Integrand size = 23, antiderivative size = 293 \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2-\frac {4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1+c^2 x^2\right )}{6 c^2}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \] Output:
-3/2*a*b*d^2*x/c+2/3*I*b^2*d^2*x/c-1/12*b^2*d^2*x^2-2/3*I*b^2*d^2*arctan(c *x)/c^2-3/2*b^2*d^2*x*arctan(c*x)/c-2/3*I*b*d^2*x^2*(a+b*arctan(c*x))+1/6* b*c*d^2*x^3*(a+b*arctan(c*x))+17/12*d^2*(a+b*arctan(c*x))^2/c^2+1/2*d^2*x^ 2*(a+b*arctan(c*x))^2+2/3*I*c*d^2*x^3*(a+b*arctan(c*x))^2-1/4*c^2*d^2*x^4* (a+b*arctan(c*x))^2-4/3*I*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^2+5/6* b^2*d^2*ln(c^2*x^2+1)/c^2+2/3*b^2*d^2*polylog(2,1-2/(1+I*c*x))/c^2
Time = 0.93 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.88 \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {d^2 \left (b^2+18 a b c x-8 i b^2 c x-6 a^2 c^2 x^2+8 i a b c^2 x^2+b^2 c^2 x^2-8 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 (-i+c x)^3 (i+3 c x) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (4 i+9 c x+4 i c^2 x^2-c^3 x^3\right )+a \left (-9-6 c^2 x^2-8 i c^3 x^3+3 c^4 x^4\right )+8 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-8 i a b \log \left (1+c^2 x^2\right )-10 b^2 \log \left (1+c^2 x^2\right )+8 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c^2} \] Input:
Integrate[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]
Output:
-1/12*(d^2*(b^2 + 18*a*b*c*x - (8*I)*b^2*c*x - 6*a^2*c^2*x^2 + (8*I)*a*b*c ^2*x^2 + b^2*c^2*x^2 - (8*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-I + c*x)^3*(I + 3*c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(4*I + 9* c*x + (4*I)*c^2*x^2 - c^3*x^3) + a*(-9 - 6*c^2*x^2 - (8*I)*c^3*x^3 + 3*c^4 *x^4) + (8*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (8*I)*a*b*Log[1 + c^2*x^ 2] - 10*b^2*Log[1 + c^2*x^2] + 8*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/ c^2
Time = 0.86 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-c^2 d^2 x^3 (a+b \arctan (c x))^2+2 i c d^2 x^2 (a+b \arctan (c x))^2+d^2 x (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))^2+\frac {17 d^2 (a+b \arctan (c x))^2}{12 c^2}-\frac {4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^2}+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))^2+\frac {1}{6} b c d^2 x^3 (a+b \arctan (c x))+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2-\frac {2}{3} i b d^2 x^2 (a+b \arctan (c x))-\frac {3 a b d^2 x}{2 c}-\frac {2 i b^2 d^2 \arctan (c x)}{3 c^2}-\frac {3 b^2 d^2 x \arctan (c x)}{2 c}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (c^2 x^2+1\right )}{6 c^2}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2\) |
Input:
Int[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]
Output:
(-3*a*b*d^2*x)/(2*c) + (((2*I)/3)*b^2*d^2*x)/c - (b^2*d^2*x^2)/12 - (((2*I )/3)*b^2*d^2*ArcTan[c*x])/c^2 - (3*b^2*d^2*x*ArcTan[c*x])/(2*c) - ((2*I)/3 )*b*d^2*x^2*(a + b*ArcTan[c*x]) + (b*c*d^2*x^3*(a + b*ArcTan[c*x]))/6 + (1 7*d^2*(a + b*ArcTan[c*x])^2)/(12*c^2) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^4*(a + b*ArcTan[c *x])^2)/4 - (((4*I)/3)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (5*b^2*d^2*Log[1 + c^2*x^2])/(6*c^2) + (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I *c*x)])/(3*c^2)
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 = 1.14 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.31
| method | result | size |
| parts | \(d^{2} a^{2} \left (-\frac {1}{4} c^{2} x^{4}+\frac {2}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d^{2} b^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {2 i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )}{c^{2}}+\frac {2 d^{2} a b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(384\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {2 i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )+2 d^{2} a b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(387\) |
| default | \(\frac {d^{2} a^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {2 i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {2 i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )^{2}}{4}-\frac {3 c x \arctan \left (c x \right )}{2}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\ln \left (c x +i\right )^{2}}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\ln \left (c x -i\right )^{2}}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{3}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{3}-\frac {2 i \arctan \left (c x \right ) c^{2} x^{2}}{3}-\frac {c^{2} x^{2}}{12}+\frac {5 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {2 i c x}{3}\right )+2 d^{2} a b \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(387\) |
| risch | \(\left (-\frac {d^{2} b^{2} \left (3 c^{2} x^{4}-8 i c \,x^{3}-6 x^{2}\right ) \ln \left (-i c x +1\right )}{24}-\frac {d^{2} b \left (-6 i a \,c^{4} x^{4}+2 i b \,c^{3} x^{3}-16 c^{3} x^{3} a +12 i a \,c^{2} x^{2}+8 b \,c^{2} x^{2}-18 i x b c -17 b \ln \left (-i c x +1\right )\right )}{24 c^{2}}\right ) \ln \left (i c x +1\right )+\frac {2 i d^{2} b \ln \left (c^{2} x^{2}+1\right ) a}{3 c^{2}}+\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}-\frac {3 i d^{2} b^{2} \ln \left (-i c x +1\right ) x}{4 c}-\frac {2 d^{2} c a b \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{6}+\frac {i d^{2} b a \ln \left (-i c x +1\right ) x^{2}}{2}+\frac {d^{2} x^{2} a^{2}}{2}+\frac {2 i b^{2} d^{2} x}{3 c}+\frac {2 i d^{2} c \,x^{3} a^{2}}{3}+\frac {2 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 )}{3 c^{2}}-\frac {2 d^{2} b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{2}}+\frac {d^{2} c b a \,x^{3}}{6}-\frac {3 d^{2} b^{2}}{4 c^{2}}+\frac {3 d^{2} b \arctan \left (c x \right ) a}{2 c^{2}}-\frac {b^{2} d^{2} x^{2}}{12}+\frac {17 d^{2} a^{2}}{12 c^{2}}-\frac {d^{2} c^{2} a^{2} x^{4}}{4}-\frac {i d^{2} c^{2} b a \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {2 i d^{2} a b \,x^{2}}{3}-\frac {215 i d^{2} b^{2} \arctan \left (c x \right )}{288 c^{2}}+\frac {d^{2} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {d^{2} b^{2} \left (3 c^{4} x^{4}-8 i c^{3} x^{3}-6 c^{2} x^{2}-1\right ) \ln \left (i c x +1\right )^{2}}{48 c^{2}}-\frac {7 i d^{2} b a}{3 c^{2}}+\frac {2 d^{2} b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}-\frac {17 d^{2} b^{2} \ln \left (-i c x +1\right )^{2}}{48 c^{2}}-\frac {23 d^{2} b^{2} \ln \left (-i c x +1\right )}{288 c^{2}}-\frac {d^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {d^{2} b^{2} \ln \left (-i c x +1\right ) x^{2}}{3}+\frac {503 b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{576 c^{2}}-\frac {3 a b \,d^{2} x}{2 c}\) | \(674\) |
Input:
int(x*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d^2*a^2*(-1/4*c^2*x^4+2/3*I*c*x^3+1/2*x^2)+d^2*b^2/c^2*(-1/4*c^4*x^4*arcta n(c*x)^2-2/3*I*arctan(c*x)+1/2*c^2*x^2*arctan(c*x)^2+2/3*I*arctan(c*x)^2*c ^3*x^3+1/6*c^3*x^3*arctan(c*x)+2/3*I*ln(c^2*x^2+1)*arctan(c*x)+3/4*arctan( c*x)^2-3/2*c*x*arctan(c*x)+1/3*ln(c*x+I)*ln(c^2*x^2+1)-1/3*ln(c*x-I)*ln(c^ 2*x^2+1)-1/6*ln(c*x+I)^2-1/3*ln(c*x+I)*ln(1/2*I*(c*x-I))+1/6*ln(c*x-I)^2+1 /3*ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/3*dilog(1/2*I*(c*x-I))+1/3*dilog(-1/2*I* (c*x+I))-2/3*I*arctan(c*x)*c^2*x^2-1/12*c^2*x^2+5/6*ln(c^2*x^2+1)+2/3*I*c* x)+2*d^2*a*b/c^2*(-1/4*c^4*x^4*arctan(c*x)+2/3*I*arctan(c*x)*c^3*x^3+1/2*c ^2*x^2*arctan(c*x)-3/4*c*x+1/12*c^3*x^3-1/3*I*c^2*x^2+1/3*I*ln(c^2*x^2+1)+ 3/4*arctan(c*x))
\[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
1/48*(3*b^2*c^2*d^2*x^4 - 8*I*b^2*c*d^2*x^3 - 6*b^2*d^2*x^2)*log(-(c*x + I )/(c*x - I))^2 + integral(-1/12*(12*a^2*c^4*d^2*x^5 - 24*I*a^2*c^3*d^2*x^4 - 24*I*a^2*c*d^2*x^2 - 12*a^2*d^2*x - (-12*I*a*b*c^4*d^2*x^5 - 3*(8*a*b - I*b^2)*c^3*d^2*x^4 + 8*b^2*c^2*d^2*x^3 - 6*(4*a*b + I*b^2)*c*d^2*x^2 + 12 *I*a*b*d^2*x)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x*(d+I*c*d*x)**2*(a+b*atan(c*x))**2,x)
Output:
Timed out
\[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
-1/4*a^2*c^2*d^2*x^4 + 2/3*I*a^2*c*d^2*x^3 + 1/2*b^2*d^2*x^2*arctan(c*x)^2 - 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*c^2*d^2 + 2/3*I*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4) )*a*b*c*d^2 + 1/2*a^2*d^2*x^2 + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/ c^3))*a*b*d^2 - 1/2*(2*c*(x/c^2 - arctan(c*x)/c^3)*arctan(c*x) + (arctan(c *x)^2 - log(c^2*x^2 + 1))/c^2)*b^2*d^2 - 1/48*(3*b^2*c^2*d^2*x^4 - 8*I*b^2 *c*d^2*x^3)*arctan(c*x)^2 + 1/48*(-3*I*b^2*c^2*d^2*x^4 - 8*b^2*c*d^2*x^3)* arctan(c*x)*log(c^2*x^2 + 1) + 1/192*(3*b^2*c^2*d^2*x^4 - 8*I*b^2*c*d^2*x^ 3)*log(c^2*x^2 + 1)^2 - integrate(-1/48*(22*b^2*c^3*d^2*x^4*arctan(c*x) - 36*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*arctan(c*x)^2 - 3*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*log(c^2*x^2 + 1)^2 - (3*b^2*c^4*d^2*x^5 - 8*b^2*c^2*d^2 *x^3 - 24*(b^2*c^3*d^2*x^4 + b^2*c*d^2*x^2)*arctan(c*x))*log(c^2*x^2 + 1)) /(c^2*x^2 + 1), x) + I*integrate(1/48*(72*(b^2*c^3*d^2*x^4 + b^2*c*d^2*x^2 )*arctan(c*x)^2 + 6*(b^2*c^3*d^2*x^4 + b^2*c*d^2*x^2)*log(c^2*x^2 + 1)^2 + 2*(3*b^2*c^4*d^2*x^5 - 8*b^2*c^2*d^2*x^3)*arctan(c*x) + (11*b^2*c^3*d^2*x ^4 + 12*(b^2*c^4*d^2*x^5 + b^2*c^2*d^2*x^3)*arctan(c*x))*log(c^2*x^2 + 1)) /(c^2*x^2 + 1), x)
\[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((I*c*d*x + d)^2*(b*arctan(c*x) + a)^2*x, x)
Timed out. \[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \] Input:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i)^2,x)
Output:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i)^2, x)
\[ \int x (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\frac {d^{2} \left (-3 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} x^{4}+8 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} i \,x^{3}+6 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} x^{2}+8 \mathit {atan} \left (c x \right )^{2} b^{2} c i x +9 \mathit {atan} \left (c x \right )^{2} b^{2}-6 \mathit {atan} \left (c x \right ) a b \,c^{4} x^{4}+16 \mathit {atan} \left (c x \right ) a b \,c^{3} i \,x^{3}+12 \mathit {atan} \left (c x \right ) a b \,c^{2} x^{2}+18 \mathit {atan} \left (c x \right ) a b +2 \mathit {atan} \left (c x \right ) b^{2} c^{3} x^{3}-8 \mathit {atan} \left (c x \right ) b^{2} c^{2} i \,x^{2}-18 \mathit {atan} \left (c x \right ) b^{2} c x -8 \mathit {atan} \left (c x \right ) b^{2} i -8 \left (\int \mathit {atan} \left (c x \right )^{2}d x \right ) b^{2} c i +8 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b i +10 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2}-3 a^{2} c^{4} x^{4}+8 a^{2} c^{3} i \,x^{3}+6 a^{2} c^{2} x^{2}+2 a b \,c^{3} x^{3}-8 a b \,c^{2} i \,x^{2}-18 a b c x -b^{2} c^{2} x^{2}+8 b^{2} c i x \right )}{12 c^{2}} \] Input:
int(x*(d+I*c*d*x)^2*(a+b*atan(c*x))^2,x)
Output:
(d**2*( - 3*atan(c*x)**2*b**2*c**4*x**4 + 8*atan(c*x)**2*b**2*c**3*i*x**3 + 6*atan(c*x)**2*b**2*c**2*x**2 + 8*atan(c*x)**2*b**2*c*i*x + 9*atan(c*x)* *2*b**2 - 6*atan(c*x)*a*b*c**4*x**4 + 16*atan(c*x)*a*b*c**3*i*x**3 + 12*at an(c*x)*a*b*c**2*x**2 + 18*atan(c*x)*a*b + 2*atan(c*x)*b**2*c**3*x**3 - 8* atan(c*x)*b**2*c**2*i*x**2 - 18*atan(c*x)*b**2*c*x - 8*atan(c*x)*b**2*i - 8*int(atan(c*x)**2,x)*b**2*c*i + 8*log(c**2*x**2 + 1)*a*b*i + 10*log(c**2* x**2 + 1)*b**2 - 3*a**2*c**4*x**4 + 8*a**2*c**3*i*x**3 + 6*a**2*c**2*x**2 + 2*a*b*c**3*x**3 - 8*a*b*c**2*i*x**2 - 18*a*b*c*x - b**2*c**2*x**2 + 8*b* *2*c*i*x))/(12*c**2)