Optimal. Leaf size=30 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0457435, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x^2]/(1 + x^2),x]
[Out]
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Rubi in Sympy [A] time = 10.5398, size = 24, normalized size = 0.8 \[ - \operatorname{asin}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- x^{2} + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)**(1/2)/(x**2+1),x)
[Out]
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Mathematica [A] time = 0.0423309, size = 30, normalized size = 1. \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x^2]/(1 + x^2),x]
[Out]
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Maple [A] time = 0.024, size = 33, normalized size = 1.1 \[ -\arcsin \left ( x \right ) -\sqrt{2}\arctan \left ({\frac{x\sqrt{2}}{{x}^{2}-1}\sqrt{-{x}^{2}+1}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)^(1/2)/(x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212727, size = 84, normalized size = 2.8 \[ \sqrt{2} \arctan \left (\frac{x^{2} + \sqrt{-x^{2} + 1} - 1}{\sqrt{2} \sqrt{-x^{2} + 1} x - \sqrt{2} x}\right ) + 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)**(1/2)/(x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.265262, size = 128, normalized size = 4.27 \[ -\frac{1}{2} \, \pi{\rm sign}\left (x\right ) + \frac{1}{2} \, \sqrt{2}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x^2 + 1),x, algorithm="giac")
[Out]