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

Optimal. Leaf size=16 \[ \frac{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right ) \]

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2

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Rubi [A]  time = 0.0260297, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2

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Rubi in Sympy [A]  time = 4.93511, size = 14, normalized size = 0.88 \[ \frac{F\left (\operatorname{asin}{\left (\sqrt{2} x \right )}\middle | - \frac{1}{2}\right )}{2} \]

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]

elliptic_f(asin(sqrt(2)*x), -1/2)/2

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Mathematica [A]  time = 0.0389256, size = 16, normalized size = 1. \[ \frac{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2

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Maple [A]  time = 0.03, size = 15, normalized size = 0.9 \[{\frac{{\it EllipticF} \left ( x\sqrt{2},{\frac{i}{2}}\sqrt{2} \right ) }{2}} \]

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*EllipticF(x*2^(1/2),1/2*I*2^(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. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} + 1} \sqrt{-4 \, x^{2} + 2}}, x\right ) \]

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/(sqrt(x^2 + 1)*sqrt(-4*x^2 + 2)), 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)