Integrand size = 25, antiderivative size = 356 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=-\frac {a b x}{c^3 d}-\frac {i b^2 x}{3 c^3 d}+\frac {i b^2 \arctan (c x)}{3 c^4 d}-\frac {b^2 x \arctan (c x)}{c^3 d}+\frac {i b x^2 (a+b \arctan (c x))}{3 c^2 d}-\frac {5 (a+b \arctan (c x))^2}{6 c^4 d}+\frac {i x (a+b \arctan (c x))^2}{c^3 d}+\frac {x^2 (a+b \arctan (c x))^2}{2 c^2 d}-\frac {i x^3 (a+b \arctan (c x))^2}{3 c d}+\frac {8 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^4 d}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d}-\frac {4 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^4 d}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d} \]
-a*b*x/c^3/d-1/3*I*b^2*x/c^3/d+1/3*I*b^2*arctan(c*x)/c^4/d-b^2*x*arctan(c* x)/c^3/d+1/3*I*b*x^2*(a+b*arctan(c*x))/c^2/d-5/6*(a+b*arctan(c*x))^2/c^4/d +I*x*(a+b*arctan(c*x))^2/c^3/d+1/2*x^2*(a+b*arctan(c*x))^2/c^2/d-1/3*I*x^3 *(a+b*arctan(c*x))^2/c/d+8/3*I*b*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^4/d+( a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^4/d+1/2*b^2*ln(c^2*x^2+1)/c^4/d-4/3*b ^2*polylog(2,1-2/(1+I*c*x))/c^4/d+I*b*(a+b*arctan(c*x))*polylog(2,1-2/(1+I *c*x))/c^4/d+1/2*b^2*polylog(3,1-2/(1+I*c*x))/c^4/d
Time = 1.03 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {i a^2 x}{c^3 d}+\frac {a^2 x^2}{2 c^2 d}-\frac {i a^2 x^3}{3 c d}-\frac {i a^2 \arctan (c x)}{c^4 d}-\frac {a^2 \log \left (1+c^2 x^2\right )}{2 c^4 d}-\frac {i a b \left (-3 i c x-8 c x \arctan (c x)+6 \arctan (c x)^2+\left (1+c^2 x^2\right ) (-1+3 i \arctan (c x)+2 c x \arctan (c x))+6 i \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-8 \log \left (\frac {1}{\sqrt {1+c^2 x^2}}\right )+3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{3 c^4 d}-\frac {i b^2 \left (2 c x-6 i c x \arctan (c x)-2 \left (1+c^2 x^2\right ) \arctan (c x)+8 i \arctan (c x)^2-8 c x \arctan (c x)^2+3 i \left (1+c^2 x^2\right ) \arctan (c x)^2+2 c x \left (1+c^2 x^2\right ) \arctan (c x)^2+4 \arctan (c x)^3-16 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+6 i \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-6 i \log \left (\frac {1}{\sqrt {1+c^2 x^2}}\right )+(8 i+6 \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 i \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{6 c^4 d} \]
(I*a^2*x)/(c^3*d) + (a^2*x^2)/(2*c^2*d) - ((I/3)*a^2*x^3)/(c*d) - (I*a^2*A rcTan[c*x])/(c^4*d) - (a^2*Log[1 + c^2*x^2])/(2*c^4*d) - ((I/3)*a*b*((-3*I )*c*x - 8*c*x*ArcTan[c*x] + 6*ArcTan[c*x]^2 + (1 + c^2*x^2)*(-1 + (3*I)*Ar cTan[c*x] + 2*c*x*ArcTan[c*x]) + (6*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan [c*x])] - 8*Log[1/Sqrt[1 + c^2*x^2]] + 3*PolyLog[2, -E^((2*I)*ArcTan[c*x]) ]))/(c^4*d) - ((I/6)*b^2*(2*c*x - (6*I)*c*x*ArcTan[c*x] - 2*(1 + c^2*x^2)* ArcTan[c*x] + (8*I)*ArcTan[c*x]^2 - 8*c*x*ArcTan[c*x]^2 + (3*I)*(1 + c^2*x ^2)*ArcTan[c*x]^2 + 2*c*x*(1 + c^2*x^2)*ArcTan[c*x]^2 + 4*ArcTan[c*x]^3 - 16*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (6*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (6*I)*Log[1/Sqrt[1 + c^2*x^2]] + (8*I + 6*ArcTa n[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (3*I)*PolyLog[3, -E^((2*I)*Ar cTan[c*x])]))/(c^4*d)
Time = 3.83 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.32, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5401, 27, 5361, 5401, 5361, 5401, 5345, 5379, 5451, 2009, 5361, 262, 216, 5419, 5455, 5379, 2849, 2752, 5529, 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.
\(\displaystyle \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))^2}{d (i c x+1)}dx}{c}-\frac {i \int x^2 (a+b \arctan (c x))^2dx}{c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))^2}{i c x+1}dx}{c d}-\frac {i \int x^2 (a+b \arctan (c x))^2dx}{c d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))^2}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))^2}{i c x+1}dx}{c}-\frac {i \int x (a+b \arctan (c x))^2dx}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))^2}{i c x+1}dx}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \int \frac {x^2 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \int \frac {(a+b \arctan (c x))^2}{i c x+1}dx}{c}-\frac {i \int (a+b \arctan (c x))^2dx}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \int \frac {x^2 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \int \frac {(a+b \arctan (c x))^2}{i c x+1}dx}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \int \frac {x^2 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \int \frac {x^2 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 5451 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {\int (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5419 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5455 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5529 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}\right )\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}\right )\right )}{c d}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 i b \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{4 c}\right )\right )}{c}-\frac {i \left (x (a+b \arctan (c x))^2-2 b c \left (-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}\right )\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))^2-b c \left (\frac {a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}}{c^2}-\frac {(a+b \arctan (c x))^2}{2 b c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}\right )\right )}{c d}\) |
((-I)*((x^3*(a + b*ArcTan[c*x])^2)/3 - (2*b*c*(((x^2*(a + b*ArcTan[c*x]))/ 2 - (b*c*(x/c^2 - ArcTan[c*x]/c^3))/2)/c^2 - (((-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*Poly Log[2, 1 - 2/(1 + I*c*x)])/c)/c)/c^2))/3))/(c*d) + (I*(((-I)*((x^2*(a + b* ArcTan[c*x])^2)/2 - b*c*(-1/2*(a + b*ArcTan[c*x])^2/(b*c^3) + (a*x + b*x*A rcTan[c*x] - (b*Log[1 + c^2*x^2])/(2*c))/c^2)))/c + (I*(((-I)*(x*(a + b*Ar cTan[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*PolyLog[2, 1 - 2/(1 + I*c*x) ])/c)/c)))/c + (I*((I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c - (2*I)* b*(((-1/2*I)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (b*Pol yLog[3, 1 - 2/(1 + I*c*x)])/(4*c))))/c))/c))/(c*d)
3.1.95.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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[(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 = 47.77 (sec) , antiderivative size = 1105, normalized size of antiderivative = 3.10
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1105\) |
default | \(\text {Expression too large to display}\) | \(1105\) |
parts | \(\text {Expression too large to display}\) | \(1149\) |
1/c^4*(4/3*I/d*a*b+I/d*a*b*ln(-1/2*I*(c*x+I))*ln(c*x-I)+1/2*a^2/d*c^2*x^2- 1/2*a^2/d*ln(c^2*x^2+1)+I*a^2/d*c*x+b^2/d*(-I*Pi*csgn((1+I*c*x)^2/(c^2*x^2 +1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-1/3*I*arctan(c*x)*(c*x-I) ^2+1/2*c^2*x^2*arctan(c*x)^2-arctan(c*x)^2*ln(c*x-I)+11/6*arctan(c*x)^2+8/ 3*I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-ln(1+(1+I*c*x)^2/(c^2* x^2+1))+8/3*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+8/3*dilog(1-I*(1+I*c*x) /(c^2*x^2+1)^(1/2))+I*Pi*arctan(c*x)^2-I*arctan(c*x)*polylog(2,-(1+I*c*x)^ 2/(c^2*x^2+1))+2/3*I*arctan(c*x)*(c*x+I)*(c*x-I)-1/3*arctan(c*x)*(c*x-I)-1 /2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*arctan (c*x)^2-1/3*I*(c*x+I)-1/3*I*arctan(c*x)^2*c^3*x^3-2/3*I*arctan(c*x)^3+8/3* I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*csgn(I/(1+(1+I* c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+ 1)))^2*arctan(c*x)^2-1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+ I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2 +1)))*arctan(c*x)^2-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^ 2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+I*arctan(c*x)^2 *c*x+arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+1/2*polylog(3,-(1+I*c*x )^2/(c^2*x^2+1)))-5/24*I/d*a*b*ln(c^4*x^4+10*c^2*x^2+9)-1/2*I/d*a*b*ln(c*x -I)^2+1/d*a*b*arctan(c*x)*c^2*x^2-2/d*a*b*arctan(c*x)*ln(c*x-I)-I*a^2/d*ar ctan(c*x)-2/3*I/d*a*b*arctan(c*x)*c^3*x^3+1/3*I/d*a*b*c^2*x^2-1/d*a*b*c...
\[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{i \, c d x + d} \,d x } \]
integral(1/4*(I*b^2*x^3*log(-(c*x + I)/(c*x - I))^2 + 4*a*b*x^3*log(-(c*x + I)/(c*x - I)) - 4*I*a^2*x^3)/(c*d*x - I*d), x)
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{i \, c d x + d} \,d x } \]
-1/6*a^2*(I*(2*c^2*x^3 + 3*I*c*x^2 - 6*x)/(c^3*d) + 6*log(I*c*x + 1)/(c^4* d)) - 1/96*(16*I*(216*b^2*c^4*integrate(1/48*x^4*arctan(c*x)^2/(c^5*d*x^2 + c^3*d), x) + 18*b^2*c^4*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^5*d*x^2 + c^3*d), x) + 576*a*b*c^4*integrate(1/48*x^4*arctan(c*x)/(c^5*d*x^2 + c^ 3*d), x) + 24*b^2*c^4*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3 *d), x) + 72*b^2*c^3*integrate(1/48*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^5* d*x^2 + c^3*d), x) + 24*b^2*c^3*integrate(1/48*x^3*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) - 36*b^2*c^2*integrate(1/48*x^2*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 144*b^2*c*integrate(1/48*x*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) - 36*b^2*integrate(1/48*log(c^2*x^2 + 1)^2/(c^5*d*x^2 + c^3*d), x) - b^ 2*arctan(c*x)^3/(c^4*d))*c^4*d - 96*c^4*d*integrate(1/48*(12*(3*b^2*c^2*x^ 3 - 2*b^2*x)*arctan(c*x)^2 + 3*(b^2*c^2*x^3 - 2*b^2*x)*log(c^2*x^2 + 1)^2 - 4*(2*b^2*c^3*x^4 - 24*a*b*c^2*x^3 - 3*b^2*c*x^2)*arctan(c*x) - 2*(6*b^2* c^3*x^4*arctan(c*x) - b^2*c^2*x^3 - 6*b^2*x)*log(c^2*x^2 + 1))/(c^4*d*x^2 + c^2*d), x) + 24*I*b^2*arctan(c*x)^3 - 3*b^2*log(c^2*x^2 + 1)^3 - 4*(-2*I *b^2*c^3*x^3 + 3*b^2*c^2*x^2 + 6*I*b^2*c*x)*arctan(c*x)^2 + (-2*I*b^2*c^3* x^3 + 3*b^2*c^2*x^2 + 6*I*b^2*c*x + 6*I*b^2*arctan(c*x))*log(c^2*x^2 + 1)^ 2 - 4*(3*b^2*arctan(c*x)^2 + (2*b^2*c^3*x^3 + 3*I*b^2*c^2*x^2 - 6*b^2*c*x) *arctan(c*x))*log(c^2*x^2 + 1))/(c^4*d)
\[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{i \, c d x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+i c d x} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]