3.432 \(\int \frac{\sqrt{1-x}}{\sqrt{-x} \sqrt{1+x}} \, dx\)

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

[Out]

-2*EllipticE[ArcSin[Sqrt[-x]], -1]

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Rubi [A]  time = 0.030591, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]/(Sqrt[-x]*Sqrt[1 + x]),x]

[Out]

-2*EllipticE[ArcSin[Sqrt[-x]], -1]

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Rubi in Sympy [A]  time = 4.58742, size = 14, normalized size = 1.17 \[ - 2 E\left (\operatorname{asin}{\left (\sqrt{- x} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*elliptic_e(asin(sqrt(-x)), -1)

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Mathematica [B]  time = 0.0382434, size = 60, normalized size = 5. \[ -\frac{2 \sqrt{2} \sqrt{x} \sqrt{x+1} \left (F\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )-E\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )\right )}{\sqrt{-x (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x]/(Sqrt[-x]*Sqrt[1 + x]),x]

[Out]

(-2*Sqrt[2]*Sqrt[x]*Sqrt[1 + x]*(-EllipticE[ArcSin[Sqrt[1 - x]], 1/2] + Elliptic
F[ArcSin[Sqrt[1 - x]], 1/2]))/Sqrt[-(x*(1 + x))]

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Maple [A]  time = 0.036, size = 17, normalized size = 1.4 \[ 2\,\sqrt{2}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/2)/(-x)^(1/2)/(1+x)^(1/2),x)

[Out]

2*2^(1/2)*EllipticE((1+x)^(1/2),1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x} \sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-x + 1)/(sqrt(-x)*sqrt(x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x + 1}}{\sqrt{-x} \sqrt{x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-x + 1)/(sqrt(-x)*sqrt(x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-x + 1)/(sqrt(-x)*sqrt(x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x} \sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-x + 1)/(sqrt(-x)*sqrt(x + 1)), x)