Integrand size = 25, antiderivative size = 410 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d} \]
-3/2*b*(a+b*arctan(c*x))^2/c^3/d+3/2*I*b*x*(a+b*arctan(c*x))^2/c^2/d+1/2*I *(a+b*arctan(c*x))^3/c^3/d+x*(a+b*arctan(c*x))^3/c^2/d-1/2*I*x^2*(a+b*arct an(c*x))^3/c/d+3*I*b^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3/d+3*b*(a+b*ar ctan(c*x))^2*ln(2/(1+I*c*x))/c^3/d-I*(a+b*arctan(c*x))^3*ln(2/(1+I*c*x))/c ^3/d-3/2*b^3*polylog(2,1-2/(1+I*c*x))/c^3/d+3*I*b^2*(a+b*arctan(c*x))*poly log(2,1-2/(1+I*c*x))/c^3/d+3/2*b*(a+b*arctan(c*x))^2*polylog(2,1-2/(1+I*c* x))/c^3/d+3/2*b^3*polylog(3,1-2/(1+I*c*x))/c^3/d-3/2*I*b^2*(a+b*arctan(c*x ))*polylog(3,1-2/(1+I*c*x))/c^3/d-3/4*b^3*polylog(4,1-2/(1+I*c*x))/c^3/d
Time = 1.03 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {i \left (4 i a^3 c x-6 a^2 b c x+2 a^3 c^2 x^2-4 i a^3 \arctan (c x)+6 a^2 b \arctan (c x)+12 i a^2 b c x \arctan (c x)-12 a b^2 c x \arctan (c x)+6 a^2 b c^2 x^2 \arctan (c x)-12 i a^2 b \arctan (c x)^2+18 a b^2 \arctan (c x)^2+6 i b^3 \arctan (c x)^2+12 i a b^2 c x \arctan (c x)^2-6 b^3 c x \arctan (c x)^2+6 a b^2 c^2 x^2 \arctan (c x)^2-8 i a b^2 \arctan (c x)^3+6 b^3 \arctan (c x)^3+4 i b^3 c x \arctan (c x)^3+2 b^3 c^2 x^2 \arctan (c x)^3-2 i b^3 \arctan (c x)^4+12 a^2 b \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-12 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+12 a b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+12 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+4 b^3 \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )-2 a^3 \log \left (1+c^2 x^2\right )-6 i a^2 b \log \left (1+c^2 x^2\right )+6 a b^2 \log \left (1+c^2 x^2\right )-6 i b (a+i b+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 b^2 (a+i b+b \arctan (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right )}{4 c^3 d} \]
((-1/4*I)*((4*I)*a^3*c*x - 6*a^2*b*c*x + 2*a^3*c^2*x^2 - (4*I)*a^3*ArcTan[ c*x] + 6*a^2*b*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] - 12*a*b^2*c*x*A rcTan[c*x] + 6*a^2*b*c^2*x^2*ArcTan[c*x] - (12*I)*a^2*b*ArcTan[c*x]^2 + 18 *a*b^2*ArcTan[c*x]^2 + (6*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c *x]^2 - 6*b^3*c*x*ArcTan[c*x]^2 + 6*a*b^2*c^2*x^2*ArcTan[c*x]^2 - (8*I)*a* b^2*ArcTan[c*x]^3 + 6*b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 + 2* b^3*c^2*x^2*ArcTan[c*x]^3 - (2*I)*b^3*ArcTan[c*x]^4 + 12*a^2*b*ArcTan[c*x] *Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I )*ArcTan[c*x])] - 12*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*a *b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*ArcTan[c*x] ^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 4*b^3*ArcTan[c*x]^3*Log[1 + E^((2*I)*A rcTan[c*x])] - 2*a^3*Log[1 + c^2*x^2] - (6*I)*a^2*b*Log[1 + c^2*x^2] + 6*a *b^2*Log[1 + c^2*x^2] - (6*I)*b*(a + I*b + b*ArcTan[c*x])^2*PolyLog[2, -E^ ((2*I)*ArcTan[c*x])] + 6*b^2*(a + I*b + b*ArcTan[c*x])*PolyLog[3, -E^((2*I )*ArcTan[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcTan[c*x])]))/(c^3*d)
Time = 3.74 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {5401, 27, 5361, 5401, 5345, 5379, 5451, 5345, 5419, 5455, 5379, 2849, 2752, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx\) |
| \(\Big \downarrow \) 5401 |
| \(\displaystyle \frac {i \int \frac {x (a+b \arctan (c x))^3}{d (i c x+1)}dx}{c}-\frac {i \int x (a+b \arctan (c x))^3dx}{c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \int \frac {x (a+b \arctan (c x))^3}{i c x+1}dx}{c d}-\frac {i \int x (a+b \arctan (c x))^3dx}{c d}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {i \int \frac {x (a+b \arctan (c x))^3}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c d}\) |
| \(\Big \downarrow \) 5401 |
| \(\displaystyle \frac {i \left (\frac {i \int \frac {(a+b \arctan (c x))^3}{i c x+1}dx}{c}-\frac {i \int (a+b \arctan (c x))^3dx}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c d}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {i \left (\frac {i \int \frac {(a+b \arctan (c x))^3}{i c x+1}dx}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c d}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \int \frac {x^2 (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c d}\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (\frac {\int (a+b \arctan (c x))^2dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (\frac {x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}-\frac {(a+b \arctan (c x))^3}{3 b c^3}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\int \frac {(a+b \arctan (c x))^2}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \left (i b \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}\right )}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 5533 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}\right )}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )}{c^2}\right )\right )}{c d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c}-3 i b \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c}\right )-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^3-3 b c \left (-\frac {i (a+b \arctan (c x))^3}{3 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{4 c}\right )}{c}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^3-\frac {3}{2} b c \left (-\frac {(a+b \arctan (c x))^3}{3 b c^3}+\frac {x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )}{c^2}\right )\right )}{c d}\) |
((-I)*((x^2*(a + b*ArcTan[c*x])^3)/2 - (3*b*c*(-1/3*(a + b*ArcTan[c*x])^3/ (b*c^3) + (x*(a + b*ArcTan[c*x])^2 - 2*b*c*(((-1/2*I)*(a + b*ArcTan[c*x])^ 2)/(b*c^2) - (((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + ((I/2)*b*PolyLo g[2, 1 - 2/(1 + I*c*x)])/c)/c))/c^2))/2))/(c*d) + (I*(((-I)*(x*(a + b*ArcT an[c*x])^3 - 3*b*c*(((-1/3*I)*(a + b*ArcTan[c*x])^3)/(b*c^2) - (((a + b*Ar cTan[c*x])^2*Log[2/(1 + I*c*x)])/c - 2*b*(((-1/2*I)*(a + b*ArcTan[c*x])*Po lyLog[2, 1 - 2/(1 + I*c*x)])/c - (b*PolyLog[3, 1 - 2/(1 + I*c*x)])/(4*c))) /c)))/c + (I*((I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/c - (3*I)*b*((( -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])*PolyLog[3, 1 - 2/(1 + I*c*x)])/c + (b*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c)))))/c))/(c*d)
3.2.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 ]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + ( e_.)*(x_)), x_Symbol] :> Simp[f/e Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p , x], x] - Simp[d*(f/e) 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] && GtQ[m, 0]
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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[m, 1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 14.23 (sec) , antiderivative size = 1359, normalized size of antiderivative = 3.31
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1359\) |
| default | \(\text {Expression too large to display}\) | \(1359\) |
| parts | \(\text {Expression too large to display}\) | \(1406\) |
1/c^3*(a^3/d*c*x-3/2*I*a^2*b/d*arctan(c*x)*c^2*x^2+3/2*I*a^2*b/d*c*x-a^3/d *arctan(c*x)+b^3/d*(-1/2*I*arctan(c*x)^2*(I*arctan(c*x)+arctan(c*x)*c*x-3) *(I+c*x)-1/2*arctan(c*x)^4-I*arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-3 /2*arctan(c*x)^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*arctan(c*x)*pol ylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+3/4*polylog(4,-(1+I*c*x)^2/(c^2*x^2+1))+3 *arctan(c*x)^2+3*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+3/2*polylog(2 ,-(1+I*c*x)^2/(c^2*x^2+1))-2*I*arctan(c*x)^3+3*arctan(c*x)^2*ln((1+I*c*x)^ 2/(c^2*x^2+1)+1)-3*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2*p olylog(3,-(1+I*c*x)^2/(c^2*x^2+1)))+3*a*b^2/d*(arctan(c*x)^2*c*x-1/2*I*pol ylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*arctan(c*x)^2+I*ln((1+I*c*x)^2/(c^2 *x^2+1)+1)-Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2* arctan(c*x)^2+1/2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/ (c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+Pi*arctan(c*x)^2+ 2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*arctan(c*x)*ln(1-I*(1+ I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*(c*x-I)-1/2*Pi*csgn((1+I*c*x)^2/(c ^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-1/2*Pi*csgn(I/((1+I *c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^ 2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+I*arctan(c*x)^2*ln(c*x -I)-2*I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*arctan(c*x)^2*c^2*x^2 -1/2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+...
\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
integral(-1/8*(b^3*x^2*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*x^2*log(-(c *x + I)/(c*x - I))^2 - 12*a^2*b*x^2*log(-(c*x + I)/(c*x - I)) + 8*I*a^3*x^ 2)/(c*d*x - I*d), x)
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
-1/2*a^3*((I*c*x^2 - 2*x)/(c^2*d) - 2*I*log(I*c*x + 1)/(c^3*d)) - 1/128*(1 6*b^3*arctan(c*x)^4 - b^3*log(c^2*x^2 + 1)^4 + 4*(384*b^3*c^3*integrate(1/ 64*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) - 32*b^3*c^3 *integrate(1/64*x^3*log(c^2*x^2 + 1)^3/(c^4*d*x^2 + c^2*d), x) + 384*b^3*c ^3*integrate(1/64*x^3*arctan(c*x)^2/(c^4*d*x^2 + c^2*d), x) - 96*b^3*c^3*i ntegrate(1/64*x^3*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 1792*b^3*c^ 2*integrate(1/64*x^2*arctan(c*x)^3/(c^4*d*x^2 + c^2*d), x) - 192*b^3*c^2*i ntegrate(1/64*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 6144*a*b^2*c^2*integrate(1/64*x^2*arctan(c*x)^2/(c^4*d*x^2 + c^2*d), x) - 384*b^3*c^2*integrate(1/64*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) - 6144*a^2*b*c^2*integrate(1/64*x^2*arctan(c*x)/(c^4*d*x^2 + c^ 2*d), x) + 384*b^3*c*integrate(1/64*x*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^4* d*x^2 + c^2*d), x) + 96*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^3/(c^4*d*x ^2 + c^2*d), x) + 768*b^3*c*integrate(1/64*x*arctan(c*x)^2/(c^4*d*x^2 + c^ 2*d), x) - 192*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2* d), x) - 192*b^3*integrate(1/64*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 3*b^3*arctan(c*x)^4/(c^3*d))*c^3*d - 128*I*c^3*d*integrate( -1/64*(192*a^2*b*c^3*x^3*arctan(c*x) + 8*(7*b^3*c^3*x^3 - 3*b^3*c*x)*arcta n(c*x)^3 - (b^3*c^2*x^2 + 3*b^3)*log(c^2*x^2 + 1)^3 + 12*(16*a*b^2*c^3*x^3 + b^3*c^2*x^2)*arctan(c*x)^2 - 3*(b^3*c^2*x^2 - 2*(b^3*c^3*x^3 - b^3*c...
\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]