Integrand size = 22, antiderivative size = 432 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {141 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^3 x}-\frac {3 a^4 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {141 a^4 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {205 a^3 \arctan (a x)^2}{128 c^3}-\frac {a \arctan (a x)^2}{2 c^3 x^2}+\frac {3 a^3 \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {33 a^3 \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac {10 i a^3 \arctan (a x)^3}{3 c^3}-\frac {\arctan (a x)^3}{3 c^3 x^3}+\frac {3 a^2 \arctan (a x)^3}{c^3 x}+\frac {a^4 x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^4 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {35 a^3 \arctan (a x)^4}{32 c^3}+\frac {a^3 \log (x)}{c^3}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^3}-\frac {10 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {10 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}-\frac {5 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c^3} \]
-3/128*a^3/c^3/(a^2*x^2+1)^2-141/128*a^3/c^3/(a^2*x^2+1)-a^2*arctan(a*x)/c ^3/x-3/32*a^4*x*arctan(a*x)/c^3/(a^2*x^2+1)^2-141/64*a^4*x*arctan(a*x)/c^3 /(a^2*x^2+1)-205/128*a^3*arctan(a*x)^2/c^3-1/2*a*arctan(a*x)^2/c^3/x^2+3/1 6*a^3*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+33/16*a^3*arctan(a*x)^2/c^3/(a^2*x^2 +1)+10/3*I*a^3*arctan(a*x)^3/c^3-1/3*arctan(a*x)^3/c^3/x^3+3*a^2*arctan(a* x)^3/c^3/x+1/4*a^4*x*arctan(a*x)^3/c^3/(a^2*x^2+1)^2+11/8*a^4*x*arctan(a*x )^3/c^3/(a^2*x^2+1)+35/32*a^3*arctan(a*x)^4/c^3+a^3*ln(x)/c^3-1/2*a^3*ln(a ^2*x^2+1)/c^3-10*a^3*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^3+10*I*a^3*arctan(a *x)*polylog(2,-1+2/(1-I*a*x))/c^3-5*a^3*polylog(3,-1+2/(1-I*a*x))/c^3
Time = 1.16 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.70 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a^3 \left (\frac {5 i \pi ^3}{12}-\frac {\arctan (a x)}{a x}-\frac {1}{2} \arctan (a x)^2-\frac {\arctan (a x)^2}{2 a^2 x^2}-\frac {10}{3} i \arctan (a x)^3-\frac {\arctan (a x)^3}{3 a^3 x^3}+\frac {3 \arctan (a x)^3}{a x}+\frac {35}{32} \arctan (a x)^4-\frac {9}{16} \cos (2 \arctan (a x))+\frac {9}{8} \arctan (a x)^2 \cos (2 \arctan (a x))-\frac {3 \cos (4 \arctan (a x))}{1024}+\frac {3}{128} \arctan (a x)^2 \cos (4 \arctan (a x))-10 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+\log (a x)-\frac {1}{2} \log \left (1+a^2 x^2\right )-10 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-5 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\frac {9}{8} \arctan (a x) \sin (2 \arctan (a x))+\frac {3}{4} \arctan (a x)^3 \sin (2 \arctan (a x))-\frac {3}{256} \arctan (a x) \sin (4 \arctan (a x))+\frac {1}{32} \arctan (a x)^3 \sin (4 \arctan (a x))\right )}{c^3} \]
(a^3*(((5*I)/12)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^ 2/(2*a^2*x^2) - ((10*I)/3)*ArcTan[a*x]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + (3* ArcTan[a*x]^3)/(a*x) + (35*ArcTan[a*x]^4)/32 - (9*Cos[2*ArcTan[a*x]])/16 + (9*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 - (3*Cos[4*ArcTan[a*x]])/1024 + (3 *ArcTan[a*x]^2*Cos[4*ArcTan[a*x]])/128 - 10*ArcTan[a*x]^2*Log[1 - E^((-2*I )*ArcTan[a*x])] + Log[a*x] - Log[1 + a^2*x^2]/2 - (10*I)*ArcTan[a*x]*PolyL og[2, E^((-2*I)*ArcTan[a*x])] - 5*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (9* ArcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (3*ArcTan[a*x]^3*Sin[2*ArcTan[a*x]])/4 - (3*ArcTan[a*x]*Sin[4*ArcTan[a*x]])/256 + (ArcTan[a*x]^3*Sin[4*ArcTan[a* x]])/32))/c^3
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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^4 \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c^2 x^4 \left (a^2 x^2+1\right )^2}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c^3 x^2 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^3}dx}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^3}dx\right )}{c^3}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5431 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (-\frac {3}{8} \left (\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (-\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx+\int \frac {\arctan (a x)^3}{x^4}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx+a \int \frac {\arctan (a x)^2}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\int \frac {\arctan (a x)^3}{x^2}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx\right )+a \left (\int \frac {\arctan (a x)^2}{x^3}dx-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}+\frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx+a \left (-\left (a^2 \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}+\frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}+\frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}+\frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )\right )}{c^3}+\frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )\right )\right )}{c^3}+\frac {-a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\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 )+\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\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}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \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 )\right )\right )\right )}{c^3}+\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 )-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^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\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 )-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx+\int \frac {\arctan (a x)^3}{x^2}dx-\left (a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\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}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \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 )\right )\right )\right )}{c^3}+\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 )-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^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}-a^2 \left (-\frac {\arctan (a x)^3}{x}-a^2 \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\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\right )-a^2 \left (-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}+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 )\right )+a \left (-\left (\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\right )+\left (-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )\right ) a-\frac {\arctan (a x)^2}{2 x^2}\right )}{c^3}-\frac {a^2 \left (-\frac {\arctan (a x)^3}{x}-a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {\arctan (a x)^2}{4 a}+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {\arctan (a x)^2}{4 a}+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )\right )\right )-a^2 \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\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\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-\frac {\arctan (a x)^3}{3 x^3}-a^2 \left (-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}-a^2 \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a^2 \left (-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}+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 )\right )+a \left (-\left (\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\right )+\left (-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )\right ) a-\frac {\arctan (a x)^2}{2 x^2}\right )}{c^3}-\frac {a^2 \left (-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}-a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {\arctan (a x)^2}{4 a}+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {\arctan (a x)^2}{4 a}+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )\right )\right )-a^2 \left (\frac {\arctan (a x)^4}{8 a}+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )}{c^3}\) |
3.5.11.3.1 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_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol ] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x ^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 136.48 (sec) , antiderivative size = 2062, normalized size of antiderivative = 4.77
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2062\) |
| default | \(\text {Expression too large to display}\) | \(2062\) |
| parts | \(\text {Expression too large to display}\) | \(2069\) |
a^3*(-1/3/c^3*arctan(a*x)^3/a^3/x^3+3/c^3*arctan(a*x)^3/a/x+11/8/c^3*arcta n(a*x)^3/(a^2*x^2+1)^2*a^3*x^3+13/8/c^3*arctan(a*x)^3/(a^2*x^2+1)^2*a*x+35 /8/c^3*arctan(a*x)^4-1/8/c^3*(105/4*arctan(a*x)^4+205/16*arctan(a*x)^2+80* arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+80*arctan(a*x)^2*ln((1+I*a *x)/(a^2*x^2+1)^(1/2)+1)+80*arctan(a*x)^2*ln(2)+3/128*cos(4*arctan(a*x))+1 60*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+160*polylog(3,(1+I*a*x)/(a^2*x^ 2+1)^(1/2))-9/4*(I+a*x)/(a*x-I)-9/4*(a*x-I)/(I+a*x)-3/2*arctan(a*x)^2/(a^2 *x^2+1)^2-8*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-8*ln((1+I*a*x)/(a^2*x^2+1)^( 1/2)+1)-33/2*arctan(a*x)^2/(a^2*x^2+1)-40*arctan(a*x)^2*ln(a^2*x^2+1)+3/32 *arctan(a*x)*sin(4*arctan(a*x))-80/3*I*arctan(a*x)^3+80*arctan(a*x)^2*ln(a *x)-80*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-160*I*arctan(a*x)*polyl og(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+40*I*Pi*arctan(a*x)^2+4*arctan(a*x)^2/a ^2/x^2-160*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+80*arctan( a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-40*I*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)^2)^2*arctan(a*x)^2+4*arctan(a* x)*(I*a*x-(a^2*x^2+1)^(1/2)+1)/a/x+40*I*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))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1 )/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2-20*I*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))*csgn(I*(1+I*a*x)^2/(a^2*x ^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2+40*I*Pi*csgn(I*((1+I...
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{10} + 3 a^{4} x^{8} + 3 a^{2} x^{6} + x^{4}}\, dx}{c^{3}} \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]