3.88 \(\int \frac{1}{\sqrt{1-2 x-x^2}} \, dx\)

Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac{x+1}{\sqrt{2}}\right ) \]

[Out]

ArcSin[(1 + x)/Sqrt[2]]

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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]

ArcSin[(1 + x)/Sqrt[2]]

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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]

atan(-(-2*x - 2)/(2*sqrt(-x**2 - 2*x + 1)))

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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]

-ArcSin[(-1 - x)/Sqrt[2]]

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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]

arcsin(1/2*(1+x)*2^(1/2))

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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]

-arcsin(-1/2*sqrt(2)*(x + 1))

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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]

-2*arctan((sqrt(-x^2 - 2*x + 1) - 1)/x)

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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]

Integral(1/sqrt(-x**2 - 2*x + 1), x)

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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]

arcsin(1/2*sqrt(2)*(x + 1))