Integrand size = 22, antiderivative size = 250 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)}{c^2 x}+\frac {a^3 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^2}{4 c^2}-\frac {\arctan (a x)^2}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {2 i a^2 \arctan (a x)^3}{3 c^2}+\frac {a^2 \log (x)}{c^2}-\frac {a^2 \log \left (1+a^2 x^2\right )}{2 c^2}-\frac {2 a^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {2 i a^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c^2} \]
1/4*a^2/c^2/(a^2*x^2+1)-a*arctan(a*x)/c^2/x+1/2*a^3*x*arctan(a*x)/c^2/(a^2 *x^2+1)-1/4*a^2*arctan(a*x)^2/c^2-1/2*arctan(a*x)^2/c^2/x^2-1/2*a^2*arctan (a*x)^2/c^2/(a^2*x^2+1)+2/3*I*a^2*arctan(a*x)^3/c^2+a^2*ln(x)/c^2-1/2*a^2* ln(a^2*x^2+1)/c^2-2*a^2*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^2+2*I*a^2*arctan (a*x)*polylog(2,-1+2/(1-I*a*x))/c^2-a^2*polylog(3,-1+2/(1-I*a*x))/c^2
Time = 1.04 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.64 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (-2 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+\frac {1}{24} \left (2 i \pi ^3-16 i \arctan (a x)^3+3 \cos (2 \arctan (a x))+6 \arctan (a x)^2 \left (-2-\frac {2}{a^2 x^2}-\cos (2 \arctan (a x))-8 \log \left (1-e^{-2 i \arctan (a x)}\right )\right )+24 \log (a x)-12 \log \left (1+a^2 x^2\right )+\frac {6 \arctan (a x) (-4+a x \sin (2 \arctan (a x)))}{a x}\right )\right )}{c^2} \]
(a^2*((-2*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, E ^((-2*I)*ArcTan[a*x])] + ((2*I)*Pi^3 - (16*I)*ArcTan[a*x]^3 + 3*Cos[2*ArcT an[a*x]] + 6*ArcTan[a*x]^2*(-2 - 2/(a^2*x^2) - Cos[2*ArcTan[a*x]] - 8*Log[ 1 - E^((-2*I)*ArcTan[a*x])]) + 24*Log[a*x] - 12*Log[1 + a^2*x^2] + (6*ArcT an[a*x]*(-4 + a*x*Sin[2*ArcTan[a*x]]))/(a*x))/24))/c^2
Time = 2.99 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.46, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {5501, 27, 5453, 5361, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 5501, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2}{x^3 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{c x^3 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)^2}{c^2 x \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^3 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^3}dx-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}+\frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c^2}\) |
-((a^2*((-1/3*I)*ArcTan[a*x]^3 - a^2*(-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) + 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) + (-1/2*ArcTan[a*x]^2/x^2 + a*(-(ArcTan[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
3.3.97.3.1 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 = 85.27 (sec) , antiderivative size = 4110, normalized size of antiderivative = 16.44
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(4110\) |
| default | \(\text {Expression too large to display}\) | \(4110\) |
| parts | \(\text {Expression too large to display}\) | \(5354\) |
a^2*(-1/2*arctan(a*x)^2/c^2/(a^2*x^2+1)+1/c^2*arctan(a*x)^2*ln(a^2*x^2+1)- 1/2/c^2*arctan(a*x)^2/a^2/x^2-2/c^2*arctan(a*x)^2*ln(a*x)-1/c^2*(2*arctan( a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2 *x^2+1)-1)-1/48/a/x/(I+a*x)*(-24*a^2*arctan(a*x)*x^2-24*I*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)^3*Pi*arctan(a*x)^2*a*x-3*a ^3*x^3-48*I*arctan(a*x)*a*x-96*ln(2)*arctan(a*x)^2*a*x-96*ln(2)*arctan(a*x )^2*a^3*x^3-48*arctan(a*x)+9*a*x-12*a^3*arctan(a*x)^2*x^3-12*a*arctan(a*x) ^2*x+24*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*cs gn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arc tan(a*x)^2*a^3*x^3-48*I*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) )*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)-I)*arctan(a*x)^2*a^3*x^3+24*I*csgn(I*( 1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/( a^2*x^2+1))*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2*a*x-48* I*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))*Pi*csgn(I*(1+I*a*x)^ 2/(a^2*x^2+1)-I)*arctan(a*x)^2*a*x+32*I*arctan(a*x)^3*a^3*x^3+32*I*arctan( a*x)^3*a*x+48*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))*Pi*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*arctan(a*x)^2*a^3*...
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]