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

Optimal. Leaf size=9 \[ 2 \sqrt{x+1} \]

[Out]

2*Sqrt[1 + x]

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Rubi [A]  time = 0.00546179, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ 2 \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]/Sqrt[1 - x^2],x]

[Out]

2*Sqrt[1 + x]

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Rubi in Sympy [A]  time = 1.40581, size = 7, normalized size = 0.78 \[ 2 \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1)

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Mathematica [B]  time = 0.0125821, size = 25, normalized size = 2.78 \[ \frac{2 \sqrt{1-x} (x+1)}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - x]*(1 + x))/Sqrt[1 - x^2]

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Maple [B]  time = 0.003, size = 22, normalized size = 2.4 \[ 2\,{\frac{ \left ( 1+x \right ) \sqrt{1-x}}{\sqrt{-{x}^{2}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.688524, size = 9, normalized size = 1. \[ 2 \, \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1)

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Fricas [A]  time = 0.267257, size = 31, normalized size = 3.44 \[ -\frac{2 \,{\left (x^{2} - 1\right )}}{\sqrt{-x^{2} + 1} \sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264204, size = 18, normalized size = 2. \[ -2 \, \sqrt{2} + 2 \, \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(2) + 2*sqrt(x + 1)