Integrand size = 22, antiderivative size = 242 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a^2}{3 c^2 x}-\frac {a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {7 a^3 \arctan (a x)}{12 c^2}-\frac {a \arctan (a x)}{3 c^2 x^2}+\frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {7 i a^3 \arctan (a x)^2}{3 c^2}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)^2}{c^2 x}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^3}{6 c^2}-\frac {14 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}+\frac {7 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^2} \]
-1/3*a^2/c^2/x-1/4*a^4*x/c^2/(a^2*x^2+1)-7/12*a^3*arctan(a*x)/c^2-1/3*a*ar ctan(a*x)/c^2/x^2+1/2*a^3*arctan(a*x)/c^2/(a^2*x^2+1)+7/3*I*a^3*arctan(a*x )^2/c^2-1/3*arctan(a*x)^2/c^2/x^3+2*a^2*arctan(a*x)^2/c^2/x+1/2*a^4*x*arct an(a*x)^2/c^2/(a^2*x^2+1)+5/6*a^3*arctan(a*x)^3/c^2-14/3*a^3*arctan(a*x)*l n(2-2/(1-I*a*x))/c^2+7/3*I*a^3*polylog(2,-1+2/(1-I*a*x))/c^2
Time = 0.40 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.69 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {20 a^3 x^3 \arctan (a x)^3+2 a x \arctan (a x) \left (-4-4 a^2 x^2+3 a^2 x^2 \cos (2 \arctan (a x))-56 a^2 x^2 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+56 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-a^2 x^2 (8+3 a x \sin (2 \arctan (a x)))+\arctan (a x)^2 \left (-8+48 a^2 x^2+56 i a^3 x^3+6 a^3 x^3 \sin (2 \arctan (a x))\right )}{24 c^2 x^3} \]
(20*a^3*x^3*ArcTan[a*x]^3 + 2*a*x*ArcTan[a*x]*(-4 - 4*a^2*x^2 + 3*a^2*x^2* Cos[2*ArcTan[a*x]] - 56*a^2*x^2*Log[1 - E^((2*I)*ArcTan[a*x])]) + (56*I)*a ^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])] - a^2*x^2*(8 + 3*a*x*Sin[2*ArcTan [a*x]]) + ArcTan[a*x]^2*(-8 + 48*a^2*x^2 + (56*I)*a^3*x^3 + 6*a^3*x^3*Sin[ 2*ArcTan[a*x]]))/(24*c^2*x^3)
Time = 3.44 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.69, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.045, Rules used = {5501, 27, 5453, 5361, 5453, 5361, 264, 216, 5419, 5459, 5403, 2897, 5501, 5427, 5453, 5361, 5419, 5459, 5403, 2897, 5465, 215, 216}
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.
| \(\displaystyle \int \frac {\arctan (a x)^2}{x^4 \left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{c x^4 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)^2}{c^2 x^2 \left (a^2 x^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^4 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^4}dx-a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx\right )+\frac {2}{3} a \int \frac {\arctan (a x)}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-\left (a^2 \left (\int \frac {\arctan (a x)^2}{x^2}dx-a^2 \int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )\right )+\frac {2}{3} a \left (\int \frac {\arctan (a x)}{x^3}dx-a^2 \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )\right )+\frac {2}{3} a \left (a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{2 x^2}\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )\right )+\frac {2}{3} a \left (a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )+\frac {1}{2} a \left (a^2 \left (-\int \frac {1}{a^2 x^2+1}dx\right )-\frac {1}{x}\right )-\frac {\arctan (a x)}{2 x^2}\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )\right )+\frac {2}{3} a \left (a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-\left (a^2 \left (2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )+\frac {2}{3} a \left (a^2 \left (-\int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-\left (a^2 \left (2 a \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )+\frac {2}{3} a \left (-\left (a^2 \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )\right )}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle -\frac {a^2 \left (-\left (a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )\right )-a^2 \int \frac {\arctan (a x)^2}{a^2 x^2+1}dx+\int \frac {\arctan (a x)^2}{x^2}dx\right )}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle -\frac {a^2 \left (-\left (a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )\right )-a^2 \int \frac {\arctan (a x)^2}{a^2 x^2+1}dx+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle -\frac {a^2 \left (-a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}+\frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )-\frac {\arctan (a x)}{2 x^2}+\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )\right )-\left (a^2 \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )-\frac {\arctan (a x)^2}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )+2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )}{c^2}\) |
(-1/3*ArcTan[a*x]^2/x^3 - a^2*(-(ArcTan[a*x]^2/x) - (a*ArcTan[a*x]^3)/3 + 2*a*((-1/2*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2, -1 + 2/(1 - I*a*x)]/2))) + (2*a*(-1/2*ArcTan[a*x]/x^2 + (a*(-x ^(-1) - a*ArcTan[a*x]))/2 - a^2*((-1/2*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a *x]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2, -1 + 2/(1 - I*a*x)]/2))))/3)/c^2 - (a^2*(-(ArcTan[a*x]^2/x) - (a*ArcTan[a*x]^3)/3 - a^2*((x*ArcTan[a*x]^2)/( 2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/2*ArcTan[a*x]/(a^2*(1 + a^2 *x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2*a))) + 2*a*((-1/2*I) *ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2, - 1 + 2/(1 - I*a*x)]/2))))/c^2
3.3.98.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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]
Time = 1.16 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.45
| method | result | size |
| derivativedivides | \(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )^{2}}{c^{2} a x}+\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}+5 \arctan \left (a x \right )^{3}}{3 c^{2}}\right )\) | \(351\) |
| default | \(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )^{2}}{c^{2} a x}+\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}+5 \arctan \left (a x \right )^{3}}{3 c^{2}}\right )\) | \(351\) |
| parts | \(\frac {a^{4} x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} x^{3}}+\frac {2 a^{2} \arctan \left (a x \right )^{2}}{c^{2} x}-\frac {2 \left (\frac {5 a^{3} \arctan \left (a x \right )^{3}}{2}+\frac {a^{3} \left (\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}\right )}{2}\right )}{3 c^{2}}\) | \(358\) |
a^3*(-1/3/c^2*arctan(a*x)^2/a^3/x^3+2/c^2*arctan(a*x)^2/a/x+1/2*a*x*arctan (a*x)^2/c^2/(a^2*x^2+1)+5/2*arctan(a*x)^3/c^2-1/3/c^2*(arctan(a*x)/a^2/x^2 +14*arctan(a*x)*ln(a*x)-3/2*arctan(a*x)/(a^2*x^2+1)-7*arctan(a*x)*ln(a^2*x ^2+1)+7*I*ln(a*x)*ln(1+I*a*x)-7*I*ln(a*x)*ln(1-I*a*x)+7*I*dilog(1+I*a*x)-7 *I*dilog(1-I*a*x)-7/2*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln( a*x-I)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)+7/2*I*(ln(I+a*x)*ln(a^2*x^2+1)- dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)+1/a/x+3/ 4*a*x/(a^2*x^2+1)+7/4*arctan(a*x)+5*arctan(a*x)^3))
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, dx}{c^{2}} \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]