Integrand size = 20, antiderivative size = 183 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {3 a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {35 a^3 \arctan (a x)^2}{16 c^3}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {5 a^3 \log \left (1+a^2 x^2\right )}{3 c^3} \] Output:
-1/6*a/c^3/x^2+1/16*a^3/c^3/(a^2*x^2+1)^2+11/16*a^3/c^3/(a^2*x^2+1)-1/3*ar ctan(a*x)/c^3/x^3+3*a^2*arctan(a*x)/c^3/x+1/4*a^4*x*arctan(a*x)/c^3/(a^2*x ^2+1)^2+11/8*a^4*x*arctan(a*x)/c^3/(a^2*x^2+1)+35/16*a^3*arctan(a*x)^2/c^3 -10/3*a^3*ln(x)/c^3+5/3*a^3*ln(a^2*x^2+1)/c^3
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {2 \left (-8+56 a^2 x^2+175 a^4 x^4+105 a^6 x^6\right ) \arctan (a x)+105 a^3 x^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2+a x \left (-8+20 a^2 x^2+25 a^4 x^4-160 \left (a x+a^3 x^3\right )^2 \log (x)+80 \left (a x+a^3 x^3\right )^2 \log \left (1+a^2 x^2\right )\right )}{48 c^3 x^3 \left (1+a^2 x^2\right )^2} \] Input:
Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),x]
Output:
(2*(-8 + 56*a^2*x^2 + 175*a^4*x^4 + 105*a^6*x^6)*ArcTan[a*x] + 105*a^3*x^3 *(1 + a^2*x^2)^2*ArcTan[a*x]^2 + a*x*(-8 + 20*a^2*x^2 + 25*a^4*x^4 - 160*( a*x + a^3*x^3)^2*Log[x] + 80*(a*x + a^3*x^3)^2*Log[1 + a^2*x^2]))/(48*c^3* x^3*(1 + a^2*x^2)^2)
Leaf count is larger than twice the leaf count of optimal. \(412\) vs. \(2(183)=366\).
Time = 3.16 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.25, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5501, 27, 5501, 5431, 5427, 241, 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.
| \(\displaystyle \int \frac {\arctan (a x)}{x^4 \left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{c^2 x^4 \left (a^2 x^2+1\right )^2}dx}{c}-a^2 \int \frac {\arctan (a x)}{c^3 x^2 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^3}dx}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx\right )}{c^3}\) |
| \(\Big \downarrow \) 5431 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)}{x^4 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx+\int \frac {\arctan (a x)}{x^4}dx}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx+\frac {1}{3} a \int \frac {1}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx+\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx+\frac {1}{6} a \int \left (\frac {a^4}{a^2 x^2+1}-\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx\right )-a^2 \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {-a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {-a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\int \frac {\arctan (a x)}{x^2}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\int \frac {\arctan (a x)}{x^2}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )\right )-a^2 \left (-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\arctan (a x)}{3 x^3}}{c^3}-\frac {a^2 \left (-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )}{c^3}\) |
Input:
Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),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^2*(1/(16* a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(4*(1 + a^2*x^2)^2) + (3*(1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a)))/4) + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2))/c^3) + (-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^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*(Lo g[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^3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((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])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol ] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x ^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(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]
Time = 0.93 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {33}{2 \left (a^{2} x^{2}+1\right )}-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-40 \ln \left (a^{2} x^{2}+1\right )+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\) | \(161\) |
| default | \(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {33}{2 \left (a^{2} x^{2}+1\right )}-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-40 \ln \left (a^{2} x^{2}+1\right )+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\) | \(161\) |
| parts | \(\frac {11 \arctan \left (a x \right ) a^{6} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a^{4} x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 a^{3} \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} x^{3}}+\frac {3 a^{2} \arctan \left (a x \right )}{c^{3} x}-\frac {\frac {105 a^{3} \arctan \left (a x \right )^{2}}{16}+\frac {a^{3} \left (-\frac {33}{2 \left (a^{2} x^{2}+1\right )}-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-40 \ln \left (a^{2} x^{2}+1\right )+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )\right )}{8}}{3 c^{3}}\) | \(168\) |
| parallelrisch | \(-\frac {-105 a^{7} \arctan \left (a x \right )^{2} x^{7}+160 \ln \left (x \right ) a^{7} x^{7}-80 \ln \left (a^{2} x^{2}+1\right ) x^{7} a^{7}+4 a^{7} x^{7}-210 a^{6} \arctan \left (a x \right ) x^{6}-210 a^{5} \arctan \left (a x \right )^{2} x^{5}+320 \ln \left (x \right ) a^{5} x^{5}-160 \ln \left (a^{2} x^{2}+1\right ) x^{5} a^{5}-17 a^{5} x^{5}-350 x^{4} \arctan \left (a x \right ) a^{4}-105 a^{3} \arctan \left (a x \right )^{2} x^{3}+160 \ln \left (x \right ) a^{3} x^{3}-80 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-16 a^{3} x^{3}-112 x^{2} a^{2} \arctan \left (a x \right )+8 a x +16 \arctan \left (a x \right )}{48 x^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}\) | \(217\) |
| risch | \(-\frac {35 a^{3} \ln \left (i a x +1\right )^{2}}{64 c^{3}}+\frac {\left (105 a^{7} x^{7} \ln \left (-i a x +1\right )-210 i a^{6} x^{6}+210 a^{5} x^{5} \ln \left (-i a x +1\right )-350 i a^{4} x^{4}+105 a^{3} x^{3} \ln \left (-i a x +1\right )-112 i a^{2} x^{2}+16 i\right ) \ln \left (i a x +1\right )}{96 x^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {105 a^{7} x^{7} \ln \left (-i a x +1\right )^{2}+640 \ln \left (x \right ) a^{7} x^{7}-320 \ln \left (3 a^{2} x^{2}+3\right ) a^{7} x^{7}-224 i a^{2} x^{2} \ln \left (-i a x +1\right )+210 a^{5} x^{5} \ln \left (-i a x +1\right )^{2}+1280 \ln \left (x \right ) a^{5} x^{5}-640 \ln \left (3 a^{2} x^{2}+3\right ) a^{5} x^{5}-700 i a^{4} \ln \left (-i a x +1\right ) x^{4}-100 a^{5} x^{5}+105 a^{3} \ln \left (-i a x +1\right )^{2} x^{3}+640 \ln \left (x \right ) a^{3} x^{3}-320 \ln \left (3 a^{2} x^{2}+3\right ) a^{3} x^{3}-420 i a^{6} x^{6} \ln \left (-i a x +1\right )-80 a^{3} x^{3}+32 i \ln \left (-i a x +1\right )+32 a x}{192 x^{3} \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2}}\) | \(373\) |
Input:
int(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
a^3*(11/8/c^3*arctan(a*x)/(a^2*x^2+1)^2*a^3*x^3+13/8*a*x*arctan(a*x)/c^3/( a^2*x^2+1)^2+35/8*arctan(a*x)^2/c^3-1/3/c^3*arctan(a*x)/a^3/x^3+3/c^3*arct an(a*x)/a/x-1/24/c^3*(-33/2/(a^2*x^2+1)-3/2/(a^2*x^2+1)^2-40*ln(a^2*x^2+1) +4/a^2/x^2+80*ln(a*x)+105/2*arctan(a*x)^2))
Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {25 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 105 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 8 \, a x + 2 \, {\left (105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8\right )} \arctan \left (a x\right ) + 80 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{48 \, {\left (a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )}} \] Input:
integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="fricas")
Output:
1/48*(25*a^5*x^5 + 20*a^3*x^3 + 105*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*arctan (a*x)^2 - 8*a*x + 2*(105*a^6*x^6 + 175*a^4*x^4 + 56*a^2*x^2 - 8)*arctan(a* x) + 80*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*log(a^2*x^2 + 1) - 160*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*log(x))/(a^4*c^3*x^7 + 2*a^2*c^3*x^5 + c^3*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 724 vs. \(2 (177) = 354\).
Time = 1.76 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.96 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {160 a^{7} x^{7} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{7} x^{7} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{7} x^{7} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{6} x^{6} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {320 a^{5} x^{5} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {160 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {25 a^{5} x^{5}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {350 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {160 a^{3} x^{3} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {20 a^{3} x^{3}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {112 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {8 a x}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {16 \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} 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)**3,x)
Output:
Piecewise((-160*a**7*x**7*log(x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 80*a**7*x**7*log(x**2 + a**(-2))/(48*a**4*c**3*x**7 + 96*a **2*c**3*x**5 + 48*c**3*x**3) + 105*a**7*x**7*atan(a*x)**2/(48*a**4*c**3*x **7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 210*a**6*x**6*atan(a*x)/(48*a**4 *c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) - 320*a**5*x**5*log(x)/(48* a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 160*a**5*x**5*log(x** 2 + a**(-2))/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 210* a**5*x**5*atan(a*x)**2/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x* *3) + 25*a**5*x**5/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 350*a**4*x**4*atan(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3 *x**3) - 160*a**3*x**3*log(x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48* c**3*x**3) + 80*a**3*x**3*log(x**2 + a**(-2))/(48*a**4*c**3*x**7 + 96*a**2 *c**3*x**5 + 48*c**3*x**3) + 105*a**3*x**3*atan(a*x)**2/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 20*a**3*x**3/(48*a**4*c**3*x**7 + 9 6*a**2*c**3*x**5 + 48*c**3*x**3) + 112*a**2*x**2*atan(a*x)/(48*a**4*c**3*x **7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) - 8*a*x/(48*a**4*c**3*x**7 + 96*a* *2*c**3*x**5 + 48*c**3*x**3) - 16*atan(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c **3*x**5 + 48*c**3*x**3), Ne(a, 0)), (0, True))
Time = 0.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.22 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{24} \, {\left (\frac {105 \, a^{3} \arctan \left (a x\right )}{c^{3}} + \frac {105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8}{a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (25 \, a^{4} x^{4} + 20 \, a^{2} x^{2} - 105 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 80 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 8\right )} a}{48 \, {\left (a^{4} c^{3} x^{6} + 2 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )}} \] Input:
integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="maxima")
Output:
1/24*(105*a^3*arctan(a*x)/c^3 + (105*a^6*x^6 + 175*a^4*x^4 + 56*a^2*x^2 - 8)/(a^4*c^3*x^7 + 2*a^2*c^3*x^5 + c^3*x^3))*arctan(a*x) + 1/48*(25*a^4*x^4 + 20*a^2*x^2 - 105*(a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*arctan(a*x)^2 + 80*(a^ 6*x^6 + 2*a^4*x^4 + a^2*x^2)*log(a^2*x^2 + 1) - 160*(a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*log(x) - 8)*a/(a^4*c^3*x^6 + 2*a^2*c^3*x^4 + c^3*x^2)
\[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}} \,d x } \] Input:
integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
integrate(arctan(a*x)/((a^2*c*x^2 + c)^3*x^4), x)
Time = 0.85 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {25\,a^5\,x^4}{2}+10\,a^3\,x^2-4\,a}{24\,a^4\,c^3\,x^6+48\,a^2\,c^3\,x^4+24\,c^3\,x^2}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {7\,x^2}{3\,c^3}-\frac {1}{3\,a^2\,c^3}+\frac {175\,a^2\,x^4}{24\,c^3}+\frac {35\,a^4\,x^6}{8\,c^3}\right )}{2\,x^5+\frac {x^3}{a^2}+a^2\,x^7}+\frac {5\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c^3}-\frac {10\,a^3\,\ln \left (x\right )}{3\,c^3}+\frac {35\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{16\,c^3} \] Input:
int(atan(a*x)/(x^4*(c + a^2*c*x^2)^3),x)
Output:
(10*a^3*x^2 - 4*a + (25*a^5*x^4)/2)/(24*c^3*x^2 + 48*a^2*c^3*x^4 + 24*a^4* c^3*x^6) + (atan(a*x)*((7*x^2)/(3*c^3) - 1/(3*a^2*c^3) + (175*a^2*x^4)/(24 *c^3) + (35*a^4*x^6)/(8*c^3)))/(2*x^5 + x^3/a^2 + a^2*x^7) + (5*a^3*log(a^ 2*x^2 + 1))/(3*c^3) - (10*a^3*log(x))/(3*c^3) + (35*a^3*atan(a*x)^2)/(16*c ^3)
Time = 0.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.18 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {210 \mathit {atan} \left (a x \right )^{2} a^{7} x^{7}+420 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}+210 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+420 \mathit {atan} \left (a x \right ) a^{6} x^{6}+700 \mathit {atan} \left (a x \right ) a^{4} x^{4}+224 \mathit {atan} \left (a x \right ) a^{2} x^{2}-32 \mathit {atan} \left (a x \right )+160 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{7} x^{7}+320 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{5} x^{5}+160 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-320 \,\mathrm {log}\left (x \right ) a^{7} x^{7}-640 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-320 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-25 a^{7} x^{7}+15 a^{3} x^{3}-16 a x}{96 c^{3} x^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int(atan(a*x)/x^4/(a^2*c*x^2+c)^3,x)
Output:
(210*atan(a*x)**2*a**7*x**7 + 420*atan(a*x)**2*a**5*x**5 + 210*atan(a*x)** 2*a**3*x**3 + 420*atan(a*x)*a**6*x**6 + 700*atan(a*x)*a**4*x**4 + 224*atan (a*x)*a**2*x**2 - 32*atan(a*x) + 160*log(a**2*x**2 + 1)*a**7*x**7 + 320*lo g(a**2*x**2 + 1)*a**5*x**5 + 160*log(a**2*x**2 + 1)*a**3*x**3 - 320*log(x) *a**7*x**7 - 640*log(x)*a**5*x**5 - 320*log(x)*a**3*x**3 - 25*a**7*x**7 + 15*a**3*x**3 - 16*a*x)/(96*c**3*x**3*(a**4*x**4 + 2*a**2*x**2 + 1))