Integrand size = 16, antiderivative size = 18 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x-\frac {8}{3} \arctan \left (\frac {x}{2}\right )+\frac {\arctan (x)}{3} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1136, 1180, 209} \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=-\frac {8}{3} \arctan \left (\frac {x}{2}\right )+\frac {\arctan (x)}{3}+x \]
[In]
[Out]
Rule 209
Rule 1136
Rule 1180
Rubi steps \begin{align*} \text {integral}& = x-\int \frac {4+5 x^2}{4+5 x^2+x^4} \, dx \\ & = x+\frac {1}{3} \int \frac {1}{1+x^2} \, dx-\frac {16}{3} \int \frac {1}{4+x^2} \, dx \\ & = x-\frac {8}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tan ^{-1}(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x+\frac {8}{3} \arctan \left (\frac {2}{x}\right )+\frac {\arctan (x)}{3} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
default | \(x -\frac {8 \arctan \left (\frac {x}{2}\right )}{3}+\frac {\arctan \left (x \right )}{3}\) | \(13\) |
risch | \(x -\frac {8 \arctan \left (\frac {x}{2}\right )}{3}+\frac {\arctan \left (x \right )}{3}\) | \(13\) |
parallelrisch | \(x +\frac {i \ln \left (x +i\right )}{6}-\frac {i \ln \left (x -i\right )}{6}+\frac {4 i \ln \left (x -2 i\right )}{3}-\frac {4 i \ln \left (x +2 i\right )}{3}\) | \(35\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x - \frac {8 \operatorname {atan}{\left (\frac {x}{2} \right )}}{3} + \frac {\operatorname {atan}{\left (x \right )}}{3} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
[In]
[Out]
Time = 8.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{4+5 x^2+x^4} \, dx=x-\frac {8\,\mathrm {atan}\left (\frac {x}{2}\right )}{3}+\frac {\mathrm {atan}\left (x\right )}{3} \]
[In]
[Out]