Integrand size = 20, antiderivative size = 211 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {c x^2}{20 a}+\frac {c x \arctan (a x)}{10 a^2}+\frac {1}{10} c x^3 \arctan (a x)-\frac {c \arctan (a x)^2}{20 a^3}-\frac {c x^2 \arctan (a x)^2}{5 a}-\frac {3}{20} a c x^4 \arctan (a x)^2-\frac {2 i c \arctan (a x)^3}{15 a^3}+\frac {1}{3} c x^3 \arctan (a x)^3+\frac {1}{5} a^2 c x^5 \arctan (a x)^3-\frac {2 c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {2 i c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a^3} \]
-1/20*c*x^2/a+1/10*c*x*arctan(a*x)/a^2+1/10*c*x^3*arctan(a*x)-1/20*c*arcta n(a*x)^2/a^3-1/5*c*x^2*arctan(a*x)^2/a-3/20*a*c*x^4*arctan(a*x)^2-2/15*I*c *arctan(a*x)^3/a^3+1/3*c*x^3*arctan(a*x)^3+1/5*a^2*c*x^5*arctan(a*x)^3-2/5 *c*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3-2/5*I*c*arctan(a*x)*polylog(2,1-2/(1+ I*a*x))/a^3-1/5*c*polylog(3,1-2/(1+I*a*x))/a^3
Time = 0.49 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (-3-3 a^2 x^2+6 a x \arctan (a x)+6 a^3 x^3 \arctan (a x)-3 \arctan (a x)^2-12 a^2 x^2 \arctan (a x)^2-9 a^4 x^4 \arctan (a x)^2+8 i \arctan (a x)^3+20 a^3 x^3 \arctan (a x)^3+12 a^5 x^5 \arctan (a x)^3-24 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{60 a^3} \]
(c*(-3 - 3*a^2*x^2 + 6*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 3*ArcTan[ a*x]^2 - 12*a^2*x^2*ArcTan[a*x]^2 - 9*a^4*x^4*ArcTan[a*x]^2 + (8*I)*ArcTan [a*x]^3 + 20*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^5*ArcTan[a*x]^3 - 24*ArcTan[ a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^( (2*I)*ArcTan[a*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(60*a^3)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(211)=422\).
Time = 3.73 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.38, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5485, 5361, 5451, 5361, 5451, 5345, 240, 5361, 243, 49, 2009, 5419, 5451, 5345, 240, 5419, 5455, 5379, 5529, 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.
| \(\displaystyle \int x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right ) \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int x^4 \arctan (a x)^3dx+c \int x^2 \arctan (a x)^3dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \int \frac {x^5 \arctan (a x)^2}{a^2 x^2+1}dx\right )+c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \int \frac {x^3 \arctan (a x)^2}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\int x \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\int x^3 \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \int \frac {x^4 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {\int \arctan (a x)dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\int x^2 \arctan (a x)dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int x \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\int x^2 \arctan (a x)dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int x \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\int x^2 \arctan (a x)dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int x \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{3} a \int \frac {x^3}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \int \frac {x^2}{a^2 x^2+1}dx^2}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (a^2 x^2+1\right )}\right )dx^2}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \int \frac {x^2 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {\int \arctan (a x)dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {\int \arctan (a x)dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\int \frac {\arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle c \left (\frac {1}{3} x^3 \arctan (a x)^3-a \left (\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}}{a^2}\right )\right )+a^2 c \left (\frac {1}{5} x^5 \arctan (a x)^3-\frac {3}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)^2-\frac {1}{2} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)-\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )}{a^2}-\frac {\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)^2-a \left (\frac {x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}}{a^2}-\frac {\arctan (a x)^2}{2 a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}}{a^2}}{a^2}\right )\right )\) |
c*((x^3*ArcTan[a*x]^3)/3 - a*(((x^2*ArcTan[a*x]^2)/2 - a*(-1/2*ArcTan[a*x] ^2/a^3 + (x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a))/a^2))/a^2 - (((-1/3*I)*A rcTan[a*x]^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*A rcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x) ]/(4*a)))/a)/a^2)) + a^2*c*((x^5*ArcTan[a*x]^3)/5 - (3*a*(((x^4*ArcTan[a*x ]^2)/4 - (a*(((x^3*ArcTan[a*x])/3 - (a*(x^2/a^2 - Log[1 + a^2*x^2]/a^4))/6 )/a^2 - (-1/2*ArcTan[a*x]^2/a^3 + (x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)) /a^2)/a^2))/2)/a^2 - (((x^2*ArcTan[a*x]^2)/2 - a*(-1/2*ArcTan[a*x]^2/a^3 + (x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a))/a^2))/a^2 - (((-1/3*I)*ArcTan[a* x]^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a* x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)) )/a)/a^2)/a^2))/5)
3.4.64.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[m, 1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 46.49 (sec) , antiderivative size = 900, normalized size of antiderivative = 4.27
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(900\) |
| default | \(\text {Expression too large to display}\) | \(900\) |
| parts | \(\text {Expression too large to display}\) | \(902\) |
1/a^3*(1/5*c*arctan(a*x)^3*a^5*x^5+1/3*c*arctan(a*x)^3*a^3*x^3-1/5*c*(3/4* a^4*arctan(a*x)^2*x^4+x^2*arctan(a*x)^2*a^2-arctan(a*x)^2*ln(a^2*x^2+1)+2* arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/12*I*(-6*Pi*arctan(a*x)^2* csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+12*Pi*arctan(a*x)^2*csgn(I*((1+I*a *x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))-6*Pi*arctan( a*x)^2*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+6*Pi*arctan(a*x)^2*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))-12*Pi*arctan(a*x)^2*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+6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x )^2/(a^2*x^2+1))^3+6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn (I/((1+I*a*x)^2/(a^2*x^2+1)+1)^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)-6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*c sgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-6*Pi*arctan (a*x)^2*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+6*Pi*arctan(a*x)^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)^3+8*arctan(a*x)^3+3*I*a^2*x^2-6 *I*arctan(a*x)*a*x+3*I*arctan(a*x)^2-6*I*arctan(a*x)*a^3*x^3+24*I*ln(2)*ar ctan(a*x)^2+3*I)-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+polyl og(3,-(1+I*a*x)^2/(a^2*x^2+1))))
\[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=c \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
\[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
1/120*(3*a^2*c*x^5 + 5*c*x^3)*arctan(a*x)^3 - 1/160*(3*a^2*c*x^5 + 5*c*x^3 )*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/160*(140*(a^4*c*x^6 + 2*a^2 *c*x^4 + c*x^2)*arctan(a*x)^3 - 4*(3*a^3*c*x^5 + 5*a*c*x^3)*arctan(a*x)^2 + 4*(3*a^4*c*x^6 + 5*a^2*c*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (3*a^3*c*x^ 5 + 5*a*c*x^3 + 15*(a^4*c*x^6 + 2*a^2*c*x^4 + c*x^2)*arctan(a*x))*log(a^2* x^2 + 1)^2)/(a^2*x^2 + 1), x)
\[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]