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

Optimal. Leaf size=24 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0531847, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.53792, size = 26, normalized size = 1.08 \[ - \frac{2 \sqrt{- x} E\left (\operatorname{asin}{\left (\sqrt{- x} \right )}\middle | -1\right )}{\sqrt{x}} \]

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*sqrt(-x)*elliptic_e(asin(sqrt(-x)), -1)/sqrt(x)

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Mathematica [A]  time = 0.0671116, size = 38, normalized size = 1.58 \[ -2 \sqrt{2} \left (F\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )-E\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 25, normalized size = 1. \[ 2\,{\frac{\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) }{\sqrt{x}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-x + 1)/(sqrt(x + 1)*sqrt(x)), 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. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{x + 1} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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