3.127 \(\int \frac{x^2 (a+b \tan ^{-1}(c x))^3}{d+i c d x} \, dx\)

Optimal. Leaf size=410 \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d}-\frac{3 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{1+i c x}\right )}{4 c^3 d}+\frac{3 i b^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{3 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d}-\frac{i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.861898, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {4866, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 4884, 4994, 6610, 4998} \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d}-\frac{3 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{1+i c x}\right )}{4 c^3 d}+\frac{3 i b^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{3 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d}-\frac{i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcTan[c*x])^3)/(d + I*c*d*x),x]

[Out]

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

Rule 4866

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

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4916

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

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

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

Rule 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 4998

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a
+ b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 -
(2*I)/(I - c*x))^2, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{d+i c d x} \, dx &=\frac{i \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{d+i c d x} \, dx}{c}-\frac{i \int x \left (a+b \tan ^{-1}(c x)\right )^3 \, dx}{c d}\\ &=-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}-\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{d+i c d x} \, dx}{c^2}+\frac{(3 i b) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 d}+\frac{\int \left (a+b \tan ^{-1}(c x)\right )^3 \, dx}{c^2 d}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{(3 i b) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c^2 d}-\frac{(3 i b) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c^2 d}+\frac{(3 i b) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac{(3 b) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c d}\\ &=\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{(3 b) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2 d}-\frac{\left (3 b^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac{\left (3 i b^2\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d}\\ &=-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{\left (3 i b^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^2 d}-\frac{\left (6 b^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}+\frac{\left (3 i b^3\right ) \int \frac{\text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^2 d}\\ &=-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+i c x}\right )}{4 c^3 d}-\frac{\left (3 i b^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac{\left (3 i b^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}\\ &=-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+i c x}\right )}{4 c^3 d}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^3 d}\\ &=-\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d}+\frac{3 i b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c^3 d}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d}-\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+i c x}\right )}{c^3 d}-\frac{3 b^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3 d}+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}+\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3 d}-\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+i c x}\right )}{4 c^3 d}\\ \end{align*}

Mathematica [A]  time = 0.965524, size = 541, normalized size = 1.32 \[ -\frac{i \left (6 b^2 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right ) \left (a+b \tan ^{-1}(c x)+i b\right )-6 i b \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right ) \left (a+b \tan ^{-1}(c x)+i b\right )^2+3 i b^3 \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(c x)}\right )-6 i a^2 b \log \left (c^2 x^2+1\right )+6 a^2 b c^2 x^2 \tan ^{-1}(c x)-6 a^2 b c x-12 i a^2 b \tan ^{-1}(c x)^2+6 a^2 b \tan ^{-1}(c x)+12 i a^2 b c x \tan ^{-1}(c x)+12 a^2 b \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+2 a^3 c^2 x^2-2 a^3 \log \left (c^2 x^2+1\right )+4 i a^3 c x-4 i a^3 \tan ^{-1}(c x)+6 a b^2 \log \left (c^2 x^2+1\right )+6 a b^2 c^2 x^2 \tan ^{-1}(c x)^2-8 i a b^2 \tan ^{-1}(c x)^3+18 a b^2 \tan ^{-1}(c x)^2+12 i a b^2 c x \tan ^{-1}(c x)^2-12 a b^2 c x \tan ^{-1}(c x)+12 a b^2 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+24 i a b^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+2 b^3 c^2 x^2 \tan ^{-1}(c x)^3-2 i b^3 \tan ^{-1}(c x)^4+6 b^3 \tan ^{-1}(c x)^3+4 i b^3 c x \tan ^{-1}(c x)^3+6 i b^3 \tan ^{-1}(c x)^2-6 b^3 c x \tan ^{-1}(c x)^2+4 b^3 \tan ^{-1}(c x)^3 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+12 i b^3 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-12 b^3 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{4 c^3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/4)*((4*I)*a^3*c*x - 6*a^2*b*c*x + 2*a^3*c^2*x^2 - (4*I)*a^3*ArcTan[c*x] + 6*a^2*b*ArcTan[c*x] + (12*I)*a^
2*b*c*x*ArcTan[c*x] - 12*a*b^2*c*x*ArcTan[c*x] + 6*a^2*b*c^2*x^2*ArcTan[c*x] - (12*I)*a^2*b*ArcTan[c*x]^2 + 18
*a*b^2*ArcTan[c*x]^2 + (6*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c*x]^2 - 6*b^3*c*x*ArcTan[c*x]^2 + 6*
a*b^2*c^2*x^2*ArcTan[c*x]^2 - (8*I)*a*b^2*ArcTan[c*x]^3 + 6*b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 +
2*b^3*c^2*x^2*ArcTan[c*x]^3 - (2*I)*b^3*ArcTan[c*x]^4 + 12*a^2*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] +
(24*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] +
12*a*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])
] + 4*b^3*ArcTan[c*x]^3*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*a^3*Log[1 + c^2*x^2] - (6*I)*a^2*b*Log[1 + c^2*x^2]
+ 6*a*b^2*Log[1 + c^2*x^2] - (6*I)*b*(a + I*b + b*ArcTan[c*x])^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*b^2*(
a + I*b + b*ArcTan[c*x])*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcTan[c*x])]))/(
c^3*d)

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Maple [C]  time = 2.217, size = 1725, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2/c^3*a*b^2/d*arctan(c*x)^2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I/((1+I*c*x)^2/(
c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*Pi+3/2/c^3*a*b^2/d*arctan(c*x)^2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1
+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*Pi-3/2/c^3*a*b^2/d*arctan(c*x)^2*csgn((1+I*c*x
)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn((1+I*c*x)^2/(c^2*x^2+1))*Pi-3/c^3*a*b^2/d*Pi*csgn((1+I*c*x
)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/c^2*b^3/d*arctan(c*x)^3*x+1/2*I/c^3*a^3/d*ln(c^
2*x^2+1)-3/2*I/c^3*b^3/d*arctan(c*x)^3-1/2*I/c*a^3/d*x^2-3/c^3*a*b^2/d*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2
*x^2+1))+3/2/c^3*b*a^2/d*ln(c*x-I)*ln(-1/2*I*(c*x+I))+6/c^3*a*b^2/d*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(
1/2))+6/c^3*a*b^2/d*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/c^2*b*a^2/d*arctan(c*x)*x+3/c^2*a*b^2/d*
arctan(c*x)^2*x+3/c^3*a*b^2/d*Pi*arctan(c*x)^2-3/2*I/c^3*a*b^2/d*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+3*I/c^3*a
*b^2/d*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-I/c^3*b^3/d*arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-3/2*I/c^3*b^3/d*a
rctan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+3*I/c^3*b^3/d*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-3*I/c^3
*b^3/d*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2*I/c^2*b*a^2/d*x-1/2*I/c*b^3/d*arctan(c*x)^3*x^2+3/2
*I/c^2*b^3/d*arctan(c*x)^2*x-9/4*I/c^3*b*a^2/d*arctan(c*x)+3/8*I/c^3*b*a^2/d*arctan(1/6*c^3*x^3+7/6*c*x)-3/8*I
/c^3*b*a^2/d*arctan(1/2*c*x)+3/4*I/c^3*b*a^2/d*arctan(1/2*c*x-1/2*I)-6*I/c^3*a*b^2/d*dilog(1-I*(1+I*c*x)/(c^2*
x^2+1)^(1/2))-9/2*I/c^3*a*b^2/d*arctan(c*x)^2-6*I/c^3*a*b^2/d*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2/c^3*a
*b^2/d*arctan(c*x)^2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*Pi-3*I/c^3*a*b^2/d*arctan(c*x
)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+3*I/c^3*a*b^2/d*arctan(c*x)^2*ln(c*x-I)+3*I/c^3*b*a^2/d*arctan(c*x)*ln(c*x
-I)+3*I/c^2*a*b^2/d*arctan(c*x)*x-3/2*I/c*a*b^2/d*arctan(c*x)^2*x^2-3/2*I/c*b*a^2/d*arctan(c*x)*x^2+3/2/c^3*b*
a^2/d*dilog(-1/2*I*(c*x+I))-3/4/c^3*b*a^2/d*ln(c*x-I)^2-2/c^3*a*b^2/d*arctan(c*x)^3-3/16/c^3*b*a^2/d*ln(c^4*x^
4+10*c^2*x^2+9)+3/c^3*a*b^2/d*arctan(c*x)-9/8/c^3*b*a^2/d*ln(c^2*x^2+1)-3/2/c^3*b^3/d*arctan(c*x)^2*polylog(2,
-(1+I*c*x)^2/(c^2*x^2+1))+3/c^3*b^3/d*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+3/2/c^3*b^3/d*polylog(3,-(1+
I*c*x)^2/(c^2*x^2+1))+3/4/c^3*b^3/d*polylog(4,-(1+I*c*x)^2/(c^2*x^2+1))-1/2/c^3*b^3/d*arctan(c*x)^4-1/c^3*a^3/
d*arctan(c*x)+3/2/c^3*b^3/d*arctan(c*x)^2+3/2/c^3*b^3/d*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2/c^3*b*a^2/d+1/
c^2*a^3/d*x

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Maxima [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^2*(a+b*arctan(c*x))^3/(d+I*c*d*x),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{3} x^{2} \log \left (-\frac{c x + i}{c x - i}\right )^{3} - 6 i \, a b^{2} x^{2} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 12 \, a^{2} b x^{2} \log \left (-\frac{c x + i}{c x - i}\right ) + 8 i \, a^{3} x^{2}}{8 \, c d x - 8 i \, d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b^3*x^2*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*x^2*log(-(c*x + I)/(c*x - I))^2 - 12*a^2*b*x^2*log(
-(c*x + I)/(c*x - I)) + 8*I*a^3*x^2)/(8*c*d*x - 8*I*d), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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