\(\int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx\) [132]

Optimal result

Integrand size = 25, antiderivative size = 414 \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=-\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}+\frac {3 b^2 c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d} \] Output:

-3/2*I*b*c^2*(a+b*arctan(c*x))^2/d-3/2*b*c*(a+b*arctan(c*x))^2/d/x-3/2*c^2 
*(a+b*arctan(c*x))^3/d-1/2*(a+b*arctan(c*x))^3/d/x^2+I*c*(a+b*arctan(c*x)) 
^3/d/x+3*b^2*c^2*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d-3*I*b*c^2*(a+b*arct 
an(c*x))^2*ln(2-2/(1-I*c*x))/d-c^2*(a+b*arctan(c*x))^3*ln(2-2/(1+I*c*x))/d 
-3/2*I*b^3*c^2*polylog(2,-1+2/(1-I*c*x))/d-3*b^2*c^2*(a+b*arctan(c*x))*pol 
ylog(2,-1+2/(1-I*c*x))/d-3/2*I*b*c^2*(a+b*arctan(c*x))^2*polylog(2,-1+2/(1 
+I*c*x))/d-3/2*I*b^3*c^2*polylog(3,-1+2/(1-I*c*x))/d-3/2*b^2*c^2*(a+b*arct 
an(c*x))*polylog(3,-1+2/(1+I*c*x))/d+3/4*I*b^3*c^2*polylog(4,-1+2/(1+I*c*x 
))/d
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\frac {-\frac {a^3}{x^2}+\frac {2 i a^3 c}{x}+2 i a^3 c^2 \arctan (c x)-2 a^3 c^2 \log (x)+a^3 c^2 \log \left (1+c^2 x^2\right )+\frac {3 i a^2 b \left (2 c^2 x^2 \arctan (c x)^2+\arctan (c x) \left (i+2 c x+i c^2 x^2+2 i c^2 x^2 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+c x \left (i-2 c x \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )+c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{x^2}+6 a b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {\arctan (c x)}{c x}-\frac {3}{2} \arctan (c x)^2-\frac {\arctan (c x)^2}{2 c^2 x^2}+\frac {i \arctan (c x)^2}{c x}-\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-2 i \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+\log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-\operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )\right )+2 b^3 c^2 \left (-\frac {\pi ^3}{8}+\frac {i \pi ^4}{64}-\frac {3}{2} i \arctan (c x)^2-\frac {3 \arctan (c x)^2}{2 c x}+\arctan (c x)^3+\frac {i \arctan (c x)^3}{c x}-\frac {\left (1+c^2 x^2\right ) \arctan (c x)^3}{2 c^2 x^2}-3 i \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )+3 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+\frac {3}{2} (2-i \arctan (c x)) \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-\frac {3}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-\frac {3}{2} (i+\arctan (c x)) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )\right )}{2 d} \] Input:

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

Output:

(-(a^3/x^2) + ((2*I)*a^3*c)/x + (2*I)*a^3*c^2*ArcTan[c*x] - 2*a^3*c^2*Log[ 
x] + a^3*c^2*Log[1 + c^2*x^2] + ((3*I)*a^2*b*(2*c^2*x^2*ArcTan[c*x]^2 + Ar 
cTan[c*x]*(I + 2*c*x + I*c^2*x^2 + (2*I)*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c 
*x])]) + c*x*(I - 2*c*x*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + c^2*x^2*PolyLog[2, 
 E^((2*I)*ArcTan[c*x])]))/x^2 + 6*a*b^2*c^2*((I/24)*Pi^3 - ArcTan[c*x]/(c* 
x) - (3*ArcTan[c*x]^2)/2 - ArcTan[c*x]^2/(2*c^2*x^2) + (I*ArcTan[c*x]^2)/( 
c*x) - ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*L 
og[1 - E^((2*I)*ArcTan[c*x])] + Log[(c*x)/Sqrt[1 + c^2*x^2]] - I*ArcTan[c* 
x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - PolyLog[2, E^((2*I)*ArcTan[c*x])] 
- PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2) + 2*b^3*c^2*(-1/8*Pi^3 + (I/64)*Pi 
^4 - ((3*I)/2)*ArcTan[c*x]^2 - (3*ArcTan[c*x]^2)/(2*c*x) + ArcTan[c*x]^3 + 
 (I*ArcTan[c*x]^3)/(c*x) - ((1 + c^2*x^2)*ArcTan[c*x]^3)/(2*c^2*x^2) - (3* 
I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^3*Log[1 - E 
^((-2*I)*ArcTan[c*x])] + 3*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (3 
*(2 - I*ArcTan[c*x])*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])])/2 - ( 
(3*I)/2)*PolyLog[2, E^((2*I)*ArcTan[c*x])] - (3*(I + ArcTan[c*x])*PolyLog[ 
3, E^((-2*I)*ArcTan[c*x])])/2 + ((3*I)/4)*PolyLog[4, E^((-2*I)*ArcTan[c*x] 
)]))/(2*d)
 

Rubi [A] (verified)

Time = 4.01 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {5405, 27, 5361, 5405, 5361, 5403, 5453, 5361, 5419, 5459, 5403, 2897, 5527, 5529, 5533, 7164}

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.

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

\(\Big \downarrow \) 5405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5405

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 2897

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

\(\Big \downarrow \) 5527

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

\(\Big \downarrow \) 5529

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

\(\Big \downarrow \) 5533

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

(-1/2*(a + b*ArcTan[c*x])^3/x^2 + (3*b*c*(-((a + b*ArcTan[c*x])^2/x) - (c* 
(a + b*ArcTan[c*x])^3)/(3*b) + 2*b*c*(((-1/2*I)*(a + b*ArcTan[c*x])^2)/b + 
 I*((-I)*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - (b*PolyLog[2, -1 + 2 
/(1 - I*c*x)])/2))))/2)/d - (I*c*(-((a + b*ArcTan[c*x])^3/x) + 3*b*c*(((-1 
/3*I)*(a + b*ArcTan[c*x])^3)/b + I*((-I)*(a + b*ArcTan[c*x])^2*Log[2 - 2/( 
1 - I*c*x)] + (2*I)*b*c*(((I/2)*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - 
 I*c*x)])/c - (b*PolyLog[3, -1 + 2/(1 - I*c*x)])/(4*c)))) - I*c*((a + b*Ar 
cTan[c*x])^3*Log[2 - 2/(1 + I*c*x)] - 3*b*c*(((-1/2*I)*(a + b*ArcTan[c*x]) 
^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/c + I*b*(((I/2)*(a + b*ArcTan[c*x])*Pol 
yLog[3, -1 + 2/(1 + I*c*x)])/c + (b*PolyLog[4, -1 + 2/(1 + I*c*x)])/(4*c)) 
))))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 5361

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

rule 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 
 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* 
d^2 + e^2, 0]
 

rule 5405

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x 
] - Simp[e/(d*f)   Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^p/(d + e*x)), x], 
 x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] 
&& LtQ[m, -1]
 

rule 5419

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, 
c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
 

rule 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

rule 5459

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si 
mp[I/d   Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
 

rule 5527

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*p*(I/2)   Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2* 
d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I + c*x)))^2, 0]
 

rule 5529

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*p*(I/2)   Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ 
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c 
^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
 

rule 5533

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. 
)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/(2* 
c*d)), x] - Simp[b*p*(I/2)   Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k + 1 
, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && 
EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 42.46 (sec) , antiderivative size = 2450, normalized size of antiderivative = 5.92

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2450\)
default	\(\text {Expression too large to display}\)	\(2450\)
parts	\(\text {Expression too large to display}\)	\(2457\)

Input:

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

Output:

c^2*(1/2*a^3/d*ln(c^2*x^2+1)+I*a^3/d*arctan(c*x)-1/2*a^3/d/c^2/x^2+I*a^3/d 
/c/x-a^3/d*ln(c*x)+b^3/d*(-6*I*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I 
*arctan(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*arctan(c*x)^3-3*I 
*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2 
*x^2+1)^(1/2))-3*I*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*arctan(c*x)*po 
lylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(3,-(1+I*c*x)/(c^2*x^2+1) 
^(1/2))+3*arctan(c*x)*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(3,(1+I 
*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2 
))-6*arctan(c*x)*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*arctan(c*x)^2* 
(I*arctan(c*x)+3*I*c*x+c*x*arctan(c*x))*(c*x+I)/c^2/x^2-6*arctan(c*x)*poly 
log(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2-6*arctan(c*x)*polylo 
g(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2) 
)+3*arctan(c*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*arctan(c*x)^4-3*I* 
arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^3*ln(1-(1+I*c* 
x)/(c^2*x^2+1)^(1/2))+3*I*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1 
/2)))+3*a*b^2/d*(-1/2/c^2/x^2*arctan(c*x)^2-1/2*I*Pi*csgn(I*((1+I*c*x)^2/( 
c^2*x^2+1)-1))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^ 
2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2-3/2*arctan(c*x)^2-1 
/2*arctan(c*x)*(I*c*x-(c^2*x^2+1)^(1/2)+1)/c/x-ln(c*x)*arctan(c*x)^2+arcta 
n(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-arctan(c*x)^2*ln(1-(1+I*c*x)/(c^...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=- \frac {i \left (\int \frac {a^{3}}{c x^{4} - i x^{3}}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx\right )}{d} \] Input:

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

Output:

-I*(Integral(a**3/(c*x**4 - I*x**3), x) + Integral(b**3*atan(c*x)**3/(c*x* 
*4 - I*x**3), x) + Integral(3*a*b**2*atan(c*x)**2/(c*x**4 - I*x**3), x) + 
Integral(3*a**2*b*atan(c*x)/(c*x**4 - I*x**3), x))/d
 

Maxima [F]

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

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

Output:

1/2*(2*c^2*log(I*c*x + 1)/d - 2*c^2*log(x)/d + (2*I*c*x - 1)/(d*x^2))*a^3 
- 1/512*(-64*I*b^3*c^2*x^2*arctan(c*x)^4 + 4*I*b^3*c^2*x^2*log(c^2*x^2 + 1 
)^4 + I*(48*b^3*c^2*arctan(c*x)^4/d - 6144*b^3*c^4*integrate(1/64*x^4*arct 
an(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) - 3*b^3*c^2*log(c^2*x^2 
 + 1)^4/d + 3072*b^3*c^3*integrate(1/64*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2 
/(c^2*d*x^5 + d*x^3), x) - 12288*b^3*c^3*integrate(1/64*x^3*arctan(c*x)*lo 
g(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) + 6144*b^3*c^2*integrate(1/64*x^2*a 
rctan(c*x)^2/(c^2*d*x^5 + d*x^3), x) - 1536*b^3*c^2*integrate(1/64*x^2*log 
(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + 28672*b^3*c*integrate(1/64*x*arc 
tan(c*x)^3/(c^2*d*x^5 + d*x^3), x) + 3072*b^3*c*integrate(1/64*x*arctan(c* 
x)*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + 98304*a*b^2*c*integrate(1/ 
64*x*arctan(c*x)^2/(c^2*d*x^5 + d*x^3), x) - 6144*b^3*c*integrate(1/64*x*a 
rctan(c*x)*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) + 98304*a^2*b*c*integr 
ate(1/64*x*arctan(c*x)/(c^2*d*x^5 + d*x^3), x) + 6144*b^3*integrate(1/64*a 
rctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) - 512*b^3*integrate( 
1/64*log(c^2*x^2 + 1)^3/(c^2*d*x^5 + d*x^3), x))*d*x^2 - 64*(192*b^3*c^4*i 
ntegrate(1/64*x^4*arctan(c*x)^3/(c^2*d*x^5 + d*x^3), x) + 48*b^3*c^4*integ 
rate(1/64*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + b^3 
*c^2*arctan(c*x)^3/d + 96*b^3*c^3*integrate(1/64*x^3*arctan(c*x)^2*log(c^2 
*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) + 24*b^3*c^3*integrate(1/64*x^3*log(c...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*atan(c*x)**3*a*b**2*c**2*i*x**2 - 3*atan(c*x)**2*a**2*b*c**2*i*x**2 + 1 
2*atan(c*x)**2*a*b**2*c**2*x**2 + 6*atan(c*x)**2*a*b**2*c*i*x + 2*atan(c*x 
)*a**3*c**2*i*x**2 - 3*atan(c*x)*a**2*b*c**2*x**2 - 3*atan(c*x)*a**2*b + 1 
2*atan(c*x)*a*b**2*c*x + 6*int(atan(c*x)/(c**3*i*x**5 + c**2*x**4 + c*i*x* 
*3 + x**2),x)*a**2*b*c*i*x**2 - 12*int(atan(c*x)/(c**3*i*x**5 + c**2*x**4 
+ c*i*x**3 + x**2),x)*a*b**2*c*x**2 + 2*int(atan(c*x)**3/(c**3*i*x**4 + c* 
*2*x**3 + c*i*x**2 + x),x)*b**3*c**2*x**2 + 6*int(atan(c*x)**2/(c**3*i*x** 
5 + c**2*x**4 + c*i*x**3 + x**2),x)*a*b**2*c*i*x**2 + 12*int(( - atan(c*x) 
)/(c**3*x**5 - c**2*i*x**4 + c*x**3 - i*x**2),x)*a**2*b*c*x**2 + 24*int(( 
- atan(c*x))/(c**3*x**5 - c**2*i*x**4 + c*x**3 - i*x**2),x)*a*b**2*c*i*x** 
2 - 2*int(( - atan(c*x)**3)/(c**3*d*i*x**6 + c**2*d*x**5 + c*d*i*x**4 + d* 
x**3),x)*b**3*d*x**2 - 6*int(( - atan(c*x)**2)/(c**3*d*i*x**6 + c**2*d*x** 
5 + c*d*i*x**4 + d*x**3),x)*a*b**2*d*x**2 - 6*int(( - atan(c*x)*x)/(c**3*x 
**3 - c**2*i*x**2 + c*x - i),x)*a**2*b*c**4*i*x**2 + 12*int(( - atan(c*x)* 
x)/(c**3*x**3 - c**2*i*x**2 + c*x - i),x)*a*b**2*c**4*x**2 + log(c**2*x**2 
 + 1)*a**3*c**2*x**2 + 6*log(c**2*x**2 + 1)*a*b**2*c**2*x**2 - 2*log(x)*a* 
*3*c**2*x**2 - 12*log(x)*a*b**2*c**2*x**2 + 2*a**3*c*i*x - a**3 - 3*a**2*b 
*c*x)/(2*d*x**2)