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

Optimal. Leaf size=438 \[ -\frac{26 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{35 c^4}-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{52 i b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{35 c^4}+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}-\frac{11 b^2 d^3 \log \left (c^2 x^2+1\right )}{10 c^4}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}-\frac{1}{105} i b^2 c d^3 x^5+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4 \]

[Out]

(3*a*b*d^3*x)/(2*c^3) - (((122*I)/105)*b^2*d^3*x)/c^3 + (7*b^2*d^3*x^2)/(20*c^2) + (((44*I)/315)*b^2*d^3*x^3)/
c - (b^2*d^3*x^4)/20 - (I/105)*b^2*c*d^3*x^5 + (((122*I)/105)*b^2*d^3*ArcTan[c*x])/c^4 + (3*b^2*d^3*x*ArcTan[c
*x])/(2*c^3) + (((26*I)/35)*b*d^3*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d^3*x^3*(a + b*ArcTan[c*x]))/(2*c) - ((13*
I)/35)*b*d^3*x^4*(a + b*ArcTan[c*x]) + (b*c*d^3*x^5*(a + b*ArcTan[c*x]))/5 + (I/21)*b*c^2*d^3*x^6*(a + b*ArcTa
n[c*x]) - (209*d^3*(a + b*ArcTan[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTan[c*x])^2)/4 + ((3*I)/5)*c*d^3*x^5*
(a + b*ArcTan[c*x])^2 - (c^2*d^3*x^6*(a + b*ArcTan[c*x])^2)/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x])^2 + (((5
2*I)/35)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (11*b^2*d^3*Log[1 + c^2*x^2])/(10*c^4) - (26*b^2*
d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(35*c^4)

________________________________________________________________________________________

Rubi [A]  time = 1.36562, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 62, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4876, 4852, 4916, 266, 43, 4846, 260, 4884, 302, 203, 321, 4920, 4854, 2402, 2315} \[ -\frac{26 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{35 c^4}-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{52 i b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{35 c^4}+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}-\frac{11 b^2 d^3 \log \left (c^2 x^2+1\right )}{10 c^4}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}-\frac{1}{105} i b^2 c d^3 x^5+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b*d^3*x)/(2*c^3) - (((122*I)/105)*b^2*d^3*x)/c^3 + (7*b^2*d^3*x^2)/(20*c^2) + (((44*I)/315)*b^2*d^3*x^3)/
c - (b^2*d^3*x^4)/20 - (I/105)*b^2*c*d^3*x^5 + (((122*I)/105)*b^2*d^3*ArcTan[c*x])/c^4 + (3*b^2*d^3*x*ArcTan[c
*x])/(2*c^3) + (((26*I)/35)*b*d^3*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d^3*x^3*(a + b*ArcTan[c*x]))/(2*c) - ((13*
I)/35)*b*d^3*x^4*(a + b*ArcTan[c*x]) + (b*c*d^3*x^5*(a + b*ArcTan[c*x]))/5 + (I/21)*b*c^2*d^3*x^6*(a + b*ArcTa
n[c*x]) - (209*d^3*(a + b*ArcTan[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTan[c*x])^2)/4 + ((3*I)/5)*c*d^3*x^5*
(a + b*ArcTan[c*x])^2 - (c^2*d^3*x^6*(a + b*ArcTan[c*x])^2)/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x])^2 + (((5
2*I)/35)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (11*b^2*d^3*Log[1 + c^2*x^2])/(10*c^4) - (26*b^2*
d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(35*c^4)

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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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]

Rubi steps

\begin{align*} \int x^3 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-3 c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 i c d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (3 c^2 d^3\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x^6 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{5} \left (6 i b c^2 d^3\right ) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\left (b c^3 d^3\right ) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{7} \left (2 i b c^4 d^3\right ) \int \frac{x^7 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{5} \left (6 i b d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{5} \left (6 i b d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{\left (b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac{\left (b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}+\left (b c d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{7} \left (2 i b c^2 d^3\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{7} \left (2 i b c^2 d^3\right ) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac{3}{10} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} \left (2 i b d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{7} \left (2 i b d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{6} \left (b^2 d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{\left (b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{\left (b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac{\left (6 i b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac{\left (6 i b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac{\left (b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}+\frac{\left (b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c}+\frac{1}{10} \left (3 i b^2 c d^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx-\frac{1}{5} \left (b^2 c^2 d^3\right ) \int \frac{x^5}{1+c^2 x^2} \, dx-\frac{1}{21} \left (i b^2 c^3 d^3\right ) \int \frac{x^6}{1+c^2 x^2} \, dx\\ &=\frac{a b d^3 x}{2 c^3}+\frac{3 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{17 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{12} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{1}{3} \left (b^2 d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{\left (6 i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^3}+\frac{\left (b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3}-\frac{\left (b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^3}+\frac{\left (b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}+\frac{\left (2 i b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c^2}-\frac{\left (2 i b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c^2}-\frac{\left (3 i b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx}{5 c}+\frac{1}{14} \left (i b^2 c d^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx+\frac{1}{10} \left (3 i b^2 c d^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{1}{10} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{21} \left (i b^2 c^3 d^3\right ) \int \left (\frac{1}{c^6}-\frac{x^2}{c^4}+\frac{x^4}{c^2}-\frac{1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{3 a b d^3 x}{2 c^3}-\frac{199 i b^2 d^3 x}{210 c^3}+\frac{73 i b^2 d^3 x^3}{630 c}-\frac{1}{105} i b^2 c d^3 x^5+\frac{b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{6 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{5 c^4}+\frac{1}{12} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{6} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{\left (2 i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{7 c^3}+\frac{\left (i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{21 c^3}+\frac{\left (3 i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{10 c^3}+\frac{\left (3 i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^3}-\frac{\left (6 i b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^3}+\frac{\left (b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{c^3}-\frac{\left (b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{2 c^2}-\frac{\left (i b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx}{7 c}+\frac{1}{14} \left (i b^2 c d^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{1}{10} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a b d^3 x}{2 c^3}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{11 b^2 d^3 x^2}{60 c^2}+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4-\frac{1}{105} i b^2 c d^3 x^5+\frac{199 i b^2 d^3 \tan ^{-1}(c x)}{210 c^4}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{52 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{35 c^4}-\frac{13 b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^4}+\frac{1}{6} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{5 c^4}+\frac{\left (i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{14 c^3}+\frac{\left (i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{7 c^3}-\frac{\left (2 i b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{7 c^3}-\frac{\left (b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac{3 a b d^3 x}{2 c^3}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4-\frac{1}{105} i b^2 c d^3 x^5+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{52 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{35 c^4}-\frac{11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac{3 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{5 c^4}-\frac{\left (2 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{7 c^4}\\ &=\frac{3 a b d^3 x}{2 c^3}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4-\frac{1}{105} i b^2 c d^3 x^5+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{26 i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{35 c^2}-\frac{b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{2 c}-\frac{13}{35} i b d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{21} i b c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{209 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{52 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{35 c^4}-\frac{11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac{26 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{35 c^4}\\ \end{align*}

Mathematica [A]  time = 1.73905, size = 408, normalized size = 0.93 \[ \frac{d^3 \left (936 b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-180 i a^2 c^7 x^7-630 a^2 c^6 x^6+756 i a^2 c^5 x^5+315 a^2 c^4 x^4+60 i a b c^6 x^6+252 a b c^5 x^5-468 i a b c^4 x^4-630 a b c^3 x^3+936 i a b c^2 x^2-936 i a b \log \left (c^2 x^2+1\right )+6 b \tan ^{-1}(c x) \left (3 a \left (-20 i c^7 x^7-70 c^6 x^6+84 i c^5 x^5+35 c^4 x^4-105\right )+b \left (10 i c^6 x^6+42 c^5 x^5-78 i c^4 x^4-105 c^3 x^3+156 i c^2 x^2+315 c x+244 i\right )+312 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+1890 a b c x+1464 i a b-12 i b^2 c^5 x^5-63 b^2 c^4 x^4+176 i b^2 c^3 x^3+441 b^2 c^2 x^2-1386 b^2 \log \left (c^2 x^2+1\right )+9 b^2 (c x-i)^4 \left (-20 i c^3 x^3+10 c^2 x^2+4 i c x-1\right ) \tan ^{-1}(c x)^2-1464 i b^2 c x+504 b^2\right )}{1260 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*((1464*I)*a*b + 504*b^2 + 1890*a*b*c*x - (1464*I)*b^2*c*x + (936*I)*a*b*c^2*x^2 + 441*b^2*c^2*x^2 - 630*a
*b*c^3*x^3 + (176*I)*b^2*c^3*x^3 + 315*a^2*c^4*x^4 - (468*I)*a*b*c^4*x^4 - 63*b^2*c^4*x^4 + (756*I)*a^2*c^5*x^
5 + 252*a*b*c^5*x^5 - (12*I)*b^2*c^5*x^5 - 630*a^2*c^6*x^6 + (60*I)*a*b*c^6*x^6 - (180*I)*a^2*c^7*x^7 + 9*b^2*
(-I + c*x)^4*(-1 + (4*I)*c*x + 10*c^2*x^2 - (20*I)*c^3*x^3)*ArcTan[c*x]^2 + 6*b*ArcTan[c*x]*(b*(244*I + 315*c*
x + (156*I)*c^2*x^2 - 105*c^3*x^3 - (78*I)*c^4*x^4 + 42*c^5*x^5 + (10*I)*c^6*x^6) + 3*a*(-105 + 35*c^4*x^4 + (
84*I)*c^5*x^5 - 70*c^6*x^6 - (20*I)*c^7*x^7) + (312*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (936*I)*a*b*Log[1 +
c^2*x^2] - 1386*b^2*Log[1 + c^2*x^2] + 936*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(1260*c^4)

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Maple [A]  time = 0.096, size = 750, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/35*I*d^3*a*b*x^4-13/35/c^4*d^3*b^2*dilog(-1/2*I*(c*x+I))+13/70/c^4*d^3*b^2*ln(c*x+I)^2+1/4*d^3*b^2*arctan(
c*x)^2*x^4-1/2*c^2*d^3*a^2*x^6-13/70/c^4*d^3*b^2*ln(c*x-I)^2-3/4/c^4*d^3*b^2*arctan(c*x)^2+13/35/c^4*d^3*b^2*d
ilog(1/2*I*(c*x-I))+26/35*I/c^2*d^3*b^2*arctan(c*x)*x^2-26/35*I/c^4*d^3*b^2*arctan(c*x)*ln(c^2*x^2+1)-26/35*I/
c^4*d^3*a*b*ln(c^2*x^2+1)+26/35*I/c^2*d^3*a*b*x^2-c^2*d^3*a*b*arctan(c*x)*x^6+1/21*I*c^2*d^3*a*b*x^6+1/21*I*c^
2*d^3*b^2*arctan(c*x)*x^6+3/5*I*c*d^3*b^2*arctan(c*x)^2*x^5-1/7*I*c^3*d^3*b^2*arctan(c*x)^2*x^7-1/2*c^2*d^3*b^
2*arctan(c*x)^2*x^6+13/35/c^4*d^3*b^2*ln(c*x-I)*ln(c^2*x^2+1)-13/35/c^4*d^3*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))-3
/2/c^4*d^3*a*b*arctan(c*x)+1/5*c*d^3*b^2*arctan(c*x)*x^5-1/2/c*d^3*b^2*arctan(c*x)*x^3-13/35*I*d^3*b^2*arctan(
c*x)*x^4-13/35/c^4*d^3*b^2*ln(c*x+I)*ln(c^2*x^2+1)+13/35/c^4*d^3*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))+1/2*d^3*a*b*a
rctan(c*x)*x^4-1/7*I*c^3*d^3*a^2*x^7+3/5*I*c*d^3*a^2*x^5+1/5*c*d^3*a*b*x^5-1/2/c*d^3*a*b*x^3-1/20*b^2*d^3*x^4+
1/4*d^3*a^2*x^4-122/105*I*b^2*d^3*x/c^3-1/105*I*b^2*c*d^3*x^5+3/2*a*b*d^3*x/c^3+3/2*b^2*d^3*x*arctan(c*x)/c^3+
44/315*I*b^2*d^3*x^3/c+122/105*I*b^2*d^3*arctan(c*x)/c^4-2/7*I*c^3*d^3*a*b*arctan(c*x)*x^7+6/5*I*c*d^3*a*b*arc
tan(c*x)*x^5-11/10*b^2*d^3*ln(c^2*x^2+1)/c^4+7/20*b^2*d^3*x^2/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*I*a^2*c^3*d^3*x^7 - 1/2*a^2*c^2*d^3*x^6 + 3/5*I*a^2*c*d^3*x^5 + 1/4*b^2*d^3*x^4*arctan(c*x)^2 - 1/42*I*(1
2*x^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*a*b*c^3*d^3 + 1/4*a^2*d^
3*x^4 - 1/15*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a*b*c^2*d^3 +
3/10*I*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*a*b*c*d^3 + 1/6*(3*x^4*arctan(
c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*d^3 - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c
^5)*arctan(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b^2*d^3 - 1/1120*(40*I*b^2*c^3*d^3*x^7
+ 140*b^2*c^2*d^3*x^6 - 168*I*b^2*c*d^3*x^5)*arctan(c*x)^2 + 1/1120*(40*b^2*c^3*d^3*x^7 - 140*I*b^2*c^2*d^3*x
^6 - 168*b^2*c*d^3*x^5)*arctan(c*x)*log(c^2*x^2 + 1) - 1/1120*(-10*I*b^2*c^3*d^3*x^7 - 35*b^2*c^2*d^3*x^6 + 42
*I*b^2*c*d^3*x^5)*log(c^2*x^2 + 1)^2 - I*integrate(1/560*(420*(b^2*c^5*d^3*x^8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d
^3*x^4)*arctan(c*x)^2 + 35*(b^2*c^5*d^3*x^8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d^3*x^4)*log(c^2*x^2 + 1)^2 - 12*(15
*b^2*c^4*d^3*x^7 - 14*b^2*c^2*d^3*x^5)*arctan(c*x) + 2*(10*b^2*c^5*d^3*x^8 - 77*b^2*c^3*d^3*x^6 - 210*(b^2*c^4
*d^3*x^7 + b^2*c^2*d^3*x^5)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate(1/560*(1260*(b^2*c^4*
d^3*x^7 + b^2*c^2*d^3*x^5)*arctan(c*x)^2 + 105*(b^2*c^4*d^3*x^7 + b^2*c^2*d^3*x^5)*log(c^2*x^2 + 1)^2 + 4*(10*
b^2*c^5*d^3*x^8 - 77*b^2*c^3*d^3*x^6)*arctan(c*x) + 2*(45*b^2*c^4*d^3*x^7 - 42*b^2*c^2*d^3*x^5 + 70*(b^2*c^5*d
^3*x^8 - 2*b^2*c^3*d^3*x^6 - 3*b^2*c*d^3*x^4)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{560} \,{\left (20 i \, b^{2} c^{3} d^{3} x^{7} + 70 \, b^{2} c^{2} d^{3} x^{6} - 84 i \, b^{2} c d^{3} x^{5} - 35 \, b^{2} d^{3} x^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{-140 i \, a^{2} c^{5} d^{3} x^{8} - 420 \, a^{2} c^{4} d^{3} x^{7} + 280 i \, a^{2} c^{3} d^{3} x^{6} - 280 \, a^{2} c^{2} d^{3} x^{5} + 420 i \, a^{2} c d^{3} x^{4} + 140 \, a^{2} d^{3} x^{3} +{\left (140 \, a b c^{5} d^{3} x^{8} +{\left (-420 i \, a b - 20 \, b^{2}\right )} c^{4} d^{3} x^{7} - 70 \,{\left (4 \, a b - i \, b^{2}\right )} c^{3} d^{3} x^{6} +{\left (-280 i \, a b + 84 \, b^{2}\right )} c^{2} d^{3} x^{5} - 35 \,{\left (12 \, a b + i \, b^{2}\right )} c d^{3} x^{4} + 140 i \, a b d^{3} x^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{140 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/560*(20*I*b^2*c^3*d^3*x^7 + 70*b^2*c^2*d^3*x^6 - 84*I*b^2*c*d^3*x^5 - 35*b^2*d^3*x^4)*log(-(c*x + I)/(c*x -
I))^2 + integral(1/140*(-140*I*a^2*c^5*d^3*x^8 - 420*a^2*c^4*d^3*x^7 + 280*I*a^2*c^3*d^3*x^6 - 280*a^2*c^2*d^3
*x^5 + 420*I*a^2*c*d^3*x^4 + 140*a^2*d^3*x^3 + (140*a*b*c^5*d^3*x^8 + (-420*I*a*b - 20*b^2)*c^4*d^3*x^7 - 70*(
4*a*b - I*b^2)*c^3*d^3*x^6 + (-280*I*a*b + 84*b^2)*c^2*d^3*x^5 - 35*(12*a*b + I*b^2)*c*d^3*x^4 + 140*I*a*b*d^3
*x^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), 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**3*(d+I*c*d*x)**3*(a+b*atan(c*x))**2,x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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