Integrand size = 15, antiderivative size = 14 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x+2 \log (1-x)-\log (x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1607, 908} \[ \int \frac {1+x^2}{-x+x^2} \, dx=x+2 \log (1-x)-\log (x) \]
[In]
[Out]
Rule 908
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x^2}{(-1+x) x} \, dx \\ & = \int \left (1+\frac {2}{-1+x}-\frac {1}{x}\right ) \, dx \\ & = x+2 \log (1-x)-\log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x+2 \log (1-x)-\log (x) \]
[In]
[Out]
Time = 0.79 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(x -\ln \left (x \right )+2 \ln \left (x -1\right )\) | \(13\) |
norman | \(x -\ln \left (x \right )+2 \ln \left (x -1\right )\) | \(13\) |
risch | \(x -\ln \left (x \right )+2 \ln \left (x -1\right )\) | \(13\) |
parallelrisch | \(x -\ln \left (x \right )+2 \ln \left (x -1\right )\) | \(13\) |
meijerg | \(2 \ln \left (1-x \right )-\ln \left (x \right )-i \pi +x\) | \(19\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x - \log {\left (x \right )} + 2 \log {\left (x - 1 \right )} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^2}{-x+x^2} \, dx=x+2\,\ln \left (x-1\right )-\ln \left (x\right ) \]
[In]
[Out]