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

Optimal. Leaf size=149 \[ -\frac{3 i b^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x^2}\right )}{4 c^2}-\frac{3 b^2 \log \left (\frac{2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 c^2}+\frac{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^3}{4 c^2}-\frac{3 i b \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{4 c^2}+\frac{1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3-\frac{3 b x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{4 c} \]

[Out]

(((-3*I)/4)*b*(a + b*ArcTan[c*x^2])^2)/c^2 - (3*b*x^2*(a + b*ArcTan[c*x^2])^2)/(4*c) + (a + b*ArcTan[c*x^2])^3
/(4*c^2) + (x^4*(a + b*ArcTan[c*x^2])^3)/4 - (3*b^2*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/(2*c^2) - (((3
*I)/4)*b^3*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/c^2

________________________________________________________________________________________

Rubi [B]  time = 4.73595, antiderivative size = 951, normalized size of antiderivative = 6.38, 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 = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2439, 2416, 2396, 2433, 2374, 6589, 2411, 43, 2334, 12, 14, 2301, 6742, 2395, 2394, 2393, 2391, 2375, 2317, 2430, 2425} \[ \frac{3}{32} i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (i c x^2+1\right ) x^4+\frac{3}{32} i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (i c x^2+1\right ) x^4+\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (i c x^2+1\right ) x^2}{8 c}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{i b^3 \left (i c x^2+1\right )^2 \log ^3\left (i c x^2+1\right )}{32 c^2}+\frac{i b^3 \left (i c x^2+1\right ) \log ^3\left (i c x^2+1\right )}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}-\frac{3 i b^3 \left (i c x^2+1\right ) \log ^2\left (i c x^2+1\right )}{16 c^2}-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (i c x^2+1\right )}{32 c^2}+\frac{3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac{3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (i c x^2+1\right )\right )}{32 c^2}+\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (i c x^2+1\right )\right )}{32 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (i c x^2+1\right )\right )}{8 c^2}-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (i c x^2+1\right )}{32 c^2}-\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (i c x^2+1\right )}{8 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right )}{8 c^2}-\frac{3 i b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (i c x^2+1\right )\right )}{8 c^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^3*(a + b*ArcTan[c*x^2])^3,x]

[Out]

(((3*I)/64)*b^2*(1 - I*c*x^2)^2*((2*I)*a - b*Log[1 - I*c*x^2]))/c^2 + (((3*I)/64)*b*(1 - I*c*x^2)^2*((2*I)*a -
b*Log[1 - I*c*x^2])^2)/c^2 + (3*b^2*(1 - I*c*x^2)^2*(2*a + I*b*Log[1 - I*c*x^2]))/(64*c^2) - (((3*I)/16)*b*(1
- I*c*x^2)*(2*a + I*b*Log[1 - I*c*x^2])^2)/c^2 + (((3*I)/64)*b*(1 - I*c*x^2)^2*(2*a + I*b*Log[1 - I*c*x^2])^2
)/c^2 + ((1 - I*c*x^2)*(2*a + I*b*Log[1 - I*c*x^2])^3)/(16*c^2) - ((1 - I*c*x^2)^2*(2*a + I*b*Log[1 - I*c*x^2]
)^3)/(32*c^2) - (((3*I)/8)*b^2*((2*I)*a - b*Log[1 - I*c*x^2])*Log[(1 + I*c*x^2)/2])/c^2 + (((3*I)/32)*b*((2*I)
*a - b*Log[1 - I*c*x^2])^2*Log[(1 + I*c*x^2)/2])/c^2 + (((3*I)/32)*b*(2*a + I*b*Log[1 - I*c*x^2])^2*Log[(1 + I
*c*x^2)/2])/c^2 - (((3*I)/8)*b^3*Log[(1 - I*c*x^2)/2]*Log[1 + I*c*x^2])/c^2 + (3*b^2*x^2*((2*I)*a - b*Log[1 -
I*c*x^2])*Log[1 + I*c*x^2])/(8*c) + ((3*I)/32)*b*x^4*((2*I)*a - b*Log[1 - I*c*x^2])^2*Log[1 + I*c*x^2] - (((3*
I)/32)*b*(2*a + I*b*Log[1 - I*c*x^2])^2*Log[1 + I*c*x^2])/c^2 - (((3*I)/16)*b^3*(1 + I*c*x^2)*Log[1 + I*c*x^2]
^2)/c^2 + ((3*I)/32)*b^2*x^4*((2*I)*a - b*Log[1 - I*c*x^2])*Log[1 + I*c*x^2]^2 - (3*b^2*(2*a + I*b*Log[1 - I*c
*x^2])*Log[1 + I*c*x^2]^2)/(32*c^2) + ((I/16)*b^3*(1 + I*c*x^2)*Log[1 + I*c*x^2]^3)/c^2 - ((I/32)*b^3*(1 + I*c
*x^2)^2*Log[1 + I*c*x^2]^3)/c^2 + (((3*I)/8)*b^3*PolyLog[2, (1 - I*c*x^2)/2])/c^2 - (((3*I)/16)*b^2*((2*I)*a -
b*Log[1 - I*c*x^2])*PolyLog[2, (1 - I*c*x^2)/2])/c^2 - (3*b^2*(2*a + I*b*Log[1 - I*c*x^2])*PolyLog[2, (1 - I*
c*x^2)/2])/(16*c^2) - (((3*I)/8)*b^3*PolyLog[2, (1 + I*c*x^2)/2])/c^2

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 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 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 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 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 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 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^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3+\frac{3}{8} i b x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3}{8} i b^2 x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{1}{8} i b^3 x^3 \log ^3\left (1+i c x^2\right )\right ) \, dx\\ &=\frac{1}{8} \int x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3 \, dx+\frac{1}{8} (3 i b) \int x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right ) \, dx+\frac{1}{8} \left (i b^3\right ) \int x^3 \log ^3\left (1+i c x^2\right ) \, dx\\ &=\frac{1}{16} \operatorname{Subst}\left (\int x (2 a+i b \log (1-i c x))^3 \, dx,x,x^2\right )+\frac{1}{16} (3 i b) \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x))^2 \log (1+i c x) \, dx,x,x^2\right )-\frac{1}{16} \left (3 i b^2\right ) \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \log ^2(1+i c x) \, dx,x,x^2\right )+\frac{1}{16} \left (i b^3\right ) \operatorname{Subst}\left (\int x \log ^3(1+i c x) \, dx,x,x^2\right )\\ &=\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{1}{16} \operatorname{Subst}\left (\int \left (-\frac{i (2 a+i b \log (1-i c x))^3}{c}+\frac{i (1-i c x) (2 a+i b \log (1-i c x))^3}{c}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{i \log ^3(1+i c x)}{c}-\frac{i (1+i c x) \log ^3(1+i c x)}{c}\right ) \, dx,x,x^2\right )+\frac{1}{32} (3 b c) \operatorname{Subst}\left (\int \frac{x^2 (-2 i a+b \log (1-i c x))^2}{1+i c x} \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1+i c x} \, dx,x,x^2\right )+\frac{1}{32} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2 \log ^2(1+i c x)}{1-i c x} \, dx,x,x^2\right )\\ &=\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^3 \, dx,x,x^2\right )}{16 c}+\frac{i \operatorname{Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^3 \, dx,x,x^2\right )}{16 c}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(1+i c x) \, dx,x,x^2\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int (1+i c x) \log ^3(1+i c x) \, dx,x,x^2\right )}{16 c}+\frac{1}{32} (3 b c) \operatorname{Subst}\left (\int \left (\frac{(-2 i a+b \log (1-i c x))^2}{c^2}-\frac{i x (-2 i a+b \log (1-i c x))^2}{c}+\frac{i (-2 i a+b \log (1-i c x))^2}{c^2 (-i+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{x (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{c^2 (-i+c x)}+\frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{x (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}-\frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{c^2 (i+c x)}+\frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^2\right )+\frac{1}{32} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log ^2(1+i c x)}{c^2}+\frac{i x \log ^2(1+i c x)}{c}-\frac{i \log ^2(1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{1}{32} (3 i b) \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac{1}{32} \left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^2\right )+\frac{\operatorname{Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\operatorname{Subst}\left (\int x (2 a+i b \log (x))^3 \, dx,x,1-i c x^2\right )}{16 c^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac{(3 i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2}{-i+c x} \, dx,x,x^2\right )}{32 c}+\frac{(3 b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{16 c}-2 \frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^2\right )}{16 c}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{i+c x} \, dx,x,x^2\right )}{32 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac{1}{32} (3 i b) \operatorname{Subst}\left (\int \left (-\frac{i (-2 i a+b \log (1-i c x))^2}{c}+\frac{i (1-i c x) (-2 i a+b \log (1-i c x))^2}{c}\right ) \, dx,x,x^2\right )+\frac{1}{32} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{i \log ^2(1+i c x)}{c}-\frac{i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,x^2\right )-2 \left (-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac{1}{16} \left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{x (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{x \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )\right )+\frac{(3 i b) \operatorname{Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}+\frac{(3 i b) \operatorname{Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}-\frac{(3 i b) \operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (2-x)) \log (x)}{x} \, dx,x,1+i c x^2\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x) \log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )}{16 c}\\ &=\frac{3 i b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{32 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac{9 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{3 i b^3 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{64 c^2}-\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-2 \left (-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac{1}{16} \left (3 i b^2\right ) \operatorname{Subst}\left (\int \left (-\frac{i (-2 i a+b \log (1-i c x))}{c}+\frac{-2 i a+b \log (1-i c x)}{c (-i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{i \log (1+i c x)}{c}+\frac{\log (1+i c x)}{c (i+c x)}\right ) \, dx,x,x^2\right )\right )-\frac{(3 i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{2-x} \, dx,x,1-i c x^2\right )}{32 c^2}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-2 i+i x)\right ) (-2 i a+b \log (x))}{x} \, dx,x,1-i c x^2\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{32 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{8 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+i c x^2\right )}{32 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (2 i-i x)\right ) \log (x)}{x} \, dx,x,1+i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{8 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}+\frac{(3 b) \operatorname{Subst}\left (\int (1-i c x) (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac{9 i a b^2 x^2}{8 c}+\frac{9 b^3 x^2}{16 c}-\frac{3 i b^3 \left (1-i c x^2\right )^2}{128 c^2}+\frac{3 i b^3 \left (1+i c x^2\right )^2}{128 c^2}+\frac{3 i b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{32 c^2}+\frac{3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{9 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac{3 i b^3 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{64 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac{9 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{3 i b^3 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{64 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 i b^3 \log \left (1+i c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}-\frac{(3 i b) \operatorname{Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}+\frac{(3 i b) \operatorname{Subst}\left (\int x (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{8 c^2}-2 \left (-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{16 c}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac{9 i a b^2 x^2}{8 c}+\frac{9 b^3 x^2}{8 c}-\frac{3 i b^3 \left (1-i c x^2\right )^2}{128 c^2}+\frac{3 i b^3 \left (1+i c x^2\right )^2}{128 c^2}-\frac{9 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{9 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac{3 i b^3 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{64 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 i b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int x (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{32 c^2}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{16 c^2}-2 \left (\frac{3 i a b^2 x^2}{8 c}+\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac{3 i a b^2 x^2}{4 c}+\frac{15 b^3 x^2}{16 c}-\frac{9 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac{3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-2 \left (\frac{3 i a b^2 x^2}{8 c}+\frac{3 b^3 x^2}{16 c}+\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}+\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}\right )+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}\\ &=\frac{3 i a b^2 x^2}{4 c}+\frac{3 b^3 x^2}{4 c}-\frac{3 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c^2}+\frac{3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac{3 i b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac{3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac{\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac{\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac{3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac{3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}+\frac{3}{32} i b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac{3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac{i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac{i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac{3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-2 \left (\frac{3 i a b^2 x^2}{8 c}+\frac{3 b^3 x^2}{8 c}-\frac{3 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac{3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}+\frac{3 i b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac{3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac{3 i b^3 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac{3 i b^3 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.311331, size = 170, normalized size = 1.14 \[ \frac{3 i b^3 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )+a \left (a c x^2 \left (a c x^2-3 b\right )+3 b^2 \log \left (c^2 x^4+1\right )\right )+3 b^2 \tan ^{-1}\left (c x^2\right )^2 \left (a c^2 x^4+a+b \left (-c x^2+i\right )\right )+3 b \tan ^{-1}\left (c x^2\right ) \left (a \left (a c^2 x^4+a-2 b c x^2\right )-2 b^2 \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )\right )+b^3 \left (c^2 x^4+1\right ) \tan ^{-1}\left (c x^2\right )^3}{4 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*b^2*(a + a*c^2*x^4 + b*(I - c*x^2))*ArcTan[c*x^2]^2 + b^3*(1 + c^2*x^4)*ArcTan[c*x^2]^3 + 3*b*ArcTan[c*x^2]
*(a*(a - 2*b*c*x^2 + a*c^2*x^4) - 2*b^2*Log[1 + E^((2*I)*ArcTan[c*x^2])]) + a*(a*c*x^2*(-3*b + a*c*x^2) + 3*b^
2*Log[1 + c^2*x^4]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])])/(4*c^2)

________________________________________________________________________________________

Maple [C]  time = 0.733, size = 690, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a^2*b/c^2*arctan(c*x^2)-3/4/c*b*a^2*x^2+3/4/c^2*a*b^2*ln(c^2*x^4+1)-3/16/c^2*a*b^2*ln(1-I*c*x^2)^2+3/16*b^
3/c*x^2*ln(1-I*c*x^2)^2+1/32*I*b^3*(c^2*x^4+1)/c^2*ln(1+I*c*x^2)^3+3/4*I/c*b^2*Sum((ln(x-_alpha)*ln(1-I*c*x^2)
+2*c*(-1/2*ln(x-_alpha)*(ln((-1/2-1/2*I)*(I*(I/c)^(1/2)-(I/c)^(1/2)-x+_alpha)/(I/c)^(1/2))+ln((1/2-1/2*I)*(I*(
I/c)^(1/2)+(I/c)^(1/2)+x-_alpha)/(I/c)^(1/2)))/c-1/2*(dilog((-1/2-1/2*I)*(I*(I/c)^(1/2)-(I/c)^(1/2)-x+_alpha)/
(I/c)^(1/2))+dilog((1/2-1/2*I)*(I*(I/c)^(1/2)+(I/c)^(1/2)+x-_alpha)/(I/c)^(1/2)))/c))*b/c,_alpha=RootOf(c*_Z^2
-RootOf(_Z^2+1,index=1)))-3/16*a*b^2*x^4*ln(1-I*c*x^2)^2+3/16*I/c^2*b^3*ln(1-I*c*x^2)^2-3/4*I/c*a*b^2*x^2*ln(1
-I*c*x^2)-1/32*I*b^3*x^4*ln(1-I*c*x^2)^3-3/32*b^2*(I*x^4*b*ln(1-I*c*x^2)*c^2+2*a*c^2*x^4-2*b*c*x^2+I*b*ln(1-I*
c*x^2)+2*I*b+2*a)/c^2*ln(1+I*c*x^2)^2-1/32*I*b^3/c^2*ln(1-I*c*x^2)^3+3/8*I*a^2*b*x^4*ln(1-I*c*x^2)+1/4*x^4*a^3
+(3/32*I*b^3*(c^2*x^4+1)/c^2*ln(1-I*c*x^2)^2+3/8*b^2*x^2*(a*c*x^2-b)/c*ln(1-I*c*x^2)-3/8*I*b*(a^2*c^2*x^4-2*a*
b*c*x^2+b^2*ln(1-I*c*x^2)+I*ln(1-I*c*x^2)*a*b)/c^2)*ln(1+I*c*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{4} \, a b^{2} x^{4} \arctan \left (c x^{2}\right )^{2} + \frac{1}{4} \, a^{3} x^{4} + \frac{3}{4} \,{\left (x^{4} \arctan \left (c x^{2}\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\arctan \left (c x^{2}\right )}{c^{3}}\right )}\right )} a^{2} b - \frac{3}{4} \,{\left (2 \, c{\left (\frac{x^{2}}{c^{2}} - \frac{\arctan \left (c x^{2}\right )}{c^{3}}\right )} \arctan \left (c x^{2}\right ) + \frac{\arctan \left (c x^{2}\right )^{2} - \log \left (4 \, c^{5} x^{4} + 4 \, c^{3}\right )}{c^{2}}\right )} a b^{2} + \frac{1}{128} \,{\left (4 \, x^{4} \arctan \left (c x^{2}\right )^{3} - 3 \, x^{4} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2} + 128 \, \int \frac{12 \, c^{2} x^{7} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right ) - 12 \, c x^{5} \arctan \left (c x^{2}\right )^{2} + 56 \,{\left (c^{2} x^{7} + x^{3}\right )} \arctan \left (c x^{2}\right )^{3} + 3 \,{\left (c x^{5} + 2 \,{\left (c^{2} x^{7} + x^{3}\right )} \arctan \left (c x^{2}\right )\right )} \log \left (c^{2} x^{4} + 1\right )^{2}}{64 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a*b^2*x^4*arctan(c*x^2)^2 + 1/4*a^3*x^4 + 3/4*(x^4*arctan(c*x^2) - c*(x^2/c^2 - arctan(c*x^2)/c^3))*a^2*b
- 3/4*(2*c*(x^2/c^2 - arctan(c*x^2)/c^3)*arctan(c*x^2) + (arctan(c*x^2)^2 - log(4*c^5*x^4 + 4*c^3))/c^2)*a*b^2
+ 1/128*(4*x^4*arctan(c*x^2)^3 - 3*x^4*arctan(c*x^2)*log(c^2*x^4 + 1)^2 + 128*integrate(1/64*(12*c^2*x^7*arct
an(c*x^2)*log(c^2*x^4 + 1) - 12*c*x^5*arctan(c*x^2)^2 + 56*(c^2*x^7 + x^3)*arctan(c*x^2)^3 + 3*(c*x^5 + 2*(c^2
*x^7 + x^3)*arctan(c*x^2))*log(c^2*x^4 + 1)^2)/(c^2*x^4 + 1), x))*b^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*atan(c*x**2))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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