Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac {1+x}{\sqrt {2}}\right ) \]
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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 = {633, 222}
\begin {gather*} \text {ArcSin}\left (\frac {x+1}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx &=-\frac {\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(10)=20\).
time = 0.05, size = 23, normalized size = 2.30 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-2 x-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 10, normalized size = 1.00
method | result | size |
default | \(\arcsin \left (\frac {\left (1+x \right ) \sqrt {2}}{2}\right )\) | \(10\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x -\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-2 x +1}\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.24, size = 11, normalized size = 1.10 \begin {gather*} -\arcsin \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\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. 21 vs.
\(2 (9) = 18\).
time = 0.99, size = 21, normalized size = 2.10 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 1} - 1}{x}\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} - 2 x + 1}}\, 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. 27 vs.
\(2 (9) = 18\).
time = 0.46, size = 27, normalized size = 2.70 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 1} {\left (x + 1\right )} + \arcsin \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \end {gather*}
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 \begin {gather*} \mathrm {asin}\left (\frac {\sqrt {8}\,\left (2\,x+2\right )}{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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