3.111 \(\int \frac{x^4 (a+b \tan ^{-1}(c x))^2}{(d+i c d x)^3} \, dx\)

Optimal. Leaf size=462 \[ -\frac{6 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}-\frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i a b x}{c^4 d^3}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (-c x+i)}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (-c x+i)^2}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (-c x+i)}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (-c x+i)^2}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{6 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}+\frac{6 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3}+\frac{i b^2 \log \left (c^2 x^2+1\right )}{2 c^5 d^3}-\frac{29 b^2}{16 c^5 d^3 (-c x+i)}+\frac{i b^2}{16 c^5 d^3 (-c x+i)^2}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}+\frac{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3} \]

[Out]

((-I)*a*b*x)/(c^4*d^3) + ((I/16)*b^2)/(c^5*d^3*(I - c*x)^2) - (29*b^2)/(16*c^5*d^3*(I - c*x)) + (29*b^2*ArcTan
[c*x])/(16*c^5*d^3) - (I*b^2*x*ArcTan[c*x])/(c^4*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c^5*d^3*(I - c*x)^2) - (((1
5*I)/4)*b*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) - (((5*I)/8)*(a + b*ArcTan[c*x])^2)/(c^5*d^3) - (3*x*(a + b
*ArcTan[c*x])^2)/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^5*
d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)) - (6*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)]
)/(c^5*d^3) + ((6*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((I/2)*b^2*Log[1 + c^2*x^2])/(c^5*d
^3) - ((3*I)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) - (6*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c
*x)])/(c^5*d^3) + ((3*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.832922, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4864, 4862, 627, 44, 203, 4994, 6610} \[ -\frac{6 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}-\frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i a b x}{c^4 d^3}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (-c x+i)}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (-c x+i)^2}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (-c x+i)}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (-c x+i)^2}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{6 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}+\frac{6 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3}+\frac{i b^2 \log \left (c^2 x^2+1\right )}{2 c^5 d^3}-\frac{29 b^2}{16 c^5 d^3 (-c x+i)}+\frac{i b^2}{16 c^5 d^3 (-c x+i)^2}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}+\frac{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]

[Out]

((-I)*a*b*x)/(c^4*d^3) + ((I/16)*b^2)/(c^5*d^3*(I - c*x)^2) - (29*b^2)/(16*c^5*d^3*(I - c*x)) + (29*b^2*ArcTan
[c*x])/(16*c^5*d^3) - (I*b^2*x*ArcTan[c*x])/(c^4*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c^5*d^3*(I - c*x)^2) - (((1
5*I)/4)*b*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) - (((5*I)/8)*(a + b*ArcTan[c*x])^2)/(c^5*d^3) - (3*x*(a + b
*ArcTan[c*x])^2)/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^5*
d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)) - (6*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)]
)/(c^5*d^3) + ((6*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((I/2)*b^2*Log[1 + c^2*x^2])/(c^5*d
^3) - ((3*I)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) - (6*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c
*x)])/(c^5*d^3) + ((3*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^3)

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

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 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 260

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

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 4864

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

Rule 4862

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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]

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{(d+i c d x)^3} \, dx &=\int \left (-\frac{3 \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (-i+c x)^3}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (-i+c x)^2}-\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (-i+c x)}\right ) \, dx\\ &=\frac{i \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^3} \, dx}{c^4 d^3}-\frac{(6 i) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{c^4 d^3}-\frac{3 \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^4 d^3}+\frac{4 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{c^4 d^3}+\frac{i \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^3 d^3}\\ &=-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{(i b) \int \left (-\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^3}+\frac{a+b \tan ^{-1}(c x)}{4 (-i+c x)^2}-\frac{a+b \tan ^{-1}(c x)}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}-\frac{(12 i b) \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^4 d^3}+\frac{(8 b) \int \left (-\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}+\frac{(6 b) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d^3}-\frac{(i b) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac{3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{(i b) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{4 c^4 d^3}-\frac{(i b) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{4 c^4 d^3}-\frac{(i b) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^4 d^3}+\frac{(i b) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^4 d^3}-\frac{(4 i b) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^4 d^3}+\frac{(4 i b) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^4 d^3}+\frac{b \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{2 c^4 d^3}-\frac{(6 b) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^4 d^3}+\frac{\left (6 b^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3}\\ &=-\frac{i a b x}{c^4 d^3}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)^2}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{\left (i b^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 c^4 d^3}-\frac{\left (i b^2\right ) \int \tan ^{-1}(c x) \, dx}{c^4 d^3}-\frac{\left (4 i b^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^4 d^3}+\frac{b^2 \int \frac{1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 c^4 d^3}+\frac{\left (6 b^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3}\\ &=-\frac{i a b x}{c^4 d^3}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)^2}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^5 d^3}+\frac{\left (i b^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{4 c^4 d^3}-\frac{\left (4 i b^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{c^4 d^3}+\frac{b^2 \int \frac{1}{(-i+c x)^3 (i+c x)} \, dx}{4 c^4 d^3}+\frac{\left (i b^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c^3 d^3}\\ &=-\frac{i a b x}{c^4 d^3}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)^2}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac{3 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{\left (i b^2\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^4 d^3}-\frac{\left (4 i b^2\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}+\frac{b^2 \int \left (-\frac{i}{2 (-i+c x)^3}+\frac{1}{4 (-i+c x)^2}-\frac{1}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^4 d^3}\\ &=-\frac{i a b x}{c^4 d^3}+\frac{i b^2}{16 c^5 d^3 (i-c x)^2}-\frac{29 b^2}{16 c^5 d^3 (i-c x)}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)^2}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac{3 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{16 c^4 d^3}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{8 c^4 d^3}+\frac{\left (2 b^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{c^4 d^3}\\ &=-\frac{i a b x}{c^4 d^3}+\frac{i b^2}{16 c^5 d^3 (i-c x)^2}-\frac{29 b^2}{16 c^5 d^3 (i-c x)}+\frac{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)^2}-\frac{15 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^5 d^3 (i-c x)}-\frac{5 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{i x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5 d^3 (i-c x)^2}+\frac{4 \left (a+b \tan ^{-1}(c x)\right )^2}{c^5 d^3 (i-c x)}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{6 i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac{3 i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}-\frac{6 b \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{c^5 d^3}\\ \end{align*}

Mathematica [A]  time = 2.50212, size = 578, normalized size = 1.25 \[ \frac{a b \left (96 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+48 \log \left (c^2 x^2+1\right )+4 i \tan ^{-1}(c x) \left (4 c^2 x^2+24 i c x+48 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+14 i \sin \left (2 \tan ^{-1}(c x)\right )-i \sin \left (4 \tan ^{-1}(c x)\right )-14 \cos \left (2 \tan ^{-1}(c x)\right )+\cos \left (4 \tan ^{-1}(c x)\right )+4\right )-16 i c x+192 \tan ^{-1}(c x)^2+28 i \sin \left (2 \tan ^{-1}(c x)\right )-i \sin \left (4 \tan ^{-1}(c x)\right )-28 \cos \left (2 \tan ^{-1}(c x)\right )+\cos \left (4 \tan ^{-1}(c x)\right )\right )+16 i b^2 \left (\left (3-6 i \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+\frac{1}{2} \log \left (c^2 x^2+1\right )+\frac{1}{2} \left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2-4 i \tan ^{-1}(c x)^3+3 i c x \tan ^{-1}(c x)^2+3 \tan ^{-1}(c x)^2-c x \tan ^{-1}(c x)+6 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+6 i \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\frac{7}{4} i \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-\frac{1}{8} i \tan ^{-1}(c x)^2 \sin \left (4 \tan ^{-1}(c x)\right )+\frac{7}{4} \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-\frac{1}{16} \tan ^{-1}(c x) \sin \left (4 \tan ^{-1}(c x)\right )-\frac{7}{8} i \sin \left (2 \tan ^{-1}(c x)\right )+\frac{1}{64} i \sin \left (4 \tan ^{-1}(c x)\right )+\frac{1}{8} \tan ^{-1}(c x)^2 \cos \left (4 \tan ^{-1}(c x)\right )-\frac{1}{16} i \tan ^{-1}(c x) \cos \left (4 \tan ^{-1}(c x)\right )-\frac{7}{8} \left (2 \tan ^{-1}(c x)^2-2 i \tan ^{-1}(c x)-1\right ) \cos \left (2 \tan ^{-1}(c x)\right )-\frac{1}{64} \cos \left (4 \tan ^{-1}(c x)\right )\right )+8 i a^2 c^2 x^2-48 i a^2 \log \left (c^2 x^2+1\right )-48 a^2 c x-\frac{64 a^2}{c x-i}-\frac{8 i a^2}{(c x-i)^2}+96 a^2 \tan ^{-1}(c x)}{16 c^5 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]

[Out]

(-48*a^2*c*x + (8*I)*a^2*c^2*x^2 - ((8*I)*a^2)/(-I + c*x)^2 - (64*a^2)/(-I + c*x) + 96*a^2*ArcTan[c*x] - (48*I
)*a^2*Log[1 + c^2*x^2] + a*b*((-16*I)*c*x + 192*ArcTan[c*x]^2 - 28*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 4
8*Log[1 + c^2*x^2] + 96*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (28*I)*Sin[2*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*(4
+ (24*I)*c*x + 4*c^2*x^2 - 14*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + E^((2*I)*ArcTan[c*x])] + (1
4*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) - I*Sin[4*ArcTan[c*x]]) + (16*I)*b^2*(-(c*x*ArcTan[c*x]) + 3*A
rcTan[c*x]^2 + (3*I)*c*x*ArcTan[c*x]^2 + ((1 + c^2*x^2)*ArcTan[c*x]^2)/2 - (4*I)*ArcTan[c*x]^3 - (7*(-1 - (2*I
)*ArcTan[c*x] + 2*ArcTan[c*x]^2)*Cos[2*ArcTan[c*x]])/8 - Cos[4*ArcTan[c*x]]/64 - (I/16)*ArcTan[c*x]*Cos[4*ArcT
an[c*x]] + (ArcTan[c*x]^2*Cos[4*ArcTan[c*x]])/8 + (6*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*ArcTan[
c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + Log[1 + c^2*x^2]/2 + (3 - (6*I)*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcT
an[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - ((7*I)/8)*Sin[2*ArcTan[c*x]] + (7*ArcTan[c*x]*Sin[2*ArcTan[
c*x]])/4 + ((7*I)/4)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (I/64)*Sin[4*ArcTan[c*x]] - (ArcTan[c*x]*Sin[4*ArcTan[
c*x]])/16 - (I/8)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(16*c^5*d^3)

________________________________________________________________________________________

Maple [C]  time = 1.728, size = 1618, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x)

[Out]

6/c^5*b^2/d^3*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+3/c^5*b^2/d^3*Pi*cs
gn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-8/c^5*a*b/d^3*arctan(c*x)/(c*x-I)-6/c^
5*a*b/d^3*ln(c*x-I)*ln(-1/2*I*(c*x+I))+7/4/c^4*b^2/d^3*arctan(c*x)/(c*x-I)*x+1/16/c^3*b^2/d^3*arctan(c*x)/(c*x
-I)^2*x^2-6/c^4*a*b/d^3*arctan(c*x)*x+43/8*I/c^5*a*b/d^3*arctan(c*x)+15/4*I/c^5*a*b/d^3/(c*x-I)-1/2*I/c^5*b^2/
d^3*arctan(c*x)^2/(c*x-I)^2+7/4*I/c^5*b^2/d^3*arctan(c*x)/(c*x-I)+6*I/c^5*b^2/d^3*arctan(c*x)^2*ln(2*I*(1+I*c*
x)^2/(c^2*x^2+1))-6*I/c^5*b^2/d^3*arctan(c*x)^2*ln(c*x-I)+5/16*I/c^5*a*b/d^3*arctan(1/2*c*x)-5/8*I/c^5*a*b/d^3
*arctan(1/2*c*x-1/2*I)-5/16*I/c^5*a*b/d^3*arctan(1/6*c^3*x^3+7/6*c*x)-7*I/c^4*b^2/d^3/(8*c*x-8*I)*x-1/64*I/c^3
*b^2/d^3/(c*x-I)^2*x^2+1/2*I/c^3*b^2/d^3*arctan(c*x)^2*x^2-1/4/c^5*a*b/d^3/(c*x-I)^2+43/16/c^5*a*b/d^3*ln(c^2*
x^2+1)+3/c^5*a*b/d^3*ln(c*x-I)^2-6/c^5*a*b/d^3*dilog(-1/2*I*(c*x+I))-6/c^5*b^2/d^3*Pi*arctan(c*x)^2+6/c^5*b^2/
d^3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-6/c^5*b^2/d^3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/
2))-6/c^5*b^2/d^3*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/16/c^5*b^2/d^3*arctan(c*x)/(c*x-I)^2-1/2*I
/c^5*a^2/d^3/(c*x-I)^2-3*I/c^5*a^2/d^3*ln(c^2*x^2+1)+1/2*I/c^3*a^2/d^3*x^2-I/c^5*b^2/d^3*ln((1+I*c*x)^2/(c^2*x
^2+1)+1)+43/8*I/c^5*b^2/d^3*arctan(c*x)^2+1/64*I/c^5*b^2/d^3/(c*x-I)^2+3*I/c^5*b^2/d^3*polylog(3,-(1+I*c*x)^2/
(c^2*x^2+1))+6*I/c^5*b^2/d^3*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I/c^5*b^2/d^3*dilog(1+I*(1+I*c*x)/(c^2*x
^2+1)^(1/2))-1/c^5*a*b/d^3-3/c^4*a^2/d^3*x+4/c^5*b^2/d^3*arctan(c*x)^3-b^2*arctan(c*x)/c^5/d^3+3/c^5*b^2/d^3*P
i*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/
(c^2*x^2+1)+1))*arctan(c*x)^2-I/c^5*a*b/d^3*arctan(c*x)/(c*x-I)^2+1/8*I/c^4*b^2/d^3*arctan(c*x)/(c*x-I)^2*x-3/
c^5*b^2/d^3*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*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+3/c^5*b^2/d^3*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*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+I/c^3*a*b/d^3*arctan(c*x)*x^2-12*I/c^5*a*b/d^3*arctan(c*x)*ln(c*x-I)-I*a*b*x/c^4/d^3
-I*b^2*x*arctan(c*x)/c^4/d^3-4/c^5*b^2/d^3*arctan(c*x)^2/(c*x-I)+1/32/c^4*b^2/d^3/(c*x-I)^2*x-3/c^4*b^2/d^3*ar
ctan(c*x)^2*x+5/32/c^5*a*b/d^3*ln(c^4*x^4+10*c^2*x^2+9)+7/c^5*b^2/d^3/(8*c*x-8*I)-4/c^5*a^2/d^3/(c*x-I)+6/c^5*
a^2/d^3*arctan(c*x)

________________________________________________________________________________________

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^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-i \, b^{2} x^{4} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 4 \, a b x^{4} \log \left (-\frac{c x + i}{c x - i}\right ) + 4 i \, a^{2} x^{4}}{4 \, c^{3} d^{3} x^{3} - 12 i \, c^{2} d^{3} x^{2} - 12 \, c d^{3} x + 4 i \, d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="fricas")

[Out]

integral((-I*b^2*x^4*log(-(c*x + I)/(c*x - I))^2 - 4*a*b*x^4*log(-(c*x + I)/(c*x - I)) + 4*I*a^2*x^4)/(4*c^3*d
^3*x^3 - 12*I*c^2*d^3*x^2 - 12*c*d^3*x + 4*I*d^3), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (i \, c d x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x) + a)^2*x^4/(I*c*d*x + d)^3, x)