Optimal. Leaf size=28 \[ \sqrt {-4 x+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {-4 x+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {654, 634, 212}
\begin {gather*} \sqrt {x^2-4 x}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-4 x+x^2}} \, dx &=\sqrt {-4 x+x^2}+2 \int \frac {1}{\sqrt {-4 x+x^2}} \, dx\\ &=\sqrt {-4 x+x^2}+4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-4 x+x^2}}\right )\\ &=\sqrt {-4 x+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {-4 x+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 1.50 \begin {gather*} \frac {(-4+x) x+4 \sqrt {-4+x} \sqrt {x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-4+x}{x}}}\right )}{\sqrt {(-4+x) x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 26, normalized size = 0.93
method | result | size |
default | \(\sqrt {x^{2}-4 x}+2 \ln \left (x -2+\sqrt {x^{2}-4 x}\right )\) | \(26\) |
trager | \(\sqrt {x^{2}-4 x}-2 \ln \left (2-x +\sqrt {x^{2}-4 x}\right )\) | \(28\) |
risch | \(\frac {x \left (x -4\right )}{\sqrt {x \left (x -4\right )}}+2 \ln \left (x -2+\sqrt {x^{2}-4 x}\right )\) | \(29\) |
meijerg | \(\frac {4 i \sqrt {-\mathrm {signum}\left (x -4\right )}\, \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x -4\right )}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 29, normalized size = 1.04 \begin {gather*} \sqrt {x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 27, normalized size = 0.96 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt {x^{2} - 4 \, x} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x \left (x - 4\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 28, normalized size = 1.00 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x} + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 23, normalized size = 0.82 \begin {gather*} 2\,\ln \left (x+\sqrt {x\,\left (x-4\right )}-2\right )+\sqrt {x^2-4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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