Integrand size = 25, antiderivative size = 402 \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2-\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {28 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac {14 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \]
11/6*I*a*b*d^3*x/c^2+37/30*b^2*d^3*x/c^2-11/18*I*b*d^3*x^3*(a+b*arctan(c*x ))-1/10*b^2*d^3*x^3+61/180*I*b^2*d^3*x^2/c-37/30*b^2*d^3*arctan(c*x)/c^3-1 13/90*I*b^2*d^3*ln(c^2*x^2+1)/c^3-14/15*b*d^3*x^2*(a+b*arctan(c*x))/c-1/60 *I*b^2*c*d^3*x^4+3/10*b*c*d^3*x^4*(a+b*arctan(c*x))-14/15*I*b^2*d^3*polylo g(2,1-2/(1+I*c*x))/c^3-37/20*I*d^3*(a+b*arctan(c*x))^2/c^3+1/3*d^3*x^3*(a+ b*arctan(c*x))^2+1/15*I*b*c^2*d^3*x^5*(a+b*arctan(c*x))-3/5*c^2*d^3*x^5*(a +b*arctan(c*x))^2+11/6*I*b^2*d^3*x*arctan(c*x)/c^2-28/15*b*d^3*(a+b*arctan (c*x))*ln(2/(1+I*c*x))/c^3-1/6*I*c^3*d^3*x^6*(a+b*arctan(c*x))^2+3/4*I*c*d ^3*x^4*(a+b*arctan(c*x))^2
Time = 1.13 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.92 \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (-162 a b+64 i b^2+330 i a b c x+222 b^2 c x-168 a b c^2 x^2+61 i b^2 c^2 x^2+60 a^2 c^3 x^3-110 i a b c^3 x^3-18 b^2 c^3 x^3+135 i a^2 c^4 x^4+54 a b c^4 x^4-3 i b^2 c^4 x^4-108 a^2 c^5 x^5+12 i a b c^5 x^5-30 i a^2 c^6 x^6+3 b^2 (-i+c x)^4 \left (i+4 c x-10 i c^2 x^2\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (-111+165 i c x-84 c^2 x^2-55 i c^3 x^3+27 c^4 x^4+6 i c^5 x^5\right )+3 a \left (-55 i+20 c^3 x^3+45 i c^4 x^4-36 c^5 x^5-10 i c^6 x^6\right )-168 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+168 a b \log \left (1+c^2 x^2\right )-226 i b^2 \log \left (1+c^2 x^2\right )+168 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{180 c^3} \]
(d^3*(-162*a*b + (64*I)*b^2 + (330*I)*a*b*c*x + 222*b^2*c*x - 168*a*b*c^2* x^2 + (61*I)*b^2*c^2*x^2 + 60*a^2*c^3*x^3 - (110*I)*a*b*c^3*x^3 - 18*b^2*c ^3*x^3 + (135*I)*a^2*c^4*x^4 + 54*a*b*c^4*x^4 - (3*I)*b^2*c^4*x^4 - 108*a^ 2*c^5*x^5 + (12*I)*a*b*c^5*x^5 - (30*I)*a^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*( I + 4*c*x - (10*I)*c^2*x^2)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(-111 + (16 5*I)*c*x - 84*c^2*x^2 - (55*I)*c^3*x^3 + 27*c^4*x^4 + (6*I)*c^5*x^5) + 3*a *(-55*I + 20*c^3*x^3 + (45*I)*c^4*x^4 - 36*c^5*x^5 - (10*I)*c^6*x^6) - 168 *b*Log[1 + E^((2*I)*ArcTan[c*x])]) + 168*a*b*Log[1 + c^2*x^2] - (226*I)*b^ 2*Log[1 + c^2*x^2] + (168*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(180 *c^3)
Time = 1.32 (sec) , antiderivative size = 402, 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.
| \(\displaystyle \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-i c^3 d^3 x^5 (a+b \arctan (c x))^2-3 c^2 d^3 x^4 (a+b \arctan (c x))^2+3 i c d^3 x^3 (a+b \arctan (c x))^2+d^3 x^2 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} i c^3 d^3 x^6 (a+b \arctan (c x))^2-\frac {37 i d^3 (a+b \arctan (c x))^2}{20 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{15 c^3}-\frac {3}{5} c^2 d^3 x^5 (a+b \arctan (c x))^2+\frac {1}{15} i b c^2 d^3 x^5 (a+b \arctan (c x))+\frac {3}{4} i c d^3 x^4 (a+b \arctan (c x))^2+\frac {3}{10} b c d^3 x^4 (a+b \arctan (c x))+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))^2-\frac {11}{18} i b d^3 x^3 (a+b \arctan (c x))-\frac {14 b d^3 x^2 (a+b \arctan (c x))}{15 c}+\frac {11 i a b d^3 x}{6 c^2}-\frac {37 b^2 d^3 \arctan (c x)}{30 c^3}+\frac {11 i b^2 d^3 x \arctan (c x)}{6 c^2}-\frac {14 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{15 c^3}+\frac {37 b^2 d^3 x}{30 c^2}-\frac {113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}-\frac {1}{60} i b^2 c d^3 x^4+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3\) |
(((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d ^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c *x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3* b*c*d^3*x^4*(a + b*ArcTan[c*x]))/10 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c *x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*ArcTa n[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5* (a + b*ArcTan[c*x])^2)/5 - (I/6)*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b *d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90)*b^2* d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c *x)])/c^3
3.1.85.3.1 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 = 2.84 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(d^{3} a^{2} \left (-\frac {1}{6} i c^{3} x^{6}-\frac {3}{5} c^{2} x^{5}+\frac {3}{4} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )}{c^{3}}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\) | \(488\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (-\frac {1}{6} i c^{6} x^{6}-\frac {3}{5} c^{5} x^{5}+\frac {3}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\) | \(491\) |
| default | \(\frac {d^{3} a^{2} \left (-\frac {1}{6} i c^{6} x^{6}-\frac {3}{5} c^{5} x^{5}+\frac {3}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{3} \left (-\frac {113 i \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {11 i \arctan \left (c x \right ) c^{3} x^{3}}{18}+\frac {11 i \arctan \left (c x \right ) c x}{6}+\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {61 i c^{2} x^{2}}{180}-\frac {14 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {14 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {11 i \arctan \left (c x \right )^{2}}{12}+\frac {3 i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {37 c x}{30}-\frac {i c^{4} x^{4}}{60}-\frac {c^{3} x^{3}}{10}-\frac {i \arctan \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {37 \arctan \left (c x \right )}{30}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {3 c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {3 i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {11 i c x}{12}+\frac {i c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}-\frac {11 i c^{3} x^{3}}{36}-\frac {7 c^{2} x^{2}}{15}+\frac {7 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 i \arctan \left (c x \right )}{12}\right )}{c^{3}}\) | \(491\) |
| risch | \(-\frac {b^{2} d^{3} x^{3}}{10}+\frac {37 b^{2} d^{3} x}{30 c^{2}}-\frac {37 b^{2} d^{3} \arctan \left (c x \right )}{30 c^{3}}+\frac {3 a b c \,d^{3} x^{4}}{10}-\frac {3 a^{2} c^{2} d^{3} x^{5}}{5}-\frac {337 a b \,d^{3}}{90 c^{3}}+\frac {a^{2} d^{3} x^{3}}{3}+\frac {14 a b \,d^{3} \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {14 d^{3} a b \,x^{2}}{15 c}-\frac {11 i a b \,d^{3} \arctan \left (c x \right )}{6 c^{3}}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{6}}{6}-\frac {7 i b^{2} d^{3} \ln \left (-i c x +1\right ) x^{2}}{15 c}+\frac {3 i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{4}}{20}+\frac {i b \,d^{3} c^{2} x^{5} a}{15}+\frac {i b^{2} d^{3} \left (10 c^{6} x^{6}-36 i c^{5} x^{5}-45 c^{4} x^{4}+20 i c^{3} x^{3}-1\right ) \ln \left (i c x +1\right )^{2}}{240 c^{3}}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {i d^{3} a b \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{6}}{24}+\frac {14 i b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{15 c^{3}}-\frac {14 i 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 )}{15 c^{3}}+\frac {11 i a b \,d^{3} x}{6 c^{2}}+\left (-\frac {i b^{2} d^{3} \left (10 c^{3} x^{6}-36 i c^{2} x^{5}-45 c \,x^{4}+20 i x^{3}\right ) \ln \left (-i c x +1\right )}{120}-\frac {b \,d^{3} \left (60 a \,c^{6} x^{6}-216 i a \,c^{5} x^{5}-12 b \,c^{5} x^{5}+54 i b \,c^{4} x^{4}-270 a \,c^{4} x^{4}+120 i a \,c^{3} x^{3}+110 b \,c^{3} x^{3}-168 i b \,c^{2} x^{2}+333 i b \ln \left (-i c x +1\right )-330 x b c \right )}{360 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {3 i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{5}}{5}-\frac {i b^{2} c \,d^{3} x^{4}}{60}-\frac {113 i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{90 c^{3}}+\frac {61 i b^{2} d^{3} x^{2}}{180 c}-\frac {11 i b \,d^{3} x^{3} a}{18}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{5}}{30}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {37 i d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{80 c^{3}}-\frac {11 d^{3} b^{2} \ln \left (-i c x +1\right ) x}{12 c^{2}}-\frac {14 i b^{2} d^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{15 c^{3}}-\frac {i a^{2} c^{3} d^{3} x^{6}}{6}+\frac {3 i d^{3} c \,x^{4} a^{2}}{4}+\frac {76 i b^{2} d^{3}}{45 c^{3}}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+\frac {11 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{3}}{36}-\frac {37 i d^{3} a^{2}}{20 c^{3}}\) | \(868\) |
d^3*a^2*(-1/6*I*c^3*x^6-3/5*c^2*x^5+3/4*I*c*x^4+1/3*x^3)+b^2*d^3/c^3*(-113 /90*I*ln(c^2*x^2+1)-3/5*arctan(c*x)^2*c^5*x^5+1/15*I*arctan(c*x)*c^5*x^5+1 /3*c^3*x^3*arctan(c*x)^2-11/18*I*arctan(c*x)*c^3*x^3+11/6*I*arctan(c*x)*c* x+3/10*c^4*x^4*arctan(c*x)+61/180*I*c^2*x^2-14/15*c^2*x^2*arctan(c*x)+14/1 5*arctan(c*x)*ln(c^2*x^2+1)+7/15*I*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2*I*( c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)-7/15*I*(ln(c*x+I)*ln (c^2*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I) ^2)-11/12*I*arctan(c*x)^2+3/4*I*arctan(c*x)^2*c^4*x^4+37/30*c*x-1/60*I*c^4 *x^4-1/10*c^3*x^3-1/6*I*arctan(c*x)^2*c^6*x^6-37/30*arctan(c*x))+2*a*d^3*b /c^3*(-1/6*I*arctan(c*x)*c^6*x^6-3/5*c^5*x^5*arctan(c*x)+3/4*I*arctan(c*x) *c^4*x^4+1/3*c^3*x^3*arctan(c*x)+11/12*I*c*x+1/30*I*c^5*x^5+3/20*c^4*x^4-1 1/36*I*c^3*x^3-7/15*c^2*x^2+7/15*ln(c^2*x^2+1)-11/12*I*arctan(c*x))
\[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/240*(10*I*b^2*c^3*d^3*x^6 + 36*b^2*c^2*d^3*x^5 - 45*I*b^2*c*d^3*x^4 - 20 *b^2*d^3*x^3)*log(-(c*x + I)/(c*x - I))^2 + integral(1/60*(-60*I*a^2*c^5*d ^3*x^7 - 180*a^2*c^4*d^3*x^6 + 120*I*a^2*c^3*d^3*x^5 - 120*a^2*c^2*d^3*x^4 + 180*I*a^2*c*d^3*x^3 + 60*a^2*d^3*x^2 + (60*a*b*c^5*d^3*x^7 - 10*(18*I*a *b + b^2)*c^4*d^3*x^6 - 12*(10*a*b - 3*I*b^2)*c^3*d^3*x^5 - 15*(8*I*a*b - 3*b^2)*c^2*d^3*x^4 - 20*(9*a*b + I*b^2)*c*d^3*x^3 + 60*I*a*b*d^3*x^2)*log( -(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
-1/6*I*a^2*c^3*d^3*x^6 - 3/5*a^2*c^2*d^3*x^5 + 3/4*I*a^2*c*d^3*x^4 - 1/45* I*(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^3*d^3 - 3/10*(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^2*d^3 + 1/3*a^2*d^3*x^3 + 1/2*I*(3*x^4* arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*c*d^3 + 1/3 *(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d^3 + 1/240* (-10*I*b^2*c^3*d^3*x^6 - 36*b^2*c^2*d^3*x^5 + 45*I*b^2*c*d^3*x^4 + 20*b^2* d^3*x^3)*arctan(c*x)^2 + 1/240*(10*b^2*c^3*d^3*x^6 - 36*I*b^2*c^2*d^3*x^5 - 45*b^2*c*d^3*x^4 + 20*I*b^2*d^3*x^3)*arctan(c*x)*log(c^2*x^2 + 1) - 1/96 0*(-10*I*b^2*c^3*d^3*x^6 - 36*b^2*c^2*d^3*x^5 + 45*I*b^2*c*d^3*x^4 + 20*b^ 2*d^3*x^3)*log(c^2*x^2 + 1)^2 - I*integrate(1/240*(180*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*arctan(c*x)^2 + 15*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*log(c^2*x^2 + 1)^2 - 2*(46*b^2*c^4*d ^3*x^6 - 65*b^2*c^2*d^3*x^4)*arctan(c*x) + (10*b^2*c^5*d^3*x^7 - 81*b^2*c^ 3*d^3*x^5 + 20*b^2*c*d^3*x^3 - 60*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate (1/240*(180*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^2)*arctan(c *x)^2 + 15*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^2)*log(c^2*x ^2 + 1)^2 + 2*(10*b^2*c^5*d^3*x^7 - 81*b^2*c^3*d^3*x^5 + 20*b^2*c*d^3*x^3) *arctan(c*x) + (46*b^2*c^4*d^3*x^6 - 65*b^2*c^2*d^3*x^4 + 60*(b^2*c^5*d...
\[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]