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

Optimal. Leaf size=31 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right ) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

1/2*(sqrt(2)*ln(sqrt(2) + sqrt(x + 1)) - sqrt(2)*ln(-sqrt(2) + sqrt(x + 1)))*i