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

Optimal. Leaf size=154 \[ -\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x^3}\right )}{9 c^3}-\frac{i \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{9 c^3}-\frac{2 b \log \left (\frac{2}{1+i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{9 c^3}+\frac{1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2-\frac{b x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{9 c}+\frac{b^2 x^3}{9 c^2}-\frac{b^2 \tan ^{-1}\left (c x^3\right )}{9 c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.39352, antiderivative size = 647, normalized size of antiderivative = 4.2, 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 = {5035, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ \frac{i b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{i a b x^3}{9 c^2}+\frac{1}{108} i b \left (\frac{2 i \left (1-i c x^3\right )^3}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}+\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )-\frac{i b \log \left (\frac{1}{2} \left (1+i c x^3\right )\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )}{18 c^3}+\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{54} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )-\frac{1}{18} b x^9 \log \left (1+i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{36 c}+\frac{19 b^2 x^3}{108 c^2}+\frac{i b^2 \left (1-i c x^3\right )^3}{162 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2}{24 c^3}-\frac{i b^2 \log ^2\left (1-i c x^3\right )}{36 c^3}-\frac{i b^2 \log ^2\left (1+i c x^3\right )}{36 c^3}+\frac{i b^2 \log \left (-c x^3+i\right )}{18 c^3}+\frac{i b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{18 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{18 c^3}-\frac{i b^2 \log \left (c x^3+i\right )}{108 c^3}-\frac{5 i b^2 x^6}{216 c}-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )+\frac{i b^2 x^6 \log \left (1+i c x^3\right )}{18 c}+\frac{b^2 x^9}{162} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^8*(a + b*ArcTan[c*x^3])^2,x]

[Out]

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

Rule 5035

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*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 \tan ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^8 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{2} b x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{4} b^2 x^8 \log ^2\left (1+i c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^8 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \, dx+\frac{1}{2} b \int x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right ) \, dx-\frac{1}{4} b^2 \int x^8 \log ^2\left (1+i c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int x^2 (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )+\frac{1}{6} b \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int x^2 \log ^2(1+i c x) \, dx,x,x^3\right )\\ &=\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )-\frac{1}{18} (i b c) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{x^3 (2 a+i b \log (1-i c x))}{1-i c x} \, dx,x,x^3\right )+\frac{1}{18} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )+\frac{1}{18} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+i c x)}{1+i c x} \, dx,x,x^3\right )\\ &=\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )-\frac{1}{18} (i b) \operatorname{Subst}\left (\int \frac{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3 (2 a+i b \log (x))}{x} \, dx,x,1-i c x^3\right )-\frac{1}{18} (i b c) \operatorname{Subst}\left (\int \left (\frac{i (-2 i a+b \log (1-i c x))}{c^3}+\frac{x (-2 i a+b \log (1-i c x))}{c^2}-\frac{i x^2 (-2 i a+b \log (1-i c x))}{c}-\frac{-2 i a+b \log (1-i c x)}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{18} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{i \log (1+i c x)}{c^3}+\frac{x \log (1+i c x)}{c^2}-\frac{i x^2 \log (1+i c x)}{c}-\frac{\log (1+i c x)}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{18} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{i \log (1+i c x)}{c^3}+\frac{x \log (1+i c x)}{c^2}+\frac{i x^2 \log (1+i c x)}{c}-\frac{\log (1+i c x)}{c^3 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{108} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )-\frac{1}{18} b \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )-\frac{1}{18} b^2 \operatorname{Subst}\left (\int -\frac{i \left (x \left (18-9 x+2 x^2\right )-6 \log (x)\right )}{6 c^3 x} \, dx,x,1-i c x^3\right )+\frac{(i b) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )}{18 c^2}+\frac{b \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{18 c^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,x^3\right )}{18 c^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^3\right )}{18 c^2}-\frac{(i b) \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{18 c}+2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,x^3\right )}{18 c}\\ &=-\frac{i a b x^3}{9 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{36 c}+\frac{1}{54} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{108} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{18 c^3}-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )+\frac{1}{36} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^3\right )+2 \left (\frac{i b^2 x^6 \log \left (1+i c x^3\right )}{36 c}+\frac{1}{36} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,x^3\right )\right )+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{x \left (18-9 x+2 x^2\right )-6 \log (x)}{x} \, dx,x,1-i c x^3\right )}{108 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i c x^3\right )}{18 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^3\right )}{18 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )}{18 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )}{18 c^2}-\frac{1}{54} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3}{1-i c x} \, dx,x,x^3\right )\\ &=-\frac{i a b x^3}{9 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{36 c}+\frac{1}{54} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{108} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{18 c^3}-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{i b^2 \log ^2\left (1+i c x^3\right )}{36 c^3}-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )+2 \left (\frac{i b^2 x^6 \log \left (1+i c x^3\right )}{36 c}+\frac{1}{36} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )\right )+\frac{1}{36} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \left (18-9 x+2 x^2-\frac{6 \log (x)}{x}\right ) \, dx,x,1-i c x^3\right )}{108 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{18 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{18 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{18 c^3}-\frac{1}{54} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{i}{c^3}+\frac{x}{c^2}+\frac{i x^2}{c}-\frac{1}{c^3 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{i a b x^3}{9 c^2}+\frac{13 b^2 x^3}{108 c^2}+\frac{i b^2 x^6}{216 c}+\frac{b^2 x^9}{162}-\frac{i b^2 \left (1-i c x^3\right )^2}{24 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3}{162 c^3}+\frac{i b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{18 c^3}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{36 c}+\frac{1}{54} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{108} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{18 c^3}-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{i b^2 \log ^2\left (1+i c x^3\right )}{36 c^3}-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )+2 \left (\frac{b^2 x^3}{36 c^2}-\frac{i b^2 x^6}{72 c}+\frac{i b^2 \log \left (i-c x^3\right )}{36 c^3}+\frac{i b^2 x^6 \log \left (1+i c x^3\right )}{36 c}\right )-\frac{i b^2 \log \left (i+c x^3\right )}{108 c^3}+\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i c x^3\right )}{18 c^3}\\ &=-\frac{i a b x^3}{9 c^2}+\frac{13 b^2 x^3}{108 c^2}+\frac{i b^2 x^6}{216 c}+\frac{b^2 x^9}{162}-\frac{i b^2 \left (1-i c x^3\right )^2}{24 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3}{162 c^3}+\frac{i b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{18 c^3}-\frac{i b^2 \log ^2\left (1-i c x^3\right )}{36 c^3}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{36 c}+\frac{1}{54} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{1}{36} x^9 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{108} i b \left (2 a+i b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{i b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{18 c^3}-\frac{1}{18} b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{i b^2 \log ^2\left (1+i c x^3\right )}{36 c^3}-\frac{1}{36} b^2 x^9 \log ^2\left (1+i c x^3\right )+2 \left (\frac{b^2 x^3}{36 c^2}-\frac{i b^2 x^6}{72 c}+\frac{i b^2 \log \left (i-c x^3\right )}{36 c^3}+\frac{i b^2 x^6 \log \left (1+i c x^3\right )}{36 c}\right )-\frac{i b^2 \log \left (i+c x^3\right )}{108 c^3}+\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{18 c^3}-\frac{i b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{18 c^3}\\ \end{align*}

Mathematica [A]  time = 0.292919, size = 141, normalized size = 0.92 \[ \frac{i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (c x^3\right )}\right )+a^2 c^3 x^9-a b c^2 x^6+a b \log \left (c^2 x^6+1\right )-b \tan ^{-1}\left (c x^3\right ) \left (-2 a c^3 x^9+b c^2 x^6+2 b \log \left (1+e^{2 i \tan ^{-1}\left (c x^3\right )}\right )+b\right )+b^2 \left (c^3 x^9+i\right ) \tan ^{-1}\left (c x^3\right )^2+b^2 c x^3}{9 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^8*(a + b*ArcTan[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*(I + c^3*x^9)*ArcTan[c*x^3]^2 - b*ArcTan[c*x^3]*(b + b*c^2*x^6 -
2*a*c^3*x^9 + 2*b*Log[1 + E^((2*I)*ArcTan[c*x^3])]) + a*b*Log[1 + c^2*x^6] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan
[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\arctan \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*arctan(c*x^3))^2,x)

[Out]

int(x^8*(a+b*arctan(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} \arctan \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}{144} \,{\left (4 \, x^{9} \arctan \left (c x^{3}\right )^{2} - x^{9} \log \left (c^{2} x^{6} + 1\right )^{2} + 144 \, \int \frac{4 \, c^{2} x^{14} \log \left (c^{2} x^{6} + 1\right ) - 8 \, c x^{11} \arctan \left (c x^{3}\right ) + 36 \,{\left (c^{2} x^{14} + x^{8}\right )} \arctan \left (c x^{3}\right )^{2} + 3 \,{\left (c^{2} x^{14} + x^{8}\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{48 \,{\left (c^{2} x^{6} + 1\right )}}\,{d x}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^2*x^9 + 1/9*(2*x^9*arctan(c*x^3) - (x^6/c^2 - log(c^2*x^6 + 1)/c^4)*c)*a*b + 1/144*(4*x^9*arctan(c*x^3)^
2 - x^9*log(c^2*x^6 + 1)^2 + 144*integrate(1/48*(4*c^2*x^14*log(c^2*x^6 + 1) - 8*c*x^11*arctan(c*x^3) + 36*(c^
2*x^14 + x^8)*arctan(c*x^3)^2 + 3*(c^2*x^14 + x^8)*log(c^2*x^6 + 1)^2)/(c^2*x^6 + 1), 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} \arctan \left (c x^{3}\right )^{2} + 2 \, a b x^{8} \arctan \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*arctan(c*x^3))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \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*arctan(c*x^3))^2,x, algorithm="giac")

[Out]

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