Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {93 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {93 a^2 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac {93 a \arctan (a x)^2}{128 c^3}-\frac {3 a \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {21 a \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^3}{c^3}-\frac {\arctan (a x)^3}{c^3 x}-\frac {a^2 x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {7 a^2 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac {15 a \arctan (a x)^4}{32 c^3}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]
3/128*a/c^3/(a^2*x^2+1)^2+93/128*a/c^3/(a^2*x^2+1)+3/32*a^2*x*arctan(a*x)/ c^3/(a^2*x^2+1)^2+93/64*a^2*x*arctan(a*x)/c^3/(a^2*x^2+1)+93/128*a*arctan( a*x)^2/c^3-3/16*a*arctan(a*x)^2/c^3/(a^2*x^2+1)^2-21/16*a*arctan(a*x)^2/c^ 3/(a^2*x^2+1)-I*a*arctan(a*x)^3/c^3-arctan(a*x)^3/c^3/x-1/4*a^2*x*arctan(a *x)^3/c^3/(a^2*x^2+1)^2-7/8*a^2*x*arctan(a*x)^3/c^3/(a^2*x^2+1)-15/32*a*ar ctan(a*x)^4/c^3+3*a*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^3-3*I*a*arctan(a*x)* polylog(2,-1+2/(1-I*a*x))/c^3+3/2*a*polylog(3,-1+2/(1-I*a*x))/c^3
Time = 0.43 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.70 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}-\frac {a x \arctan (a x)^3}{1+a^2 x^2}-\frac {15}{32} \arctan (a x)^4+\frac {3}{8} \cos (2 \arctan (a x))-\frac {3}{4} \arctan (a x)^2 \cos (2 \arctan (a x))+\frac {3 \cos (4 \arctan (a x))}{1024}-\frac {3}{128} \arctan (a x)^2 \cos (4 \arctan (a x))+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+\frac {3}{4} \arctan (a x) \sin (2 \arctan (a x))+\frac {3}{256} \arctan (a x) \sin (4 \arctan (a x))-\frac {1}{32} \arctan (a x)^3 \sin (4 \arctan (a x))\right )}{c^3} \]
(a*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) - (a*x*ArcTan[a* x]^3)/(1 + a^2*x^2) - (15*ArcTan[a*x]^4)/32 + (3*Cos[2*ArcTan[a*x]])/8 - ( 3*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/4 + (3*Cos[4*ArcTan[a*x]])/1024 - (3*A rcTan[a*x]^2*Cos[4*ArcTan[a*x]])/128 + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*A rcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*P olyLog[3, E^((-2*I)*ArcTan[a*x])])/2 + (3*ArcTan[a*x]*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
Time = 4.12 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.62, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5501, 27, 5435, 5427, 5431, 5427, 241, 5465, 5427, 241, 5501, 5427, 5453, 5361, 5419, 5459, 5403, 5465, 5427, 241, 5527, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c^2 x^2 \left (a^2 x^2+1\right )^2}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c^3 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^3}dx}{c^3}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5431 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (-\frac {3}{8} \left (\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (-\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-\left (a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )\right )-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx+\int \frac {\arctan (a x)^3}{x^2}dx}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {-a^2 \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {-a^2 \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {a^2 \left (\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^3}+\frac {-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}}{c^3}\) |
-((a^2*((3*ArcTan[a*x]^2)/(16*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^3)/(4*(1 + a^2*x^2)^2) - (3*(1/(16*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(4*(1 + a^ 2*x^2)^2) + (3*(1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a)))/4))/8 + (3*((x*ArcTan[a*x]^3)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*(-1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4 *a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a ))/a))/2))/4))/c^3) + (-(ArcTan[a*x]^3/x) - (a*ArcTan[a*x]^4)/4 - a^2*((x* ArcTan[a*x]^3)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*(-1/2*ArcTan [a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2* (1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/2) + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*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^3
3.5.9.3.1 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[(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[((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]
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 = 82.41 (sec) , antiderivative size = 1799, normalized size of antiderivative = 5.42
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1799\) |
| default | \(\text {Expression too large to display}\) | \(1799\) |
| parts | \(\text {Expression too large to display}\) | \(1802\) |
a*(-7/8/c^3*arctan(a*x)^3/(a^2*x^2+1)^2*a^3*x^3-9/8/c^3*arctan(a*x)^3/(a^2 *x^2+1)^2*a*x-15/8/c^3*arctan(a*x)^4-1/c^3*arctan(a*x)^3/a/x-3/8/c^3*(-15/ 4*arctan(a*x)^4+1/2*arctan(a*x)^2/(a^2*x^2+1)^2+4*arctan(a*x)^2*ln(a^2*x^2 +1)+7/2*arctan(a*x)^2/(a^2*x^2+1)-8*arctan(a*x)^2*ln(a*x)-8*arctan(a*x)^2* ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+16*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2* x^2+1)^(1/2))+8/3*I*arctan(a*x)^3-I*arctan(a*x)*(a*x-I)/(I+a*x)+1/2*(I+a*x )/(a*x-I)+1/2*(a*x-I)/(I+a*x)+8*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1 )-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+16*I*arctan(a*x)*polyl og(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-16*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/ 2))-8*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*arctan(a*x)*(I+a*x )/(a*x-I)-16*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/16*(32*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)+1)^2)-32*I*P i*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-64 *I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2-32*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)+32*I*Pi*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)^2+64*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/ (a^2*x^2+1)+1))^3-32*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-64*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)/((1+I*...
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{8} + 3 a^{4} x^{6} + 3 a^{2} x^{4} + x^{2}}\, dx}{c^{3}} \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]
-1/16384*(2400*(a^5*x^5 + 2*a^3*x^3 + a*x)*arctan(a*x)^4 - 90*(a^5*x^5 + 2 *a^3*x^3 + a*x)*log(a^2*x^2 + 1)^4 + 256*(15*a^4*x^4 + 25*a^2*x^2 + 8)*arc tan(a*x)^3 - 48*(15*(a^5*x^5 + 2*a^3*x^3 + a*x)*arctan(a*x)^2 + 4*(15*a^4* x^4 + 25*a^2*x^2 + 8)*arctan(a*x))*log(a^2*x^2 + 1)^2 - (a^4*c^3*x^5 + 2*a ^2*c^3*x^3 + c^3*x)*(360*((8*a^2*x^2 + 7)*a^2/(a^12*c^3*x^4 + 2*a^10*c^3*x ^2 + a^8*c^3) + 2*(4*a^2*x^2 + 3)*log(a^2*x^2 + 1)/(a^10*c^3*x^4 + 2*a^8*c ^3*x^2 + a^6*c^3))*a^7 - 2949120*a^7*integrate(1/1024*x^7*arctan(a*x)^2*lo g(a^2*x^2 + 1)/(a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x) - 737280*a^7*integrate(1/1024*x^7*log(a^2*x^2 + 1)^3/(a^6*c^3*x^8 + 3*a^4 *c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x) + 360*(2*a^2*x^2 + 1)*a^5*log(a^2* x^2 + 1)^3/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) + 5898240*a^6*integrate (1/1024*x^6*arctan(a*x)^3/(a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c ^3*x^2), x) + 1474560*a^6*integrate(1/1024*x^6*arctan(a*x)*log(a^2*x^2 + 1 )^2/(a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x) - 11796480 *a^6*integrate(1/1024*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^6*c^3*x^8 + 3*a^ 4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x) + 720*(2*a^2*x^2 + 1)*a^5*log(a^2 *x^2 + 1)^2/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) + 270*(((16*a^2*x^2 + 15)*a^2/(a^14*c^3*x^4 + 2*a^12*c^3*x^2 + a^10*c^3) + 2*(8*a^2*x^2 + 7)*log (a^2*x^2 + 1)/(a^12*c^3*x^4 + 2*a^10*c^3*x^2 + a^8*c^3))*a^4 + 2*(4*a^2*x^ 2 + 3)*a^2*log(a^2*x^2 + 1)^2/(a^10*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3))...
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]