Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {619, 216} \[ -\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+x-x^2}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,1-2 x\right )\right )\\ &=-\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 21, normalized size = 1.75 \[ -2 \tan ^{-1}\left (\frac {\sqrt {-x^2+x+2}}{x+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 30, normalized size = 2.50 \[ -\arctan \left (\frac {\sqrt {-x^{2} + x + 2} {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - x - 2\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 6, normalized size = 0.50 \[ \arcsin \left (\frac {2}{3} \, x - \frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 7, normalized size = 0.58
method | result | size |
default | \(\arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )\) | \(7\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{2}+x +2}\right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 8, normalized size = 0.67 \[ -\arcsin \left (-\frac {2}{3} \, x + \frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 6, normalized size = 0.50 \[ \mathrm {asin}\left (\frac {2\,x}{3}-\frac {1}{3}\right ) \]
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 \[ \int \frac {1}{\sqrt {- x^{2} + x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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