Integrand size = 13, antiderivative size = 31 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {4}{3} \log \left (2-\sqrt {2+x}\right )+\frac {2}{3} \log \left (1+\sqrt {2+x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {646, 31} \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {4}{3} \log \left (2-\sqrt {x+2}\right )+\frac {2}{3} \log \left (\sqrt {x+2}+1\right ) \]
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Rule 31
Rule 646
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{-2-x+x^2} \, dx,x,\sqrt {2+x}\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {2+x}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{-2+x} \, dx,x,\sqrt {2+x}\right ) \\ & = \frac {4}{3} \log \left (2-\sqrt {2+x}\right )+\frac {2}{3} \log \left (1+\sqrt {2+x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {4}{3} \log \left (-2+\sqrt {2+x}\right )+\frac {2}{3} \log \left (1+\sqrt {2+x}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 \ln \left (1+\sqrt {2+x}\right )}{3}+\frac {4 \ln \left (\sqrt {2+x}-2\right )}{3}\) | \(22\) |
trager | \(\frac {\ln \left (6 \sqrt {2+x}\, x^{2}-x^{3}+16 \sqrt {2+x}\, x -15 x^{2}+8 \sqrt {2+x}-24 x -12\right )}{3}\) | \(44\) |
default | \(\frac {\ln \left (1+x \right )}{3}+\frac {2 \ln \left (-2+x \right )}{3}+\frac {\ln \left (1+\sqrt {2+x}\right )}{3}-\frac {\ln \left (\sqrt {2+x}-1\right )}{3}+\frac {2 \ln \left (\sqrt {2+x}-2\right )}{3}-\frac {2 \ln \left (\sqrt {2+x}+2\right )}{3}\) | \(54\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {2}{3} \, \log \left (\sqrt {x + 2} + 1\right ) + \frac {4}{3} \, \log \left (\sqrt {x + 2} - 2\right ) \]
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Time = 0.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\log {\left (x - \sqrt {x + 2} \right )} + \frac {\log {\left (2 \sqrt {x + 2} - 4 \right )}}{3} - \frac {\log {\left (2 \sqrt {x + 2} + 2 \right )}}{3} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {2}{3} \, \log \left (\sqrt {x + 2} + 1\right ) + \frac {4}{3} \, \log \left (\sqrt {x + 2} - 2\right ) \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {2}{3} \, \log \left (\sqrt {x + 2} + 1\right ) + \frac {4}{3} \, \log \left ({\left | \sqrt {x + 2} - 2 \right |}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x-\sqrt {2+x}} \, dx=\frac {2\,\ln \left (\frac {2\,\sqrt {x+2}}{3}+\frac {2}{3}\right )}{3}+\frac {4\,\ln \left (\frac {4}{3}-\frac {2\,\sqrt {x+2}}{3}\right )}{3} \]
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