Optimal. Leaf size=13 \[ 2 F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
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Rubi [A] time = 0.0511525, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ 2 F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]
[Out]
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Rubi in Sympy [A] time = 13.1367, size = 14, normalized size = 1.08 \[ - E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) + 2 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0223889, size = 12, normalized size = 0.92 \[ -i E\left (\left .i \sinh ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x^2]/Sqrt[1 + x^2],x]
[Out]
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Maple [A] time = 0.011, size = 14, normalized size = 1.1 \[ -{\it EllipticE} \left ( x,i \right ) +2\,{\it EllipticF} \left ( x,i \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)^(1/2)/(x^2+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/sqrt(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{2} + 1}}{\sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/sqrt(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/sqrt(x^2 + 1),x, algorithm="giac")
[Out]