3.253 \(\int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]

[Out]

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

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Rubi [A]  time = 0.0525185, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 - x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.84701, size = 32, normalized size = 0.89 \[ \frac{\sqrt{x^{2} + 1} F\left (\operatorname{asin}{\left (\sqrt{2} x \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{- x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)

[Out]

sqrt(x**2 + 1)*elliptic_f(asin(sqrt(2)*x), -1/2)/(2*sqrt(-x**2 - 1))

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Mathematica [A]  time = 0.0449215, size = 36, normalized size = 1. \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[2 - 4*x^2]*Sqrt[-1 - x^2]),x]

[Out]

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

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Maple [A]  time = 0.026, size = 34, normalized size = 0.9 \[{{\frac{i}{2}}{\it EllipticF} \left ( ix,i\sqrt{2} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-4*x^2+2)^(1/2)/(-x^2-1)^(1/2),x)

[Out]

1/2*I*EllipticF(I*x,I*2^(1/2))*2^(1/2)/(x^2+1)^(1/2)*(-x^2-1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-4 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{2} - 1} \sqrt{-4 \, x^{2} + 2}}{2 \,{\left (2 \, x^{4} + x^{2} - 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)),x, algorithm="fricas")

[Out]

integral(1/2*sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)/(2*x^4 + x^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-4*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-4 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-4*x^2 + 2)), x)