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

Optimal. Leaf size=11 \[ -2 \sqrt{1-x} \]

[Out]

-2*Sqrt[1 - x]

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Rubi [A]  time = 0.00611616, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -2 \sqrt{1-x} \]

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 = 1.25145, size = 8, normalized size = 0.73 \[ - 2 \sqrt{- x + 1} \]

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 = 0.0123075, size = 23, normalized size = 2.09 \[ \frac{2 (x-1) \sqrt{x+1}}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + x)*Sqrt[1 + x])/Sqrt[1 - x^2]

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Maple [B]  time = 0.003, size = 20, normalized size = 1.8 \[ 2\,{\frac{ \left ( -1+x \right ) \sqrt{1+x}}{\sqrt{-{x}^{2}+1}}} \]

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 = 0.689151, size = 16, normalized size = 1.45 \[ \frac{2 \,{\left (x - 1\right )}}{\sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x - 1)/sqrt(-x + 1)

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Fricas [A]  time = 0.261909, size = 28, normalized size = 2.55 \[ \frac{2 \,{\left (x^{2} - 1\right )}}{\sqrt{-x^{2} + 1} \sqrt{x + 1}} \]

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 = 0.265007, size = 20, normalized size = 1.82 \[ 2 \, \sqrt{2} - 2 \, \sqrt{-x + 1} \]

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)