Optimal. Leaf size=24 \[ -\sqrt {2 x-x^2}-2 \sin ^{-1}(1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, 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*} -\sqrt {2 x-x^2}-2 \sin ^{-1}(1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {1+x}{\sqrt {2 x-x^2}} \, dx &=-\sqrt {2 x-x^2}+2 \int \frac {1}{\sqrt {2 x-x^2}} \, dx\\ &=-\sqrt {2 x-x^2}-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,2-2 x\right )\\ &=-\sqrt {2 x-x^2}-2 \sin ^{-1}(1-x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 1.79 \begin {gather*} \frac {(-2+x) x+4 \sqrt {-2+x} \sqrt {x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-2+x}{x}}}\right )}{\sqrt {-((-2+x) x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in comparison} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 21, normalized size = 0.88
method | result | size |
default | \(2 \arcsin \left (-1+x \right )-\sqrt {-x^{2}+2 x}\) | \(21\) |
risch | \(\frac {x \left (-2+x \right )}{\sqrt {-x \left (-2+x \right )}}+2 \arcsin \left (-1+x \right )\) | \(21\) |
trager | \(-\sqrt {-x^{2}+2 x}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )\) | \(45\) |
meijerg | \(2 \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )+\frac {2 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {1-\frac {x}{2}}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )\right )}{\sqrt {\pi }}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 22, normalized size = 0.92 \begin {gather*} -\sqrt {-x^{2} + 2 \, x} - 2 \, \arcsin \left (-x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 32, normalized size = 1.33 \begin {gather*} -\sqrt {-x^{2} + 2 \, x} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 2 \, 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 + 1}{\sqrt {- x \left (x - 2\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 20, normalized size = 0.83 \begin {gather*} -\sqrt {-x^{2}+2 x}+2 \arcsin \left (x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 20, normalized size = 0.83 \begin {gather*} 2\,\mathrm {asin}\left (x-1\right )-\sqrt {2\,x-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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