Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac {141 \arctan (a x)}{256 c^3}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^3} \]
3/128*a*x/c^3/(a^2*x^2+1)^2+141/256*a*x/c^3/(a^2*x^2+1)+141/256*arctan(a*x )/c^3-3/32*arctan(a*x)/c^3/(a^2*x^2+1)^2-33/32*arctan(a*x)/c^3/(a^2*x^2+1) -3/16*a*x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2-33/32*a*x*arctan(a*x)^2/c^3/(a^2 *x^2+1)-11/32*arctan(a*x)^3/c^3+1/4*arctan(a*x)^3/c^3/(a^2*x^2+1)^2+1/2*ar ctan(a*x)^3/c^3/(a^2*x^2+1)-1/4*I*arctan(a*x)^4/c^3+arctan(a*x)^3*ln(2-2/( 1-I*a*x))/c^3-3/2*I*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^3+3/2*arctan (a*x)*polylog(3,-1+2/(1-I*a*x))/c^3+3/4*I*polylog(4,-1+2/(1-I*a*x))/c^3
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.63 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {-16 i \pi ^4+256 i \arctan (a x)^4-576 \arctan (a x) \cos (2 \arctan (a x))+384 \arctan (a x)^3 \cos (2 \arctan (a x))-12 \arctan (a x) \cos (4 \arctan (a x))+32 \arctan (a x)^3 \cos (4 \arctan (a x))+1024 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+1536 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+1536 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-768 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+288 \sin (2 \arctan (a x))-576 \arctan (a x)^2 \sin (2 \arctan (a x))+3 \sin (4 \arctan (a x))-24 \arctan (a x)^2 \sin (4 \arctan (a x))}{1024 c^3} \]
((-16*I)*Pi^4 + (256*I)*ArcTan[a*x]^4 - 576*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 384*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 12*ArcTan[a*x]*Cos[4*ArcTan[a*x] ] + 32*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] + 1024*ArcTan[a*x]^3*Log[1 - E^((- 2*I)*ArcTan[a*x])] + (1536*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a* x])] + 1536*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (768*I)*PolyL og[4, E^((-2*I)*ArcTan[a*x])] + 288*Sin[2*ArcTan[a*x]] - 576*ArcTan[a*x]^2 *Sin[2*ArcTan[a*x]] + 3*Sin[4*ArcTan[a*x]] - 24*ArcTan[a*x]^2*Sin[4*ArcTan [a*x]])/(1024*c^3)
Time = 3.19 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.55, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5501, 27, 5465, 5435, 215, 215, 216, 5427, 5465, 215, 216, 5501, 5459, 5403, 5465, 5427, 5465, 215, 216, 5527, 5531, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c^2 x \left (a^2 x^2+1\right )^2}dx}{c}-a^2 \int \frac {x \arctan (a x)^3}{c^3 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^3}dx}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^3}dx}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx-\frac {1}{8} \int \frac {1}{\left (a^2 x^2+1\right )^3}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \int \frac {1}{\left (a^2 x^2+1\right )^2}dx-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+i \int \frac {\arctan (a x)^3}{x (a x+i)}dx-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \int \frac {x \arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {3 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {3 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {3 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 5531 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a}-\frac {\arctan (a x)^3}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{c^3}+\frac {-a^2 \left (\frac {3 \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{2 a}-\frac {\arctan (a x)^3}{2 a^2 \left (a^2 x^2+1\right )}\right )+i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {\operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 a}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{4} i \arctan (a x)^4}{c^3}\) |
-((a^2*(-1/4*ArcTan[a*x]^3/(a^2*(1 + a^2*x^2)^2) + (3*(ArcTan[a*x]/(8*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^2)/(4*(1 + a^2*x^2)^2) + (-1/4*x/(1 + a^2* x^2)^2 - (3*(x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a)))/4)/8 + (3*((x*ArcTa n[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))))/4)) /(4*a)))/c^3) + ((-1/4*I)*ArcTan[a*x]^4 - a^2*(-1/2*ArcTan[a*x]^3/(a^2*(1 + a^2*x^2)) + (3*((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)) + ArcTa n[a*x]/(2*a))/(2*a))))/(2*a)) + I*((-I)*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x )] + (3*I)*a*(((I/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - I*( ((-1/2*I)*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/a + PolyLog[4, -1 + 2/(1 - I*a*x)]/(4*a)))))/c^3
3.5.8.3.1 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[((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)^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_)*((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_.)/((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[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. )*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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 = 93.26 (sec) , antiderivative size = 1841, normalized size of antiderivative = 5.55
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1841\) |
| default | \(\text {Expression too large to display}\) | \(1841\) |
| parts | \(\text {Expression too large to display}\) | \(2278\) |
1/c^3*arctan(a*x)^3*ln(a*x)+1/2*arctan(a*x)^3/c^3/(a^2*x^2+1)-1/2/c^3*arct an(a*x)^3*ln(a^2*x^2+1)+1/4*arctan(a*x)^3/c^3/(a^2*x^2+1)^2-3/4/c^3*(-4/3* arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*(I+a*x)/(16*a*x-16*I)-3/ 8*arctan(a*x)*(a*x-I)/(I+a*x)-3*I*(a*x-I)/(16*a*x+16*I)+4*I*arctan(a*x)^2* polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/3*I*arctan(a*x)^4-3/8*arctan(a*x) *(I+a*x)/(a*x-I)+3*I*arctan(a*x)^2*(a*x-I)/(8*a*x+8*I)-8*I*polylog(4,(1+I* a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan(a*x)^2*(I+a*x)/(8*a*x-8*I)+4/3*arctan(a *x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-4/3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^ 2+1)^(1/2)+1)-8*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-8*I*po lylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-4/3*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+4*I*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))- 8*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/24*(-8*I*Pi*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))+16*I*Pi*csg n(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))+16*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))^3-8*I*Pi*csgn(I*( 1+I*a*x)^2/(a^2*x^2+1))^3+16*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn (I*(1+I*a*x)^2/(a^2*x^2+1))^2+16*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))^3+8*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*c sgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-16*I*Pi*...
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]