\(\int \frac {\arctan (a x)}{x^4 (c+a^2 c x^2)^2} \, dx\) [191]

Optimal result

Integrand size = 20, antiderivative size = 136 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)}{c^2 x}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \] Output:

-1/6*a/c^2/x^2+1/4*a^3/c^2/(a^2*x^2+1)-1/3*arctan(a*x)/c^2/x^3+2*a^2*arcta 
n(a*x)/c^2/x+1/2*a^4*x*arctan(a*x)/c^2/(a^2*x^2+1)+5/4*a^3*arctan(a*x)^2/c 
^2-7/3*a^3*ln(x)/c^2+7/6*a^3*ln(a^2*x^2+1)/c^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}+\frac {\left (-2+10 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)}{6 c^2 x^3 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \] Input:

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

Output:

-1/6*a/(c^2*x^2) + a^3/(4*c^2*(1 + a^2*x^2)) + ((-2 + 10*a^2*x^2 + 15*a^4* 
x^4)*ArcTan[a*x])/(6*c^2*x^3*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^2)/(4*c^2 
) - (7*a^3*Log[x])/(3*c^2) + (7*a^3*Log[1 + a^2*x^2])/(6*c^2)
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.52, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5501, 27, 5453, 5361, 243, 54, 2009, 5453, 5361, 243, 47, 14, 16, 5419, 5501, 5427, 241, 5453, 5361, 243, 47, 14, 16, 5419}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )+\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )+\frac {1}{6} a \int \left (\frac {a^4}{a^2 x^2+1}-\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {-\left (a^2 \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-\left (a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )}{c^2}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )}{c^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\int \frac {\arctan (a x)}{x^2}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )\right )}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^2}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^2}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^2}-\frac {a^2 \left (-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )}{c^2}\)

Input:

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

Output:

-((a^2*(-(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 - a^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*(Log[x^2 
] - Log[1 + a^2*x^2]))/2))/c^2) + (-1/3*ArcTan[a*x]/x^3 - a^2*(-(ArcTan[a* 
x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) + (a*(- 
x^(-2) - a^2*Log[x^2] + a^2*Log[1 + a^2*x^2]))/6)/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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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 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]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\)	\(121\)
default	\(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\)	\(121\)
parts	\(\frac {a^{4} x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} x^{3}}+\frac {2 a^{2} \arctan \left (a x \right )}{c^{2} x}-\frac {\frac {15 a^{3} \arctan \left (a x \right )^{2}}{4}+\frac {a^{3} \left (-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )\right )}{2}}{3 c^{2}}\)	\(128\)
parallelrisch	\(-\frac {-15 a^{5} \arctan \left (a x \right )^{2} x^{5}+28 \ln \left (x \right ) a^{5} x^{5}-14 \ln \left (a^{2} x^{2}+1\right ) x^{5} a^{5}-3 a^{5} x^{5}-30 x^{4} \arctan \left (a x \right ) a^{4}-15 a^{3} \arctan \left (a x \right )^{2} x^{3}+28 \ln \left (x \right ) a^{3} x^{3}-14 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-4 a^{3} x^{3}-20 x^{2} a^{2} \arctan \left (a x \right )+2 a x +4 \arctan \left (a x \right )}{12 x^{3} c^{2} \left (a^{2} x^{2}+1\right )}\)	\(155\)
risch	\(-\frac {5 a^{3} \ln \left (i a x +1\right )^{2}}{16 c^{2}}+\frac {\left (15 a^{5} x^{5} \ln \left (-i a x +1\right )-30 i a^{4} x^{4}+15 a^{3} x^{3} \ln \left (-i a x +1\right )-20 i a^{2} x^{2}+4 i\right ) \ln \left (i a x +1\right )}{24 x^{3} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {15 a^{5} x^{5} \ln \left (-i a x +1\right )^{2}+112 \ln \left (x \right ) a^{5} x^{5}-56 \ln \left (-3 a^{2} x^{2}-3\right ) a^{5} x^{5}-60 i a^{4} \ln \left (-i a x +1\right ) x^{4}+15 a^{3} \ln \left (-i a x +1\right )^{2} x^{3}+112 \ln \left (x \right ) a^{3} x^{3}-56 \ln \left (-3 a^{2} x^{2}-3\right ) a^{3} x^{3}-40 i a^{2} x^{2} \ln \left (-i a x +1\right )-4 a^{3} x^{3}+8 i \ln \left (-i a x +1\right )+8 a x}{48 \left (a x +i\right ) x^{3} c^{2} \left (a x -i\right )}\)	\(276\)

Input:

int(arctan(a*x)/x^4/(a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
 

Output:

a^3*(1/2*a*x*arctan(a*x)/c^2/(a^2*x^2+1)+5/2*arctan(a*x)^2/c^2-1/3/c^2*arc 
tan(a*x)/a^3/x^3+2/c^2*arctan(a*x)/a/x-1/6/c^2*(-7*ln(a^2*x^2+1)-3/2/(a^2* 
x^2+1)+1/a^2/x^2+14*ln(a*x)+15/2*arctan(a*x)^2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^{3} x^{3} + 15 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, {\left (15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2\right )} \arctan \left (a x\right ) + 14 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{12 \, {\left (a^{2} c^{2} x^{5} + c^{2} x^{3}\right )}} \] Input:

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

Output:

1/12*(a^3*x^3 + 15*(a^5*x^5 + a^3*x^3)*arctan(a*x)^2 - 2*a*x + 2*(15*a^4*x 
^4 + 10*a^2*x^2 - 2)*arctan(a*x) + 14*(a^5*x^5 + a^3*x^3)*log(a^2*x^2 + 1) 
 - 28*(a^5*x^5 + a^3*x^3)*log(x))/(a^2*c^2*x^5 + c^2*x^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (129) = 258\).

Time = 1.02 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.66 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\begin {cases} - \frac {28 a^{5} x^{5} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {30 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {28 a^{3} x^{3} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {a^{3} x^{3}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {20 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {2 a x}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {4 \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-28*a**5*x**5*log(x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 14*a* 
*5*x**5*log(x**2 + a**(-2))/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 15*a**5*x 
**5*atan(a*x)**2/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 30*a**4*x**4*atan(a* 
x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) - 28*a**3*x**3*log(x)/(12*a**2*c**2* 
x**5 + 12*c**2*x**3) + 14*a**3*x**3*log(x**2 + a**(-2))/(12*a**2*c**2*x**5 
 + 12*c**2*x**3) + 15*a**3*x**3*atan(a*x)**2/(12*a**2*c**2*x**5 + 12*c**2* 
x**3) + a**3*x**3/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 20*a**2*x**2*atan(a 
*x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) - 2*a*x/(12*a**2*c**2*x**5 + 12*c** 
2*x**3) - 4*atan(a*x)/(12*a**2*c**2*x**5 + 12*c**2*x**3), Ne(a, 0)), (0, T 
rue))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.18 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{6} \, {\left (\frac {15 \, a^{3} \arctan \left (a x\right )}{c^{2}} + \frac {15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2}{a^{2} c^{2} x^{5} + c^{2} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 14 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 2\right )} a}{12 \, {\left (a^{2} c^{2} x^{4} + c^{2} x^{2}\right )}} \] Input:

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

Output:

1/6*(15*a^3*arctan(a*x)/c^2 + (15*a^4*x^4 + 10*a^2*x^2 - 2)/(a^2*c^2*x^5 + 
 c^2*x^3))*arctan(a*x) + 1/12*(a^2*x^2 - 15*(a^4*x^4 + a^2*x^2)*arctan(a*x 
)^2 + 14*(a^4*x^4 + a^2*x^2)*log(a^2*x^2 + 1) - 28*(a^4*x^4 + a^2*x^2)*log 
(x) - 2)*a/(a^2*c^2*x^4 + c^2*x^2)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {5\,x^2}{3\,c^2}-\frac {1}{3\,a^2\,c^2}+\frac {5\,a^2\,x^4}{2\,c^2}\right )}{x^5+\frac {x^3}{a^2}}-\frac {a-\frac {a^3\,x^2}{2}}{6\,a^2\,c^2\,x^4+6\,c^2\,x^2}+\frac {7\,a^3\,\ln \left (a^2\,x^2+1\right )}{6\,c^2}-\frac {7\,a^3\,\ln \left (x\right )}{3\,c^2}+\frac {5\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,c^2} \] Input:

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

Output:

(atan(a*x)*((5*x^2)/(3*c^2) - 1/(3*a^2*c^2) + (5*a^2*x^4)/(2*c^2)))/(x^5 + 
 x^3/a^2) - (a - (a^3*x^2)/2)/(6*c^2*x^2 + 6*a^2*c^2*x^4) + (7*a^3*log(a^2 
*x^2 + 1))/(6*c^2) - (7*a^3*log(x))/(3*c^2) + (5*a^3*atan(a*x)^2)/(4*c^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {15 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}+15 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+30 \mathit {atan} \left (a x \right ) a^{4} x^{4}+20 \mathit {atan} \left (a x \right ) a^{2} x^{2}-4 \mathit {atan} \left (a x \right )+14 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{5} x^{5}+14 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-28 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-28 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-a^{5} x^{5}-2 a x}{12 c^{2} x^{3} \left (a^{2} x^{2}+1\right )} \] Input:

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

Output:

(15*atan(a*x)**2*a**5*x**5 + 15*atan(a*x)**2*a**3*x**3 + 30*atan(a*x)*a**4 
*x**4 + 20*atan(a*x)*a**2*x**2 - 4*atan(a*x) + 14*log(a**2*x**2 + 1)*a**5* 
x**5 + 14*log(a**2*x**2 + 1)*a**3*x**3 - 28*log(x)*a**5*x**5 - 28*log(x)*a 
**3*x**3 - a**5*x**5 - 2*a*x)/(12*c**2*x**3*(a**2*x**2 + 1))