(36*c^4) + (b^2*x^9*Log[1 + c*x^3])/(36*c) + (b^2*(1 +
c*x^3)*Log[1 + c*x^3])/(12*c^4) + (b^2*Log[(1 - c*x^3)/2]*Log[1 + c*x^3])/(24*c^4) + (b*x^12*(2*a - b*Log[1 -
c*x^3])*Log[1 + c*x^3])/24 - (b^2*Log[1 + c*x^3]^2)/(48*c^4) + (b^2*x^12*Log[1 + c*x^3]^2)/48 + (b^2*PolyLog[
2, (1 - c*x^3)/2])/(24*c^4) + (b^2*PolyLog[2, (1 + c*x^3)/2])/(24*c^4)

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^{11} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^{11} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{2} b x^{11} \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^{11} \log ^2\left (1+c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^{11} \left (2 a-b \log \left (1-c x^3\right )\right )^2 \, dx-\frac{1}{2} b \int x^{11} \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^{11} \log ^2\left (1+c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int x^3 (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int x^3 (-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^3 \log ^2(1+c x) \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac{1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )-\frac{1}{24} (b c) \operatorname{Subst}\left (\int \frac{x^4 (2 a-b \log (1-c x))}{1-c x} \, dx,x,x^3\right )+\frac{1}{24} (b c) \operatorname{Subst}\left (\int \frac{x^4 (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+c x)}{1-c x} \, dx,x,x^3\right )-\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac{1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+\frac{1}{24} b \operatorname{Subst}\left (\int \frac{\left (\frac{1}{c}-\frac{x}{c}\right )^4 (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )+\frac{1}{24} (b c) \operatorname{Subst}\left (\int \left (-\frac{-2 a+b \log (1-c x)}{c^4}+\frac{x (-2 a+b \log (1-c x))}{c^3}-\frac{x^2 (-2 a+b \log (1-c x))}{c^2}+\frac{x^3 (-2 a+b \log (1-c x))}{c}+\frac{-2 a+b \log (1-c x)}{c^4 (1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c^4}-\frac{x \log (1+c x)}{c^3}-\frac{x^2 \log (1+c x)}{c^2}-\frac{x^3 \log (1+c x)}{c}-\frac{\log (1+c x)}{c^4 (-1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c^4}+\frac{x \log (1+c x)}{c^3}-\frac{x^2 \log (1+c x)}{c^2}+\frac{x^3 \log (1+c x)}{c}+\frac{\log (1+c x)}{c^4 (1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac{48 \left (1-c x^3\right )}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{3 \left (1-c x^3\right )^4}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\right )+\frac{1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+\frac{1}{24} b \operatorname{Subst}\left (\int x^3 (-2 a+b \log (1-c x)) \, dx,x,x^3\right )+\frac{1}{24} b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{12 c^4 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 )}{24 c^3}+\frac{b \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{24 c^3}+2 \frac{b^2 \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{24 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{24 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^3\right )}{24 c^3}+\frac{b \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{24 c^2}-\frac{b \operatorname{Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{24 c}+2 \frac{b^2 \operatorname{Subst}\left (\int x^2 \log (1+c x) \, dx,x,x^3\right )}{24 c}\\ &=\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac{b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac{1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac{48 \left (1-c x^3\right )}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{3 \left (1-c x^3\right )^4}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\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 )}{24 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac{1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )-\frac{1}{72} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1-c x} \, dx,x,x^3\right )+2 \left (\frac{b^2 x^9 \log \left (1+c x^3\right )}{72 c}-\frac{1}{72} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1+c x} \, dx,x,x^3\right )\right )+\frac{b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{x} \, dx,x,1-c x^3\right )}{288 c^4}+2 \frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{24 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^3\right )}{24 c^4}-\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 )}{24 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{24 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 )}{24 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-c x} \, dx,x,x^3\right )}{48 c}+\frac{1}{96} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-c x} \, dx,x,x^3\right )\\ &=\frac{a b x^3}{12 c^3}-\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac{b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac{1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac{48 \left (1-c x^3\right )}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{3 \left (1-c x^3\right )^4}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\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 )}{24 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac{1}{24} b x^{12} \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 )}{48 c^4}+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac{b^2 x^3}{24 c^3}+\frac{b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )-\frac{1}{72} b^2 \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 )+2 \left (\frac{b^2 x^9 \log \left (1+c x^3\right )}{72 c}-\frac{1}{72} b^2 \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 )\right )+\frac{b^2 \operatorname{Subst}\left (\int \left (-48+36 x-16 x^2+3 x^3+\frac{12 \log (x)}{x}\right ) \, dx,x,1-c x^3\right )}{288 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{24 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{24 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{24 c^4}+\frac{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 )}{48 c}+\frac{1}{96} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^3}-\frac{x^2}{c^2}-\frac{x^3}{c}-\frac{1}{c^4 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{a b x^3}{12 c^3}+\frac{55 b^2 x^3}{288 c^3}-\frac{5 b^2 x^6}{576 c^2}+\frac{b^2 x^9}{864 c}-\frac{b^2 x^{12}}{384}+\frac{b^2 \left (1-c x^3\right )^2}{16 c^4}-\frac{b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac{b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac{5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}-\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac{b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac{1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac{48 \left (1-c x^3\right )}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{3 \left (1-c x^3\right )^4}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\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 )}{24 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac{1}{24} b x^{12} \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 )}{48 c^4}+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac{b^2 x^3}{72 c^3}+\frac{b^2 x^6}{144 c^2}-\frac{b^2 x^9}{216 c}+\frac{b^2 \log \left (1+c x^3\right )}{72 c^4}+\frac{b^2 x^9 \log \left (1+c x^3\right )}{72 c}\right )+2 \left (-\frac{b^2 x^3}{24 c^3}+\frac{b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x^3\right )}{24 c^4}\\ &=\frac{a b x^3}{12 c^3}+\frac{55 b^2 x^3}{288 c^3}-\frac{5 b^2 x^6}{576 c^2}+\frac{b^2 x^9}{864 c}-\frac{b^2 x^{12}}{384}+\frac{b^2 \left (1-c x^3\right )^2}{16 c^4}-\frac{b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac{b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac{5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}+\frac{b^2 \log ^2\left (1-c x^3\right )}{48 c^4}-\frac{b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac{b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac{1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac{48 \left (1-c x^3\right )}{c^4}-\frac{36 \left (1-c x^3\right )^2}{c^4}+\frac{16 \left (1-c x^3\right )^3}{c^4}-\frac{3 \left (1-c x^3\right )^4}{c^4}-\frac{12 \log \left (1-c x^3\right )}{c^4}\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 )}{24 c^4}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac{1}{24} b x^{12} \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 )}{48 c^4}+\frac{1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac{b^2 x^3}{72 c^3}+\frac{b^2 x^6}{144 c^2}-\frac{b^2 x^9}{216 c}+\frac{b^2 \log \left (1+c x^3\right )}{72 c^4}+\frac{b^2 x^9 \log \left (1+c x^3\right )}{72 c}\right )+2 \left (-\frac{b^2 x^3}{24 c^3}+\frac{b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{24 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0761196, size = 146, normalized size = 1.17 \[ \frac{3 a^2 c^4 x^{12}+2 a b c^3 x^9+2 b c x^3 \tanh ^{-1}\left (c x^3\right ) \left (3 a c^3 x^9+b \left (c^2 x^6+3\right )\right )+6 a b c x^3+b (3 a+4 b) \log \left (1-c x^3\right )-3 a b \log \left (c x^3+1\right )+b^2 c^2 x^6+3 b^2 \left (c^4 x^{12}-1\right ) \tanh ^{-1}\left (c x^3\right )^2+4 b^2 \log \left (c x^3+1\right )}{36 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*ArcTanh[c*x^3])^2,x]

[Out]

(6*a*b*c*x^3 + b^2*c^2*x^6 + 2*a*b*c^3*x^9 + 3*a^2*c^4*x^12 + 2*b*c*x^3*(3*a*c^3*x^9 + b*(3 + c^2*x^6))*ArcTan
h[c*x^3] + 3*b^2*(-1 + c^4*x^12)*ArcTanh[c*x^3]^2 + b*(3*a + 4*b)*Log[1 - c*x^3] - 3*a*b*Log[1 + c*x^3] + 4*b^
2*Log[1 + c*x^3])/(36*c^4)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{11} \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^11*(a+b*arctanh(c*x^3))^2,x)

[Out]

int(x^11*(a+b*arctanh(c*x^3))^2,x)

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Maxima [A]  time = 0.985566, size = 293, normalized size = 2.34 \begin{align*} \frac{1}{12} \, b^{2} x^{12} \operatorname{artanh}\left (c x^{3}\right )^{2} + \frac{1}{12} \, a^{2} x^{12} + \frac{1}{36} \,{\left (6 \, x^{12} \operatorname{artanh}\left (c x^{3}\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{4}} - \frac{3 \, \log \left (c x^{3} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x^{3} - 1\right )}{c^{5}}\right )}\right )} a b + \frac{1}{144} \,{\left (4 \, c{\left (\frac{2 \,{\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{4}} - \frac{3 \, \log \left (c x^{3} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x^{3} - 1\right )}{c^{5}}\right )} \operatorname{artanh}\left (c x^{3}\right ) + \frac{4 \, c^{2} x^{6} - 2 \,{\left (3 \, \log \left (c x^{3} - 1\right ) - 8\right )} \log \left (c x^{3} + 1\right ) + 3 \, \log \left (c x^{3} + 1\right )^{2} + 3 \, \log \left (c x^{3} - 1\right )^{2} + 16 \, \log \left (c x^{3} - 1\right )}{c^{4}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arctanh(c*x^3))^2,x, algorithm="maxima")

[Out]

1/12*b^2*x^12*arctanh(c*x^3)^2 + 1/12*a^2*x^12 + 1/36*(6*x^12*arctanh(c*x^3) + c*(2*(c^2*x^9 + 3*x^3)/c^4 - 3*
log(c*x^3 + 1)/c^5 + 3*log(c*x^3 - 1)/c^5))*a*b + 1/144*(4*c*(2*(c^2*x^9 + 3*x^3)/c^4 - 3*log(c*x^3 + 1)/c^5 +
3*log(c*x^3 - 1)/c^5)*arctanh(c*x^3) + (4*c^2*x^6 - 2*(3*log(c*x^3 - 1) - 8)*log(c*x^3 + 1) + 3*log(c*x^3 + 1
)^2 + 3*log(c*x^3 - 1)^2 + 16*log(c*x^3 - 1))/c^4)*b^2

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Fricas [A]  time = 1.85454, size = 381, normalized size = 3.05 \begin{align*} \frac{12 \, a^{2} c^{4} x^{12} + 8 \, a b c^{3} x^{9} + 4 \, b^{2} c^{2} x^{6} + 24 \, a b c x^{3} + 3 \,{\left (b^{2} c^{4} x^{12} - b^{2}\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \,{\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{3} + 1\right ) + 4 \,{\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{3} - 1\right ) + 4 \,{\left (3 \, a b c^{4} x^{12} + b^{2} c^{3} x^{9} + 3 \, b^{2} c x^{3}\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{144 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arctanh(c*x^3))^2,x, algorithm="fricas")

[Out]

1/144*(12*a^2*c^4*x^12 + 8*a*b*c^3*x^9 + 4*b^2*c^2*x^6 + 24*a*b*c*x^3 + 3*(b^2*c^4*x^12 - b^2)*log(-(c*x^3 + 1
)/(c*x^3 - 1))^2 - 4*(3*a*b - 4*b^2)*log(c*x^3 + 1) + 4*(3*a*b + 4*b^2)*log(c*x^3 - 1) + 4*(3*a*b*c^4*x^12 + b
^2*c^3*x^9 + 3*b^2*c*x^3)*log(-(c*x^3 + 1)/(c*x^3 - 1)))/c^4

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

[Out]

Timed out

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Giac [A]  time = 1.28881, size = 236, normalized size = 1.89 \begin{align*} \frac{1}{12} \, a^{2} x^{12} + \frac{a b x^{9}}{18 \, c} + \frac{b^{2} x^{6}}{36 \, c^{2}} + \frac{1}{48} \,{\left (b^{2} x^{12} - \frac{b^{2}}{c^{4}}\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )^{2} + \frac{a b x^{3}}{6 \, c^{3}} + \frac{1}{36} \,{\left (3 \, a b x^{12} + \frac{b^{2} x^{9}}{c} + \frac{3 \, b^{2} x^{3}}{c^{3}}\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) - \frac{{\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{3} + 1\right )}{36 \, c^{4}} + \frac{{\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{3} - 1\right )}{36 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arctanh(c*x^3))^2,x, algorithm="giac")

[Out]

1/12*a^2*x^12 + 1/18*a*b*x^9/c + 1/36*b^2*x^6/c^2 + 1/48*(b^2*x^12 - b^2/c^4)*log(-(c*x^3 + 1)/(c*x^3 - 1))^2
+ 1/6*a*b*x^3/c^3 + 1/36*(3*a*b*x^12 + b^2*x^9/c + 3*b^2*x^3/c^3)*log(-(c*x^3 + 1)/(c*x^3 - 1)) - 1/36*(3*a*b
- 4*b^2)*log(c*x^3 + 1)/c^4 + 1/36*(3*a*b + 4*b^2)*log(c*x^3 - 1)/c^4