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

Optimal. Leaf size=13 \[ 2 F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-EllipticE[ArcSin[x], -1] + 2*EllipticF[ArcSin[x], -1]

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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]

-EllipticE[ArcSin[x], -1] + 2*EllipticF[ArcSin[x], -1]

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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]

-elliptic_e(asin(x), -1) + 2*elliptic_f(asin(x), -1)

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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]

(-I)*EllipticE[I*ArcSinh[x], -1]

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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]

-EllipticE(x,I)+2*EllipticF(x,I)

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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]

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

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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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)**(1/2)/(x**2+1)**(1/2),x)

[Out]

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

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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]

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