Optimal. Leaf size=14 \[ -2 \log \left (1+\sqrt {4-x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {31}
\begin {gather*} -2 \log \left (\sqrt {4-x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rubi steps
\begin {align*} \int \frac {1}{4+\sqrt {4-x}-x} \, dx &=-\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*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -2 \log \left (1+\sqrt {4-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 18, normalized size = 1.29
method | result | size |
derivativedivides | \(-2 \ln \left (1+\sqrt {4-x}\right )\) | \(13\) |
default | \(-\ln \left (x -3\right )-2 \arctanh \left (\sqrt {4-x}\right )\) | \(18\) |
trager | \(-\ln \left (2 \sqrt {4-x}+5-x \right )\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs.
\(2 (12) = 24\).
time = 1.83, size = 32, normalized size = 2.29 \begin {gather*} \log {\left (2 \sqrt {4 - x} \right )} - \log {\left (2 \sqrt {4 - x} + 2 \right )} - \log {\left (x - \sqrt {4 - x} - 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 12, normalized size = 0.86 \begin {gather*} -2 \, \log \left (\sqrt {-x + 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 12, normalized size = 0.86 \begin {gather*} -2\,\ln \left (\sqrt {4-x}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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