\(\int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx\) [40]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {b c d^4}{20 x^4}-\frac {i b c^2 d^4}{3 x^3}+\frac {11 b c^3 d^4}{10 x^2}+\frac {3 i b c^4 d^4}{x}-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {16}{5} b c^5 d^4 \log (x)-\frac {16}{5} b c^5 d^4 \log (i+c x) \]

[Out]

-1/20*b*c*d^4/x^4-1/3*I*b*c^2*d^4/x^3+11/10*b*c^3*d^4/x^2+3*I*b*c^4*d^4/x-1/5*d^4*(1+I*c*x)^5*(a+b*arctan(c*x)
)/x^5+16/5*b*c^5*d^4*ln(x)-16/5*b*c^5*d^4*ln(I+c*x)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {37, 4992, 12, 90} \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {16}{5} b c^5 d^4 \log (x)-\frac {16}{5} b c^5 d^4 \log (c x+i)+\frac {3 i b c^4 d^4}{x}+\frac {11 b c^3 d^4}{10 x^2}-\frac {i b c^2 d^4}{3 x^3}-\frac {b c d^4}{20 x^4} \]

[In]

Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]

[Out]

-1/20*(b*c*d^4)/x^4 - ((I/3)*b*c^2*d^4)/x^3 + (11*b*c^3*d^4)/(10*x^2) + ((3*I)*b*c^4*d^4)/x - (d^4*(1 + I*c*x)
^5*(a + b*ArcTan[c*x]))/(5*x^5) + (16*b*c^5*d^4*Log[x])/5 - (16*b*c^5*d^4*Log[I + c*x])/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 4992

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}-(b c) \int -\frac {i d^4 (i-c x)^4}{5 x^5 (i+c x)} \, dx \\ & = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {1}{5} \left (i b c d^4\right ) \int \frac {(i-c x)^4}{x^5 (i+c x)} \, dx \\ & = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {1}{5} \left (i b c d^4\right ) \int \left (-\frac {i}{x^5}+\frac {5 c}{x^4}+\frac {11 i c^2}{x^3}-\frac {15 c^3}{x^2}-\frac {16 i c^4}{x}+\frac {16 i c^5}{i+c x}\right ) \, dx \\ & = -\frac {b c d^4}{20 x^4}-\frac {i b c^2 d^4}{3 x^3}+\frac {11 b c^3 d^4}{10 x^2}+\frac {3 i b c^4 d^4}{x}-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {16}{5} b c^5 d^4 \log (x)-\frac {16}{5} b c^5 d^4 \log (i+c x) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.63 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {d^4 \left (20 i b c^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )+3 \left (4 a+20 i a c x+b c x-40 a c^2 x^2-40 i a c^3 x^3-22 b c^3 x^3+20 a c^4 x^4+4 b \left (1+5 i c x-10 c^2 x^2-10 i c^3 x^3+5 c^4 x^4\right ) \arctan (c x)-40 i b c^4 x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )-64 b c^5 x^5 \log (x)+32 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )\right )}{60 x^5} \]

[In]

Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]

[Out]

-1/60*(d^4*((20*I)*b*c^2*x^2*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 3*(4*a + (20*I)*a*c*x + b*c*x - 40
*a*c^2*x^2 - (40*I)*a*c^3*x^3 - 22*b*c^3*x^3 + 20*a*c^4*x^4 + 4*b*(1 + (5*I)*c*x - 10*c^2*x^2 - (10*I)*c^3*x^3
+ 5*c^4*x^4)*ArcTan[c*x] - (40*I)*b*c^4*x^4*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - 64*b*c^5*x^5*Log[x]
+ 32*b*c^5*x^5*Log[1 + c^2*x^2])))/x^5

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.50

method	result	size
parts	\(d^{4} a \left (\frac {2 i c^{3}}{x^{2}}-\frac {i c}{x^{4}}-\frac {1}{5 x^{5}}-\frac {c^{4}}{x}+\frac {2 c^{2}}{x^{3}}\right )+d^{4} b \,c^{5} \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{c x}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )\right )\)	\(175\)
derivativedivides	\(c^{5} \left (d^{4} a \left (-\frac {1}{5 c^{5} x^{5}}-\frac {i}{c^{4} x^{4}}+\frac {2}{c^{3} x^{3}}-\frac {1}{c x}+\frac {2 i}{c^{2} x^{2}}\right )+d^{4} b \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{c x}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )\right )\right )\)	\(181\)
default	\(c^{5} \left (d^{4} a \left (-\frac {1}{5 c^{5} x^{5}}-\frac {i}{c^{4} x^{4}}+\frac {2}{c^{3} x^{3}}-\frac {1}{c x}+\frac {2 i}{c^{2} x^{2}}\right )+d^{4} b \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{c x}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )\right )\right )\)	\(181\)
parallelrisch	\(\frac {180 i x^{4} b \,c^{4} d^{4}+120 i x^{3} \arctan \left (c x \right ) b \,c^{3} d^{4}-96 b \,c^{5} d^{4} \ln \left (c^{2} x^{2}+1\right ) x^{5}+192 b \,c^{5} d^{4} \ln \left (x \right ) x^{5}-66 b \,c^{5} d^{4} x^{5}+180 i c^{5} b \,d^{4} \arctan \left (c x \right ) x^{5}-60 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{4}-60 i x \arctan \left (c x \right ) b c \,d^{4}-60 a \,c^{4} d^{4} x^{4}-60 i a c \,d^{4} x +66 b \,c^{3} d^{4} x^{3}-120 i x^{5} a \,c^{5} d^{4}+120 x^{2} \arctan \left (c x \right ) b \,c^{2} d^{4}+120 i x^{3} a \,c^{3} d^{4}+120 x^{2} d^{4} c^{2} a -20 i x^{2} b \,c^{2} d^{4}-3 b c \,d^{4} x -12 b \,d^{4} \arctan \left (c x \right )-12 d^{4} a}{60 x^{5}}\)	\(255\)
risch	\(\frac {i d^{4} b \left (5 c^{4} x^{4}-10 i c^{3} x^{3}-10 c^{2} x^{2}+5 i c x +1\right ) \ln \left (i c x +1\right )}{10 x^{5}}-\frac {d^{4} \left (186 b \,c^{5} \ln \left (-c x -i\right ) x^{5}+6 b \,c^{5} \ln \left (c x -i\right ) x^{5}-192 b \,c^{5} \ln \left (-x \right ) x^{5}-60 i b \,x^{2} \ln \left (-i c x +1\right ) c^{2}+20 i b \,c^{2} x^{2}+60 a \,c^{4} x^{4}+30 i b \,c^{4} x^{4} \ln \left (-i c x +1\right )+60 b \,c^{3} x^{3} \ln \left (-i c x +1\right )-180 i b \,c^{4} x^{4}-66 b \,c^{3} x^{3}-120 i a \,c^{3} x^{3}-120 c^{2} x^{2} a +60 i x a c -30 b c x \ln \left (-i c x +1\right )+6 i b \ln \left (-i c x +1\right )+3 x b c +12 a \right )}{60 x^{5}}\)	\(256\)

[In]

int((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x,method=_RETURNVERBOSE)

[Out]

d^4*a*(2*I*c^3/x^2-I*c/x^4-1/5/x^5-c^4/x+2*c^2/x^3)+d^4*b*c^5*(-1/5/c^5/x^5*arctan(c*x)-I*arctan(c*x)/c^4/x^4+
2*arctan(c*x)/c^3/x^3-1/c/x*arctan(c*x)+2*I*arctan(c*x)/c^2/x^2-1/3*I/c^3/x^3+3*I/c/x-1/20/c^4/x^4+11/10/c^2/x
^2+16/5*ln(c*x)-8/5*ln(c^2*x^2+1)+3*I*arctan(c*x))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (97) = 194\).

Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.73 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {192 \, b c^{5} d^{4} x^{5} \log \left (x\right ) - 186 \, b c^{5} d^{4} x^{5} \log \left (\frac {c x + i}{c}\right ) - 6 \, b c^{5} d^{4} x^{5} \log \left (\frac {c x - i}{c}\right ) - 60 \, {\left (a - 3 i \, b\right )} c^{4} d^{4} x^{4} - 6 \, {\left (-20 i \, a - 11 \, b\right )} c^{3} d^{4} x^{3} + 20 \, {\left (6 \, a - i \, b\right )} c^{2} d^{4} x^{2} - 3 \, {\left (20 i \, a + b\right )} c d^{4} x - 12 \, a d^{4} - 6 \, {\left (5 i \, b c^{4} d^{4} x^{4} + 10 \, b c^{3} d^{4} x^{3} - 10 i \, b c^{2} d^{4} x^{2} - 5 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{60 \, x^{5}} \]

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="fricas")

[Out]

1/60*(192*b*c^5*d^4*x^5*log(x) - 186*b*c^5*d^4*x^5*log((c*x + I)/c) - 6*b*c^5*d^4*x^5*log((c*x - I)/c) - 60*(a
- 3*I*b)*c^4*d^4*x^4 - 6*(-20*I*a - 11*b)*c^3*d^4*x^3 + 20*(6*a - I*b)*c^2*d^4*x^2 - 3*(20*I*a + b)*c*d^4*x -
12*a*d^4 - 6*(5*I*b*c^4*d^4*x^4 + 10*b*c^3*d^4*x^3 - 10*I*b*c^2*d^4*x^2 - 5*b*c*d^4*x + I*b*d^4)*log(-(c*x +
I)/(c*x - I)))/x^5

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. 366 vs. \(2 (114) = 228\).

Time = 38.18 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.13 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {16 b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x \right )}}{5} - \frac {b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x - 10395 i b^{2} c^{10} d^{8} \right )}}{10} - \frac {31 b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x + 10395 i b^{2} c^{10} d^{8} \right )}}{10} + \frac {- 12 a d^{4} + x^{4} \left (- 60 a c^{4} d^{4} + 180 i b c^{4} d^{4}\right ) + x^{3} \cdot \left (120 i a c^{3} d^{4} + 66 b c^{3} d^{4}\right ) + x^{2} \cdot \left (120 a c^{2} d^{4} - 20 i b c^{2} d^{4}\right ) + x \left (- 60 i a c d^{4} - 3 b c d^{4}\right )}{60 x^{5}} + \frac {\left (- 5 i b c^{4} d^{4} x^{4} - 10 b c^{3} d^{4} x^{3} + 10 i b c^{2} d^{4} x^{2} + 5 b c d^{4} x - i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{10 x^{5}} + \frac {\left (5 i b c^{4} d^{4} x^{4} + 10 b c^{3} d^{4} x^{3} - 10 i b c^{2} d^{4} x^{2} - 5 b c d^{4} x + i b d^{4}\right ) \log {\left (i c x + 1 \right )}}{10 x^{5}} \]

[In]

integrate((d+I*c*d*x)**4*(a+b*atan(c*x))/x**6,x)

[Out]

16*b*c**5*d**4*log(10395*b**2*c**11*d**8*x)/5 - b*c**5*d**4*log(10395*b**2*c**11*d**8*x - 10395*I*b**2*c**10*d
**8)/10 - 31*b*c**5*d**4*log(10395*b**2*c**11*d**8*x + 10395*I*b**2*c**10*d**8)/10 + (-12*a*d**4 + x**4*(-60*a
*c**4*d**4 + 180*I*b*c**4*d**4) + x**3*(120*I*a*c**3*d**4 + 66*b*c**3*d**4) + x**2*(120*a*c**2*d**4 - 20*I*b*c
**2*d**4) + x*(-60*I*a*c*d**4 - 3*b*c*d**4))/(60*x**5) + (-5*I*b*c**4*d**4*x**4 - 10*b*c**3*d**4*x**3 + 10*I*b
*c**2*d**4*x**2 + 5*b*c*d**4*x - I*b*d**4)*log(-I*c*x + 1)/(10*x**5) + (5*I*b*c**4*d**4*x**4 + 10*b*c**3*d**4*
x**3 - 10*I*b*c**2*d**4*x**2 - 5*b*c*d**4*x + I*b*d**4)*log(I*c*x + 1)/(10*x**5)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (97) = 194\).

Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.35 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c^{4} d^{4} + 2 i \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{3} d^{4} - {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{2} d^{4} - \frac {a c^{4} d^{4}}{x} + \frac {1}{3} i \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c d^{4} - \frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{4} + \frac {2 i \, a c^{3} d^{4}}{x^{2}} + \frac {2 \, a c^{2} d^{4}}{x^{3}} - \frac {i \, a c d^{4}}{x^{4}} - \frac {a d^{4}}{5 \, x^{5}} \]

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="maxima")

[Out]

-1/2*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*b*c^4*d^4 + 2*I*((c*arctan(c*x) + 1/x)*c + arctan(c*x
)/x^2)*b*c^3*d^4 - ((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*b*c^2*d^4 - a*c^4*d^4
/x + 1/3*I*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x^4)*b*c*d^4 - 1/20*((2*c^4*log(c^2*x^
2 + 1) - 2*c^4*log(x^2) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*d^4 + 2*I*a*c^3*d^4/x^2 + 2*a*c^2*d^4/
x^3 - I*a*c*d^4/x^4 - 1/5*a*d^4/x^5

Giac [F]

\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{6}} \,d x } \]

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.59 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {d^4\,\left (192\,b\,c^5\,\ln \left (x\right )-96\,b\,c^5\,\ln \left (c^2\,x^2+1\right )+b\,c^5\,\mathrm {atan}\left (c\,x\right )\,180{}\mathrm {i}\right )}{60}-\frac {\frac {d^4\,\left (12\,a+12\,b\,\mathrm {atan}\left (c\,x\right )\right )}{60}+\frac {d^4\,x\,\left (a\,c\,60{}\mathrm {i}+3\,b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{60}-\frac {d^4\,x^2\,\left (120\,a\,c^2+120\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,20{}\mathrm {i}\right )}{60}+\frac {d^4\,x^4\,\left (60\,a\,c^4+60\,b\,c^4\,\mathrm {atan}\left (c\,x\right )-b\,c^4\,180{}\mathrm {i}\right )}{60}-\frac {d^4\,x^3\,\left (a\,c^3\,120{}\mathrm {i}+66\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,120{}\mathrm {i}\right )}{60}}{x^5} \]

[In]

int(((a + b*atan(c*x))*(d + c*d*x*1i)^4)/x^6,x)

[Out]

(d^4*(b*c^5*atan(c*x)*180i - 96*b*c^5*log(c^2*x^2 + 1) + 192*b*c^5*log(x)))/60 - ((d^4*(12*a + 12*b*atan(c*x))
)/60 + (d^4*x*(a*c*60i + 3*b*c + b*c*atan(c*x)*60i))/60 - (d^4*x^2*(120*a*c^2 - b*c^2*20i + 120*b*c^2*atan(c*x
)))/60 + (d^4*x^4*(60*a*c^4 - b*c^4*180i + 60*b*c^4*atan(c*x)))/60 - (d^4*x^3*(a*c^3*120i + 66*b*c^3 + b*c^3*a
tan(c*x)*120i))/60)/x^5