Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]
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Rubi [A] time = 0.0066545, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {619, 216} \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{5-4 x-x^2}} \, dx &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,-4-2 x\right )\right )\\ &=-\sin ^{-1}\left (\frac{1}{3} (-2-x)\right )\\ \end{align*}
Mathematica [A] time = 0.0060933, size = 12, normalized size = 1. \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 7, normalized size = 0.6 \begin{align*} \arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40417, size = 11, normalized size = 0.92 \begin{align*} -\arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10494, size = 74, normalized size = 6.17 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\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} - 4 x + 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06393, size = 8, normalized size = 0.67 \begin{align*} \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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