Integrand size = 23, antiderivative size = 205 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {3 b d^3 x}{4 c^3}+\frac {13 i b d^3 x^2}{35 c^2}-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} i b c^2 d^3 x^6-\frac {3 b d^3 \arctan (c x)}{4 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {13 i b d^3 \log \left (1+c^2 x^2\right )}{35 c^4} \] Output:
3/4*b*d^3*x/c^3+13/35*I*b*d^3*x^2/c^2-1/4*b*d^3*x^3/c-13/70*I*b*d^3*x^4+1/ 10*b*c*d^3*x^5+1/42*I*b*c^2*d^3*x^6-3/4*b*d^3*arctan(c*x)/c^4+1/4*d^3*x^4* (a+b*arctan(c*x))+3/5*I*c*d^3*x^5*(a+b*arctan(c*x))-1/2*c^2*d^3*x^6*(a+b*a rctan(c*x))-1/7*I*c^3*d^3*x^7*(a+b*arctan(c*x))-13/35*I*b*d^3*ln(c^2*x^2+1 )/c^4
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.75 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {d^3 \left (3 a c^4 x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )+b c x \left (315+156 i c x-105 c^2 x^2-78 i c^3 x^3+42 c^4 x^4+10 i c^5 x^5\right )+3 b \left (-105+35 c^4 x^4+84 i c^5 x^5-70 c^6 x^6-20 i c^7 x^7\right ) \arctan (c x)-156 i b \log \left (1+c^2 x^2\right )\right )}{420 c^4} \] Input:
Integrate[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]
Output:
(d^3*(3*a*c^4*x^4*(35 + (84*I)*c*x - 70*c^2*x^2 - (20*I)*c^3*x^3) + b*c*x* (315 + (156*I)*c*x - 105*c^2*x^2 - (78*I)*c^3*x^3 + 42*c^4*x^4 + (10*I)*c^ 5*x^5) + 3*b*(-105 + 35*c^4*x^4 + (84*I)*c^5*x^5 - 70*c^6*x^6 - (20*I)*c^7 *x^7)*ArcTan[c*x] - (156*I)*b*Log[1 + c^2*x^2]))/(420*c^4)
Time = 0.48 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5407, 27, 2333, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5407 |
| \(\displaystyle -b c \int \frac {d^3 x^4 \left (-20 i c^3 x^3-70 c^2 x^2+84 i c x+35\right )}{140 \left (c^2 x^2+1\right )}dx-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{140} b c d^3 \int \frac {x^4 \left (-20 i c^3 x^3-70 c^2 x^2+84 i c x+35\right )}{c^2 x^2+1}dx-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle -\frac {1}{140} b c d^3 \int \left (-20 i c x^5-70 x^4+\frac {104 i x^3}{c}+\frac {105 x^2}{c^2}-\frac {104 i x}{c^3}+\frac {104 i c x+105}{c^4 \left (c^2 x^2+1\right )}-\frac {105}{c^4}\right )dx-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))-\frac {1}{140} b c d^3 \left (\frac {105 \arctan (c x)}{c^5}-\frac {105 x}{c^4}-\frac {52 i x^2}{c^3}+\frac {35 x^3}{c^2}+\frac {52 i \log \left (c^2 x^2+1\right )}{c^5}-\frac {10}{3} i c x^6+\frac {26 i x^4}{c}-14 x^5\right )\) |
Input:
Int[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]
Output:
(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]) - (b*c*d^3*((-105*x)/c^4 - ((52*I)*x^2)/c^3 + (35*x^3)/c^2 + ((26*I)*x ^4)/c - 14*x^5 - ((10*I)/3)*c*x^6 + (105*ArcTan[c*x])/c^5 + ((52*I)*Log[1 + c^2*x^2])/c^5))/140
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[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]) )
Time = 0.76 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.79
| method | result | size |
| parts | \(d^{3} a \left (-\frac {1}{7} i c^{3} x^{7}-\frac {1}{2} c^{2} x^{6}+\frac {3}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{6} x^{6} \arctan \left (c x \right )}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(162\) |
| derivativedivides | \(\frac {d^{3} a \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{6} x^{6} \arctan \left (c x \right )}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(168\) |
| default | \(\frac {d^{3} a \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{6} x^{6} \arctan \left (c x \right )}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(168\) |
| parallelrisch | \(-\frac {-252 i c^{5} b \,d^{3} \arctan \left (c x \right ) x^{5}-156 i x^{2} b \,c^{2} d^{3}+60 i x^{7} a \,c^{7} d^{3}+210 b \,c^{6} d^{3} \arctan \left (c x \right ) x^{6}-10 i x^{6} b \,c^{6} d^{3}+210 a \,c^{6} d^{3} x^{6}+78 i x^{4} b \,c^{4} d^{3}-42 b \,c^{5} d^{3} x^{5}-252 i x^{5} a \,c^{5} d^{3}-105 d^{3} b \arctan \left (c x \right ) x^{4} c^{4}-105 a \,c^{4} d^{3} x^{4}+105 b \,c^{3} d^{3} x^{3}+60 i c^{7} b \,d^{3} \arctan \left (c x \right ) x^{7}+156 i b \,d^{3} \ln \left (c^{2} x^{2}+1\right )-315 b c \,d^{3} x +315 b \,d^{3} \arctan \left (c x \right )}{420 c^{4}}\) | \(221\) |
| risch | \(-\frac {d^{3} b \left (20 c^{3} x^{7}-70 i c^{2} x^{6}-84 c \,x^{5}+35 i x^{4}\right ) \ln \left (i c x +1\right )}{280}+\frac {d^{3} c^{3} b \,x^{7} \ln \left (-i c x +1\right )}{14}-\frac {13 i d^{3} b \ln \left (11025 c^{2} x^{2}+11025\right )}{35 c^{4}}-\frac {d^{3} c^{2} a \,x^{6}}{2}+\frac {i d^{3} x^{4} b \ln \left (-i c x +1\right )}{8}-\frac {3 d^{3} c b \,x^{5} \ln \left (-i c x +1\right )}{10}-\frac {i d^{3} c^{3} a \,x^{7}}{7}+\frac {b c \,d^{3} x^{5}}{10}+\frac {i b \,c^{2} d^{3} x^{6}}{42}+\frac {d^{3} a \,x^{4}}{4}-\frac {13 i b \,d^{3} x^{4}}{70}+\frac {13 i b \,d^{3} x^{2}}{35 c^{2}}-\frac {b \,d^{3} x^{3}}{4 c}+\frac {3 i d^{3} c \,x^{5} a}{5}+\frac {3 b \,d^{3} x}{4 c^{3}}-\frac {3 b \,d^{3} \arctan \left (c x \right )}{4 c^{4}}-\frac {i d^{3} c^{2} x^{6} b \ln \left (-i c x +1\right )}{4}\) | \(270\) |
Input:
int(x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
Output:
d^3*a*(-1/7*I*c^3*x^7-1/2*c^2*x^6+3/5*I*c*x^5+1/4*x^4)+d^3*b/c^4*(-1/7*I*a rctan(c*x)*c^7*x^7-1/2*c^6*x^6*arctan(c*x)+3/5*I*arctan(c*x)*c^5*x^5+1/4*c ^4*x^4*arctan(c*x)+3/4*c*x+1/42*I*c^6*x^6+1/10*c^5*x^5-13/70*I*c^4*x^4-1/4 *c^3*x^3+13/35*I*c^2*x^2-13/35*I*ln(c^2*x^2+1)-3/4*arctan(c*x))
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.99 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {-120 i \, a c^{7} d^{3} x^{7} - 20 \, {\left (21 \, a - i \, b\right )} c^{6} d^{3} x^{6} - 84 \, {\left (-6 i \, a - b\right )} c^{5} d^{3} x^{5} + 6 \, {\left (35 \, a - 26 i \, b\right )} c^{4} d^{3} x^{4} - 210 \, b c^{3} d^{3} x^{3} + 312 i \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 627 i \, b d^{3} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b d^{3} \log \left (\frac {c x - i}{c}\right ) + 3 \, {\left (20 \, b c^{7} d^{3} x^{7} - 70 i \, b c^{6} d^{3} x^{6} - 84 \, b c^{5} d^{3} x^{5} + 35 i \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{840 \, c^{4}} \] Input:
integrate(x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x, algorithm="fricas")
Output:
1/840*(-120*I*a*c^7*d^3*x^7 - 20*(21*a - I*b)*c^6*d^3*x^6 - 84*(-6*I*a - 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*l og((c*x - I)/c) + 3*(20*b*c^7*d^3*x^7 - 70*I*b*c^6*d^3*x^6 - 84*b*c^5*d^3* x^5 + 35*I*b*c^4*d^3*x^4)*log(-(c*x + I)/(c*x - I)))/c^4
Time = 2.90 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.60 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=- \frac {i a c^{3} d^{3} x^{7}}{7} - \frac {b d^{3} x^{3}}{4 c} + \frac {13 i b d^{3} x^{2}}{35 c^{2}} + \frac {3 b d^{3} x}{4 c^{3}} - \frac {b d^{3} \left (- \frac {i \log {\left (353 b c d^{3} x - 353 i b d^{3} \right )}}{280} + \frac {351 i \log {\left (353 b c d^{3} x + 353 i b d^{3} \right )}}{560}\right )}{c^{4}} - x^{6} \left (\frac {a c^{2} d^{3}}{2} - \frac {i b c^{2} d^{3}}{42}\right ) - x^{5} \left (- \frac {3 i a c d^{3}}{5} - \frac {b c d^{3}}{10}\right ) - x^{4} \left (- \frac {a d^{3}}{4} + \frac {13 i b d^{3}}{70}\right ) + \left (- \frac {b c^{3} d^{3} x^{7}}{14} + \frac {i b c^{2} d^{3} x^{6}}{4} + \frac {3 b c d^{3} x^{5}}{10} - \frac {i b d^{3} x^{4}}{8}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (40 b c^{7} d^{3} x^{7} - 140 i b c^{6} d^{3} x^{6} - 168 b c^{5} d^{3} x^{5} + 70 i b c^{4} d^{3} x^{4} - 67 i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{560 c^{4}} \] Input:
integrate(x**3*(d+I*c*d*x)**3*(a+b*atan(c*x)),x)
Output:
-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) - b*d**3*(-I*log(353*b*c*d**3*x - 353*I*b*d**3)/280 + 3 51*I*log(353*b*c*d**3*x + 353*I*b*d**3)/560)/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) + (40*b*c**7*d**3*x**7 - 14 0*I*b*c**6*d**3*x**6 - 168*b*c**5*d**3*x**5 + 70*I*b*c**4*d**3*x**4 - 67*I *b*d**3)*log(-I*c*x + 1)/(560*c**4)
Time = 0.11 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.27 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {1}{7} i \, a c^{3} d^{3} x^{7} - \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {3}{5} i \, a c d^{3} x^{5} - \frac {1}{84} i \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac {1}{4} \, a d^{3} x^{4} - \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac {3}{20} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} \] Input:
integrate(x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x, algorithm="maxima")
Output:
-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
Time = 0.14 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {120 i \, b c^{7} d^{3} x^{7} \arctan \left (c x\right ) + 120 i \, a c^{7} d^{3} x^{7} + 420 \, b c^{6} d^{3} x^{6} \arctan \left (c x\right ) + 420 \, a c^{6} d^{3} x^{6} - 20 i \, b c^{6} d^{3} x^{6} - 504 i \, b c^{5} d^{3} x^{5} \arctan \left (c x\right ) - 504 i \, a c^{5} d^{3} x^{5} - 84 \, b c^{5} d^{3} x^{5} - 210 \, b c^{4} d^{3} x^{4} \arctan \left (c x\right ) - 210 \, a c^{4} d^{3} x^{4} + 156 i \, b c^{4} d^{3} x^{4} + 210 \, b c^{3} d^{3} x^{3} - 312 i \, b c^{2} d^{3} x^{2} - 630 \, b c d^{3} x - 3 i \, b d^{3} \log \left (i \, c x + 1\right ) + 627 i \, b d^{3} \log \left (-i \, c x + 1\right )}{840 \, c^{4}} \] Input:
integrate(x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x, algorithm="giac")
Output:
-1/840*(120*I*b*c^7*d^3*x^7*arctan(c*x) + 120*I*a*c^7*d^3*x^7 + 420*b*c^6* d^3*x^6*arctan(c*x) + 420*a*c^6*d^3*x^6 - 20*I*b*c^6*d^3*x^6 - 504*I*b*c^5 *d^3*x^5*arctan(c*x) - 504*I*a*c^5*d^3*x^5 - 84*b*c^5*d^3*x^5 - 210*b*c^4* d^3*x^4*arctan(c*x) - 210*a*c^4*d^3*x^4 + 156*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 - 3*I*b*d^3*log(I*c*x + 1) + 627*I*b*d^3*log(-I*c*x + 1))/c^4
Time = 1.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.91 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {\frac {d^3\,\left (315\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,156{}\mathrm {i}\right )}{420}+\frac {b\,c^3\,d^3\,x^3}{4}-\frac {3\,b\,c\,d^3\,x}{4}-\frac {b\,c^2\,d^3\,x^2\,13{}\mathrm {i}}{35}}{c^4}+\frac {d^3\,\left (105\,a\,x^4+105\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,78{}\mathrm {i}\right )}{420}-\frac {c^3\,d^3\,\left (a\,x^7\,60{}\mathrm {i}+b\,x^7\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{420}+\frac {c\,d^3\,\left (a\,x^5\,252{}\mathrm {i}+42\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,252{}\mathrm {i}\right )}{420}-\frac {c^2\,d^3\,\left (210\,a\,x^6+210\,b\,x^6\,\mathrm {atan}\left (c\,x\right )-b\,x^6\,10{}\mathrm {i}\right )}{420} \] Input:
int(x^3*(a + b*atan(c*x))*(d + c*d*x*1i)^3,x)
Output:
(d^3*(105*a*x^4 - b*x^4*78i + 105*b*x^4*atan(c*x)))/420 - ((d^3*(315*b*ata n(c*x) + b*log(c^2*x^2 + 1)*156i))/420 - (b*c^2*d^3*x^2*13i)/35 + (b*c^3*d ^3*x^3)/4 - (3*b*c*d^3*x)/4)/c^4 - (c^3*d^3*(a*x^7*60i + b*x^7*atan(c*x)*6 0i))/420 + (c*d^3*(a*x^5*252i + 42*b*x^5 + b*x^5*atan(c*x)*252i))/420 - (c ^2*d^3*(210*a*x^6 - b*x^6*10i + 210*b*x^6*atan(c*x)))/420
Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {d^{3} \left (-60 \mathit {atan} \left (c x \right ) b \,c^{7} i \,x^{7}-210 \mathit {atan} \left (c x \right ) b \,c^{6} x^{6}+252 \mathit {atan} \left (c x \right ) b \,c^{5} i \,x^{5}+105 \mathit {atan} \left (c x \right ) b \,c^{4} x^{4}-315 \mathit {atan} \left (c x \right ) b -156 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b i -60 a \,c^{7} i \,x^{7}-210 a \,c^{6} x^{6}+252 a \,c^{5} i \,x^{5}+105 a \,c^{4} x^{4}+10 b \,c^{6} i \,x^{6}+42 b \,c^{5} x^{5}-78 b \,c^{4} i \,x^{4}-105 b \,c^{3} x^{3}+156 b \,c^{2} i \,x^{2}+315 b c x \right )}{420 c^{4}} \] Input:
int(x^3*(d+I*c*d*x)^3*(a+b*atan(c*x)),x)
Output:
(d**3*( - 60*atan(c*x)*b*c**7*i*x**7 - 210*atan(c*x)*b*c**6*x**6 + 252*ata n(c*x)*b*c**5*i*x**5 + 105*atan(c*x)*b*c**4*x**4 - 315*atan(c*x)*b - 156*l og(c**2*x**2 + 1)*b*i - 60*a*c**7*i*x**7 - 210*a*c**6*x**6 + 252*a*c**5*i* x**5 + 105*a*c**4*x**4 + 10*b*c**6*i*x**6 + 42*b*c**5*x**5 - 78*b*c**4*i*x **4 - 105*b*c**3*x**3 + 156*b*c**2*i*x**2 + 315*b*c*x))/(420*c**4)