Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]
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Rubi [A] time = 0.0061394, antiderivative size = 12, normalized size of antiderivative = 1., 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 619
Rule 216
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*}
Mathematica [A] time = 0.0051869, size = 12, normalized size = 1. \[ -\sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 7, normalized size = 0.6 \begin{align*} \arcsin \left ( -{\frac{1}{3}}+{\frac{2\,x}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43113, size = 11, normalized size = 0.92 \begin{align*} -\arcsin \left (-\frac{2}{3} \, x + \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78112, size = 77, normalized size = 6.42 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} + x + 2}{\left (2 \, x - 1\right )}}{2 \,{\left (x^{2} - x - 2\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{2} + x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06304, size = 8, normalized size = 0.67 \begin{align*} \arcsin \left (\frac{2}{3} \, x - \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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