Integrand size = 23, antiderivative size = 216 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2} \]
1/2*I*b^2/c^2/d^2/(I-c*x)-1/2*I*b^2*arctan(c*x)/c^2/d^2-b*(a+b*arctan(c*x) )/c^2/d^2/(I-c*x)+1/2*(a+b*arctan(c*x))^2/c^2/d^2-I*(a+b*arctan(c*x))^2/c^ 2/d^2/(I-c*x)+(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^2/d^2+I*b*(a+b*arctan( c*x))*polylog(2,1-2/(1+I*c*x))/c^2/d^2+1/2*b^2*polylog(3,1-2/(1+I*c*x))/c^ 2/d^2
Time = 0.96 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.39 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {\frac {12 i a^2}{-i+c x}-12 i a^2 \arctan (c x)-6 a^2 \log \left (1+c^2 x^2\right )-6 i a b \left (4 \arctan (c x)^2-\cos (2 \arctan (c x))+2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-2 i \arctan (c x) \left (\cos (2 \arctan (c x))-2 \log \left (1+e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))\right )+i \sin (2 \arctan (c x))\right )+b^2 \left (-8 i \arctan (c x)^3+3 \cos (2 \arctan (c x))+6 i \arctan (c x) \cos (2 \arctan (c x))-6 \arctan (c x)^2 \cos (2 \arctan (c x))+12 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-12 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-3 i \sin (2 \arctan (c x))+6 \arctan (c x) \sin (2 \arctan (c x))+6 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{12 c^2 d^2} \]
(((12*I)*a^2)/(-I + c*x) - (12*I)*a^2*ArcTan[c*x] - 6*a^2*Log[1 + c^2*x^2] - (6*I)*a*b*(4*ArcTan[c*x]^2 - Cos[2*ArcTan[c*x]] + 2*PolyLog[2, -E^((2*I )*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*(Cos[2*ArcTan[c*x]] - 2*Log[1 + E^((2* I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]]) + I*Sin[2*ArcTan[c*x]]) + b^2*((- 8*I)*ArcTan[c*x]^3 + 3*Cos[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]*Cos[2*ArcTan [c*x]] - 6*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 12*ArcTan[c*x]^2*Log[1 + E^( (2*I)*ArcTan[c*x])] - (12*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x]) ] + 6*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - (3*I)*Sin[2*ArcTan[c*x]] + 6*Ar cTan[c*x]*Sin[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(1 2*c^2*d^2)
Time = 0.52 (sec) , antiderivative size = 216, 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.
| \(\displaystyle \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-\frac {(a+b \arctan (c x))^2}{c d^2 (c x-i)}-\frac {i (a+b \arctan (c x))^2}{c d^2 (c x-i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (-c x+i)}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (-c x+i)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}+\frac {i b^2}{2 c^2 d^2 (-c x+i)}\) |
((I/2)*b^2)/(c^2*d^2*(I - c*x)) - ((I/2)*b^2*ArcTan[c*x])/(c^2*d^2) - (b*( a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(2*c^2*d^2 ) - (I*(a + b*ArcTan[c*x])^2)/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])^2 *Log[2/(1 + I*c*x)])/(c^2*d^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2 /(1 + I*c*x)])/(c^2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d^2)
3.2.6.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.03 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.97
| method | result | size |
| derivativedivides | \(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\) | \(857\) |
| default | \(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\) | \(857\) |
| parts | \(-\frac {i a^{2}}{d^{2} c^{2} \left (-c x +i\right )}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{2}}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2} c^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c^{2}}+\frac {i a b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2} c^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2} c^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2} c^{2}}+\frac {a b}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2} c^{2}}-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2} c^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2} c^{2}}\) | \(902\) |
1/c^2*(-1/4*I*a*b/d^2*ln(c^2*x^2+1)-1/2*a^2/d^2*ln(c^2*x^2+1)-1/2*I*a*b/d^ 2*ln(c*x-I)^2+b^2/d^2*(I*arctan(c*x)^2/(c*x-I)-arctan(c*x)^2*ln(c*x-I)+arc tan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-2/3*I*arctan(c*x)^3+1/2*I*Pi*cs gn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x )^2/(c^2*x^2+1)))^2*arctan(c*x)^2-1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+ 1)))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c* x)^2/(c^2*x^2+1)))*arctan(c*x)^2-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+ (1+I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x ^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan( c*x)^2-I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*ar ctan(c*x)^2+1/2*arctan(c*x)^2+I*Pi*arctan(c*x)^2-2*I*arctan(c*x)*(c*x+I)/( 4*c*x-4*I)-1/4*(c*x+I)/(c*x-I)-I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x ^2+1))+1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1)))+I*a*b/d^2*dilog(-1/2*I*(c* x+I))-2*a*b/d^2*arctan(c*x)*ln(c*x-I)+1/8*I*a*b/d^2*ln(c^4*x^4+10*c^2*x^2+ 9)-1/4*a*b/d^2*arctan(1/2*c*x)+1/4*a*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x)+1/2 *a*b/d^2*arctan(1/2*c*x-1/2*I)+a*b/d^2/(c*x-I)+I*a*b/d^2*ln(-1/2*I*(c*x+I) )*ln(c*x-I)+1/2*a*b/d^2*arctan(c*x)+I*a^2/d^2/(c*x-I)-I*a^2/d^2*arctan(c*x )+2*I*a*b/d^2*arctan(c*x)/(c*x-I))
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
integral(1/4*(b^2*x*log(-(c*x + I)/(c*x - I))^2 - 4*I*a*b*x*log(-(c*x + I) /(c*x - I)) - 4*a^2*x)/(c^2*d^2*x^2 - 2*I*c*d^2*x - d^2), x)
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
a^2*(I/(c^3*d^2*x - I*c^2*d^2) - log(c*x - I)/(c^2*d^2)) - 1/32*(-8*I*b^2* arctan(c*x)^2 - 8*(-I*b^2*c*x - b^2)*arctan(c*x)^3 - (b^2*c*x - I*b^2)*log (c^2*x^2 + 1)^3 - 2*(-I*b^2 + (-I*b^2*c*x - b^2)*arctan(c*x))*log(c^2*x^2 + 1)^2 - (2*b^2*c^3*(c^2/(c^9*d^2*x^2 + c^7*d^2) + log(c^2*x^2 + 1)/(c^7*d ^2*x^2 + c^5*d^2)) - 640*b^2*c^3*integrate(1/16*x^3*arctan(c*x)^2/(c^5*d^2 *x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 96*b^2*c^3*integrate(1/16*x^3*log(c^2* x^2 + 1)^2/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 1024*a*b*c^3*integr ate(1/16*x^3*arctan(c*x)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 256*b ^2*c^2*integrate(1/16*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^5*d^2*x^4 + 2*c^ 3*d^2*x^2 + c*d^2), x) + 256*b^2*c^2*integrate(1/16*x^2*arctan(c*x)/(c^5*d ^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) + 16*(c*(x/(c^5*d^2*x^2 + c^3*d^2) + a rctan(c*x)/(c^4*d^2)) - 2*arctan(c*x)/(c^5*d^2*x^2 + c^3*d^2))*a*b*c + 128 *b^2*c*integrate(1/16*x*arctan(c*x)^2/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2 ), x) + b^2*c*log(c^2*x^2 + 1)^2/(c^5*d^2*x^2 + c^3*d^2) + 256*b^2*integra te(1/16*arctan(c*x)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x))*(c^3*d^2*x - I*c^2*d^2) - 32*(I*c^3*d^2*x + c^2*d^2)*integrate(-1/8*(32*a*b*c^2*x^2*a rctan(c*x) - b^2*log(c^2*x^2 + 1)^2 + 4*(2*b^2*c^2*x^2 - b^2)*arctan(c*x)^ 2 - 2*(b^2*c^2*x^2 + b^2 + (b^2*c^3*x^3 - b^2*c*x)*arctan(c*x))*log(c^2*x^ 2 + 1))/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) + 4*(2*b^2*arctan(c*x) - (b^2*c*x - I*b^2)*arctan(c*x)^2)*log(c^2*x^2 + 1))/(c^3*d^2*x - I*c^2*...
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]