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

Optimal. Leaf size=205 \[ -\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{42} i b c^2 d^3 x^6+\frac{13 i b d^3 x^2}{35 c^2}-\frac{13 i b d^3 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac{3 b d^3 x}{4 c^3}-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{10} b c d^3 x^5-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4 \]

[Out]

(3*b*d^3*x)/(4*c^3) + (((13*I)/35)*b*d^3*x^2)/c^2 - (b*d^3*x^3)/(4*c) - ((13*I)/70)*b*d^3*x^4 + (b*c*d^3*x^5)/
10 + (I/42)*b*c^2*d^3*x^6 - (3*b*d^3*ArcTan[c*x])/(4*c^4) + (d^3*x^4*(a + b*ArcTan[c*x]))/4 + ((3*I)/5)*c*d^3*
x^5*(a + b*ArcTan[c*x]) - (c^2*d^3*x^6*(a + b*ArcTan[c*x]))/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x]) - (((13*
I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4

________________________________________________________________________________________

Rubi [A]  time = 0.183864, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {43, 4872, 12, 1802, 635, 203, 260} \[ -\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{42} i b c^2 d^3 x^6+\frac{13 i b d^3 x^2}{35 c^2}-\frac{13 i b d^3 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac{3 b d^3 x}{4 c^3}-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{10} b c d^3 x^5-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d^3*x)/(4*c^3) + (((13*I)/35)*b*d^3*x^2)/c^2 - (b*d^3*x^3)/(4*c) - ((13*I)/70)*b*d^3*x^4 + (b*c*d^3*x^5)/
10 + (I/42)*b*c^2*d^3*x^6 - (3*b*d^3*ArcTan[c*x])/(4*c^4) + (d^3*x^4*(a + b*ArcTan[c*x]))/4 + ((3*I)/5)*c*d^3*
x^5*(a + b*ArcTan[c*x]) - (c^2*d^3*x^6*(a + b*ArcTan[c*x]))/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x]) - (((13*
I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4

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 4872

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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 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]

Rubi steps

\begin{align*} \int x^3 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )}{140 \left (1+c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{140} \left (b c d^3\right ) \int \frac{x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{140} \left (b c d^3\right ) \int \left (-\frac{105}{c^4}-\frac{104 i x}{c^3}+\frac{105 x^2}{c^2}+\frac{104 i x^3}{c}-70 x^4-20 i c x^5+\frac{105+104 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 i b d^3 x^2}{35 c^2}-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} i b c^2 d^3 x^6+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (b d^3\right ) \int \frac{105+104 i c x}{1+c^2 x^2} \, dx}{140 c^3}\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 i b d^3 x^2}{35 c^2}-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} i b c^2 d^3 x^6+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (3 b d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}-\frac{\left (26 i b d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{35 c^2}\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 i b d^3 x^2}{35 c^2}-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} i b c^2 d^3 x^6-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} i c d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} c^2 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{7} i c^3 d^3 x^7 \left (a+b \tan ^{-1}(c x)\right )-\frac{13 i b d^3 \log \left (1+c^2 x^2\right )}{35 c^4}\\ \end{align*}

Mathematica [A]  time = 0.105529, size = 248, normalized size = 1.21 \[ -\frac{1}{7} i a c^3 d^3 x^7-\frac{1}{2} a c^2 d^3 x^6+\frac{3}{5} i a c d^3 x^5+\frac{1}{4} a d^3 x^4+\frac{1}{42} i b c^2 d^3 x^6+\frac{13 i b d^3 x^2}{35 c^2}-\frac{13 i b d^3 \log \left (c^2 x^2+1\right )}{35 c^4}-\frac{1}{7} i b c^3 d^3 x^7 \tan ^{-1}(c x)-\frac{1}{2} b c^2 d^3 x^6 \tan ^{-1}(c x)+\frac{3 b d^3 x}{4 c^3}-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{10} b c d^3 x^5-\frac{b d^3 x^3}{4 c}+\frac{3}{5} i b c d^3 x^5 \tan ^{-1}(c x)+\frac{1}{4} b d^3 x^4 \tan ^{-1}(c x)-\frac{13}{70} i b d^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d^3*x)/(4*c^3) + (((13*I)/35)*b*d^3*x^2)/c^2 - (b*d^3*x^3)/(4*c) + (a*d^3*x^4)/4 - ((13*I)/70)*b*d^3*x^4
+ ((3*I)/5)*a*c*d^3*x^5 + (b*c*d^3*x^5)/10 - (a*c^2*d^3*x^6)/2 + (I/42)*b*c^2*d^3*x^6 - (I/7)*a*c^3*d^3*x^7 -
(3*b*d^3*ArcTan[c*x])/(4*c^4) + (b*d^3*x^4*ArcTan[c*x])/4 + ((3*I)/5)*b*c*d^3*x^5*ArcTan[c*x] - (b*c^2*d^3*x^6
*ArcTan[c*x])/2 - (I/7)*b*c^3*d^3*x^7*ArcTan[c*x] - (((13*I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 209, normalized size = 1. \begin{align*} -{\frac{i}{7}}{c}^{3}{d}^{3}b\arctan \left ( cx \right ){x}^{7}-{\frac{{c}^{2}{d}^{3}a{x}^{6}}{2}}-{\frac{{\frac{13\,i}{35}}b{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{4}}}+{\frac{{d}^{3}a{x}^{4}}{4}}-{\frac{13\,i}{70}}b{d}^{3}{x}^{4}-{\frac{{c}^{2}{d}^{3}b\arctan \left ( cx \right ){x}^{6}}{2}}+{\frac{3\,i}{5}}c{d}^{3}b\arctan \left ( cx \right ){x}^{5}+{\frac{{d}^{3}b\arctan \left ( cx \right ){x}^{4}}{4}}+{\frac{3\,{d}^{3}bx}{4\,{c}^{3}}}+{\frac{{\frac{13\,i}{35}}b{d}^{3}{x}^{2}}{{c}^{2}}}+{\frac{bc{d}^{3}{x}^{5}}{10}}+{\frac{i}{42}}b{c}^{2}{d}^{3}{x}^{6}-{\frac{{d}^{3}b{x}^{3}}{4\,c}}-{\frac{i}{7}}{c}^{3}{d}^{3}a{x}^{7}+{\frac{3\,i}{5}}c{d}^{3}a{x}^{5}-{\frac{3\,b{d}^{3}\arctan \left ( cx \right ) }{4\,{c}^{4}}} \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)),x)

[Out]

-1/7*I*c^3*d^3*b*arctan(c*x)*x^7-1/2*c^2*d^3*a*x^6-13/35*I*b*d^3*ln(c^2*x^2+1)/c^4+1/4*d^3*a*x^4-13/70*I*b*d^3
*x^4-1/2*c^2*d^3*b*arctan(c*x)*x^6+3/5*I*c*d^3*b*arctan(c*x)*x^5+1/4*d^3*b*arctan(c*x)*x^4+3/4*b*d^3*x/c^3+13/
35*I*b*d^3*x^2/c^2+1/10*b*c*d^3*x^5+1/42*I*b*c^2*d^3*x^6-1/4*b*d^3*x^3/c-1/7*I*c^3*d^3*a*x^7+3/5*I*c*d^3*a*x^5
-3/4*b*d^3*arctan(c*x)/c^4

________________________________________________________________________________________

Maxima [A]  time = 1.48081, size = 352, normalized size = 1.72 \begin{align*} -\frac{1}{7} i \, a c^{3} d^{3} x^{7} - \frac{1}{2} \, a c^{2} d^{3} x^{6} + \frac{3}{5} i \, a c d^{3} x^{5} - \frac{1}{84} i \,{\left (12 \, x^{7} \arctan \left (c x\right ) - c{\left (\frac{2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac{6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{30} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac{3}{20} i \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} \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)),x, algorithm="maxima")

[Out]

-1/7*I*a*c^3*d^3*x^7 - 1/2*a*c^2*d^3*x^6 + 3/5*I*a*c*d^3*x^5 - 1/84*I*(12*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))*b*c^3*d^3 + 1/4*a*d^3*x^4 - 1/30*(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))*b*c^2*d^3 + 3/20*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))*b*c*d^3 + 1/12*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)
/c^5))*b*d^3

________________________________________________________________________________________

Fricas [A]  time = 2.76209, size = 481, normalized size = 2.35 \begin{align*} \frac{-120 i \, a c^{7} d^{3} x^{7} - 20 \,{\left (21 \, a - i \, b\right )} c^{6} d^{3} x^{6} +{\left (504 i \, a + 84 \, b\right )} c^{5} d^{3} x^{5} + 6 \,{\left (35 \, a - 26 i \, b\right )} c^{4} d^{3} x^{4} - 210 \, b c^{3} d^{3} x^{3} + 312 i \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 627 i \, b d^{3} \log \left (\frac{c x + i}{c}\right ) + 3 i \, b d^{3} \log \left (\frac{c x - i}{c}\right ) +{\left (60 \, b c^{7} d^{3} x^{7} - 210 i \, b c^{6} d^{3} x^{6} - 252 \, b c^{5} d^{3} x^{5} + 105 i \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{840 \, c^{4}} \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)),x, algorithm="fricas")

[Out]

1/840*(-120*I*a*c^7*d^3*x^7 - 20*(21*a - I*b)*c^6*d^3*x^6 + (504*I*a + 84*b)*c^5*d^3*x^5 + 6*(35*a - 26*I*b)*c
^4*d^3*x^4 - 210*b*c^3*d^3*x^3 + 312*I*b*c^2*d^3*x^2 + 630*b*c*d^3*x - 627*I*b*d^3*log((c*x + I)/c) + 3*I*b*d^
3*log((c*x - I)/c) + (60*b*c^7*d^3*x^7 - 210*I*b*c^6*d^3*x^6 - 252*b*c^5*d^3*x^5 + 105*I*b*c^4*d^3*x^4)*log(-(
c*x + I)/(c*x - I)))/c^4

________________________________________________________________________________________

Sympy [A]  time = 3.45616, size = 292, normalized size = 1.42 \begin{align*} - \frac{i a c^{3} d^{3} x^{7}}{7} - \frac{b d^{3} x^{3}}{4 c} + \frac{13 i b d^{3} x^{2}}{35 c^{2}} + \frac{3 b d^{3} x}{4 c^{3}} + \frac{i b d^{3} \log{\left (x - \frac{i}{c} \right )}}{280 c^{4}} - \frac{209 i b d^{3} \log{\left (x + \frac{i}{c} \right )}}{280 c^{4}} - x^{6} \left (\frac{a c^{2} d^{3}}{2} - \frac{i b c^{2} d^{3}}{42}\right ) - x^{5} \left (- \frac{3 i a c d^{3}}{5} - \frac{b c d^{3}}{10}\right ) - x^{4} \left (- \frac{a d^{3}}{4} + \frac{13 i b d^{3}}{70}\right ) + \left (- \frac{b c^{3} d^{3} x^{7}}{14} + \frac{i b c^{2} d^{3} x^{6}}{4} + \frac{3 b c d^{3} x^{5}}{10} - \frac{i b d^{3} x^{4}}{8}\right ) \log{\left (i c x + 1 \right )} + \left (\frac{b c^{3} d^{3} x^{7}}{14} - \frac{i b c^{2} d^{3} x^{6}}{4} - \frac{3 b c d^{3} x^{5}}{10} + \frac{i b d^{3} x^{4}}{8}\right ) \log{\left (- i c x + 1 \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*atan(c*x)),x)

[Out]

-I*a*c**3*d**3*x**7/7 - b*d**3*x**3/(4*c) + 13*I*b*d**3*x**2/(35*c**2) + 3*b*d**3*x/(4*c**3) + I*b*d**3*log(x
- I/c)/(280*c**4) - 209*I*b*d**3*log(x + I/c)/(280*c**4) - x**6*(a*c**2*d**3/2 - I*b*c**2*d**3/42) - x**5*(-3*
I*a*c*d**3/5 - b*c*d**3/10) - x**4*(-a*d**3/4 + 13*I*b*d**3/70) + (-b*c**3*d**3*x**7/14 + I*b*c**2*d**3*x**6/4
+ 3*b*c*d**3*x**5/10 - I*b*d**3*x**4/8)*log(I*c*x + 1) + (b*c**3*d**3*x**7/14 - I*b*c**2*d**3*x**6/4 - 3*b*c*
d**3*x**5/10 + I*b*d**3*x**4/8)*log(-I*c*x + 1)

________________________________________________________________________________________

Giac [A]  time = 1.196, size = 301, normalized size = 1.47 \begin{align*} \frac{120 \, b c^{7} d^{3} x^{7} \arctan \left (c x\right ) + 120 \, a c^{7} d^{3} x^{7} - 420 \, b c^{6} d^{3} i x^{6} \arctan \left (c x\right ) - 420 \, a c^{6} d^{3} i x^{6} - 20 \, b c^{6} d^{3} x^{6} + 84 \, b c^{5} d^{3} i x^{5} - 504 \, b c^{5} d^{3} x^{5} \arctan \left (c x\right ) - 504 \, a c^{5} d^{3} x^{5} + 210 \, b c^{4} d^{3} i x^{4} \arctan \left (c x\right ) + 210 \, a c^{4} d^{3} i x^{4} + 156 \, b c^{4} d^{3} x^{4} - 210 \, b c^{3} d^{3} i x^{3} - 312 \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} i x - 3 \, b d^{3} \log \left (c i x + 1\right ) + 627 \, b d^{3} \log \left (-c i x + 1\right )}{840 \, c^{4} i} \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)),x, algorithm="giac")

[Out]

1/840*(120*b*c^7*d^3*x^7*arctan(c*x) + 120*a*c^7*d^3*x^7 - 420*b*c^6*d^3*i*x^6*arctan(c*x) - 420*a*c^6*d^3*i*x
^6 - 20*b*c^6*d^3*x^6 + 84*b*c^5*d^3*i*x^5 - 504*b*c^5*d^3*x^5*arctan(c*x) - 504*a*c^5*d^3*x^5 + 210*b*c^4*d^3
*i*x^4*arctan(c*x) + 210*a*c^4*d^3*i*x^4 + 156*b*c^4*d^3*x^4 - 210*b*c^3*d^3*i*x^3 - 312*b*c^2*d^3*x^2 + 630*b
*c*d^3*i*x - 3*b*d^3*log(c*i*x + 1) + 627*b*d^3*log(-c*i*x + 1))/(c^4*i)