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

Optimal. Leaf size=29 \[ -\frac{\sqrt{\frac{1}{x}+1} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0421779, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\sqrt{\frac{1}{x}+1} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.71113, size = 26, normalized size = 0.9 \[ \frac{\sqrt{x} \sqrt{1 + \frac{1}{x}} \operatorname{asin}{\left (2 x - 1 \right )}}{\sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193164, size = 41, normalized size = 1.41 \[ -\tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{x}} (2 x-1) \sqrt{1-x^2}}{2 \left (x^2-1\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 40, normalized size = 1.4 \[{\frac{x\arcsin \left ( 2\,x-1 \right ) }{1+x}\sqrt{{\frac{1+x}{x}}}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{-x \left ( -1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.862538, size = 19, normalized size = 0.66 \[ -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

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Fricas [A]  time = 0.313473, size = 46, normalized size = 1.59 \[ -\arctan \left (\frac{2 \, \sqrt{-x^{2} + 1} x \sqrt{\frac{x + 1}{x}}}{2 \, x^{2} + x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{1}{x} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(1/x + 1)/sqrt(-x^2 + 1), x)