Integrand size = 23, antiderivative size = 306 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {b c}{2 d^3 x}-\frac {i b c^2}{8 d^3 (i-c x)^2}-\frac {13 b c^2}{8 d^3 (i-c x)}+\frac {9 b c^2 \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (i-c x)^2}-\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (i-c x)}-\frac {6 a c^2 \log (x)}{d^3}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {6 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 i b c^2 \log \left (1+c^2 x^2\right )}{2 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3} \] Output:
-1/2*b*c/d^3/x-1/8*I*b*c^2/d^3/(I-c*x)^2-13/8*b*c^2/d^3/(I-c*x)+9/8*b*c^2* arctan(c*x)/d^3-1/2*(a+b*arctan(c*x))/d^3/x^2+3*I*c*(a+b*arctan(c*x))/d^3/ x+1/2*c^2*(a+b*arctan(c*x))/d^3/(I-c*x)^2-3*I*c^2*(a+b*arctan(c*x))/d^3/(I -c*x)-6*a*c^2*ln(x)/d^3-3*I*b*c^2*ln(x)/d^3-6*c^2*(a+b*arctan(c*x))*ln(2/( 1+I*c*x))/d^3+3/2*I*b*c^2*ln(c^2*x^2+1)/d^3-3*I*b*c^2*polylog(2,-I*c*x)/d^ 3+3*I*b*c^2*polylog(2,I*c*x)/d^3-3*I*b*c^2*polylog(2,1-2/(1+I*c*x))/d^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.35 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {12 b c^2 \left (\frac {1}{i-c x}-\arctan (c x)\right )+\frac {4 (a+b \arctan (c x))}{x^2}-\frac {24 i c (a+b \arctan (c x))}{x}-\frac {4 c^2 (a+b \arctan (c x))}{(-i+c x)^2}-\frac {24 i c^2 (a+b \arctan (c x))}{-i+c x}-\frac {b c^2 \left (-2 i+c x+(-i+c x)^2 \arctan (c x)\right )}{(-i+c x)^2}+\frac {4 b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )}{x}+48 a c^2 \log (x)+48 c^2 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+12 i b c^2 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+24 i b c^2 \operatorname {PolyLog}(2,-i c x)-24 i b c^2 \operatorname {PolyLog}(2,i c x)+24 i b c^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{8 d^3} \] Input:
Integrate[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^3),x]
Output:
-1/8*(12*b*c^2*((I - c*x)^(-1) - ArcTan[c*x]) + (4*(a + b*ArcTan[c*x]))/x^ 2 - ((24*I)*c*(a + b*ArcTan[c*x]))/x - (4*c^2*(a + b*ArcTan[c*x]))/(-I + c *x)^2 - ((24*I)*c^2*(a + b*ArcTan[c*x]))/(-I + c*x) - (b*c^2*(-2*I + c*x + (-I + c*x)^2*ArcTan[c*x]))/(-I + c*x)^2 + (4*b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 48*a*c^2*Log[x] + 48*c^2*(a + b*ArcTan[c*x])*Log[ (2*I)/(I - c*x)] + (12*I)*b*c^2*(2*Log[x] - Log[1 + c^2*x^2]) + (24*I)*b*c ^2*PolyLog[2, (-I)*c*x] - (24*I)*b*c^2*PolyLog[2, I*c*x] + (24*I)*b*c^2*Po lyLog[2, (I + c*x)/(-I + c*x)])/d^3
Time = 0.63 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5411, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {6 c^3 (a+b \arctan (c x))}{d^3 (c x-i)}-\frac {3 i c^3 (a+b \arctan (c x))}{d^3 (c x-i)^2}-\frac {c^3 (a+b \arctan (c x))}{d^3 (c x-i)^3}-\frac {6 c^2 (a+b \arctan (c x))}{d^3 x}+\frac {a+b \arctan (c x)}{d^3 x^3}-\frac {3 i c (a+b \arctan (c x))}{d^3 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 i c^2 (a+b \arctan (c x))}{d^3 (-c x+i)}+\frac {c^2 (a+b \arctan (c x))}{2 d^3 (-c x+i)^2}-\frac {6 c^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {a+b \arctan (c x)}{2 d^3 x^2}+\frac {3 i c (a+b \arctan (c x))}{d^3 x}-\frac {6 a c^2 \log (x)}{d^3}+\frac {9 b c^2 \arctan (c x)}{8 d^3}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{d^3}+\frac {3 i b c^2 \log \left (c^2 x^2+1\right )}{2 d^3}-\frac {13 b c^2}{8 d^3 (-c x+i)}-\frac {i b c^2}{8 d^3 (-c x+i)^2}-\frac {3 i b c^2 \log (x)}{d^3}-\frac {b c}{2 d^3 x}\) |
Input:
Int[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^3),x]
Output:
-1/2*(b*c)/(d^3*x) - ((I/8)*b*c^2)/(d^3*(I - c*x)^2) - (13*b*c^2)/(8*d^3*( I - c*x)) + (9*b*c^2*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*x^2 ) + ((3*I)*c*(a + b*ArcTan[c*x]))/(d^3*x) + (c^2*(a + b*ArcTan[c*x]))/(2*d ^3*(I - c*x)^2) - ((3*I)*c^2*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - (6*a*c ^2*Log[x])/d^3 - ((3*I)*b*c^2*Log[x])/d^3 - (6*c^2*(a + b*ArcTan[c*x])*Log [2/(1 + I*c*x)])/d^3 + (((3*I)/2)*b*c^2*Log[1 + c^2*x^2])/d^3 - ((3*I)*b*c ^2*PolyLog[2, (-I)*c*x])/d^3 + ((3*I)*b*c^2*PolyLog[2, I*c*x])/d^3 - ((3*I )*b*c^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.51 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 i a}{d^{3} c x}-\frac {6 a \ln \left (c x \right )}{d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {6 i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )-6 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {i}{8 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {9 \arctan \left (c x \right )}{8}+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {13}{8 \left (c x -i\right )}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {3 i \arctan \left (c x \right )}{c x -i}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )\right )}{d^{3}}\right )\) | \(345\) |
| default | \(c^{2} \left (-\frac {a}{2 d^{3} c^{2} x^{2}}+\frac {3 i a}{d^{3} c x}-\frac {6 a \ln \left (c x \right )}{d^{3}}+\frac {3 i a}{d^{3} \left (c x -i\right )}+\frac {a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {6 i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )-6 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {i}{8 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {9 \arctan \left (c x \right )}{8}+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {13}{8 \left (c x -i\right )}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {3 i \arctan \left (c x \right )}{c x -i}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )\right )}{d^{3}}\right )\) | \(345\) |
| parts | \(-\frac {3 i a \,c^{2}}{d^{3} \left (-c x +i\right )}+\frac {a \,c^{2}}{2 d^{3} \left (-c x +i\right )^{2}}+\frac {6 i c^{2} a \arctan \left (c x \right )}{d^{3}}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}-\frac {a}{2 d^{3} x^{2}}+\frac {3 i c a}{d^{3} x}-\frac {6 a \,c^{2} \ln \left (x \right )}{d^{3}}+\frac {b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )-6 \ln \left (c x \right ) \arctan \left (c x \right )-\frac {i}{8 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}+6 \arctan \left (c x \right ) \ln \left (c x -i\right )-3 i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {3 i \ln \left (c x -i\right )^{2}}{2}+\frac {9 \arctan \left (c x \right )}{8}+\frac {3 i \arctan \left (c x \right )}{c x}+\frac {13}{8 \left (c x -i\right )}+3 i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {3 i \arctan \left (c x \right )}{c x -i}-3 i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )\right )}{d^{3}}\) | \(354\) |
| risch | \(-\frac {a}{2 d^{3} x^{2}}+\frac {3 c^{3} b \ln \left (-i c x +1\right ) x}{4 d^{3} \left (-i c x -1\right )}+\frac {3 i c^{2} b \ln \left (-i c x +1\right )}{4 d^{3} \left (-i c x -1\right )}+\frac {3 i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{3}}-\frac {3 i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{3}}+\frac {6 i c^{2} a \arctan \left (c x \right )}{d^{3}}+\frac {3 i c a}{d^{3} x}+\frac {i c^{2} b}{8 d^{3} \left (i c x +1\right )^{2}}-\frac {7 i c^{2} b \ln \left (i c x \right )}{4 d^{3}}+\frac {i b \ln \left (i c x +1\right )}{4 d^{3} x^{2}}+\frac {3 i c^{2} b}{2 d^{3} \left (i c x +1\right )}-\frac {3 i c^{2} b \ln \left (i c x +1\right )^{2}}{2 d^{3}}-\frac {3 i c^{2} b \operatorname {dilog}\left (i c x +1\right )}{d^{3}}+\frac {15 i c^{2} b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {3 c b \ln \left (i c x +1\right )}{2 d^{3} x}+\frac {3 i c^{2} b \operatorname {dilog}\left (-i c x +1\right )}{d^{3}}-\frac {3 c b \ln \left (-i c x +1\right )}{2 d^{3} x}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{d^{3}}+\frac {i c^{2} b \ln \left (i c x +1\right )}{4 d^{3} \left (i c x +1\right )^{2}}+\frac {3 i c^{2} b \ln \left (i c x +1\right )}{2 d^{3} \left (i c x +1\right )}-\frac {3 i c^{2} b \ln \left (-i c x +1\right )}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {c^{3} b \ln \left (-i c x +1\right ) x}{8 d^{3} \left (-i c x -1\right )^{2}}-\frac {i c^{4} b \ln \left (-i c x +1\right ) x^{2}}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {5 i c^{2} b \ln \left (-i c x \right )}{4 d^{3}}+\frac {5 i c^{2} b \ln \left (-i c x +1\right )}{4 d^{3}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{3} x^{2}}-\frac {3 i c^{2} b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{3}}-\frac {i c^{2} b}{8 d^{3} \left (-i c x -1\right )}-\frac {c^{2} a}{2 d^{3} \left (-i c x -1\right )^{2}}-\frac {6 c^{2} a \ln \left (-i c x \right )}{d^{3}}+\frac {3 c^{2} a}{d^{3} \left (-i c x -1\right )}-\frac {b c}{2 d^{3} x}-\frac {15 b \,c^{2} \arctan \left (c x \right )}{16 d^{3}}\) | \(646\) |
Input:
int((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^3,x,method=_RETURNVERBOSE)
Output:
c^2*(-1/2*a/d^3/c^2/x^2+3*I*a/d^3/c/x-6*a/d^3*ln(c*x)+3*I*a/d^3/(c*x-I)+1/ 2*a/d^3/(c*x-I)^2+3*a/d^3*ln(c^2*x^2+1)+6*I*a/d^3*arctan(c*x)+b/d^3*(-1/2/ c^2/x^2*arctan(c*x)-3*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*ln(-1/2*I*(c*x+I) ))-6*ln(c*x)*arctan(c*x)-1/8*I/(c*x-I)^2+1/2*arctan(c*x)/(c*x-I)^2+6*arcta n(c*x)*ln(c*x-I)-3*I*ln(c*x)-1/2/c/x+3/2*I*ln(c*x-I)^2+9/8*arctan(c*x)+3*I *arctan(c*x)/c/x+13/8/(c*x-I)+3*I*(dilog(-I*(c*x+I))+ln(c*x)*ln(-I*(c*x+I) ))+3/2*I*ln(c^2*x^2+1)+3*I*arctan(c*x)/(c*x-I)-3*I*((ln(c*x)-ln(-I*c*x))*l n(-I*(-c*x+I))-dilog(-I*c*x))))
Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=-\frac {6 \, {\left (-16 i \, a - 3 \, b\right )} c^{3} x^{3} - 12 \, {\left (12 \, a - i \, b\right )} c^{2} x^{2} + 8 \, {\left (4 i \, a - b\right )} c x + 48 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 48 \, {\left ({\left (2 \, a + i \, b\right )} c^{4} x^{4} + 2 \, {\left (-2 i \, a + b\right )} c^{3} x^{3} - {\left (2 \, a + i \, b\right )} c^{2} x^{2}\right )} \log \left (x\right ) + 4 \, {\left (12 \, b c^{3} x^{3} - 18 i \, b c^{2} x^{2} - 4 \, b c x - i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 33 \, {\left (-i \, b c^{4} x^{4} - 2 \, b c^{3} x^{3} + i \, b c^{2} x^{2}\right )} \log \left (\frac {c x + i}{c}\right ) - 3 \, {\left ({\left (32 \, a + 5 i \, b\right )} c^{4} x^{4} - 2 \, {\left (32 i \, a - 5 \, b\right )} c^{3} x^{3} - {\left (32 \, a + 5 i \, b\right )} c^{2} x^{2}\right )} \log \left (\frac {c x - i}{c}\right ) - 8 \, a}{16 \, {\left (c^{2} d^{3} x^{4} - 2 i \, c d^{3} x^{3} - d^{3} x^{2}\right )}} \] Input:
integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^3,x, algorithm="fricas")
Output:
-1/16*(6*(-16*I*a - 3*b)*c^3*x^3 - 12*(12*a - I*b)*c^2*x^2 + 8*(4*I*a - b) *c*x + 48*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*dilog((c*x + I)/(c*x - I) + 1) + 48*((2*a + I*b)*c^4*x^4 + 2*(-2*I*a + b)*c^3*x^3 - (2*a + I*b) *c^2*x^2)*log(x) + 4*(12*b*c^3*x^3 - 18*I*b*c^2*x^2 - 4*b*c*x - I*b)*log(- (c*x + I)/(c*x - I)) + 33*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*log(( c*x + I)/c) - 3*((32*a + 5*I*b)*c^4*x^4 - 2*(32*I*a - 5*b)*c^3*x^3 - (32*a + 5*I*b)*c^2*x^2)*log((c*x - I)/c) - 8*a)/(c^2*d^3*x^4 - 2*I*c*d^3*x^3 - d^3*x^2)
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*atan(c*x))/x**3/(d+I*c*d*x)**3,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (253) = 506\).
Time = 0.14 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.93 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^3,x, algorithm="maxima")
Output:
-1/16*(33*b*c^4*x^4*arctan2(1, c*x) + 6*(b*(-11*I*arctan2(1, c*x) - 3) - 1 6*I*a)*c^3*x^3 - 3*(b*(11*arctan2(1, c*x) - 4*I) + 48*a)*c^2*x^2 + 8*(4*I* a - b)*c*x + 24*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*arctan(c*x)^2 + 6*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2*x^2)*log(c^2*x^2 + 1)^2 - 24*(b*c ^4*x^4 - 2*I*b*c^3*x^3 - b*c^2*x^2)*arctan(c*x)*log(1/4*c^2*x^2 + 1/4) + 9 6*(b*c^4*x^4 - 2*I*b*c^3*x^3 - b*c^2*x^2)*arctan(c*x)*log(c*x) + (3*(-32*I *a + 5*b)*c^4*x^4 - 6*(32*a + 21*I*b)*c^3*x^3 + 3*(32*I*a - 53*b)*c^2*x^2 + 32*I*b*c*x - 8*b)*arctan(c*x) + 48*(-I*b*c^4*x^4 - 2*b*c^3*x^3 + I*b*c^2 *x^2)*dilog(I*c*x + 1) + 48*(I*b*c^4*x^4 + 2*b*c^3*x^3 - I*b*c^2*x^2)*dilo g(1/2*I*c*x + 1/2) + 48*(I*b*c^4*x^4 + 2*b*c^3*x^3 - I*b*c^2*x^2)*dilog(-I *c*x + 1) - 12*(2*((pi + I)*b + 2*a)*c^4*x^4 - 4*((I*pi - 1)*b + 2*I*a)*c^ 3*x^3 - 2*((pi + I)*b + 2*a)*c^2*x^2 - (I*b*c^4*x^4 + 2*b*c^3*x^3 - I*b*c^ 2*x^2)*log(1/4*c^2*x^2 + 1/4))*log(c^2*x^2 + 1) + 48*((2*a + I*b)*c^4*x^4 + 2*(-2*I*a + b)*c^3*x^3 - (2*a + I*b)*c^2*x^2)*log(x) - 8*a)/(c^2*d^3*x^4 - 2*I*c*d^3*x^3 - d^3*x^2)
\[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^3,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)/((I*c*d*x + d)^3*x^3), x)
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \] Input:
int((a + b*atan(c*x))/(x^3*(d + c*d*x*1i)^3),x)
Output:
int((a + b*atan(c*x))/(x^3*(d + c*d*x*1i)^3), x)
\[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^3} \, dx=\frac {-\left (\int \frac {\mathit {atan} \left (c x \right )}{c^{3} i \,x^{6}+3 c^{2} x^{5}-3 c i \,x^{4}-x^{3}}d x \right ) b -\left (\int \frac {1}{c^{3} i \,x^{6}+3 c^{2} x^{5}-3 c i \,x^{4}-x^{3}}d x \right ) a}{d^{3}} \] Input:
int((a+b*atan(c*x))/x^3/(d+I*c*d*x)^3,x)
Output:
( - (int(atan(c*x)/(c**3*i*x**6 + 3*c**2*x**5 - 3*c*i*x**4 - x**3),x)*b + int(1/(c**3*i*x**6 + 3*c**2*x**5 - 3*c*i*x**4 - x**3),x)*a))/d**3