Integrand size = 23, antiderivative size = 103 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=-\frac {b c d^3}{12 x^3}-\frac {i b c^2 d^3}{2 x^2}+\frac {7 b c^3 d^3}{4 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{4 x^4}-2 i b c^4 d^3 \log (x)+2 i b c^4 d^3 \log (i+c x) \] Output:
-1/12*b*c*d^3/x^3-1/2*I*b*c^2*d^3/x^2+7/4*b*c^3*d^3/x-1/4*d^3*(1+I*c*x)^4* (a+b*arctan(c*x))/x^4-2*I*b*c^4*d^3*ln(x)+2*I*b*c^4*d^3*ln(I+c*x)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.60 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=\frac {d^3 \left (-b c x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )-3 i \left (-i a+4 a c x+6 i a c^2 x^2+2 b c^2 x^2-4 a c^3 x^3+b \left (-i+4 c x+6 i c^2 x^2-4 c^3 x^3\right ) \arctan (c x)+6 i b c^3 x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+8 b c^4 x^4 \log (x)-4 b c^4 x^4 \log \left (1+c^2 x^2\right )\right )\right )}{12 x^4} \] Input:
Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^5,x]
Output:
(d^3*(-(b*c*x*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)]) - (3*I)*((-I)* a + 4*a*c*x + (6*I)*a*c^2*x^2 + 2*b*c^2*x^2 - 4*a*c^3*x^3 + b*(-I + 4*c*x + (6*I)*c^2*x^2 - 4*c^3*x^3)*ArcTan[c*x] + (6*I)*b*c^3*x^3*Hypergeometric2 F1[-1/2, 1, 1/2, -(c^2*x^2)] + 8*b*c^4*x^4*Log[x] - 4*b*c^4*x^4*Log[1 + c^ 2*x^2])))/(12*x^4)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83, 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.
\(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx\) |
\(\Big \downarrow \) 5407 |
\(\displaystyle -b c \int \frac {d^3 (i-c x)^3}{4 x^4 (c x+i)}dx-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{4} b c d^3 \int \frac {(i-c x)^3}{x^4 (c x+i)}dx-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {1}{4} b c d^3 \int \left (-\frac {8 i c^4}{c x+i}+\frac {8 i c^3}{x}+\frac {7 c^2}{x^2}-\frac {4 i c}{x^3}-\frac {1}{x^4}\right )dx-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{4 x^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{4 x^4}-\frac {1}{4} b c d^3 \left (8 i c^3 \log (x)-8 i c^3 \log (c x+i)-\frac {7 c^2}{x}+\frac {2 i c}{x^2}+\frac {1}{3 x^3}\right )\) |
Input:
Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^5,x]
Output:
-1/4*(d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/x^4 - (b*c*d^3*(1/(3*x^3) + ( (2*I)*c)/x^2 - (7*c^2)/x + (8*I)*c^3*Log[x] - (8*I)*c^3*Log[I + c*x]))/4
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 = 0.57 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.43
method | result | size |
parts | \(d^{3} a \left (\frac {3 c^{2}}{2 x^{2}}-\frac {i c}{x^{3}}+\frac {i c^{3}}{x}-\frac {1}{4 x^{4}}\right )+d^{3} b \,c^{4} \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\) | \(147\) |
derivativedivides | \(c^{4} \left (d^{3} a \left (\frac {3}{2 c^{2} x^{2}}-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}\right )+d^{3} b \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\right )\) | \(153\) |
default | \(c^{4} \left (d^{3} a \left (\frac {3}{2 c^{2} x^{2}}-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}\right )+d^{3} b \left (\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}\right )\right )\) | \(153\) |
parallelrisch | \(-\frac {24 i c^{4} b \,d^{3} \ln \left (x \right ) x^{4}-12 i b \,d^{3} c^{3} \arctan \left (c x \right ) x^{3}+12 i a c \,d^{3} x -21 d^{3} b \arctan \left (c x \right ) x^{4} c^{4}+6 i x^{2} b \,c^{2} d^{3}+18 a \,c^{4} d^{3} x^{4}-12 i x^{3} a \,c^{3} d^{3}-21 b \,c^{3} d^{3} x^{3}-6 i x^{4} b \,c^{4} d^{3}-18 d^{3} b \arctan \left (c x \right ) x^{2} c^{2}-12 i c^{4} b \,d^{3} \ln \left (c^{2} x^{2}+1\right ) x^{4}-18 a \,c^{2} d^{3} x^{2}+12 i x \arctan \left (c x \right ) b c \,d^{3}+b c \,d^{3} x +3 b \,d^{3} \arctan \left (c x \right )+3 d^{3} a}{12 x^{4}}\) | \(214\) |
risch | \(\frac {\left (4 b \,c^{3} d^{3} x^{3}-6 i x^{2} b \,c^{2} d^{3}-4 b c \,d^{3} x +i d^{3} b \right ) \ln \left (i c x +1\right )}{8 x^{4}}-\frac {i d^{3} \left (-45 b \,c^{4} \ln \left (-217 c x -217 i\right ) x^{4}+48 b \,c^{4} \ln \left (-527 c x \right ) x^{4}-3 b \,c^{4} \ln \left (119 c x -119 i\right ) x^{4}-24 c^{3} x^{3} a -12 i b \,c^{3} x^{3} \ln \left (-i c x +1\right )-18 x^{2} b \ln \left (-i c x +1\right ) c^{2}+42 i b \,c^{3} x^{3}+12 b \,c^{2} x^{2}+36 i a \,c^{2} x^{2}+24 a c x +12 i b c x \ln \left (-i c x +1\right )+3 b \ln \left (-i c x +1\right )-2 i b c x -6 i a \right )}{24 x^{4}}\) | \(227\) |
Input:
int((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^5,x,method=_RETURNVERBOSE)
Output:
d^3*a*(3/2*c^2/x^2-I*c/x^3+I*c^3/x-1/4/x^4)+d^3*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)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (85) = 170\).
Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.69 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=\frac {-48 i \, b c^{4} d^{3} x^{4} \log \left (x\right ) + 45 i \, b c^{4} d^{3} x^{4} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b c^{4} d^{3} x^{4} \log \left (\frac {c x - i}{c}\right ) - 6 \, {\left (-4 i \, a - 7 \, b\right )} c^{3} d^{3} x^{3} + 12 \, {\left (3 \, a - i \, b\right )} c^{2} d^{3} x^{2} - 2 \, {\left (12 i \, a + b\right )} c d^{3} x - 6 \, a d^{3} - 3 \, {\left (4 \, b c^{3} d^{3} x^{3} - 6 i \, b c^{2} d^{3} x^{2} - 4 \, b c d^{3} x + i \, b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{24 \, x^{4}} \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^5,x, algorithm="fricas")
Output:
1/24*(-48*I*b*c^4*d^3*x^4*log(x) + 45*I*b*c^4*d^3*x^4*log((c*x + I)/c) + 3 *I*b*c^4*d^3*x^4*log((c*x - I)/c) - 6*(-4*I*a - 7*b)*c^3*d^3*x^3 + 12*(3*a - I*b)*c^2*d^3*x^2 - 2*(12*I*a + b)*c*d^3*x - 6*a*d^3 - 3*(4*b*c^3*d^3*x^ 3 - 6*I*b*c^2*d^3*x^2 - 4*b*c*d^3*x + I*b*d^3)*log(-(c*x + I)/(c*x - I)))/ x^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (99) = 198\).
Time = 17.04 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.02 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=- 2 i b c^{4} d^{3} \log {\left (3689 b^{2} c^{9} d^{6} x \right )} + \frac {i b c^{4} d^{3} \log {\left (3689 b^{2} c^{9} d^{6} x - 3689 i b^{2} c^{8} d^{6} \right )}}{8} + \frac {15 i b c^{4} d^{3} \log {\left (3689 b^{2} c^{9} d^{6} x + 3689 i b^{2} c^{8} d^{6} \right )}}{8} - \frac {3 a d^{3} + x^{3} \left (- 12 i a c^{3} d^{3} - 21 b c^{3} d^{3}\right ) + x^{2} \left (- 18 a c^{2} d^{3} + 6 i b c^{2} d^{3}\right ) + x \left (12 i a c d^{3} + b c d^{3}\right )}{12 x^{4}} + \frac {\left (- 4 b c^{3} d^{3} x^{3} + 6 i b c^{2} d^{3} x^{2} + 4 b c d^{3} x - i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{8 x^{4}} + \frac {\left (4 b c^{3} d^{3} x^{3} - 6 i b c^{2} d^{3} x^{2} - 4 b c d^{3} x + i b d^{3}\right ) \log {\left (i c x + 1 \right )}}{8 x^{4}} \] Input:
integrate((d+I*c*d*x)**3*(a+b*atan(c*x))/x**5,x)
Output:
-2*I*b*c**4*d**3*log(3689*b**2*c**9*d**6*x) + I*b*c**4*d**3*log(3689*b**2* c**9*d**6*x - 3689*I*b**2*c**8*d**6)/8 + 15*I*b*c**4*d**3*log(3689*b**2*c* *9*d**6*x + 3689*I*b**2*c**8*d**6)/8 - (3*a*d**3 + x**3*(-12*I*a*c**3*d**3 - 21*b*c**3*d**3) + x**2*(-18*a*c**2*d**3 + 6*I*b*c**2*d**3) + x*(12*I*a* c*d**3 + b*c*d**3))/(12*x**4) + (-4*b*c**3*d**3*x**3 + 6*I*b*c**2*d**3*x** 2 + 4*b*c*d**3*x - I*b*d**3)*log(-I*c*x + 1)/(8*x**4) + (4*b*c**3*d**3*x** 3 - 6*I*b*c**2*d**3*x**2 - 4*b*c*d**3*x + I*b*d**3)*log(I*c*x + 1)/(8*x**4 )
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (85) = 170\).
Time = 0.11 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.96 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=\frac {1}{2} i \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c^{3} d^{3} + \frac {3}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{2} d^{3} + \frac {1}{2} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c d^{3} + \frac {i \, a c^{3} d^{3}}{x} + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d^{3} + \frac {3 \, a c^{2} d^{3}}{2 \, x^{2}} - \frac {i \, a c d^{3}}{x^{3}} - \frac {a d^{3}}{4 \, x^{4}} \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^5,x, algorithm="maxima")
Output:
1/2*I*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*b*c^3*d^3 + 3/2* ((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*b*c^2*d^3 + 1/2*I*((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*d^3 + I*a*c^ 3*d^3/x + 1/12*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x )/x^4)*b*d^3 + 3/2*a*c^2*d^3/x^2 - I*a*c*d^3/x^3 - 1/4*a*d^3/x^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (85) = 170\).
Time = 0.14 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.77 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=-\frac {-45 i \, b c^{4} d^{3} x^{4} \log \left (i \, c x - 1\right ) - 3 i \, b c^{4} d^{3} x^{4} \log \left (-i \, c x - 1\right ) + 48 i \, b c^{4} d^{3} x^{4} \log \left (x\right ) - 24 i \, b c^{3} d^{3} x^{3} \arctan \left (c x\right ) - 24 i \, a c^{3} d^{3} x^{3} - 42 \, b c^{3} d^{3} x^{3} - 36 \, b c^{2} d^{3} x^{2} \arctan \left (c x\right ) - 36 \, a c^{2} d^{3} x^{2} + 12 i \, b c^{2} d^{3} x^{2} + 24 i \, b c d^{3} x \arctan \left (c x\right ) + 24 i \, a c d^{3} x + 2 \, b c d^{3} x + 6 \, b d^{3} \arctan \left (c x\right ) + 6 \, a d^{3}}{24 \, x^{4}} \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))/x^5,x, algorithm="giac")
Output:
-1/24*(-45*I*b*c^4*d^3*x^4*log(I*c*x - 1) - 3*I*b*c^4*d^3*x^4*log(-I*c*x - 1) + 48*I*b*c^4*d^3*x^4*log(x) - 24*I*b*c^3*d^3*x^3*arctan(c*x) - 24*I*a* c^3*d^3*x^3 - 42*b*c^3*d^3*x^3 - 36*b*c^2*d^3*x^2*arctan(c*x) - 36*a*c^2*d ^3*x^2 + 12*I*b*c^2*d^3*x^2 + 24*I*b*c*d^3*x*arctan(c*x) + 24*I*a*c*d^3*x + 2*b*c*d^3*x + 6*b*d^3*arctan(c*x) + 6*a*d^3)/x^4
Time = 0.99 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.50 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=-\frac {\frac {d^3\,\left (3\,a+3\,b\,\mathrm {atan}\left (c\,x\right )\right )}{12}+\frac {d^3\,x\,\left (a\,c\,12{}\mathrm {i}+b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,12{}\mathrm {i}\right )}{12}-\frac {d^3\,x^2\,\left (18\,a\,c^2+18\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,6{}\mathrm {i}\right )}{12}-\frac {d^3\,x^3\,\left (a\,c^3\,12{}\mathrm {i}+21\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,12{}\mathrm {i}\right )}{12}}{x^4}+\frac {d^3\,\left (21\,b\,c^4\,\mathrm {atan}\left (c\,x\right )+b\,c^4\,\ln \left (c^2\,x^2+1\right )\,12{}\mathrm {i}-b\,c^4\,\ln \left (x\right )\,24{}\mathrm {i}\right )}{12} \] Input:
int(((a + b*atan(c*x))*(d + c*d*x*1i)^3)/x^5,x)
Output:
(d^3*(21*b*c^4*atan(c*x) + b*c^4*log(c^2*x^2 + 1)*12i - b*c^4*log(x)*24i)) /12 - ((d^3*(3*a + 3*b*atan(c*x)))/12 + (d^3*x*(a*c*12i + b*c + b*c*atan(c *x)*12i))/12 - (d^3*x^2*(18*a*c^2 - b*c^2*6i + 18*b*c^2*atan(c*x)))/12 - ( d^3*x^3*(a*c^3*12i + 21*b*c^3 + b*c^3*atan(c*x)*12i))/12)/x^4
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.46 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^5} \, dx=\frac {d^{3} \left (21 \mathit {atan} \left (c x \right ) b \,c^{4} x^{4}+12 \mathit {atan} \left (c x \right ) b \,c^{3} i \,x^{3}+18 \mathit {atan} \left (c x \right ) b \,c^{2} x^{2}-12 \mathit {atan} \left (c x \right ) b c i x -3 \mathit {atan} \left (c x \right ) b +12 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{4} i \,x^{4}-24 \,\mathrm {log}\left (x \right ) b \,c^{4} i \,x^{4}+12 a \,c^{3} i \,x^{3}+18 a \,c^{2} x^{2}-12 a c i x -3 a +21 b \,c^{3} x^{3}-6 b \,c^{2} i \,x^{2}-b c x \right )}{12 x^{4}} \] Input:
int((d+I*c*d*x)^3*(a+b*atan(c*x))/x^5,x)
Output:
(d**3*(21*atan(c*x)*b*c**4*x**4 + 12*atan(c*x)*b*c**3*i*x**3 + 18*atan(c*x )*b*c**2*x**2 - 12*atan(c*x)*b*c*i*x - 3*atan(c*x)*b + 12*log(c**2*x**2 + 1)*b*c**4*i*x**4 - 24*log(x)*b*c**4*i*x**4 + 12*a*c**3*i*x**3 + 18*a*c**2* x**2 - 12*a*c*i*x - 3*a + 21*b*c**3*x**3 - 6*b*c**2*i*x**2 - b*c*x))/(12*x **4)