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

Optimal. Leaf size=30 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

integrate(sqrt(-x^2 + 1)/(x^2 + 1), x)

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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]

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

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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]

Integral(sqrt(-(x - 1)*(x + 1))/(x**2 + 1), x)

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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]

-1/2*pi*sign(x) + 1/2*sqrt(2)*(pi*sign(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2
+ 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - arctan(-1/2*x*((sqrt(-x^2 + 1) - 1
)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))