Integrand size = 23, antiderivative size = 256 \[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {3 a x}{c^4 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (i-c x)^2}-\frac {15 i b}{8 c^5 d^3 (i-c x)}+\frac {19 i b \arctan (c x)}{8 c^5 d^3}-\frac {3 b x \arctan (c x)}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))}{c^5 d^3 (i-c x)}+\frac {6 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 b \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3} \] Output:
-3*a*x/c^4/d^3-1/2*I*b*x/c^4/d^3-1/8*b/c^5/d^3/(I-c*x)^2-15/8*I*b/c^5/d^3/ (I-c*x)+19/8*I*b*arctan(c*x)/c^5/d^3-3*b*x*arctan(c*x)/c^4/d^3+1/2*I*x^2*( a+b*arctan(c*x))/c^3/d^3-1/2*I*(a+b*arctan(c*x))/c^5/d^3/(I-c*x)^2+4*(a+b* arctan(c*x))/c^5/d^3/(I-c*x)+6*I*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5/d^3 +3/2*b*ln(c^2*x^2+1)/c^5/d^3-3*b*polylog(2,1-2/(1+I*c*x))/c^5/d^3
Time = 0.74 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.92 \[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {-96 a c x+16 i a c^2 x^2-\frac {16 i a}{(-i+c x)^2}-\frac {128 a}{-i+c x}+192 a \arctan (c x)-96 i a \log \left (1+c^2 x^2\right )+b \left (-16 i c x+192 \arctan (c x)^2-28 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+48 \log \left (1+c^2 x^2\right )+96 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+28 i \sin (2 \arctan (c x))+4 i \arctan (c x) \left (4+24 i c x+4 c^2 x^2-14 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+48 \log \left (1+e^{2 i \arctan (c x)}\right )+14 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )-i \sin (4 \arctan (c x))\right )}{32 c^5 d^3} \] Input:
Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]
Output:
(-96*a*c*x + (16*I)*a*c^2*x^2 - ((16*I)*a)/(-I + c*x)^2 - (128*a)/(-I + c* x) + 192*a*ArcTan[c*x] - (96*I)*a*Log[1 + c^2*x^2] + b*((-16*I)*c*x + 192* ArcTan[c*x]^2 - 28*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + c^ 2*x^2] + 96*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (28*I)*Sin[2*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*(4 + (24*I)*c*x + 4*c^2*x^2 - 14*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + E^((2*I)*ArcTan[c*x])] + (14*I)*Sin[2*Arc Tan[c*x]] - I*Sin[4*ArcTan[c*x]]) - I*Sin[4*ArcTan[c*x]]))/(32*c^5*d^3)
Time = 0.55 (sec) , antiderivative size = 256, 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.
| \(\displaystyle \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-\frac {6 i (a+b \arctan (c x))}{c^4 d^3 (c x-i)}+\frac {4 (a+b \arctan (c x))}{c^4 d^3 (c x-i)^2}-\frac {3 (a+b \arctan (c x))}{c^4 d^3}+\frac {i (a+b \arctan (c x))}{c^4 d^3 (c x-i)^3}+\frac {i x (a+b \arctan (c x))}{c^3 d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 (a+b \arctan (c x))}{c^5 d^3 (-c x+i)}-\frac {i (a+b \arctan (c x))}{2 c^5 d^3 (-c x+i)^2}+\frac {6 i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^5 d^3}+\frac {i x^2 (a+b \arctan (c x))}{2 c^3 d^3}-\frac {3 a x}{c^4 d^3}+\frac {19 i b \arctan (c x)}{8 c^5 d^3}-\frac {3 b x \arctan (c x)}{c^4 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^5 d^3}-\frac {15 i b}{8 c^5 d^3 (-c x+i)}-\frac {b}{8 c^5 d^3 (-c x+i)^2}-\frac {i b x}{2 c^4 d^3}+\frac {3 b \log \left (c^2 x^2+1\right )}{2 c^5 d^3}\) |
Input:
Int[(x^4*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]
Output:
(-3*a*x)/(c^4*d^3) - ((I/2)*b*x)/(c^4*d^3) - b/(8*c^5*d^3*(I - c*x)^2) - ( ((15*I)/8)*b)/(c^5*d^3*(I - c*x)) + (((19*I)/8)*b*ArcTan[c*x])/(c^5*d^3) - (3*b*x*ArcTan[c*x])/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x]))/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c* x]))/(c^5*d^3*(I - c*x)) + ((6*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/ (c^5*d^3) + (3*b*Log[1 + c^2*x^2])/(2*c^5*d^3) - (3*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3)
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 = 0.49 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 a c x}{d^{3}}-\frac {i b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}+\frac {6 a \arctan \left (c x \right )}{d^{3}}-\frac {5 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}-\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {4 a}{d^{3} \left (c x -i\right )}-\frac {3 b \arctan \left (c x \right ) c x}{d^{3}}+\frac {i b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{3}}+\frac {43 i b \arctan \left (c x \right )}{16 d^{3}}-\frac {5 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}-\frac {4 b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}-\frac {b}{2 d^{3}}+\frac {5 i b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {5 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}-\frac {6 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}-\frac {i b c x}{2 d^{3}}+\frac {15 i b}{8 d^{3} \left (c x -i\right )}-\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}-\frac {b}{8 d^{3} \left (c x -i\right )^{2}}+\frac {43 b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {i a \,c^{2} x^{2}}{2 d^{3}}-\frac {3 b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{3}}+\frac {3 b \ln \left (c x -i\right )^{2}}{2 d^{3}}-\frac {3 b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{3}}}{c^{5}}\) | \(364\) |
| default | \(\frac {-\frac {3 a c x}{d^{3}}-\frac {i b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}+\frac {6 a \arctan \left (c x \right )}{d^{3}}-\frac {5 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}-\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {4 a}{d^{3} \left (c x -i\right )}-\frac {3 b \arctan \left (c x \right ) c x}{d^{3}}+\frac {i b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{3}}+\frac {43 i b \arctan \left (c x \right )}{16 d^{3}}-\frac {5 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}-\frac {4 b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}-\frac {b}{2 d^{3}}+\frac {5 i b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {5 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}-\frac {6 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}-\frac {i b c x}{2 d^{3}}+\frac {15 i b}{8 d^{3} \left (c x -i\right )}-\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}-\frac {b}{8 d^{3} \left (c x -i\right )^{2}}+\frac {43 b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {i a \,c^{2} x^{2}}{2 d^{3}}-\frac {3 b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{3}}+\frac {3 b \ln \left (c x -i\right )^{2}}{2 d^{3}}-\frac {3 b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{3}}}{c^{5}}\) | \(364\) |
| parts | \(-\frac {6 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{5} d^{3}}-\frac {3 a x}{c^{4} d^{3}}-\frac {5 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 c^{5} d^{3}}+\frac {6 a \arctan \left (c x \right )}{d^{3} c^{5}}+\frac {4 a}{d^{3} c^{5} \left (-c x +i\right )}+\frac {i a \,x^{2}}{2 d^{3} c^{3}}-\frac {3 b x \arctan \left (c x \right )}{c^{4} d^{3}}+\frac {15 i b}{8 c^{5} d^{3} \left (c x -i\right )}+\frac {i b \arctan \left (c x \right ) x^{2}}{2 c^{3} d^{3}}-\frac {5 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 c^{5} d^{3}}-\frac {4 b \arctan \left (c x \right )}{c^{5} d^{3} \left (c x -i\right )}-\frac {i a}{2 d^{3} c^{5} \left (-c x +i\right )^{2}}-\frac {b}{2 c^{5} d^{3}}+\frac {5 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 c^{5} d^{3}}+\frac {43 i b \arctan \left (c x \right )}{16 c^{5} d^{3}}-\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{d^{3} c^{5}}+\frac {5 i b \arctan \left (\frac {c x}{2}\right )}{32 c^{5} d^{3}}-\frac {i b \arctan \left (c x \right )}{2 c^{5} d^{3} \left (c x -i\right )^{2}}-\frac {b}{8 c^{5} d^{3} \left (c x -i\right )^{2}}+\frac {43 b \ln \left (c^{2} x^{2}+1\right )}{32 c^{5} d^{3}}-\frac {i b x}{2 c^{4} d^{3}}-\frac {3 b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{5} d^{3}}+\frac {3 b \ln \left (c x -i\right )^{2}}{2 c^{5} d^{3}}-\frac {3 b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{5} d^{3}}\) | \(425\) |
| risch | \(-\frac {2 i b}{c^{5} d^{3} \left (-c x +i\right )}-\frac {x^{2} b \ln \left (-i c x +1\right )}{4 d^{3} c^{3}}-\frac {3 b \ln \left (-i c x +1\right )}{16 d^{3} c^{5} \left (-i c x -1\right )^{2}}+\frac {3 b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{3} c^{5}}-\frac {3 b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{3} c^{5}}-\frac {b \ln \left (-i c x +1\right )}{d^{3} c^{5} \left (-i c x -1\right )}-\frac {9 b}{8 c^{5} d^{3}}+\frac {43 i b \arctan \left (c x \right )}{16 c^{5} d^{3}}-\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{d^{3} c^{5}}-\frac {i b x}{2 c^{4} d^{3}}+\frac {i a \,x^{2}}{2 d^{3} c^{3}}+\frac {6 a \arctan \left (c x \right )}{d^{3} c^{5}}-\frac {3 b \ln \left (i c x +1\right )^{2}}{2 c^{5} d^{3}}+\frac {5 b \ln \left (-i c x +1\right )}{4 d^{3} c^{5}}-\frac {b}{8 d^{3} c^{5} \left (-i c x -1\right )}-\frac {3 b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{3} c^{5}}-\frac {5 i a}{2 d^{3} c^{5}}-\frac {b}{8 c^{5} d^{3} \left (-c x +i\right )^{2}}+\frac {i a}{2 d^{3} c^{5} \left (-i c x -1\right )^{2}}+\frac {4 i a}{d^{3} c^{5} \left (-i c x -1\right )}+\frac {i b \ln \left (-i c x +1\right ) x}{d^{3} c^{4} \left (-i c x -1\right )}+\frac {i b \ln \left (-i c x +1\right ) x}{8 d^{3} c^{4} \left (-i c x -1\right )^{2}}-\frac {b \ln \left (-i c x +1\right ) x^{2}}{16 d^{3} c^{3} \left (-i c x -1\right )^{2}}-\frac {3 i b \ln \left (-i c x +1\right ) x}{2 d^{3} c^{4}}+\left (\frac {b \left (\frac {1}{2} c \,x^{2}+3 i x \right )}{2 c^{4} d^{3}}+\frac {4 i b \,d^{3} x +\frac {7 b \,d^{3}}{2 c}}{2 c^{4} d^{6} \left (c x -i\right )^{2}}\right ) \ln \left (i c x +1\right )-\frac {3 a x}{c^{4} d^{3}}+\frac {43 b \ln \left (c^{2} x^{2}+1\right )}{32 c^{5} d^{3}}\) | \(557\) |
Input:
int(x^4*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,method=_RETURNVERBOSE)
Output:
1/c^5*(-3*a/d^3*c*x-1/2*I*b/d^3*arctan(c*x)/(c*x-I)^2+6*a/d^3*arctan(c*x)- 5/16*I*b/d^3*arctan(1/2*c*x-1/2*I)-1/2*I*a/d^3/(c*x-I)^2-4*a/d^3/(c*x-I)-3 *b/d^3*arctan(c*x)*c*x+1/2*I*b/d^3*arctan(c*x)*c^2*x^2+43/16*I*b/d^3*arcta n(c*x)-5/32*I*b/d^3*arctan(1/6*c^3*x^3+7/6*c*x)-4*b/d^3*arctan(c*x)/(c*x-I )-1/2*b/d^3+5/32*I*b/d^3*arctan(1/2*c*x)+5/64*b/d^3*ln(c^4*x^4+10*c^2*x^2+ 9)-6*I*b/d^3*arctan(c*x)*ln(c*x-I)-1/2*I*b/d^3*c*x+15/8*I*b/d^3/(c*x-I)-3* I*a/d^3*ln(c^2*x^2+1)-1/8*b/d^3/(c*x-I)^2+43/32*b/d^3*ln(c^2*x^2+1)+1/2*I* a/d^3*c^2*x^2-3*b/d^3*ln(c*x-I)*ln(-1/2*I*(c*x+I))+3/2*b/d^3*ln(c*x-I)^2-3 *b/d^3*dilog(-1/2*I*(c*x+I)))
\[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{4}}{{\left (i \, c d x + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x, algorithm="fricas")
Output:
integral(-1/2*(b*x^4*log(-(c*x + I)/(c*x - I)) - 2*I*a*x^4)/(c^3*d^3*x^3 - 3*I*c^2*d^3*x^2 - 3*c*d^3*x + I*d^3), x)
Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\text {Timed out} \] Input:
integrate(x**4*(a+b*atan(c*x))/(d+I*c*d*x)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.40 \[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {8 i \, a c^{4} x^{4} - 8 \, {\left (4 \, a + i \, b\right )} c^{3} x^{3} + {\left (b {\left (5 i \, \arctan \left (1, c x\right ) - 16\right )} + 88 i \, a\right )} c^{2} x^{2} + 2 \, {\left (b {\left (5 \, \arctan \left (1, c x\right ) + 19 i\right )} - 8 \, a\right )} c x + 24 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right )^{2} + 6 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 24 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + b {\left (-5 i \, \arctan \left (1, c x\right ) + 28\right )} + {\left (8 i \, b c^{4} x^{4} - 32 \, b c^{3} x^{3} + {\left (96 \, a + 131 i \, b\right )} c^{2} x^{2} - 2 \, {\left (96 i \, a - 35 \, b\right )} c x - 96 \, a + 13 i \, b\right )} \arctan \left (c x\right ) - 48 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) - 12 \, {\left (2 \, {\left (2 i \, a - b\right )} c^{2} x^{2} + 4 \, {\left (2 \, a + i \, b\right )} c x + {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) - 4 i \, a + 2 \, b\right )} \log \left (c^{2} x^{2} + 1\right ) + 56 i \, a}{16 \, {\left (c^{7} d^{3} x^{2} - 2 i \, c^{6} d^{3} x - c^{5} d^{3}\right )}} \] Input:
integrate(x^4*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x, algorithm="maxima")
Output:
1/16*(8*I*a*c^4*x^4 - 8*(4*a + I*b)*c^3*x^3 + (b*(5*I*arctan2(1, c*x) - 16 ) + 88*I*a)*c^2*x^2 + 2*(b*(5*arctan2(1, c*x) + 19*I) - 8*a)*c*x + 24*(b*c ^2*x^2 - 2*I*b*c*x - b)*arctan(c*x)^2 + 6*(b*c^2*x^2 - 2*I*b*c*x - b)*log( c^2*x^2 + 1)^2 - 24*(I*b*c^2*x^2 + 2*b*c*x - I*b)*arctan(c*x)*log(1/4*c^2* x^2 + 1/4) + b*(-5*I*arctan2(1, c*x) + 28) + (8*I*b*c^4*x^4 - 32*b*c^3*x^3 + (96*a + 131*I*b)*c^2*x^2 - 2*(96*I*a - 35*b)*c*x - 96*a + 13*I*b)*arcta n(c*x) - 48*(b*c^2*x^2 - 2*I*b*c*x - b)*dilog(1/2*I*c*x + 1/2) - 12*(2*(2* I*a - b)*c^2*x^2 + 4*(2*a + I*b)*c*x + (b*c^2*x^2 - 2*I*b*c*x - b)*log(1/4 *c^2*x^2 + 1/4) - 4*I*a + 2*b)*log(c^2*x^2 + 1) + 56*I*a)/(c^7*d^3*x^2 - 2 *I*c^6*d^3*x - c^5*d^3)
\[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{4}}{{\left (i \, c d x + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)*x^4/(I*c*d*x + d)^3, x)
Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \] Input:
int((x^4*(a + b*atan(c*x)))/(d + c*d*x*1i)^3,x)
Output:
int((x^4*(a + b*atan(c*x)))/(d + c*d*x*1i)^3, x)
\[ \int \frac {x^4 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {-2 \left (\int \frac {\mathit {atan} \left (c x \right ) x^{4}}{c^{3} i \,x^{3}+3 c^{2} x^{2}-3 c i x -1}d x \right ) b \,c^{5}+8 \left (\int \frac {x}{c^{3} i \,x^{3}+3 c^{2} x^{2}-3 c i x -1}d x \right ) a \,c^{2} i +6 \left (\int \frac {1}{c^{3} i \,x^{3}+3 c^{2} x^{2}-3 c i x -1}d x \right ) a c -4 \,\mathrm {log}\left (c^{3} i \,x^{3}+3 c^{2} x^{2}-3 c i x -1\right ) a i +a \,c^{2} i \,x^{2}-6 a c x}{2 c^{5} d^{3}} \] Input:
int(x^4*(a+b*atan(c*x))/(d+I*c*d*x)^3,x)
Output:
( - 2*int((atan(c*x)*x**4)/(c**3*i*x**3 + 3*c**2*x**2 - 3*c*i*x - 1),x)*b* c**5 + 8*int(x/(c**3*i*x**3 + 3*c**2*x**2 - 3*c*i*x - 1),x)*a*c**2*i + 6*i nt(1/(c**3*i*x**3 + 3*c**2*x**2 - 3*c*i*x - 1),x)*a*c - 4*log(c**3*i*x**3 + 3*c**2*x**2 - 3*c*i*x - 1)*a*i + a*c**2*i*x**2 - 6*a*c*x)/(2*c**5*d**3)