Optimal. Leaf size=31 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right ) \]
[Out]
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Rubi [A] time = 0.0343201, antiderivative size = 31, 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 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 + x]*Sqrt[1 - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 4.89939, size = 27, normalized size = 0.87 \[ \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- x^{2} + 1}}{2 \sqrt{x - 1}} \right )} \]
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 = 0.0229898, size = 46, normalized size = 1.48 \[ -\frac{\sqrt{2} \sqrt{x-1} \sqrt{x+1} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{\sqrt{1-x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 + x]*Sqrt[1 - x^2]),x]
[Out]
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Maple [A] time = 0.015, size = 39, normalized size = 1.3 \[{\sqrt{2}\sqrt{-{x}^{2}+1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ){\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{-1-x}}}} \]
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 = 0.796185, size = 23, normalized size = 0.74 \[ \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 + 1)*sqrt(x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219157, size = 41, normalized size = 1.32 \[ \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x - 1}}{x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 + 1)*sqrt(x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{x - 1}}\, dx \]
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 [A] time = 0.221944, size = 47, normalized size = 1.52 \[ \frac{1}{2} \,{\left (\sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{x + 1}\right ) - \sqrt{2}{\rm ln}\left (-\sqrt{2} + \sqrt{x + 1}\right )\right )} i \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 + 1)*sqrt(x - 1)),x, algorithm="giac")
[Out]