Integrand size = 14, antiderivative size = 16 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=\frac {4}{2-x}+\log (2-x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 45} \[ \int \frac {2+x}{4-4 x+x^2} \, dx=\frac {4}{2-x}+\log (2-x) \]
[In]
[Out]
Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+x}{(-2+x)^2} \, dx \\ & = \int \left (\frac {4}{(-2+x)^2}+\frac {1}{-2+x}\right ) \, dx \\ & = \frac {4}{2-x}+\log (2-x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=-\frac {4}{-2+x}+\log (-2+x) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {4}{-2+x}+\ln \left (-2+x \right )\) | \(13\) |
norman | \(-\frac {4}{-2+x}+\ln \left (-2+x \right )\) | \(13\) |
risch | \(-\frac {4}{-2+x}+\ln \left (-2+x \right )\) | \(13\) |
meijerg | \(\frac {x}{1-\frac {x}{2}}+\ln \left (1-\frac {x}{2}\right )\) | \(17\) |
parallelrisch | \(\frac {\ln \left (-2+x \right ) x -4-2 \ln \left (-2+x \right )}{-2+x}\) | \(21\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=\frac {{\left (x - 2\right )} \log \left (x - 2\right ) - 4}{x - 2} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.50 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=\log {\left (x - 2 \right )} - \frac {4}{x - 2} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=-\frac {4}{x - 2} + \log \left (x - 2\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=-\frac {4}{x - 2} + \log \left ({\left | x - 2 \right |}\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2+x}{4-4 x+x^2} \, dx=\ln \left (x-2\right )-\frac {4}{x-2} \]
[In]
[Out]