\(\int \frac {\arctan (a x)^3}{x^4 (c+a^2 c x^2)^2} \, dx\) [403]

Optimal result

Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {3 a^4 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {7 a^3 \arctan (a x)^2}{8 c^2}-\frac {a \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^3 \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac {7 i a^3 \arctan (a x)^3}{3 c^2}-\frac {\arctan (a x)^3}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)^3}{c^2 x}+\frac {a^4 x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^4}{8 c^2}+\frac {a^3 \log (x)}{c^2}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^2}-\frac {7 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {7 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {7 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2} \] Output:

-3/8*a^3/c^2/(a^2*x^2+1)-a^2*arctan(a*x)/c^2/x-3/4*a^4*x*arctan(a*x)/c^2/( 
a^2*x^2+1)-7/8*a^3*arctan(a*x)^2/c^2-1/2*a*arctan(a*x)^2/c^2/x^2+3/4*a^3*a 
rctan(a*x)^2/c^2/(a^2*x^2+1)+7*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x)) 
/c^2-1/3*arctan(a*x)^3/c^2/x^3+2*a^2*arctan(a*x)^3/c^2/x+1/2*a^4*x*arctan( 
a*x)^3/c^2/(a^2*x^2+1)+5/8*a^3*arctan(a*x)^4/c^2+a^3*ln(x)/c^2-1/2*a^3*ln( 
a^2*x^2+1)/c^2-7*a^3*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^2+7/3*I*a^3*arctan( 
a*x)^3/c^2-7/2*a^3*polylog(3,-1+2/(1-I*a*x))/c^2
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.73 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^3 \left (\frac {7 i \pi ^3}{24}-\frac {\arctan (a x)}{a x}-\frac {1}{2} \arctan (a x)^2-\frac {\arctan (a x)^2}{2 a^2 x^2}-\frac {7}{3} i \arctan (a x)^3-\frac {\arctan (a x)^3}{3 a^3 x^3}+\frac {2 \arctan (a x)^3}{a x}+\frac {5}{8} \arctan (a x)^4-\frac {3}{16} \cos (2 \arctan (a x))+\frac {3}{8} \arctan (a x)^2 \cos (2 \arctan (a x))-7 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+\log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-7 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-\frac {7}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\frac {3}{8} \arctan (a x) \sin (2 \arctan (a x))+\frac {1}{4} \arctan (a x)^3 \sin (2 \arctan (a x))\right )}{c^2} \] Input:

Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]
 

Output:

(a^3*(((7*I)/24)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^ 
2/(2*a^2*x^2) - ((7*I)/3)*ArcTan[a*x]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + (2*A 
rcTan[a*x]^3)/(a*x) + (5*ArcTan[a*x]^4)/8 - (3*Cos[2*ArcTan[a*x]])/16 + (3 
*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 - 7*ArcTan[a*x]^2*Log[1 - E^((-2*I)*A 
rcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (7*I)*ArcTan[a*x]*PolyLog[2, 
 E^((-2*I)*ArcTan[a*x])] - (7*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 - (3*A 
rcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (ArcTan[a*x]^3*Sin[2*ArcTan[a*x]])/4))/ 
c^2
 

Rubi [A] (verified)

Time = 6.46 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.75, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {5501, 27, 5453, 5361, 5453, 5361, 5419, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 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.

\(\displaystyle \int \frac {\arctan (a x)^3}{x^4 \left (a^2 c x^2+c\right )^2} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c x^4 \left (a^2 x^2+1\right )}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c^2 x^2 \left (a^2 x^2+1\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^4}dx-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx+a \int \frac {\arctan (a x)^2}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {-a^2 \left (\int \frac {\arctan (a x)^3}{x^2}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx\right )+a \left (\int \frac {\arctan (a x)^2}{x^3}dx-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {-a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5459

\(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {a \left (-\left (a^2 \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5403

\(\displaystyle -\frac {a^2 \int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5501

\(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5427

\(\displaystyle -\frac {a^2 \left (\int \frac {\arctan (a x)^3}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle -\frac {a^2 \left (-\left (a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )\right )-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx+\int \frac {\arctan (a x)^3}{x^2}dx\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle -\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx-\frac {\arctan (a x)^3}{x}\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle -\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}+\frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}\)

\(\Big \downarrow \) 5459

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 5403

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 5527

\(\displaystyle \frac {-a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )+a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {a \left (-\left (a^2 \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )-a^2 \left (3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )-\frac {\arctan (a x)^3}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )-\frac {1}{4} a \arctan (a x)^4-\frac {\arctan (a x)^3}{x}\right )}{c^2}\)

Input:

Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]
 

Output:

-((a^2*(-(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)*ArcT 
an[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, 
-1 + 2/(1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)]/(4*a))))))/c^2) + 
(-1/3*ArcTan[a*x]^3/x^3 - a^2*(-(ArcTan[a*x]^3/x) - (a*ArcTan[a*x]^4)/4 + 
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))))) + a*(-1/2*ArcTan[a*x]^2/x^2 + a*(-(ArcTa 
n[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) - a 
^2*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] 
+ (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[ 
3, -1 + 2/(1 - I*a*x)]/(4*a))))))/c^2
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 47

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]
 

rule 241

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]
 

rule 243

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]
 

rule 5361

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]
 

rule 5403

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]
 

rule 5419

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]
 

rule 5427

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]
 

rule 5453

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]
 

rule 5459

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]
 

rule 5465

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]
 

rule 5501

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]
 

rule 5527

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]
 

rule 7164

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 9.99 (sec) , antiderivative size = 1990, normalized size of antiderivative = 5.99

\[\text {Expression too large to display}\]

Input:

int(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x)
 

Output:

a^3*(1/2/c^2*arctan(a*x)^3*a*x/(a^2*x^2+1)+5/2/c^2*arctan(a*x)^4-1/3/c^2*a 
rctan(a*x)^3/a^3/x^3+2/c^2*arctan(a*x)^3/a/x-1/2/c^2*(7/4*arctan(a*x)^2-2* 
ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-3/2*arctan(a*x)^2/(a^2*x^2+1)-2*ln(1+(1+ 
I*a*x)/(a^2*x^2+1)^(1/2))+28*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+28*pol 
ylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+14*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^ 
2+1)^(1/2))-3*I*arctan(a*x)*(a*x+I)/(8*a*x-8*I)+3*I*arctan(a*x)*(a*x-I)/(8 
*a*x+8*I)-7*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))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arc 
tan(a*x)^2+7*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*arctan(a*x)^2-7*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csg 
n(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x) 
^2+7/2*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*arctan(a*x)^2-7/2*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))*arctan(a*x)^2+7/2*I*P 
i*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 
+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-7*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*a*x)^2/(a^2*x^2+1)+1 
))^2*arctan(a*x)^2+7/2*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)*arctan(a*x)^2-14*arctan(a*x)^2*ln((1+I*a*x)^ 
2/(a^2*x^2+1)-1)+7*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2...
 

Fricas [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(arctan(a*x)^3/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x)
 

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, dx}{c^{2}} \] Input:

integrate(atan(a*x)**3/x**4/(a**2*c*x**2+c)**2,x)
 

Output:

Integral(atan(a*x)**3/(a**4*x**8 + 2*a**2*x**6 + x**4), x)/c**2
 

Maxima [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x, algorithm="maxima")
 

Output:

1/6144*(1200*(a^5*x^5 + a^3*x^3)*arctan(a*x)^4 - 45*(a^5*x^5 + a^3*x^3)*lo 
g(a^2*x^2 + 1)^4 + 128*(15*a^4*x^4 + 10*a^2*x^2 - 2)*arctan(a*x)^3 - 24*(1 
5*(a^5*x^5 + a^3*x^3)*arctan(a*x)^2 + 4*(15*a^4*x^4 + 10*a^2*x^2 - 2)*arct 
an(a*x))*log(a^2*x^2 + 1)^2 - 12*(a^2*c^2*x^5 + c^2*x^3)*(120*a^7*(a^2/(a^ 
8*c^2*x^2 + a^6*c^2) + log(a^2*x^2 + 1)/(a^6*c^2*x^2 + a^4*c^2)) - 30720*a 
^7*integrate(1/256*x^7*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c^2*x^8 + 2*a^2 
*c^2*x^6 + c^2*x^4), x) - 7680*a^7*integrate(1/256*x^7*log(a^2*x^2 + 1)^3/ 
(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 61440*a^6*integrate(1/256*x^ 
6*arctan(a*x)^3/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15360*a^6*in 
tegrate(1/256*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c^2*x^8 + 2*a^2*c^2* 
x^6 + c^2*x^4), x) - 122880*a^6*integrate(1/256*x^6*arctan(a*x)*log(a^2*x^ 
2 + 1)/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15*a^5*log(a^2*x^2 + 
1)^3/(a^4*c^2*x^2 + a^2*c^2) + 45*(2*a^4*(a^2/(a^10*c^2*x^2 + a^8*c^2) + l 
og(a^2*x^2 + 1)/(a^8*c^2*x^2 + a^6*c^2)) + a^2*log(a^2*x^2 + 1)^2/(a^6*c^2 
*x^2 + a^4*c^2))*a^5 - 30720*a^5*integrate(1/256*x^5*arctan(a*x)^2*log(a^2 
*x^2 + 1)/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 122880*a^5*integra 
te(1/256*x^5*arctan(a*x)^2/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 6 
0*a^5*log(a^2*x^2 + 1)^2/(a^4*c^2*x^2 + a^2*c^2) + 61440*a^4*integrate(1/2 
56*x^4*arctan(a*x)^3/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x) + 15360*a 
^4*integrate(1/256*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c^2*x^8 + 2*...
 

Giac [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \] Input:

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x, algorithm="giac")
 

Output:

integrate(arctan(a*x)^3/((a^2*c*x^2 + c)^2*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \] Input:

int(atan(a*x)^3/(x^4*(c + a^2*c*x^2)^2),x)
 

Output:

int(atan(a*x)^3/(x^4*(c + a^2*c*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {15 \mathit {atan} \left (a x \right )^{4} a^{5} x^{5}+15 \mathit {atan} \left (a x \right )^{4} a^{3} x^{3}+60 \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+40 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}-8 \mathit {atan} \left (a x \right )^{3}+90 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+30 \mathit {atan} \left (a x \right )^{2} a x +60 \mathit {atan} \left (a x \right ) a^{2} x^{2}+84 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{a^{4} x^{7}+2 a^{2} x^{5}+x^{3}}d x \right ) a^{3} x^{5}+84 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{a^{4} x^{7}+2 a^{2} x^{5}+x^{3}}d x \right ) a \,x^{3}+30 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{5} x^{5}+30 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-60 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-60 \,\mathrm {log}\left (x \right ) a^{3} x^{3}+30 a^{5} x^{5}}{24 c^{2} x^{3} \left (a^{2} x^{2}+1\right )} \] Input:

int(atan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x)
 

Output:

(15*atan(a*x)**4*a**5*x**5 + 15*atan(a*x)**4*a**3*x**3 + 60*atan(a*x)**3*a 
**4*x**4 + 40*atan(a*x)**3*a**2*x**2 - 8*atan(a*x)**3 + 90*atan(a*x)**2*a* 
*3*x**3 + 30*atan(a*x)**2*a*x + 60*atan(a*x)*a**2*x**2 + 84*int(atan(a*x)* 
*2/(a**4*x**7 + 2*a**2*x**5 + x**3),x)*a**3*x**5 + 84*int(atan(a*x)**2/(a* 
*4*x**7 + 2*a**2*x**5 + x**3),x)*a*x**3 + 30*log(a**2*x**2 + 1)*a**5*x**5 
+ 30*log(a**2*x**2 + 1)*a**3*x**3 - 60*log(x)*a**5*x**5 - 60*log(x)*a**3*x 
**3 + 30*a**5*x**5)/(24*c**2*x**3*(a**2*x**2 + 1))