Integrand size = 25, antiderivative size = 438 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {3 a b d^3 x}{2 c^3}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4-\frac {1}{105} i b^2 c d^3 x^5+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))-\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2+\frac {52 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{35 c^4}-\frac {11 b^2 d^3 \log \left (1+c^2 x^2\right )}{10 c^4}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{35 c^4} \]
3/2*a*b*d^3*x/c^3-1/105*I*b^2*c*d^3*x^5+7/20*b^2*d^3*x^2/c^2+1/21*I*b*c^2* d^3*x^6*(a+b*arctan(c*x))-1/20*b^2*d^3*x^4+44/315*I*b^2*d^3*x^3/c-13/35*I* b*d^3*x^4*(a+b*arctan(c*x))+3/2*b^2*d^3*x*arctan(c*x)/c^3-122/105*I*b^2*d^ 3*x/c^3-1/2*b*d^3*x^3*(a+b*arctan(c*x))/c+52/35*I*b*d^3*(a+b*arctan(c*x))* ln(2/(1+I*c*x))/c^4+1/5*b*c*d^3*x^5*(a+b*arctan(c*x))+3/5*I*c*d^3*x^5*(a+b *arctan(c*x))^2-209/140*d^3*(a+b*arctan(c*x))^2/c^4+1/4*d^3*x^4*(a+b*arcta n(c*x))^2+122/105*I*b^2*d^3*arctan(c*x)/c^4-1/2*c^2*d^3*x^6*(a+b*arctan(c* x))^2+26/35*I*b*d^3*x^2*(a+b*arctan(c*x))/c^2-1/7*I*c^3*d^3*x^7*(a+b*arcta n(c*x))^2-11/10*b^2*d^3*ln(c^2*x^2+1)/c^4-26/35*b^2*d^3*polylog(2,1-2/(1+I *c*x))/c^4
Time = 1.51 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.93 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (1464 i a b+504 b^2+1890 a b c x-1464 i b^2 c x+936 i a b c^2 x^2+441 b^2 c^2 x^2-630 a b c^3 x^3+176 i b^2 c^3 x^3+315 a^2 c^4 x^4-468 i a b c^4 x^4-63 b^2 c^4 x^4+756 i a^2 c^5 x^5+252 a b c^5 x^5-12 i b^2 c^5 x^5-630 a^2 c^6 x^6+60 i a b c^6 x^6-180 i a^2 c^7 x^7+9 b^2 (-i+c x)^4 \left (-1+4 i c x+10 c^2 x^2-20 i c^3 x^3\right ) \arctan (c x)^2+6 b \arctan (c x) \left (b \left (244 i+315 c x+156 i c^2 x^2-105 c^3 x^3-78 i c^4 x^4+42 c^5 x^5+10 i c^6 x^6\right )+3 a \left (-105+35 c^4 x^4+84 i c^5 x^5-70 c^6 x^6-20 i c^7 x^7\right )+312 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-936 i a b \log \left (1+c^2 x^2\right )-1386 b^2 \log \left (1+c^2 x^2\right )+936 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{1260 c^4} \]
(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)
Time = 1.44 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5411, 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.
| \(\displaystyle \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-i c^3 d^3 x^6 (a+b \arctan (c x))^2-3 c^2 d^3 x^5 (a+b \arctan (c x))^2+3 i c d^3 x^4 (a+b \arctan (c x))^2+d^3 x^3 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {209 d^3 (a+b \arctan (c x))^2}{140 c^4}+\frac {52 i b d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{35 c^4}-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))^2+\frac {1}{21} i b c^2 d^3 x^6 (a+b \arctan (c x))+\frac {26 i b d^3 x^2 (a+b \arctan (c x))}{35 c^2}+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))^2+\frac {1}{5} b c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))^2-\frac {13}{35} i b d^3 x^4 (a+b \arctan (c x))-\frac {b d^3 x^3 (a+b \arctan (c x))}{2 c}+\frac {3 a b d^3 x}{2 c^3}+\frac {122 i b^2 d^3 \arctan (c x)}{105 c^4}+\frac {3 b^2 d^3 x \arctan (c x)}{2 c^3}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{35 c^4}-\frac {122 i b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}-\frac {11 b^2 d^3 \log \left (c^2 x^2+1\right )}{10 c^4}-\frac {1}{105} i b^2 c d^3 x^5+\frac {44 i b^2 d^3 x^3}{315 c}-\frac {1}{20} b^2 d^3 x^4\) |
(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)/(2 0*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*ArcTan[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 + (((52*I)/3 5)*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)
3.1.84.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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])
Time = 2.93 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.17
| method | result | size |
| parts | \(d^{3} a^{2} \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 {b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )}{c^{4}}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{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}}\) | \(511\) |
| derivativedivides | \(\frac {d^{3} a^{2} \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 )+b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{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}}\) | \(514\) |
| default | \(\frac {d^{3} a^{2} \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 )+b^{2} d^{3} \left (\frac {3 c x \arctan \left (c x \right )}{2}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {11 \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {7 c^{2} x^{2}}{20}-\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {c^{4} x^{4}}{20}-\frac {122 i c x}{105}-\frac {13 \ln \left (c x -i\right )^{2}}{70}-\frac {13 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{35}+\frac {13 \ln \left (c x +i\right )^{2}}{70}+\frac {13 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{35}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {13 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {13 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{35}-\frac {13 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{35}+\frac {13 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{35}-\frac {\arctan \left (c x \right )^{2} c^{6} x^{6}}{2}-\frac {i c^{5} x^{5}}{105}-\frac {13 i \arctan \left (c x \right ) c^{4} x^{4}}{35}+\frac {122 i \arctan \left (c x \right )}{105}+\frac {44 i c^{3} x^{3}}{315}+\frac {26 i \arctan \left (c x \right ) c^{2} x^{2}}{35}+\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{21}+\frac {3 i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {26 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {i \arctan \left (c x \right )^{2} c^{7} x^{7}}{7}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{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}}\) | \(514\) |
| risch | \(-\frac {b^{2} d^{3} x^{4}}{20}+\frac {7 b^{2} d^{3} x^{2}}{20 c^{2}}-\frac {11 b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{10 c^{4}}+\frac {3 a b \,d^{3} x}{2 c^{3}}+\frac {77 b^{2} d^{3}}{45 c^{4}}+\frac {d^{3} x^{4} a^{2}}{4}-\frac {3 d^{3} b a \arctan \left (c x \right )}{2 c^{4}}-\frac {d^{3} a b \,x^{3}}{2 c}-\frac {d^{3} c^{2} a^{2} x^{6}}{2}-\frac {209 d^{3} a^{2}}{140 c^{4}}+\frac {d^{3} c b a \,x^{5}}{5}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{7}}{7}+\frac {3 i b^{2} d^{3} \ln \left (-i c x +1\right ) x}{4 c^{3}}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{7}}{28}+\frac {i b^{2} d^{3} \left (20 c^{7} x^{7}-70 i c^{6} x^{6}-84 c^{5} x^{5}+35 i c^{4} x^{4}-i\right ) \ln \left (i c x +1\right )^{2}}{560 c^{4}}+\frac {i d^{3} a b \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {26 i d^{3} b a \ln \left (c^{2} x^{2}+1\right )}{35 c^{4}}-\frac {i b^{2} d^{3} \ln \left (-i c x +1\right ) x^{3}}{4 c}+\frac {i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{5}}{10}+\frac {26 i b \,d^{3} x^{2} a}{35 c^{2}}+\frac {i b \,d^{3} c^{2} x^{6} a}{21}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{5}}{5}-\frac {13 i b \,d^{3} x^{4} a}{35}+\frac {353 i b \,d^{3} a}{105 c^{4}}+\frac {26 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{35 c^{4}}-\frac {26 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{35 c^{4}}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{6}}{42}-\frac {13 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{2}}{35 c^{2}}+\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{6}}{8}-\frac {122 i b^{2} d^{3} x}{105 c^{3}}-\frac {i b^{2} c \,d^{3} x^{5}}{105}+\frac {122 i b^{2} d^{3} \arctan \left (c x \right )}{105 c^{4}}+\frac {44 i b^{2} d^{3} x^{3}}{315 c}+\frac {3 i d^{3} c \,x^{5} a^{2}}{5}-\frac {i d^{3} c^{3} a^{2} x^{7}}{7}+\frac {13 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{4}}{70}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {209 d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{560 c^{4}}-\frac {26 b^{2} d^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{35 c^{4}}+\left (-\frac {i b^{2} d^{3} \left (20 c^{3} x^{7}-70 i c^{2} x^{6}-84 x^{5} c +35 i x^{4}\right ) \ln \left (-i c x +1\right )}{280}+\frac {b \,d^{3} \left (-120 a \,c^{7} x^{7}+420 i a \,c^{6} x^{6}+20 b \,c^{6} x^{6}-84 i b \,c^{5} x^{5}+504 a \,c^{5} x^{5}-210 i a \,c^{4} x^{4}-156 b \,c^{4} x^{4}+210 i b \,c^{3} x^{3}+312 b \,c^{2} x^{2}-630 i b c x -627 b \ln \left (-i c x +1\right )\right )}{840 c^{4}}\right ) \ln \left (i c x +1\right )-\frac {i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{6}}{2}\) | \(924\) |
d^3*a^2*(-1/7*I*c^3*x^7-1/2*c^2*x^6+3/5*I*c*x^5+1/4*x^4)+b^2*d^3/c^4*(3/2* c*x*arctan(c*x)-1/2*c^3*x^3*arctan(c*x)+1/5*c^5*x^5*arctan(c*x)-11/10*ln(c ^2*x^2+1)+7/20*c^2*x^2-3/4*arctan(c*x)^2-1/20*c^4*x^4-13/35*dilog(-1/2*I*( c*x+I))+1/4*c^4*x^4*arctan(c*x)^2+13/35*dilog(1/2*I*(c*x-I))-13/70*ln(c*x- I)^2+13/70*ln(c*x+I)^2+13/35*ln(c*x-I)*ln(c^2*x^2+1)-13/35*ln(c*x-I)*ln(-1 /2*I*(c*x+I))-13/35*ln(c*x+I)*ln(c^2*x^2+1)+13/35*ln(c*x+I)*ln(1/2*I*(c*x- I))-1/2*arctan(c*x)^2*c^6*x^6-13/35*I*arctan(c*x)*c^4*x^4+26/35*I*arctan(c *x)*c^2*x^2+1/21*I*arctan(c*x)*c^6*x^6-1/7*I*arctan(c*x)^2*c^7*x^7+3/5*I*a rctan(c*x)^2*c^5*x^5+122/105*I*arctan(c*x)-122/105*I*c*x-1/105*I*c^5*x^5+4 4/315*I*c^3*x^3-26/35*I*arctan(c*x)*ln(c^2*x^2+1))+2*a*d^3*b/c^4*(-1/7*I*a rctan(c*x)*c^7*x^7-1/2*arctan(c*x)*c^6*x^6+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))
\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
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 - 20*(21 *I*a*b + b^2)*c^4*d^3*x^7 - 70*(4*a*b - I*b^2)*c^3*d^3*x^6 - 28*(10*I*a*b - 3*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)
Timed out. \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
-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*(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))*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 - 1 5*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 + 1/280*(-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)*arctan(c*x)^2 + 1/280*(10*b^2*c^3*d^ 3*x^7 - 35*I*b^2*c^2*d^3*x^6 - 42*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) + ...
\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
Timed out. \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]