Optimal. Leaf size=26 \[ -\sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 619, 216} \begin {gather*} -\sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
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}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,4-2 x\right )\\ &=-\sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 1.04 \begin {gather*} -\sqrt {-((x-4) x)}-4 \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 36, normalized size = 1.38 \begin {gather*} -\sqrt {4 x-x^2}-4 \tan ^{-1}\left (\frac {\sqrt {4 x-x^2}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 32, normalized size = 1.23 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} + 2 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 0.88 \begin {gather*} 2 \arcsin \left (\frac {x}{2}-1\right )-\sqrt {-x^{2}+4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 2 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 22, normalized size = 0.85 \begin {gather*} 2\,\mathrm {asin}\left (\frac {x}{2}-1\right )-\sqrt {4\,x-x^2} \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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