Integrand size = 23, antiderivative size = 117 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {b c d^4}{20 x^4}-\frac {i b c^2 d^4}{3 x^3}+\frac {11 b c^3 d^4}{10 x^2}+\frac {3 i b c^4 d^4}{x}-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}+\frac {16}{5} b c^5 d^4 \log (x)-\frac {16}{5} b c^5 d^4 \log (i+c x) \] Output:
-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)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.63 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {d^4 \left (20 i b c^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )+3 \left (4 a+20 i a c x+b c x-40 a c^2 x^2-40 i a c^3 x^3-22 b c^3 x^3+20 a c^4 x^4+4 b \left (1+5 i c x-10 c^2 x^2-10 i c^3 x^3+5 c^4 x^4\right ) \arctan (c x)-40 i b c^4 x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )-64 b c^5 x^5 \log (x)+32 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )\right )}{60 x^5} \] Input:
Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]
Output:
-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
Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5407, 27, 99, 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.
| \(\displaystyle \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx\) |
| \(\Big \downarrow \) 5407 |
| \(\displaystyle -b c \int -\frac {i d^4 (i-c x)^4}{5 x^5 (c x+i)}dx-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} i b c d^4 \int \frac {(i-c x)^4}{x^5 (c x+i)}dx-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{5} i b c d^4 \int \left (\frac {16 i c^5}{c x+i}-\frac {16 i c^4}{x}-\frac {15 c^3}{x^2}+\frac {11 i c^2}{x^3}+\frac {5 c}{x^4}-\frac {i}{x^5}\right )dx-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} i b c d^4 \left (-16 i c^4 \log (x)+16 i c^4 \log (c x+i)+\frac {15 c^3}{x}-\frac {11 i c^2}{2 x^2}-\frac {5 c}{3 x^3}+\frac {i}{4 x^4}\right )-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 x^5}\) |
Input:
Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]
Output:
-1/5*(d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/x^5 + (I/5)*b*c*d^4*((I/4)/x^ 4 - (5*c)/(3*x^3) - (((11*I)/2)*c^2)/x^2 + (15*c^3)/x - (16*I)*c^4*Log[x] + (16*I)*c^4*Log[I + c*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[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]) )
Time = 0.70 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.50
| method | result | size |
| parts | \(d^{4} a \left (\frac {2 i c^{3}}{x^{2}}+\frac {2 c^{2}}{x^{3}}-\frac {c^{4}}{x}-\frac {1}{5 x^{5}}-\frac {i c}{x^{4}}\right )+d^{4} b \,c^{5} \left (-\frac {\arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}\right )\) | \(175\) |
| derivativedivides | \(c^{5} \left (d^{4} a \left (-\frac {1}{c x}-\frac {1}{5 c^{5} x^{5}}+\frac {2}{c^{3} x^{3}}+\frac {2 i}{c^{2} x^{2}}-\frac {i}{c^{4} x^{4}}\right )+d^{4} b \left (-\frac {\arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}\right )\right )\) | \(181\) |
| default | \(c^{5} \left (d^{4} a \left (-\frac {1}{c x}-\frac {1}{5 c^{5} x^{5}}+\frac {2}{c^{3} x^{3}}+\frac {2 i}{c^{2} x^{2}}-\frac {i}{c^{4} x^{4}}\right )+d^{4} b \left (-\frac {\arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {2 \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {2 i \arctan \left (c x \right )}{c^{2} x^{2}}-\frac {i \arctan \left (c x \right )}{c^{4} x^{4}}-\frac {8 \ln \left (c^{2} x^{2}+1\right )}{5}+3 i \arctan \left (c x \right )-\frac {i}{3 c^{3} x^{3}}+\frac {3 i}{c x}-\frac {1}{20 c^{4} x^{4}}+\frac {11}{10 c^{2} x^{2}}+\frac {16 \ln \left (c x \right )}{5}\right )\right )\) | \(181\) |
| parallelrisch | \(\frac {180 i x^{4} b \,c^{4} d^{4}-60 i x \arctan \left (c x \right ) b c \,d^{4}+192 b \,c^{5} d^{4} \ln \left (x \right ) x^{5}-96 b \,c^{5} d^{4} \ln \left (c^{2} x^{2}+1\right ) x^{5}-66 b \,c^{5} d^{4} x^{5}+180 i c^{5} b \,d^{4} \arctan \left (c x \right ) x^{5}-60 d^{4} b \arctan \left (c x \right ) x^{4} c^{4}-20 i x^{2} b \,c^{2} d^{4}-60 a \,c^{4} d^{4} x^{4}+120 i c^{3} b \,d^{4} \arctan \left (c x \right ) x^{3}+66 b \,c^{3} d^{4} x^{3}-120 i x^{5} a \,c^{5} d^{4}+120 d^{4} b \arctan \left (c x \right ) x^{2} c^{2}-60 i x a c \,d^{4}+120 a \,c^{2} d^{4} x^{2}+120 i x^{3} a \,c^{3} d^{4}-3 b c \,d^{4} x -12 b \,d^{4} \arctan \left (c x \right )-12 d^{4} a}{60 x^{5}}\) | \(255\) |
| risch | \(\frac {i d^{4} b \left (5 c^{4} x^{4}-10 i c^{3} x^{3}-10 c^{2} x^{2}+5 i c x +1\right ) \ln \left (i c x +1\right )}{10 x^{5}}-\frac {d^{4} \left (186 b \,c^{5} \ln \left (-c x -i\right ) x^{5}+6 b \,c^{5} \ln \left (c x -i\right ) x^{5}-192 b \,c^{5} \ln \left (-x \right ) x^{5}-180 i c^{4} x^{4} b -60 i b \,c^{2} x^{2} \ln \left (-i c x +1\right )+60 c^{4} x^{4} a +60 i a c x +60 b \,c^{3} x^{3} \ln \left (-i c x +1\right )+20 i b \,c^{2} x^{2}-66 b \,c^{3} x^{3}+30 i b \,c^{4} x^{4} \ln \left (-i c x +1\right )-120 a \,c^{2} x^{2}+6 i b \ln \left (-i c x +1\right )-30 b c x \ln \left (-i c x +1\right )-120 i a \,c^{3} x^{3}+3 b c x +12 a \right )}{60 x^{5}}\) | \(256\) |
Input:
int((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x,method=_RETURNVERBOSE)
Output:
d^4*a*(2*I*c^3/x^2+2*c^2/x^3-c^4/x-1/5/x^5-I*c/x^4)+d^4*b*c^5*(-1/c/x*arct an(c*x)-1/5/c^5/x^5*arctan(c*x)+2/c^3/x^3*arctan(c*x)+2*I*arctan(c*x)/c^2/ x^2-I*arctan(c*x)/c^4/x^4-8/5*ln(c^2*x^2+1)+3*I*arctan(c*x)-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))
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 = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.73 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {192 \, b c^{5} d^{4} x^{5} \log \left (x\right ) - 186 \, b c^{5} d^{4} x^{5} \log \left (\frac {c x + i}{c}\right ) - 6 \, b c^{5} d^{4} x^{5} \log \left (\frac {c x - i}{c}\right ) - 60 \, {\left (a - 3 i \, b\right )} c^{4} d^{4} x^{4} - 6 \, {\left (-20 i \, a - 11 \, b\right )} c^{3} d^{4} x^{3} + 20 \, {\left (6 \, a - i \, b\right )} c^{2} d^{4} x^{2} - 3 \, {\left (20 i \, a + b\right )} c d^{4} x - 12 \, a d^{4} - 6 \, {\left (5 i \, b c^{4} d^{4} x^{4} + 10 \, b c^{3} d^{4} x^{3} - 10 i \, b c^{2} d^{4} x^{2} - 5 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{60 \, x^{5}} \] Input:
integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="fricas")
Output:
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 - 1 1*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 - 1 2*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
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 = 43.47 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.13 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {16 b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x \right )}}{5} - \frac {b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x - 10395 i b^{2} c^{10} d^{8} \right )}}{10} - \frac {31 b c^{5} d^{4} \log {\left (10395 b^{2} c^{11} d^{8} x + 10395 i b^{2} c^{10} d^{8} \right )}}{10} + \frac {- 12 a d^{4} + x^{4} \left (- 60 a c^{4} d^{4} + 180 i b c^{4} d^{4}\right ) + x^{3} \cdot \left (120 i a c^{3} d^{4} + 66 b c^{3} d^{4}\right ) + x^{2} \cdot \left (120 a c^{2} d^{4} - 20 i b c^{2} d^{4}\right ) + x \left (- 60 i a c d^{4} - 3 b c d^{4}\right )}{60 x^{5}} + \frac {\left (- 5 i b c^{4} d^{4} x^{4} - 10 b c^{3} d^{4} x^{3} + 10 i b c^{2} d^{4} x^{2} + 5 b c d^{4} x - i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{10 x^{5}} + \frac {\left (5 i b c^{4} d^{4} x^{4} + 10 b c^{3} d^{4} x^{3} - 10 i b c^{2} d^{4} x^{2} - 5 b c d^{4} x + i b d^{4}\right ) \log {\left (i c x + 1 \right )}}{10 x^{5}} \] Input:
integrate((d+I*c*d*x)**4*(a+b*atan(c*x))/x**6,x)
Output:
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)
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 = 0.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.35 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c^{4} d^{4} + 2 i \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{3} d^{4} - {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{2} d^{4} - \frac {a c^{4} d^{4}}{x} + \frac {1}{3} i \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c d^{4} - \frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{4} + \frac {2 i \, a c^{3} d^{4}}{x^{2}} + \frac {2 \, a c^{2} d^{4}}{x^{3}} - \frac {i \, a c d^{4}}{x^{4}} - \frac {a d^{4}}{5 \, x^{5}} \] Input:
integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="maxima")
Output:
-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
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (97) = 194\).
Time = 0.18 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.88 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=-\frac {186 \, b c^{5} d^{4} x^{5} \log \left (c x + i\right ) + 6 \, b c^{5} d^{4} x^{5} \log \left (c x - i\right ) - 192 \, b c^{5} d^{4} x^{5} \log \left (x\right ) + 60 \, b c^{4} d^{4} x^{4} \arctan \left (c x\right ) + 60 \, a c^{4} d^{4} x^{4} - 180 i \, b c^{4} d^{4} x^{4} - 120 i \, b c^{3} d^{4} x^{3} \arctan \left (c x\right ) - 120 i \, a c^{3} d^{4} x^{3} - 66 \, b c^{3} d^{4} x^{3} - 120 \, b c^{2} d^{4} x^{2} \arctan \left (c x\right ) - 120 \, a c^{2} d^{4} x^{2} + 20 i \, b c^{2} d^{4} x^{2} + 60 i \, b c d^{4} x \arctan \left (c x\right ) + 60 i \, a c d^{4} x + 3 \, b c d^{4} x + 12 \, b d^{4} \arctan \left (c x\right ) + 12 \, a d^{4}}{60 \, x^{5}} \] Input:
integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x, algorithm="giac")
Output:
-1/60*(186*b*c^5*d^4*x^5*log(c*x + I) + 6*b*c^5*d^4*x^5*log(c*x - I) - 192 *b*c^5*d^4*x^5*log(x) + 60*b*c^4*d^4*x^4*arctan(c*x) + 60*a*c^4*d^4*x^4 - 180*I*b*c^4*d^4*x^4 - 120*I*b*c^3*d^4*x^3*arctan(c*x) - 120*I*a*c^3*d^4*x^ 3 - 66*b*c^3*d^4*x^3 - 120*b*c^2*d^4*x^2*arctan(c*x) - 120*a*c^2*d^4*x^2 + 20*I*b*c^2*d^4*x^2 + 60*I*b*c*d^4*x*arctan(c*x) + 60*I*a*c*d^4*x + 3*b*c* d^4*x + 12*b*d^4*arctan(c*x) + 12*a*d^4)/x^5
Time = 1.05 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.59 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {d^4\,\left (192\,b\,c^5\,\ln \left (x\right )-96\,b\,c^5\,\ln \left (c^2\,x^2+1\right )+b\,c^5\,\mathrm {atan}\left (c\,x\right )\,180{}\mathrm {i}\right )}{60}-\frac {\frac {d^4\,\left (12\,a+12\,b\,\mathrm {atan}\left (c\,x\right )\right )}{60}+\frac {d^4\,x\,\left (a\,c\,60{}\mathrm {i}+3\,b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{60}-\frac {d^4\,x^2\,\left (120\,a\,c^2+120\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,20{}\mathrm {i}\right )}{60}+\frac {d^4\,x^4\,\left (60\,a\,c^4+60\,b\,c^4\,\mathrm {atan}\left (c\,x\right )-b\,c^4\,180{}\mathrm {i}\right )}{60}-\frac {d^4\,x^3\,\left (a\,c^3\,120{}\mathrm {i}+66\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,120{}\mathrm {i}\right )}{60}}{x^5} \] Input:
int(((a + b*atan(c*x))*(d + c*d*x*1i)^4)/x^6,x)
Output:
(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*a tan(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*atan(c*x)*120i))/60)/x^5
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.55 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^6} \, dx=\frac {d^{4} \left (180 \mathit {atan} \left (c x \right ) b \,c^{5} i \,x^{5}-60 \mathit {atan} \left (c x \right ) b \,c^{4} x^{4}+120 \mathit {atan} \left (c x \right ) b \,c^{3} i \,x^{3}+120 \mathit {atan} \left (c x \right ) b \,c^{2} x^{2}-60 \mathit {atan} \left (c x \right ) b c i x -12 \mathit {atan} \left (c x \right ) b -96 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{5} x^{5}+192 \,\mathrm {log}\left (x \right ) b \,c^{5} x^{5}-60 a \,c^{4} x^{4}+120 a \,c^{3} i \,x^{3}+120 a \,c^{2} x^{2}-60 a c i x -12 a +180 b \,c^{4} i \,x^{4}+66 b \,c^{3} x^{3}-20 b \,c^{2} i \,x^{2}-3 b c x \right )}{60 x^{5}} \] Input:
int((d+I*c*d*x)^4*(a+b*atan(c*x))/x^6,x)
Output:
(d**4*(180*atan(c*x)*b*c**5*i*x**5 - 60*atan(c*x)*b*c**4*x**4 + 120*atan(c *x)*b*c**3*i*x**3 + 120*atan(c*x)*b*c**2*x**2 - 60*atan(c*x)*b*c*i*x - 12* atan(c*x)*b - 96*log(c**2*x**2 + 1)*b*c**5*x**5 + 192*log(x)*b*c**5*x**5 - 60*a*c**4*x**4 + 120*a*c**3*i*x**3 + 120*a*c**2*x**2 - 60*a*c*i*x - 12*a + 180*b*c**4*i*x**4 + 66*b*c**3*x**3 - 20*b*c**2*i*x**2 - 3*b*c*x))/(60*x* *5)