Integrand size = 11, antiderivative size = 19 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=\frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x) \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {36, 31} \[ \int \frac {1}{(-1+x) (2+x)} \, dx=\frac {1}{3} \log (1-x)-\frac {1}{3} \log (x+2) \]
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Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {1}{-1+x} \, dx-\frac {1}{3} \int \frac {1}{2+x} \, dx \\ & = \frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=\frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x) \]
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Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
norman | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
risch | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
parallelrisch | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=-\frac {1}{3} \, \log \left (x + 2\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=\frac {\log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x + 2 \right )}}{3} \]
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none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=-\frac {1}{3} \, \log \left (x + 2\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=-\frac {1}{3} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(-1+x) (2+x)} \, dx=\frac {\ln \left (\frac {x-1}{x+2}\right )}{3} \]
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