Optimal. Leaf size=26 \[ -\sqrt {4 x-x^2}-4 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]
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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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {654, 633, 222}
\begin {gather*} -4 \text {ArcSin}\left (1-\frac {x}{2}\right )-\sqrt {4 x-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx &=-\sqrt {4 x-x^2}+4 \int \frac {1}{\sqrt {4 x-x^2}} \, dx\\ &=-\sqrt {4 x-x^2}-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,4-2 x\right )\\ &=-\sqrt {4 x-x^2}-4 \sin ^{-1}\left (1-\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 1.65 \begin {gather*} \frac {(-4+x) x+8 \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 = 0.24, size = 23, normalized size = 0.88
method | result | size |
default | \(4 \arcsin \left (-1+\frac {x}{2}\right )-\sqrt {-x^{2}+4 x}\) | \(23\) |
risch | \(\frac {x \left (x -4\right )}{\sqrt {-x \left (x -4\right )}}+4 \arcsin \left (-1+\frac {x}{2}\right )\) | \(23\) |
meijerg | \(4 \arcsin \left (\frac {\sqrt {x}}{2}\right )+\frac {4 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }}\) | \(45\) |
trager | \(-\sqrt {-x^{2}+4 x}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+4 x}\right )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 4 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 32, normalized size = 1.23 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 8 \, \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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\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 = 4.98, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} + 4 \, \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 = 3.53, size = 22, normalized size = 0.85 \begin {gather*} 4\,\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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