Integrand size = 11, antiderivative size = 11 \[ \int \frac {2+x}{x+x^2} \, dx=2 \log (x)-\log (1+x) \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {645} \[ \int \frac {2+x}{x+x^2} \, dx=2 \log (x)-\log (x+1) \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{-1-x}+\frac {2}{x}\right ) \, dx \\ & = 2 \log (x)-\log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{x+x^2} \, dx=2 \log (x)-\log (1+x) \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
default | \(2 \ln \left (x \right )-\ln \left (1+x \right )\) | \(12\) |
norman | \(2 \ln \left (x \right )-\ln \left (1+x \right )\) | \(12\) |
meijerg | \(2 \ln \left (x \right )-\ln \left (1+x \right )\) | \(12\) |
risch | \(2 \ln \left (x \right )-\ln \left (1+x \right )\) | \(12\) |
parallelrisch | \(2 \ln \left (x \right )-\ln \left (1+x \right )\) | \(12\) |
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none
Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{x+x^2} \, dx=-\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {2+x}{x+x^2} \, dx=2 \log {\left (x \right )} - \log {\left (x + 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{x+x^2} \, dx=-\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {2+x}{x+x^2} \, dx=-\log \left ({\left | x + 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{x+x^2} \, dx=2\,\ln \left (x\right )-\ln \left (x+1\right ) \]
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