Optimal. Leaf size=27 \[ \sin ^{-1}\left (\frac {1}{2} (-x-1)\right )-\sqrt {-x^2-2 x+3} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 619, 216} \[ \sin ^{-1}\left (\frac {1}{2} (-x-1)\right )-\sqrt {-x^2-2 x+3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {3-2 x-x^2}} \, dx &=-\sqrt {3-2 x-x^2}-\int \frac {1}{\sqrt {3-2 x-x^2}} \, dx\\ &=-\sqrt {3-2 x-x^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,-2-2 x\right )\\ &=-\sqrt {3-2 x-x^2}+\sin ^{-1}\left (\frac {1}{2} (-1-x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \sin ^{-1}\left (\frac {1}{4} (-2 x-2)\right )-\sqrt {-x^2-2 x+3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 1.56 \[ -\sqrt {-x^{2} - 2 \, x + 3} + \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 23, normalized size = 0.85 \[ -\sqrt {-x^{2} - 2 \, x + 3} - \arcsin \left (\frac {1}{2} \, x + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.89 \[ -\arcsin \left (\frac {x}{2}+\frac {1}{2}\right )-\sqrt {-x^{2}-2 x +3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 21, normalized size = 0.78 \[ -\sqrt {-x^{2} - 2 \, x + 3} + \arcsin \left (-\frac {1}{2} \, x - \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 38, normalized size = 1.41 \[ -\sqrt {-x^2-2\,x+3}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2-2\,x+3}+1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 3\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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