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