Integrand size = 11, antiderivative size = 17 \[ \int \frac {2+x^2}{2+x} \, dx=-2 x+\frac {x^2}{2}+6 \log (2+x) \]
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Time = 0.01 (sec) , antiderivative size = 17, 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 = {711} \[ \int \frac {2+x^2}{2+x} \, dx=\frac {x^2}{2}-2 x+6 \log (x+2) \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (-2+x+\frac {6}{2+x}\right ) \, dx \\ & = -2 x+\frac {x^2}{2}+6 \log (2+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {2+x^2}{2+x} \, dx=-6-2 x+\frac {x^2}{2}+6 \log (2+x) \]
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Time = 0.81 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(-2 x +\frac {x^{2}}{2}+6 \ln \left (x +2\right )\) | \(16\) |
norman | \(-2 x +\frac {x^{2}}{2}+6 \ln \left (x +2\right )\) | \(16\) |
risch | \(-2 x +\frac {x^{2}}{2}+6 \ln \left (x +2\right )\) | \(16\) |
parallelrisch | \(-2 x +\frac {x^{2}}{2}+6 \ln \left (x +2\right )\) | \(16\) |
meijerg | \(6 \ln \left (1+\frac {x}{2}\right )-\frac {x \left (-\frac {3 x}{2}+6\right )}{3}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {2+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left (x + 2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {2+x^2}{2+x} \, dx=\frac {x^{2}}{2} - 2 x + 6 \log {\left (x + 2 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {2+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left (x + 2\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {2+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x + 6 \, \log \left ({\left | x + 2 \right |}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {2+x^2}{2+x} \, dx=6\,\ln \left (x+2\right )-2\,x+\frac {x^2}{2} \]
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