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

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

[Out]

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

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Rubi [A]  time = 0.0505064, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.66463, size = 29, normalized size = 0.97 \[ \frac{\sqrt{2} \sqrt{- x^{2} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -2\right )}{2 \sqrt{x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0433599, size = 30, normalized size = 1. \[ \frac{\sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.043, size = 34, normalized size = 1.1 \[{-{\frac{i}{2}}{\it EllipticF} \left ( ix\sqrt{2},{\frac{i}{2}}\sqrt{2} \right ) \sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*I*EllipticF(I*x*2^(1/2),1/2*I*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. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{x^{2} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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