Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac {x+1}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, 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} \[ \sin ^{-1}\left (\frac {x+1}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{8}}} \, dx,x,-2-2 x\right )}{2 \sqrt {2}}\\ &=\sin ^{-1}\left (\frac {1+x}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.40 \[ -\sin ^{-1}\left (\frac {-x-1}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 21, normalized size = 2.10 \[ -2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.03, size = 9, normalized size = 0.90 \[ \arcsin \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 10, normalized size = 1.00 \[ \arcsin \left (\frac {\left (x +1\right ) \sqrt {2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 11, normalized size = 1.10 \[ -\arcsin \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 11, normalized size = 1.10 \[ \mathrm {asin}\left (\frac {\sqrt {8}\,\left (2\,x+2\right )}{8}\right ) \]
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 \[ \int \frac {1}{\sqrt {- x^{2} - 2 x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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