Integrand size = 14, antiderivative size = 54 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {x^2}{2}+\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1373, 1136, 1175, 632, 210} \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {x^2}{2} \]
[In]
[Out]
Rule 210
Rule 632
Rule 1136
Rule 1175
Rule 1373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^4}{1+x^2+x^4} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{1+x^2+x^4} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right ) \\ & = \frac {x^2}{2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.81 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {x^2}{2}-\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x^2\right )}{2 \sqrt {6+6 i \sqrt {3}}}-\frac {\left (-i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x^2\right )}{2 \sqrt {6-6 i \sqrt {3}}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {x^{2}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}\) | \(43\) |
risch | \(\frac {x^{2}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {x^{2} \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {x^{6} \sqrt {3}}{3}+\frac {2 x^{2} \sqrt {3}}{3}\right )}{6}\) | \(44\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x^{2}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{6} + 2 \, x^{2}\right )}\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {x^{2}}{2} + \frac {\sqrt {3} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{2}}{3} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{6}}{3} + \frac {2 \sqrt {3} x^{2}}{3} \right )}\right )}{12} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {x^9}{1+x^4+x^8} \, dx=\frac {x^2}{2}-\frac {\sqrt {3}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^6}{3}+\frac {2\,\sqrt {3}\,x^2}{3}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^2}{3}\right )\right )}{12} \]
[In]
[Out]