Integrand size = 16, antiderivative size = 14 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \log \left (1+\sqrt {4-x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {31} \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \log \left (\sqrt {4-x}+1\right ) \]
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Rule 31
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {4-x}\right )\right ) \\ & = -2 \log \left (1+\sqrt {4-x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \log \left (1+\sqrt {4-x}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-2 \ln \left (1+\sqrt {4-x}\right )\) | \(13\) |
default | \(-\ln \left (-3+x \right )-2 \,\operatorname {arctanh}\left (\sqrt {4-x}\right )\) | \(18\) |
trager | \(-\ln \left (2 \sqrt {4-x}+5-x \right )\) | \(18\) |
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none
Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \, \log \left (\sqrt {-x + 4} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
Time = 0.65 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=\log {\left (2 \sqrt {4 - x} \right )} - \log {\left (2 \sqrt {4 - x} + 2 \right )} - \log {\left (- x + \sqrt {4 - x} + 4 \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \, \log \left (\sqrt {-x + 4} + 1\right ) \]
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none
Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2 \, \log \left (\sqrt {-x + 4} + 1\right ) \]
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Time = 17.96 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{4+\sqrt {4-x}-x} \, dx=-2\,\ln \left (\sqrt {4-x}+1\right ) \]
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