Integrand size = 30, antiderivative size = 24 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x+\log \left (\frac {6480 (2-x)^2 x^2}{(1-x)^2}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1608, 1642} \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x-2 \log (1-x)+2 \log (2-x)+2 \log (x) \]
[In]
[Out]
Rule 1608
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \int \frac {4-6 x+5 x^2-x^3}{x \left (2-3 x+x^2\right )} \, dx \\ & = \int \left (-1+\frac {2}{-2+x}-\frac {2}{-1+x}+\frac {2}{x}\right ) \, dx \\ & = -x-2 \log (1-x)+2 \log (2-x)+2 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x-2 \log (1-x)+2 \log (2-x)+2 \log (x) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
default | \(-x +2 \ln \left (-2+x \right )+2 \ln \left (x \right )-2 \ln \left (-1+x \right )\) | \(21\) |
norman | \(-x +2 \ln \left (-2+x \right )+2 \ln \left (x \right )-2 \ln \left (-1+x \right )\) | \(21\) |
risch | \(-x -2 \ln \left (-1+x \right )+2 \ln \left (x^{2}-2 x \right )\) | \(21\) |
parallelrisch | \(-x +2 \ln \left (-2+x \right )+2 \ln \left (x \right )-2 \ln \left (-1+x \right )\) | \(21\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x + 2 \, \log \left (x^{2} - 2 \, x\right ) - 2 \, \log \left (x - 1\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=- x - 2 \log {\left (x - 1 \right )} + 2 \log {\left (x^{2} - 2 x \right )} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x - 2 \, \log \left (x - 1\right ) + 2 \, \log \left (x - 2\right ) + 2 \, \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=-x - 2 \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x - 2 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 11.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {4-6 x+5 x^2-x^3}{2 x-3 x^2+x^3} \, dx=2\,\ln \left (x\,\left (x-2\right )\right )-x-2\,\ln \left (x-1\right ) \]
[In]
[Out]