Integrand size = 25, antiderivative size = 333 \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\frac {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))-\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2-\frac {16 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {8 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3} \]
1/6*I*b^2*d^2*x^2/c+19/30*b^2*d^2*x/c^2-1/3*I*b*d^2*x^3*(a+b*arctan(c*x))- 1/30*b^2*d^2*x^3-19/30*b^2*d^2*arctan(c*x)/c^3+1/2*I*c*d^2*x^4*(a+b*arctan (c*x))^2-8/15*b*d^2*x^2*(a+b*arctan(c*x))/c-31/30*I*d^2*(a+b*arctan(c*x))^ 2/c^3+1/10*b*c*d^2*x^4*(a+b*arctan(c*x))+I*b^2*d^2*x*arctan(c*x)/c^2+1/3*d ^2*x^3*(a+b*arctan(c*x))^2-2/3*I*b^2*d^2*ln(c^2*x^2+1)/c^3-1/5*c^2*d^2*x^5 *(a+b*arctan(c*x))^2-16/15*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3-8/1 5*I*b^2*d^2*polylog(2,1-2/(1+I*c*x))/c^3+I*a*b*d^2*x/c^2
Time = 0.97 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.92 \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {d^2 \left (9 a b-5 i b^2-30 i a b c x-19 b^2 c x+16 a b c^2 x^2-5 i b^2 c^2 x^2-10 a^2 c^3 x^3+10 i a b c^3 x^3+b^2 c^3 x^3-15 i a^2 c^4 x^4-3 a b c^4 x^4+6 a^2 c^5 x^5+b^2 (-i+c x)^3 \left (-1+3 i c x+6 c^2 x^2\right ) \arctan (c x)^2+b \arctan (c x) \left (b \left (19-30 i c x+16 c^2 x^2+10 i c^3 x^3-3 c^4 x^4\right )+2 a \left (15 i-10 c^3 x^3-15 i c^4 x^4+6 c^5 x^5\right )+32 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-16 a b \log \left (1+c^2 x^2\right )+20 i b^2 \log \left (1+c^2 x^2\right )-16 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{30 c^3} \]
-1/30*(d^2*(9*a*b - (5*I)*b^2 - (30*I)*a*b*c*x - 19*b^2*c*x + 16*a*b*c^2*x ^2 - (5*I)*b^2*c^2*x^2 - 10*a^2*c^3*x^3 + (10*I)*a*b*c^3*x^3 + b^2*c^3*x^3 - (15*I)*a^2*c^4*x^4 - 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + b^2*(-I + c*x)^3*( -1 + (3*I)*c*x + 6*c^2*x^2)*ArcTan[c*x]^2 + b*ArcTan[c*x]*(b*(19 - (30*I)* c*x + 16*c^2*x^2 + (10*I)*c^3*x^3 - 3*c^4*x^4) + 2*a*(15*I - 10*c^3*x^3 - (15*I)*c^4*x^4 + 6*c^5*x^5) + 32*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - 16*a* b*Log[1 + c^2*x^2] + (20*I)*b^2*Log[1 + c^2*x^2] - (16*I)*b^2*PolyLog[2, - E^((2*I)*ArcTan[c*x])]))/c^3
Time = 0.97 (sec) , antiderivative size = 333, 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.
| \(\displaystyle \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-c^2 d^2 x^4 (a+b \arctan (c x))^2+2 i c d^2 x^3 (a+b \arctan (c x))^2+d^2 x^2 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {31 i d^2 (a+b \arctan (c x))^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 (a+b \arctan (c x))^2+\frac {1}{2} i c d^2 x^4 (a+b \arctan (c x))^2+\frac {1}{10} b c d^2 x^4 (a+b \arctan (c x))+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{3} i b d^2 x^3 (a+b \arctan (c x))-\frac {8 b d^2 x^2 (a+b \arctan (c x))}{15 c}+\frac {i a b d^2 x}{c^2}-\frac {19 b^2 d^2 \arctan (c x)}{30 c^3}+\frac {i b^2 d^2 x \arctan (c x)}{c^2}-\frac {8 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{15 c^3}+\frac {19 b^2 d^2 x}{30 c^2}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3\) |
(I*a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + ((I/6)*b^2*d^2*x^2)/c - (b^2 *d^2*x^3)/30 - (19*b^2*d^2*ArcTan[c*x])/(30*c^3) + (I*b^2*d^2*x*ArcTan[c*x ])/c^2 - (8*b*d^2*x^2*(a + b*ArcTan[c*x]))/(15*c) - (I/3)*b*d^2*x^3*(a + b *ArcTan[c*x]) + (b*c*d^2*x^4*(a + b*ArcTan[c*x]))/10 - (((31*I)/30)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/2)*c*d^2 *x^4*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^5*(a + b*ArcTan[c*x])^2)/5 - (16*b *d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((2*I)/3)*b^2*d^2 *Log[1 + c^2*x^2])/c^3 - (((8*I)/15)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] )/c^3
3.1.77.3.1 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 = 2.52 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.26
| method | result | size |
| parts | \(a^{2} d^{2} \left (-\frac {1}{5} c^{2} x^{5}+\frac {1}{2} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )}{c^{3}}+\frac {2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\) | \(420\) |
| derivativedivides | \(\frac {a^{2} d^{2} \left (-\frac {1}{5} c^{5} x^{5}+\frac {1}{2} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )+2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\) | \(423\) |
| default | \(\frac {a^{2} d^{2} \left (-\frac {1}{5} c^{5} x^{5}+\frac {1}{2} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+b^{2} d^{2} \left (-\frac {\arctan \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+\frac {4 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{15}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}-\frac {4 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{15}-\frac {8 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {8 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )^{2}}{2}+\frac {19 c x}{30}-\frac {c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{2}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}-\frac {19 \arctan \left (c x \right )}{30}+i \arctan \left (c x \right ) c x +\frac {i c^{2} x^{2}}{6}\right )+2 a \,d^{2} b \left (-\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{2}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{2}+\frac {c^{4} x^{4}}{20}-\frac {i c^{3} x^{3}}{6}-\frac {4 c^{2} x^{2}}{15}+\frac {4 \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i \arctan \left (c x \right )}{2}\right )}{c^{3}}\) | \(423\) |
| risch | \(\frac {19 b^{2} d^{2} x}{30 c^{2}}-\frac {2537 b^{2} d^{2} \arctan \left (c x \right )}{3600 c^{3}}-\frac {b^{2} d^{2} x^{3}}{30}+\frac {a^{2} d^{2} x^{3}}{3}-\frac {a^{2} c^{2} d^{2} x^{5}}{5}+\frac {a b c \,d^{2} x^{4}}{10}-\frac {59 a b \,d^{2}}{30 c^{3}}+\frac {i a b \,d^{2} x}{c^{2}}-\frac {8 i d^{2} b^{2} \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 {8 i d^{2} b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{15 c^{3}}+\frac {8 a b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} x^{2} a b}{15 c}+\frac {5 i d^{2} b^{2}}{6 c^{3}}-\frac {31 i d^{2} a^{2}}{30 c^{3}}+\frac {b^{2} d^{2} \ln \left (-i c x +1\right ) x^{3}}{6}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {i a b \,d^{2} \arctan \left (c x \right )}{c^{3}}-\frac {4 i d^{2} b^{2} \ln \left (-i c x +1\right ) x^{2}}{15 c}+\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right ) x^{4}}{20}-\frac {d^{2} c a b \ln \left (-i c x +1\right ) x^{4}}{2}-\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{8}+\frac {i d^{2} a b \ln \left (-i c x +1\right ) x^{3}}{3}+\left (-\frac {b^{2} d^{2} \left (6 c^{2} x^{5}-15 i c \,x^{4}-10 x^{3}\right ) \ln \left (-i c x +1\right )}{60}-\frac {i b \,d^{2} \left (-12 a \,c^{5} x^{5}+3 b \,c^{4} x^{4}+30 i a \,c^{4} x^{4}+20 a \,c^{3} x^{3}-10 i b \,c^{3} x^{3}-16 b \,c^{2} x^{2}+31 b \ln \left (-i c x +1\right )+30 i b c x \right )}{60 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {i d^{2} c^{2} b a \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {i a^{2} c \,d^{2} x^{4}}{2}+\frac {i b^{2} d^{2} x^{2}}{6 c}-\frac {5057 i d^{2} b^{2} \ln \left (c^{2} x^{2}+1\right )}{7200 c^{3}}-\frac {i a b \,d^{2} x^{3}}{3}-\frac {8 i d^{2} b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{15 c^{3}}+\frac {257 i d^{2} b^{2} \ln \left (-i c x +1\right )}{3600 c^{3}}+\frac {31 i d^{2} b^{2} \ln \left (-i c x +1\right )^{2}}{120 c^{3}}+\frac {b^{2} d^{2} \left (6 c^{5} x^{5}-15 i c^{4} x^{4}-10 c^{3} x^{3}-i\right ) \ln \left (i c x +1\right )^{2}}{120 c^{3}}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right ) x}{2 c^{2}}+\frac {d^{2} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}\) | \(740\) |
a^2*d^2*(-1/5*c^2*x^5+1/2*I*x^4*c+1/3*x^3)+b^2*d^2/c^3*(-1/5*arctan(c*x)^2 *c^5*x^5-2/3*I*ln(c^2*x^2+1)+1/3*c^3*x^3*arctan(c*x)^2+4/15*I*(ln(c*x-I)*l n(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)+1/10*c^4*x^4*arctan(c*x)-4/15*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)-8/15*c^2*x^2*arcta n(c*x)+8/15*arctan(c*x)*ln(c^2*x^2+1)-1/2*I*arctan(c*x)^2+19/30*c*x-1/30*c ^3*x^3+1/2*I*arctan(c*x)^2*c^4*x^4-1/3*I*arctan(c*x)*c^3*x^3-19/30*arctan( c*x)+I*arctan(c*x)*c*x+1/6*I*c^2*x^2)+2*a*d^2*b/c^3*(-1/5*c^5*x^5*arctan(c *x)+1/2*I*arctan(c*x)*c^4*x^4+1/3*c^3*x^3*arctan(c*x)+1/2*I*c*x+1/20*c^4*x ^4-1/6*I*c^3*x^3-4/15*c^2*x^2+4/15*ln(c^2*x^2+1)-1/2*I*arctan(c*x))
\[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/120*(6*b^2*c^2*d^2*x^5 - 15*I*b^2*c*d^2*x^4 - 10*b^2*d^2*x^3)*log(-(c*x + I)/(c*x - I))^2 + integral(-1/30*(30*a^2*c^4*d^2*x^6 - 60*I*a^2*c^3*d^2* x^5 - 60*I*a^2*c*d^2*x^3 - 30*a^2*d^2*x^2 - (-30*I*a*b*c^4*d^2*x^6 - 6*(10 *a*b - I*b^2)*c^3*d^2*x^5 + 15*b^2*c^2*d^2*x^4 - 10*(6*a*b + I*b^2)*c*d^2* x^3 + 30*I*a*b*d^2*x^2)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
-1/5*a^2*c^2*d^2*x^5 + 1/2*I*a^2*c*d^2*x^4 - 1/10*(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^2 + 1/3*a^2*d^2 *x^3 + 1/3*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^2 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^ 4))*a*b*d^2 - 1/120*(6*b^2*c^2*d^2*x^5 - 15*I*b^2*c*d^2*x^4 - 10*b^2*d^2*x ^3)*arctan(c*x)^2 + 1/120*(-6*I*b^2*c^2*d^2*x^5 - 15*b^2*c*d^2*x^4 + 10*I* b^2*d^2*x^3)*arctan(c*x)*log(c^2*x^2 + 1) + 1/480*(6*b^2*c^2*d^2*x^5 - 15* I*b^2*c*d^2*x^4 - 10*b^2*d^2*x^3)*log(c^2*x^2 + 1)^2 - integrate(1/240*(18 0*(b^2*c^4*d^2*x^6 - b^2*d^2*x^2)*arctan(c*x)^2 + 15*(b^2*c^4*d^2*x^6 - b^ 2*d^2*x^2)*log(c^2*x^2 + 1)^2 - 4*(21*b^2*c^3*d^2*x^5 - 10*b^2*c*d^2*x^3)* arctan(c*x) + 2*(6*b^2*c^4*d^2*x^6 - 25*b^2*c^2*d^2*x^4 - 60*(b^2*c^3*d^2* x^5 + b^2*c*d^2*x^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) + I* integrate(1/120*(180*(b^2*c^3*d^2*x^5 + b^2*c*d^2*x^3)*arctan(c*x)^2 + 15* (b^2*c^3*d^2*x^5 + b^2*c*d^2*x^3)*log(c^2*x^2 + 1)^2 + 2*(6*b^2*c^4*d^2*x^ 6 - 25*b^2*c^2*d^2*x^4)*arctan(c*x) + (21*b^2*c^3*d^2*x^5 - 10*b^2*c*d^2*x ^3 + 30*(b^2*c^4*d^2*x^6 - b^2*d^2*x^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^ 2*x^2 + 1), x)
\[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \]