\(\int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx\) [115]

Optimal result

Integrand size = 22, antiderivative size = 180 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {i b^2}{16 c d^3 (i-c x)^2}+\frac {3 b^2}{16 c d^3 (i-c x)}-\frac {3 b^2 \arctan (c x)}{16 c d^3}-\frac {b (a+b \arctan (c x))}{4 c d^3 (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{4 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^2}{8 c d^3}+\frac {i (a+b \arctan (c x))^2}{2 c d^3 (1+i c x)^2} \] Output:

1/16*I*b^2/c/d^3/(I-c*x)^2+3/16*b^2/c/d^3/(I-c*x)-3/16*b^2*arctan(c*x)/c/d 
^3-1/4*b*(a+b*arctan(c*x))/c/d^3/(I-c*x)^2+1/4*I*b*(a+b*arctan(c*x))/c/d^3 
/(I-c*x)-1/8*I*(a+b*arctan(c*x))^2/c/d^3+1/2*I*(a+b*arctan(c*x))^2/c/d^3/( 
1+I*c*x)^2
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=-\frac {i \left (8 a^2+b^2 (-4-3 i c x)+4 a b (-2 i+c x)+b (i+c x) (b (-5-3 i c x)+4 a (-3 i+c x)) \arctan (c x)+2 b^2 \left (3-2 i c x+c^2 x^2\right ) \arctan (c x)^2\right )}{16 c d^3 (-i+c x)^2} \] Input:

Integrate[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^3,x]
 

Output:

((-1/16*I)*(8*a^2 + b^2*(-4 - (3*I)*c*x) + 4*a*b*(-2*I + c*x) + b*(I + c*x 
)*(b*(-5 - (3*I)*c*x) + 4*a*(-3*I + c*x))*ArcTan[c*x] + 2*b^2*(3 - (2*I)*c 
*x + c^2*x^2)*ArcTan[c*x]^2))/(c*d^3*(-I + c*x)^2)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5389, 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.

\(\displaystyle \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx\)

\(\Big \downarrow \) 5389

\(\displaystyle \frac {i (a+b \arctan (c x))^2}{2 c d^3 (1+i c x)^2}-\frac {i b \int \left (\frac {a+b \arctan (c x)}{4 d^2 \left (c^2 x^2+1\right )}-\frac {a+b \arctan (c x)}{4 d^2 (i-c x)^2}-\frac {i (a+b \arctan (c x))}{2 d^2 (i-c x)^3}\right )dx}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i (a+b \arctan (c x))^2}{2 c d^3 (1+i c x)^2}-\frac {i b \left (\frac {(a+b \arctan (c x))^2}{8 b c d^2}-\frac {a+b \arctan (c x)}{4 c d^2 (-c x+i)}-\frac {i (a+b \arctan (c x))}{4 c d^2 (-c x+i)^2}-\frac {3 i b \arctan (c x)}{16 c d^2}+\frac {3 i b}{16 c d^2 (-c x+i)}-\frac {b}{16 c d^2 (-c x+i)^2}\right )}{d}\)

Input:

Int[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^3,x]
 

Output:

((I/2)*(a + b*ArcTan[c*x])^2)/(c*d^3*(1 + I*c*x)^2) - (I*b*(-1/16*b/(c*d^2 
*(I - c*x)^2) + (((3*I)/16)*b)/(c*d^2*(I - c*x)) - (((3*I)/16)*b*ArcTan[c* 
x])/(c*d^2) - ((I/4)*(a + b*ArcTan[c*x]))/(c*d^2*(I - c*x)^2) - (a + b*Arc 
Tan[c*x])/(4*c*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(8*b*c*d^2)))/d
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5389

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S 
imp[b*c*(p/(e*(q + 1)))   Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), 
(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && 
 IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(\frac {\frac {i a^{2}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {3 i \arctan \left (c x \right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}\right )\right )}{d^{3}}+\frac {i a b \arctan \left (c x \right )}{d^{3} \left (i c x +1\right )^{2}}-\frac {i a b \arctan \left (c x \right )}{4 d^{3}}-\frac {a b}{4 d^{3} \left (c x -i\right )^{2}}-\frac {i a b}{4 d^{3} \left (c x -i\right )}}{c}\)	\(277\)
default	\(\frac {\frac {i a^{2}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {3 i \arctan \left (c x \right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}\right )\right )}{d^{3}}+\frac {i a b \arctan \left (c x \right )}{d^{3} \left (i c x +1\right )^{2}}-\frac {i a b \arctan \left (c x \right )}{4 d^{3}}-\frac {a b}{4 d^{3} \left (c x -i\right )^{2}}-\frac {i a b}{4 d^{3} \left (c x -i\right )}}{c}\)	\(277\)
parts	\(\frac {i a^{2}}{2 d^{3} \left (i c x +1\right )^{2} c}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {3 i \arctan \left (c x \right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}\right )\right )}{d^{3} c}+\frac {i a b \arctan \left (c x \right )}{c \,d^{3} \left (i c x +1\right )^{2}}-\frac {i a b \arctan \left (c x \right )}{4 c \,d^{3}}-\frac {a b}{4 c \,d^{3} \left (c x -i\right )^{2}}-\frac {i a b}{4 c \,d^{3} \left (c x -i\right )}\)	\(291\)
orering	\(\frac {\left (42 c^{5} x^{5}+27 i c^{4} x^{4}+106 c^{3} x^{3}+108 i c^{2} x^{2}+64 c x +81 i\right ) \left (a +b \arctan \left (c x \right )\right )^{2}}{64 c \left (i c d x +d \right )^{3}}-\frac {\left (-28 c^{6} x^{6}+9 i c^{5} x^{5}-93 c^{4} x^{4}+18 i c^{3} x^{3}-102 c^{2} x^{2}+9 i c x -37\right ) \left (\frac {2 \left (a +b \arctan \left (c x \right )\right ) b c}{\left (i c d x +d \right )^{3} \left (c^{2} x^{2}+1\right )}-\frac {3 i \left (a +b \arctan \left (c x \right )\right )^{2} c d}{\left (i c d x +d \right )^{4}}\right )}{64 c^{2}}-\frac {\left (-7 c x +9 i\right ) \left (c^{2} x^{2}+1\right )^{3} \left (\frac {2 b^{2} c^{2}}{\left (c^{2} x^{2}+1\right )^{2} \left (i c d x +d \right )^{3}}-\frac {12 i \left (a +b \arctan \left (c x \right )\right ) b \,c^{2} d}{\left (i c d x +d \right )^{4} \left (c^{2} x^{2}+1\right )}-\frac {4 \left (a +b \arctan \left (c x \right )\right ) b \,c^{3} x}{\left (i c d x +d \right )^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {12 \left (a +b \arctan \left (c x \right )\right )^{2} c^{2} d^{2}}{\left (i c d x +d \right )^{5}}\right )}{128 c^{3}}\)	\(335\)
risch	\(\frac {i b^{2} \left (c^{2} x^{2}-2 i c x +3\right ) \ln \left (i c x +1\right )^{2}}{32 d^{3} \left (c x -i\right )^{2} c}-\frac {\left (3 i b^{2} \ln \left (-i c x +1\right )+i \ln \left (-i c x +1\right ) b^{2} c^{2} x^{2}+2 \ln \left (-i c x +1\right ) b^{2} c x +2 b^{2} c x -4 i b^{2}+8 a b \right ) \ln \left (i c x +1\right )}{16 d^{3} \left (c x -i\right )^{2} c}-\frac {i \left (4 i \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) a b \,c^{2} x^{2}+16 i a b \ln \left (-i c x +1\right )-b^{2} c^{2} x^{2} \ln \left (-i c x +1\right )^{2}+3 \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) b^{2} c^{2} x^{2}-3 \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) b^{2} c^{2} x^{2}-6 i b^{2} c x -6 i \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) b^{2} c x +6 i \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) b^{2} c x +4 i \ln \left (-i c x +1\right ) b^{2} c x +8 \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) a b c x -8 \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) a b c x -4 i \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) a b +4 i \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) a b -4 i \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) a b \,c^{2} x^{2}+2 i b^{2} c x \ln \left (-i c x +1\right )^{2}+8 a b c x -3 b^{2} \ln \left (-i c x +1\right )^{2}-3 \ln \left (\left (-3 i b c +4 a c \right ) x +4 i a +3 b \right ) b^{2}+3 \ln \left (\left (3 i b c -4 a c \right ) x +4 i a +3 b \right ) b^{2}-16 i a b +8 b^{2} \ln \left (-i c x +1\right )+16 a^{2}-8 b^{2}\right )}{32 d^{3} \left (c x -i\right )^{2} c}\)	\(624\)

Input:

int((a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/c*(1/2*I*a^2/d^3/(1+I*c*x)^2+b^2/d^3*(1/2*I/(1+I*c*x)^2*arctan(c*x)^2-I* 
(1/8*I*arctan(c*x)*ln(c*x+I)-1/8*I*arctan(c*x)*ln(c*x-I)-1/4*I*arctan(c*x) 
/(c*x-I)^2+1/4*arctan(c*x)/(c*x-I)-1/16*(ln(c*x+I)-ln(-1/2*I*(c*x+I)))*ln( 
-1/2*I*(-c*x+I))+1/32*ln(c*x+I)^2-3/16*I*arctan(c*x)-3/16*I/(c*x-I)-1/16/( 
c*x-I)^2-1/16*ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/32*ln(c*x-I)^2))+I*a*b/d^3/(1 
+I*c*x)^2*arctan(c*x)-1/4*I*a*b/d^3*arctan(c*x)-1/4*a*b/d^3/(c*x-I)^2-1/4* 
I*a*b/d^3/(c*x-I))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=-\frac {2 \, {\left (4 i \, a b + 3 \, b^{2}\right )} c x - {\left (i \, b^{2} c^{2} x^{2} + 2 \, b^{2} c x + 3 i \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 16 i \, a^{2} + 16 \, a b - 8 i \, b^{2} - {\left ({\left (4 \, a b - 3 i \, b^{2}\right )} c^{2} x^{2} - 2 \, {\left (4 i \, a b + b^{2}\right )} c x + 12 \, a b - 5 i \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{32 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \] Input:

integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="fricas")
 

Output:

-1/32*(2*(4*I*a*b + 3*b^2)*c*x - (I*b^2*c^2*x^2 + 2*b^2*c*x + 3*I*b^2)*log 
(-(c*x + I)/(c*x - I))^2 + 16*I*a^2 + 16*a*b - 8*I*b^2 - ((4*a*b - 3*I*b^2 
)*c^2*x^2 - 2*(4*I*a*b + b^2)*c*x + 12*a*b - 5*I*b^2)*log(-(c*x + I)/(c*x 
- I)))/(c^3*d^3*x^2 - 2*I*c^2*d^3*x - c*d^3)
 

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (144) = 288\).

Time = 36.17 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.58 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=- \frac {b \left (4 a - 3 i b\right ) \log {\left (- \frac {i b \left (4 a - 3 i b\right )}{c} + x \left (4 a b - 3 i b^{2}\right ) \right )}}{32 c d^{3}} + \frac {b \left (4 a - 3 i b\right ) \log {\left (\frac {i b \left (4 a - 3 i b\right )}{c} + x \left (4 a b - 3 i b^{2}\right ) \right )}}{32 c d^{3}} + \frac {\left (- 4 a b - b^{2} c x + 2 i b^{2}\right ) \log {\left (i c x + 1 \right )}}{8 c^{3} d^{3} x^{2} - 16 i c^{2} d^{3} x - 8 c d^{3}} + \frac {\left (i b^{2} c^{2} x^{2} + 2 b^{2} c x + 3 i b^{2}\right ) \log {\left (- i c x + 1 \right )}^{2}}{32 c^{3} d^{3} x^{2} - 64 i c^{2} d^{3} x - 32 c d^{3}} + \frac {\left (i b^{2} c^{2} x^{2} + 2 b^{2} c x + 3 i b^{2}\right ) \log {\left (i c x + 1 \right )}^{2}}{32 c^{3} d^{3} x^{2} - 64 i c^{2} d^{3} x - 32 c d^{3}} + \frac {- 8 i a^{2} - 8 a b + 4 i b^{2} + x \left (- 4 i a b c - 3 b^{2} c\right )}{16 c^{3} d^{3} x^{2} - 32 i c^{2} d^{3} x - 16 c d^{3}} + \frac {\left (8 a b - i b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} - 2 b^{2} c x \log {\left (i c x + 1 \right )} + 2 b^{2} c x - 3 i b^{2} \log {\left (i c x + 1 \right )} - 4 i b^{2}\right ) \log {\left (- i c x + 1 \right )}}{16 c^{3} d^{3} x^{2} - 32 i c^{2} d^{3} x - 16 c d^{3}} \] Input:

integrate((a+b*atan(c*x))**2/(d+I*c*d*x)**3,x)
 

Output:

-b*(4*a - 3*I*b)*log(-I*b*(4*a - 3*I*b)/c + x*(4*a*b - 3*I*b**2))/(32*c*d* 
*3) + b*(4*a - 3*I*b)*log(I*b*(4*a - 3*I*b)/c + x*(4*a*b - 3*I*b**2))/(32* 
c*d**3) + (-4*a*b - b**2*c*x + 2*I*b**2)*log(I*c*x + 1)/(8*c**3*d**3*x**2 
- 16*I*c**2*d**3*x - 8*c*d**3) + (I*b**2*c**2*x**2 + 2*b**2*c*x + 3*I*b**2 
)*log(-I*c*x + 1)**2/(32*c**3*d**3*x**2 - 64*I*c**2*d**3*x - 32*c*d**3) + 
(I*b**2*c**2*x**2 + 2*b**2*c*x + 3*I*b**2)*log(I*c*x + 1)**2/(32*c**3*d**3 
*x**2 - 64*I*c**2*d**3*x - 32*c*d**3) + (-8*I*a**2 - 8*a*b + 4*I*b**2 + x* 
(-4*I*a*b*c - 3*b**2*c))/(16*c**3*d**3*x**2 - 32*I*c**2*d**3*x - 16*c*d**3 
) + (8*a*b - I*b**2*c**2*x**2*log(I*c*x + 1) - 2*b**2*c*x*log(I*c*x + 1) + 
 2*b**2*c*x - 3*I*b**2*log(I*c*x + 1) - 4*I*b**2)*log(-I*c*x + 1)/(16*c**3 
*d**3*x**2 - 32*I*c**2*d**3*x - 16*c*d**3)
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=-\frac {{\left (4 i \, a b + 3 \, b^{2}\right )} c x - 2 \, {\left (-i \, b^{2} c^{2} x^{2} - 2 \, b^{2} c x - 3 i \, b^{2}\right )} \arctan \left (c x\right )^{2} + 8 i \, a^{2} + 8 \, a b - 4 i \, b^{2} + {\left ({\left (4 i \, a b + 3 \, b^{2}\right )} c^{2} x^{2} + 2 \, {\left (4 \, a b - i \, b^{2}\right )} c x + 12 i \, a b + 5 \, b^{2}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \] Input:

integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="maxima")
 

Output:

-1/16*((4*I*a*b + 3*b^2)*c*x - 2*(-I*b^2*c^2*x^2 - 2*b^2*c*x - 3*I*b^2)*ar 
ctan(c*x)^2 + 8*I*a^2 + 8*a*b - 4*I*b^2 + ((4*I*a*b + 3*b^2)*c^2*x^2 + 2*( 
4*a*b - I*b^2)*c*x + 12*I*a*b + 5*b^2)*arctan(c*x))/(c^3*d^3*x^2 - 2*I*c^2 
*d^3*x - c*d^3)
 

Giac [F]

\[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3}} \,d x } \] Input:

integrate((a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="giac")
 

Output:

integrate((b*arctan(c*x) + a)^2/(I*c*d*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \] Input:

int((a + b*atan(c*x))^2/(d + c*d*x*1i)^3,x)
 

Output:

int((a + b*atan(c*x))^2/(d + c*d*x*1i)^3, x)
 

Reduce [F]

\[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {-\mathit {atan} \left (c x \right )^{3} b^{2}-3 \mathit {atan} \left (c x \right )^{2} a b +24 \left (\int \frac {\mathit {atan} \left (c x \right )}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) a b c i +12 \left (\int \frac {\mathit {atan} \left (c x \right )^{2}}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) b^{2} c i +9 \left (\int \frac {x^{2}}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) a^{2} c^{3} i -6 \left (\int -\frac {\mathit {atan} \left (c x \right ) x^{3}}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) a b \,c^{4}-3 \left (\int -\frac {\mathit {atan} \left (c x \right )^{2} x^{3}}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) b^{2} c^{4}-18 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) a b \,c^{2}-9 \left (\int \frac {\mathit {atan} \left (c x \right )^{2} x}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) b^{2} c^{2}+9 \left (\int \frac {1}{c^{5} x^{5}-3 c^{4} i \,x^{4}-2 c^{3} x^{3}-2 c^{2} i \,x^{2}-3 c x +i}d x \right ) a^{2} c i}{9 c \,d^{3}} \] Input:

int((a+b*atan(c*x))^2/(d+I*c*d*x)^3,x)
 

Output:

( - atan(c*x)**3*b**2 - 3*atan(c*x)**2*a*b + 24*int(atan(c*x)/(c**5*x**5 - 
 3*c**4*i*x**4 - 2*c**3*x**3 - 2*c**2*i*x**2 - 3*c*x + i),x)*a*b*c*i + 12* 
int(atan(c*x)**2/(c**5*x**5 - 3*c**4*i*x**4 - 2*c**3*x**3 - 2*c**2*i*x**2 
- 3*c*x + i),x)*b**2*c*i + 9*int(x**2/(c**5*x**5 - 3*c**4*i*x**4 - 2*c**3* 
x**3 - 2*c**2*i*x**2 - 3*c*x + i),x)*a**2*c**3*i - 6*int(( - atan(c*x)*x** 
3)/(c**5*x**5 - 3*c**4*i*x**4 - 2*c**3*x**3 - 2*c**2*i*x**2 - 3*c*x + i),x 
)*a*b*c**4 - 3*int(( - atan(c*x)**2*x**3)/(c**5*x**5 - 3*c**4*i*x**4 - 2*c 
**3*x**3 - 2*c**2*i*x**2 - 3*c*x + i),x)*b**2*c**4 - 18*int((atan(c*x)*x)/ 
(c**5*x**5 - 3*c**4*i*x**4 - 2*c**3*x**3 - 2*c**2*i*x**2 - 3*c*x + i),x)*a 
*b*c**2 - 9*int((atan(c*x)**2*x)/(c**5*x**5 - 3*c**4*i*x**4 - 2*c**3*x**3 
- 2*c**2*i*x**2 - 3*c*x + i),x)*b**2*c**2 + 9*int(1/(c**5*x**5 - 3*c**4*i* 
x**4 - 2*c**3*x**3 - 2*c**2*i*x**2 - 3*c*x + i),x)*a**2*c*i)/(9*c*d**3)