Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0278542, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(1 - x^2)^(-1)],x]
[Out]
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Rubi in Sympy [A] time = 0.693494, size = 20, normalized size = 0.74 \[ \sqrt{- x^{2} + 1} \sqrt{\frac{1}{- x^{2} + 1}} \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1/(-x**2+1))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0115194, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(1 - x^2)^(-1)],x]
[Out]
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Maple [A] time = 0.008, size = 30, normalized size = 1.1 \[ \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1/(-x^2+1))^(1/2),x)
[Out]
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Maxima [A] time = 0.7899, size = 3, normalized size = 0.11 \[ \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-1/(x^2 - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268061, size = 42, normalized size = 1.56 \[ 2 \, \arctan \left (\frac{\sqrt{-\frac{1}{x^{2} - 1}} - 1}{x \sqrt{-\frac{1}{x^{2} - 1}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-1/(x^2 - 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.9306, size = 7, normalized size = 0.26 \[ \begin{cases} \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1/(-x**2+1))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270197, size = 14, normalized size = 0.52 \[ -\arcsin \left (x\right ){\rm sign}\left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-1/(x^2 - 1)),x, algorithm="giac")
[Out]