Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \]
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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 = 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 216
Rule 619
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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 29, normalized size = 2.42 \[ -\arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 6, normalized size = 0.50 \[ \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 7, normalized size = 0.58 \[ \arcsin \left (\frac {x}{3}+\frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 8, normalized size = 0.67 \[ -\arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 6, normalized size = 0.50 \[ \mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right ) \]
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 \[ \int \frac {1}{\sqrt {- x^{2} - 4 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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