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 = {640, 619, 216} \[ -\sqrt {2 x-x^2}-2 \sin ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
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}-\operatorname {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 = 0.03, size = 27, normalized size = 1.12 \[ -\sqrt {-((x-2) x)}-4 \sin ^{-1}\left (\sqrt {1-\frac {x}{2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 32, normalized size = 1.33 \[ -\sqrt {-x^{2} + 2 \, x} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 20, normalized size = 0.83 \[ -\sqrt {-x^{2} + 2 \, x} + 2 \, \arcsin \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.88 \[ 2 \arcsin \left (x -1\right )-\sqrt {-x^{2}+2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 22, normalized size = 0.92 \[ -\sqrt {-x^{2} + 2 \, x} - 2 \, \arcsin \left (-x + 1\right ) \]
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 \[ 2\,\mathrm {asin}\left (x-1\right )-\sqrt {2\,x-x^2} \]
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 \[ \int \frac {x + 1}{\sqrt {- x \left (x - 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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