Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {3 a^4 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {7 a^3 \arctan (a x)^2}{8 c^2}-\frac {a \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^3 \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac {7 i a^3 \arctan (a x)^3}{3 c^2}-\frac {\arctan (a x)^3}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)^3}{c^2 x}+\frac {a^4 x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^4}{8 c^2}+\frac {a^3 \log (x)}{c^2}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^2}-\frac {7 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {7 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {7 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]
-3/8*a^3/c^2/(a^2*x^2+1)-a^2*arctan(a*x)/c^2/x-3/4*a^4*x*arctan(a*x)/c^2/( a^2*x^2+1)-7/8*a^3*arctan(a*x)^2/c^2-1/2*a*arctan(a*x)^2/c^2/x^2+3/4*a^3*a rctan(a*x)^2/c^2/(a^2*x^2+1)+7/3*I*a^3*arctan(a*x)^3/c^2-1/3*arctan(a*x)^3 /c^2/x^3+2*a^2*arctan(a*x)^3/c^2/x+1/2*a^4*x*arctan(a*x)^3/c^2/(a^2*x^2+1) +5/8*a^3*arctan(a*x)^4/c^2+a^3*ln(x)/c^2-1/2*a^3*ln(a^2*x^2+1)/c^2-7*a^3*a rctan(a*x)^2*ln(2-2/(1-I*a*x))/c^2+7*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I *a*x))/c^2-7/2*a^3*polylog(3,-1+2/(1-I*a*x))/c^2
Time = 1.51 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.59 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^3 \left (-7 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {1}{48} \left (14 i \pi ^3+30 \arctan (a x)^4-9 \cos (2 \arctan (a x))+6 \arctan (a x)^2 \left (-4-\frac {4}{a^2 x^2}+3 \cos (2 \arctan (a x))-56 \log \left (1-e^{-2 i \arctan (a x)}\right )\right )+48 \log (a x)-24 \log \left (1+a^2 x^2\right )-168 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+4 \arctan (a x)^3 \left (-28 i-\frac {4}{a^3 x^3}+\frac {24}{a x}+3 \sin (2 \arctan (a x))\right )-\frac {6 \arctan (a x) (8+3 a x \sin (2 \arctan (a x)))}{a x}\right )\right )}{c^2} \]
(a^3*((-7*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + ((14*I)*Pi^3 + 30*ArcTan[a*x]^4 - 9*Cos[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2*(-4 - 4/(a^2* x^2) + 3*Cos[2*ArcTan[a*x]] - 56*Log[1 - E^((-2*I)*ArcTan[a*x])]) + 48*Log [a*x] - 24*Log[1 + a^2*x^2] - 168*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 4*A rcTan[a*x]^3*(-28*I - 4/(a^3*x^3) + 24/(a*x) + 3*Sin[2*ArcTan[a*x]]) - (6* ArcTan[a*x]*(8 + 3*a*x*Sin[2*ArcTan[a*x]]))/(a*x))/48))/c^2
Time = 6.08 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.75, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {5501, 27, 5453, 5361, 5453, 5361, 5419, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 5501, 5427, 5453, 5361, 5419, 5459, 5403, 5465, 5427, 241, 5527, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^4 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c x^4 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c^2 x^2 \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx+a \int \frac {\arctan (a x)^2}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-a^2 \left (\int \frac {\arctan (a x)^3}{x^2}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx\right )+a \left (\int \frac {\arctan (a x)^2}{x^3}dx-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {a \left (-\left (a^2 \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle -\frac {a^2 \left (-\left (a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )\right )-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx+\int \frac {\arctan (a x)^3}{x^2}dx\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle -\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle -\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\) |
-((a^2*(-(ArcTan[a*x]^3/x) - (a*ArcTan[a*x]^4)/4 - a^2*((x*ArcTan[a*x]^3)/ (2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*(-1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/2) + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcT an[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)]/(4*a))))))/c^2) + (-1/3*ArcTan[a*x]^3/x^3 - a^2*(-(ArcTan[a*x]^3/x) - (a*ArcTan[a*x]^4)/4 + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog [3, -1 + 2/(1 - I*a*x)]/(4*a))))) + a*(-1/2*ArcTan[a*x]^2/x^2 + a*(-(ArcTa n[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) - a ^2*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[ 3, -1 + 2/(1 - I*a*x)]/(4*a))))))/c^2
3.5.3.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)/((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]
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_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
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]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
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 = 233.77 (sec) , antiderivative size = 4175, normalized size of antiderivative = 12.58
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(4175\) |
| default | \(\text {Expression too large to display}\) | \(4175\) |
| parts | \(\text {Expression too large to display}\) | \(4182\) |
a^3*(-1/3/c^2*arctan(a*x)^3/a^3/x^3+2/c^2*arctan(a*x)^3/a/x+1/2/c^2*arctan (a*x)^3*a*x/(a^2*x^2+1)+5/2/c^2*arctan(a*x)^4-1/2/c^2*(arctan(a*x)^2/a^2/x ^2+14*arctan(a*x)^2*ln(a*x)-3/2*arctan(a*x)^2/(a^2*x^2+1)-7*arctan(a*x)^2* ln(a^2*x^2+1)+14*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-14*arctan(a *x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-1/48/a/x/(I+a*x)*(-168*a^2*arctan(a*x) *x^2-168*I*arctan(a*x)^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*( 1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*a^3*x^3-336*I*arct an(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(1+I*a*x)/(a^2*x^2+1 )^(1/2))*a^3*x^3-168*I*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*cs gn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*a^3*x^3+168* I*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)/(a^2*x ^2+1)^(1/2))^2*a^3*x^3+336*I*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+ 1)-I)*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a *x)^2/(a^2*x^2+1)+1))^2*a^3*x^3+336*I*arctan(a*x)^2*Pi*csgn(I/((1+I*a*x)^2 /(a^2*x^2+1)+1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1 )-I/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*a^3*x^3+336*I*arctan(a*x)^2*Pi*csgn(I/( (1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2 +1)+1))*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((1+I *a*x)^2/(a^2*x^2+1)+1))^2*a^3*x^3+336*I*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^ 4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^2/(a^...
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, dx}{c^{2}} \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
1/6144*(1200*(a^5*x^5 + a^3*x^3)*arctan(a*x)^4 - 45*(a^5*x^5 + a^3*x^3)*lo g(a^2*x^2 + 1)^4 + 128*(15*a^4*x^4 + 10*a^2*x^2 - 2)*arctan(a*x)^3 - 24*(1 5*(a^5*x^5 + a^3*x^3)*arctan(a*x)^2 + 4*(15*a^4*x^4 + 10*a^2*x^2 - 2)*arct an(a*x))*log(a^2*x^2 + 1)^2 - 12*(a^2*c^2*x^5 + c^2*x^3)*(120*a^7*(a^2/(a^ 8*c^2*x^2 + a^6*c^2) + log(a^2*x^2 + 1)/(a^6*c^2*x^2 + a^4*c^2)) - 30720*a ^7*integrate(1/256*x^7*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c^2*x^8 + 2*a^2 *c^2*x^6 + c^2*x^4), x) - 7680*a^7*integrate(1/256*x^7*log(a^2*x^2 + 1)^3/ (a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 61440*a^6*integrate(1/256*x^ 6*arctan(a*x)^3/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15360*a^6*in tegrate(1/256*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c^2*x^8 + 2*a^2*c^2* x^6 + c^2*x^4), x) - 122880*a^6*integrate(1/256*x^6*arctan(a*x)*log(a^2*x^ 2 + 1)/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15*a^5*log(a^2*x^2 + 1)^3/(a^4*c^2*x^2 + a^2*c^2) + 45*(2*a^4*(a^2/(a^10*c^2*x^2 + a^8*c^2) + l og(a^2*x^2 + 1)/(a^8*c^2*x^2 + a^6*c^2)) + a^2*log(a^2*x^2 + 1)^2/(a^6*c^2 *x^2 + a^4*c^2))*a^5 - 30720*a^5*integrate(1/256*x^5*arctan(a*x)^2*log(a^2 *x^2 + 1)/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 122880*a^5*integra te(1/256*x^5*arctan(a*x)^2/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 6 0*a^5*log(a^2*x^2 + 1)^2/(a^4*c^2*x^2 + a^2*c^2) + 61440*a^4*integrate(1/2 56*x^4*arctan(a*x)^3/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15360*a ^4*integrate(1/256*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c^2*x^8 + 2*...
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]