Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac {1}{3} (-2-x)\right ) \]
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Rubi [A]
time = 0.00, 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 = {633, 222}
\begin {gather*} -\text {ArcSin}\left (\frac {1}{3} (-x-2)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {5-4 x-x^2}} \, dx &=-\left (\frac {1}{6} \text {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 = 0.06, size = 23, normalized size = 1.92 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 7, normalized size = 0.58
method | result | size |
default | \(\arcsin \left (\frac {2}{3}+\frac {x}{3}\right )\) | \(7\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-4 x +5}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.50, size = 8, normalized size = 0.67 \begin {gather*} -\arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (6) = 12\).
time = 2.73, size = 29, normalized size = 2.42 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} - 4 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (6) = 12\).
time = 1.51, size = 26, normalized size = 2.17 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \end {gather*}
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 \begin {gather*} \mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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