Optimal. Leaf size=20 \[ 2 \sqrt {x}+2 \log \left (1-\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {196, 45}
\begin {gather*} 2 \sqrt {x}+2 \log \left (1-\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=2 \text {Subst}\left (\int \frac {x}{-1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}\right ) \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}+2 \log \left (1-\sqrt {x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} 2 \sqrt {x}+2 \log \left (-1+\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 25, normalized size = 1.25
method | result | size |
derivativedivides | \(2 \sqrt {x}+2 \ln \left (\sqrt {x}-1\right )\) | \(15\) |
meijerg | \(2 \sqrt {x}+2 \ln \left (1-\sqrt {x}\right )\) | \(17\) |
trager | \(2 \sqrt {x}+\ln \left (2 \sqrt {x}-1-x \right )\) | \(18\) |
default | \(\ln \left (x -1\right )+2 \sqrt {x}+\ln \left (\sqrt {x}-1\right )-\ln \left (1+\sqrt {x}\right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} - 1\right ) - 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.63, size = 14, normalized size = 0.70 \begin {gather*} 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.75 \begin {gather*} 2 \sqrt {x} + 2 \log {\left (\sqrt {x} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 15, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} + 2 \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 14, normalized size = 0.70 \begin {gather*} 2\,\ln \left (\sqrt {x}-1\right )+2\,\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [A]
time = 1.00, size = 14, normalized size = 0.70 \begin {gather*} 2 \sqrt {x}+2 \ln \left (\sqrt {x}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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