Integrand size = 20, antiderivative size = 10 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x (1+x)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1608, 27, 75} \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x (x+1)} \]
[In]
[Out]
Rule 27
Rule 75
Rule 1608
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+4 x}{x^2 \left (1+2 x+x^2\right )} \, dx \\ & = \int \frac {2+4 x}{x^2 (1+x)^2} \, dx \\ & = -\frac {2}{x (1+x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x+x^2} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
gosper | \(-\frac {2}{x \left (x +1\right )}\) | \(11\) |
norman | \(-\frac {2}{x \left (x +1\right )}\) | \(11\) |
risch | \(-\frac {2}{x \left (x +1\right )}\) | \(11\) |
parallelrisch | \(-\frac {2}{x \left (x +1\right )}\) | \(11\) |
default | \(-\frac {2}{x}+\frac {2}{x +1}\) | \(14\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x^{2} + x} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=- \frac {2}{x^{2} + x} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x^{2} + x} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x^{2} + x} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {2+4 x}{x^2+2 x^3+x^4} \, dx=-\frac {2}{x\,\left (x+1\right )} \]
[In]
[Out]