\(\int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx\) [92]

Optimal result

Integrand size = 25, antiderivative size = 293 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=-\frac {b^2 c^2 d^3}{12 x^2}-\frac {i b^2 c^3 d^3}{x}-i b^2 c^4 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-\frac {11}{3} b^2 c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {11}{6} b^2 c^4 d^3 \log \left (1+c^2 x^2\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \] Output:

-1/12*b^2*c^2*d^3/x^2-I*b^2*c^3*d^3/x-I*b^2*c^4*d^3*arctan(c*x)-1/6*b*c*d^ 
3*(a+b*arctan(c*x))/x^3-I*b*c^2*d^3*(a+b*arctan(c*x))/x^2+7/2*b*c^3*d^3*(a 
+b*arctan(c*x))/x-1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^4-4*I*a*b*c^4* 
d^3*ln(x)-11/3*b^2*c^4*d^3*ln(x)-4*I*b*c^4*d^3*(a+b*arctan(c*x))*ln(2/(1-I 
*c*x))+11/6*b^2*c^4*d^3*ln(c^2*x^2+1)+2*b^2*c^4*d^3*polylog(2,-I*c*x)-2*b^ 
2*c^4*d^3*polylog(2,I*c*x)-2*b^2*c^4*d^3*polylog(2,1-2/(1-I*c*x))
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.10 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\frac {d^3 \left (-3 a^2-12 i a^2 c x-2 a b c x+18 a^2 c^2 x^2-12 i a b c^2 x^2-b^2 c^2 x^2+12 i a^2 c^3 x^3+42 a b c^3 x^3-12 i b^2 c^3 x^3-b^2 c^4 x^4-3 b^2 (-i+c x)^4 \arctan (c x)^2+2 b \arctan (c x) \left (b c x \left (-1-6 i c x+21 c^2 x^2-6 i c^3 x^3\right )+3 a \left (-1-4 i c x+6 c^2 x^2+4 i c^3 x^3+7 c^4 x^4\right )-24 i b c^4 x^4 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-48 i a b c^4 x^4 \log (c x)-44 b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+24 i a b c^4 x^4 \log \left (1+c^2 x^2\right )-24 b^2 c^4 x^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{12 x^4} \] Input:

Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]
 

Output:

(d^3*(-3*a^2 - (12*I)*a^2*c*x - 2*a*b*c*x + 18*a^2*c^2*x^2 - (12*I)*a*b*c^ 
2*x^2 - b^2*c^2*x^2 + (12*I)*a^2*c^3*x^3 + 42*a*b*c^3*x^3 - (12*I)*b^2*c^3 
*x^3 - b^2*c^4*x^4 - 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b 
*c*x*(-1 - (6*I)*c*x + 21*c^2*x^2 - (6*I)*c^3*x^3) + 3*a*(-1 - (4*I)*c*x + 
 6*c^2*x^2 + (4*I)*c^3*x^3 + 7*c^4*x^4) - (24*I)*b*c^4*x^4*Log[1 - E^((2*I 
)*ArcTan[c*x])]) - (48*I)*a*b*c^4*x^4*Log[c*x] - 44*b^2*c^4*x^4*Log[(c*x)/ 
Sqrt[1 + c^2*x^2]] + (24*I)*a*b*c^4*x^4*Log[1 + c^2*x^2] - 24*b^2*c^4*x^4* 
PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(12*x^4)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5409, 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.

\(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx\)

\(\Big \downarrow \) 5409

\(\displaystyle -2 b c \int \left (-\frac {2 i d^3 (a+b \arctan (c x)) c^4}{c x+i}+\frac {2 i d^3 (a+b \arctan (c x)) c^3}{x}+\frac {7 d^3 (a+b \arctan (c x)) c^2}{4 x^2}-\frac {i d^3 (a+b \arctan (c x)) c}{x^3}-\frac {d^3 (a+b \arctan (c x))}{4 x^4}\right )dx-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 b c \left (2 i c^3 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {7 c^2 d^3 (a+b \arctan (c x))}{4 x}+\frac {d^3 (a+b \arctan (c x))}{12 x^3}+\frac {i c d^3 (a+b \arctan (c x))}{2 x^2}+2 i a c^3 d^3 \log (x)+\frac {1}{2} i b c^3 d^3 \arctan (c x)-b c^3 d^3 \operatorname {PolyLog}(2,-i c x)+b c^3 d^3 \operatorname {PolyLog}(2,i c x)+b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {11}{6} b c^3 d^3 \log (x)+\frac {i b c^2 d^3}{2 x}-\frac {11}{12} b c^3 d^3 \log \left (c^2 x^2+1\right )+\frac {b c d^3}{24 x^2}\right )-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}\)

Input:

Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]
 

Output:

-1/4*(d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/x^4 - 2*b*c*((b*c*d^3)/(24* 
x^2) + ((I/2)*b*c^2*d^3)/x + (I/2)*b*c^3*d^3*ArcTan[c*x] + (d^3*(a + b*Arc 
Tan[c*x]))/(12*x^3) + ((I/2)*c*d^3*(a + b*ArcTan[c*x]))/x^2 - (7*c^2*d^3*( 
a + b*ArcTan[c*x]))/(4*x) + (2*I)*a*c^3*d^3*Log[x] + (11*b*c^3*d^3*Log[x]) 
/6 + (2*I)*c^3*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] - (11*b*c^3*d^3* 
Log[1 + c^2*x^2])/12 - b*c^3*d^3*PolyLog[2, (-I)*c*x] + b*c^3*d^3*PolyLog[ 
2, I*c*x] + b*c^3*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])
 

Defintions of rubi rules used

rule 2009

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

rule 5409

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e_ 
.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Sim 
p[(a + b*ArcTan[c*x])^p   u, x] - Simp[b*c*p   Int[ExpandIntegrand[(a + b*A 
rcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, 
f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[ 
m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
 

Maple [A] (verified)

Time = 1.94 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.71

method	result	size
parts	\(d^{3} a^{2} \left (\frac {3 c^{2}}{2 x^{2}}-\frac {i c}{x^{3}}+\frac {i c^{3}}{x}-\frac {1}{4 x^{4}}\right )+d^{3} b^{2} c^{4} \left (\frac {7 \arctan \left (c x \right )}{2 c x}+\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-i \arctan \left (c x \right )-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}+\frac {7 \arctan \left (c x \right )^{2}}{4}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {i}{c x}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 d^{3} a b \,c^{4} \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\)	\(501\)
derivativedivides	\(c^{4} \left (d^{3} a^{2} \left (\frac {3}{2 c^{2} x^{2}}-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}\right )+d^{3} b^{2} \left (\frac {7 \arctan \left (c x \right )}{2 c x}+\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-i \arctan \left (c x \right )-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}+\frac {7 \arctan \left (c x \right )^{2}}{4}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {i}{c x}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 d^{3} a b \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\right )\)	\(504\)
default	\(c^{4} \left (d^{3} a^{2} \left (\frac {3}{2 c^{2} x^{2}}-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}\right )+d^{3} b^{2} \left (\frac {7 \arctan \left (c x \right )}{2 c x}+\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-i \arctan \left (c x \right )-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}+\frac {7 \arctan \left (c x \right )^{2}}{4}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {i}{c x}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 d^{3} a b \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\right )\)	\(504\)

Input:

int((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x,method=_RETURNVERBOSE)
 

Output:

d^3*a^2*(3/2*c^2/x^2-I*c/x^3+I*c^3/x-1/4/x^4)+d^3*b^2*c^4*(7/2/c/x*arctan( 
c*x)+3/2/c^2/x^2*arctan(c*x)^2-1/6/c^3/x^3*arctan(c*x)+2*dilog(1+I*c*x)-2* 
dilog(1-I*c*x)-11/3*ln(c*x)-1/12/c^2/x^2+2*I*arctan(c*x)*ln(c^2*x^2+1)-4*I 
*arctan(c*x)*ln(c*x)+7/4*arctan(c*x)^2+11/6*ln(c^2*x^2+1)-1/4/c^4/x^4*arct 
an(c*x)^2+1/2*ln(c*x-I)^2+dilog(-1/2*I*(c*x+I))-1/2*ln(c*x+I)^2-dilog(1/2* 
I*(c*x-I))-ln(c*x-I)*ln(c^2*x^2+1)+ln(c*x-I)*ln(-1/2*I*(c*x+I))+ln(c*x+I)* 
ln(c^2*x^2+1)-ln(c*x+I)*ln(1/2*I*(c*x-I))+I*arctan(c*x)^2/c/x-I*arctan(c*x 
)^2/c^3/x^3-I*arctan(c*x)/c^2/x^2-I/c/x-I*arctan(c*x)+2*ln(c*x)*ln(1+I*c*x 
)-2*ln(c*x)*ln(1-I*c*x))+2*d^3*a*b*c^4*(3/2/c^2/x^2*arctan(c*x)-1/4/c^4/x^ 
4*arctan(c*x)-I*arctan(c*x)/c^3/x^3+I*arctan(c*x)/c/x+I*ln(c^2*x^2+1)+7/4* 
arctan(c*x)-1/2*I/c^2/x^2-2*I*ln(c*x)-1/12/c^3/x^3+7/4/c/x)
 

Fricas [F]

\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="fricas")
 

Output:

1/16*(16*x^4*integral(1/4*(-4*I*a^2*c^5*d^3*x^5 - 12*a^2*c^4*d^3*x^4 + 8*I 
*a^2*c^3*d^3*x^3 - 8*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (4*a 
*b*c^5*d^3*x^5 - 4*(3*I*a*b - b^2)*c^4*d^3*x^4 - 2*(4*a*b + 3*I*b^2)*c^3*d 
^3*x^3 - 4*(2*I*a*b + b^2)*c^2*d^3*x^2 - (12*a*b - I*b^2)*c*d^3*x + 4*I*a* 
b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^7 + x^5), x) + (-4*I*b^2*c^3*d^3* 
x^3 - 6*b^2*c^2*d^3*x^2 + 4*I*b^2*c*d^3*x + b^2*d^3)*log(-(c*x + I)/(c*x - 
 I))^2)/x^4
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\text {Timed out} \] Input:

integrate((d+I*c*d*x)**3*(a+b*atan(c*x))**2/x**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="maxima")
 

Output:

I*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*a*b*c^3*d^3 + 3*((c* 
arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*c^2*d^3 + I*((c^2*log(c^2*x^2 
+ 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a*b*c*d^3 + I*a^2*c^3* 
d^3/x + 1/6*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x 
^4)*a*b*d^3 + 1/12*(2*(3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c*arctan(c 
*x) - (3*c^2*x^2*arctan(c*x)^2 - 4*c^2*x^2*log(c^2*x^2 + 1) + 8*c^2*x^2*lo 
g(x) + 1)*c^2/x^2)*b^2*d^3 + 3/2*a^2*c^2*d^3/x^2 - I*a^2*c*d^3/x^3 - 1/4*b 
^2*d^3*arctan(c*x)^2/x^4 - 1/4*a^2*d^3/x^4 - 1/32*(8*I*(b^2*c^4*d^3*arctan 
(c*x)^3 + 4*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^6 + x 
^4), x) - 16*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^6 + x^ 
4), x) - 48*b^2*c^4*d^3*integrate(1/16*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c 
^2*x^6 + x^4), x) + 80*b^2*c^4*d^3*integrate(1/16*x^3*arctan(c*x)/(c^2*x^6 
 + x^4), x) - 96*b^2*c^3*d^3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^6 + x 
^4), x) - 8*b^2*c^3*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^6 + x 
^4), x) + 40*b^2*c^3*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^6 + x^ 
4), x) - 48*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2 
*x^6 + x^4), x) - 32*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^6 + x 
^4), x) - 144*b^2*c*d^3*integrate(1/16*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 
 12*b^2*c*d^3*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x))*x^3 - 
 8*(b^2*c^4*d^3*arctan(c*x)^2 - 16*b^2*c^5*d^3*integrate(1/16*x^4*arcta...
 

Giac [F]

\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x, algorithm="giac")
 

Output:

integrate((I*c*d*x + d)^3*(b*arctan(c*x) + a)^2/x^5, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^5} \,d x \] Input:

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^5,x)
 

Output:

int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^5, x)
 

Reduce [F]

\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\frac {d^{3} \left (-24 \mathit {atan} \left (c x \right ) a b c i x +12 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} i \,x^{3}+22 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2} c^{4} x^{4}-44 \,\mathrm {log}\left (x \right ) b^{2} c^{4} x^{4}+42 \mathit {atan} \left (c x \right ) a b \,c^{4} x^{4}-12 \mathit {atan} \left (c x \right )^{2} b^{2} c i x -12 \mathit {atan} \left (c x \right ) b^{2} c^{4} i \,x^{4}-12 \mathit {atan} \left (c x \right ) b^{2} c^{2} i \,x^{2}-12 a b \,c^{2} i \,x^{2}-2 \mathit {atan} \left (c x \right ) b^{2} c x +42 a b \,c^{3} x^{3}-2 a b c x -3 a^{2}+18 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} x^{2}+24 \mathit {atan} \left (c x \right ) a b \,c^{3} i \,x^{3}+12 a^{2} c^{3} i \,x^{3}+18 a^{2} c^{2} x^{2}+36 \mathit {atan} \left (c x \right ) a b \,c^{2} x^{2}+21 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} x^{4}-12 a^{2} c i x +42 \mathit {atan} \left (c x \right ) b^{2} c^{3} x^{3}-12 b^{2} c^{3} i \,x^{3}-48 \left (\int \frac {\mathit {atan} \left (c x \right )}{c^{2} x^{3}+x}d x \right ) b^{2} c^{4} i \,x^{4}-3 \mathit {atan} \left (c x \right )^{2} b^{2}+24 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b \,c^{4} i \,x^{4}-48 \,\mathrm {log}\left (x \right ) a b \,c^{4} i \,x^{4}-6 \mathit {atan} \left (c x \right ) a b -b^{2} c^{2} x^{2}\right )}{12 x^{4}} \] Input:

int((d+I*c*d*x)^3*(a+b*atan(c*x))^2/x^5,x)
 

Output:

(d**3*(21*atan(c*x)**2*b**2*c**4*x**4 + 12*atan(c*x)**2*b**2*c**3*i*x**3 + 
 18*atan(c*x)**2*b**2*c**2*x**2 - 12*atan(c*x)**2*b**2*c*i*x - 3*atan(c*x) 
**2*b**2 + 42*atan(c*x)*a*b*c**4*x**4 + 24*atan(c*x)*a*b*c**3*i*x**3 + 36* 
atan(c*x)*a*b*c**2*x**2 - 24*atan(c*x)*a*b*c*i*x - 6*atan(c*x)*a*b - 12*at 
an(c*x)*b**2*c**4*i*x**4 + 42*atan(c*x)*b**2*c**3*x**3 - 12*atan(c*x)*b**2 
*c**2*i*x**2 - 2*atan(c*x)*b**2*c*x - 48*int(atan(c*x)/(c**2*x**3 + x),x)* 
b**2*c**4*i*x**4 + 24*log(c**2*x**2 + 1)*a*b*c**4*i*x**4 + 22*log(c**2*x** 
2 + 1)*b**2*c**4*x**4 - 48*log(x)*a*b*c**4*i*x**4 - 44*log(x)*b**2*c**4*x* 
*4 + 12*a**2*c**3*i*x**3 + 18*a**2*c**2*x**2 - 12*a**2*c*i*x - 3*a**2 + 42 
*a*b*c**3*x**3 - 12*a*b*c**2*i*x**2 - 2*a*b*c*x - 12*b**2*c**3*i*x**3 - b* 
*2*c**2*x**2))/(12*x**4)