Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac{x+1}{\sqrt{2}}\right ) \]
[Out]
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Rubi [A] time = 0.0223867, 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 \[ \sin ^{-1}\left (\frac{x+1}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[1 - 2*x - x^2],x]
[Out]
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Rubi in Sympy [A] time = 0.689149, size = 22, normalized size = 2.2 \[ \operatorname{atan}{\left (- \frac{- 2 x - 2}{2 \sqrt{- x^{2} - 2 x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2-2*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00936366, size = 14, normalized size = 1.4 \[ -\sin ^{-1}\left (\frac{-x-1}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[1 - 2*x - x^2],x]
[Out]
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Maple [A] time = 0.026, size = 10, normalized size = 1. \[ \arcsin \left ({\frac{ \left ( 1+x \right ) \sqrt{2}}{2}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2-2*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.67626, size = 15, normalized size = 1.5 \[ -\arcsin \left (-\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-x^2 - 2*x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201066, size = 28, normalized size = 2.8 \[ -2 \, \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-x^2 - 2*x + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x^{2} - 2 x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2-2*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223129, size = 12, normalized size = 1.2 \[ \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-x^2 - 2*x + 1),x, algorithm="giac")
[Out]