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

Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

[Out]

-ArcSin[(1 - 2*x)/3]

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Rubi [A]  time = 0.0138181, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + x - x^2],x]

[Out]

-ArcSin[(1 - 2*x)/3]

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Rubi in Sympy [A]  time = 0.665824, size = 19, normalized size = 1.58 \[ - \operatorname{atan}{\left (\frac{- 2 x + 1}{2 \sqrt{- x^{2} + x + 2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+x+2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0087237, size = 12, normalized size = 1. \[ -\sin ^{-1}\left (\frac{1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + x - x^2],x]

[Out]

-ArcSin[(1 - 2*x)/3]

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Maple [A]  time = 0.006, size = 7, normalized size = 0.6 \[ \arcsin \left ( -{\frac{1}{3}}+{\frac{2\,x}{3}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+x+2)^(1/2),x)

[Out]

arcsin(-1/3+2/3*x)

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Maxima [A]  time = 1.52782, size = 11, normalized size = 0.92 \[ -\arcsin \left (-\frac{2}{3} \, x + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + x + 2),x, algorithm="maxima")

[Out]

-arcsin(-2/3*x + 1/3)

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Fricas [A]  time = 0.213481, size = 24, normalized size = 2. \[ \arctan \left (\frac{2 \, x - 1}{2 \, \sqrt{-x^{2} + x + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + x + 2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x^{2} + x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+x+2)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 + x + 2), x)

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GIAC/XCAS [A]  time = 0.212079, size = 8, normalized size = 0.67 \[ \arcsin \left (\frac{2}{3} \, x - \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + x + 2),x, algorithm="giac")

[Out]

arcsin(2/3*x - 1/3)