Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac{x+1}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0086627, antiderivative size = 10, 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} \[ \sin ^{-1}\left (\frac{x+1}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0050393, size = 14, normalized size = 1.4 \[ -\sin ^{-1}\left (\frac{-x-1}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 10, normalized size = 1. \begin{align*} \arcsin \left ({\frac{ \left ( 1+x \right ) \sqrt{2}}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43247, size = 15, normalized size = 1.5 \begin{align*} -\arcsin \left (-\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.510158, size = 55, normalized size = 5.5 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 1} - 1}{x}\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} - 2 x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09148, size = 12, normalized size = 1.2 \begin{align*} \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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