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

Optimal. Leaf size=42 \[ -\frac{\sqrt{1-\frac{1}{x^4}} x^2 F\left (\left .\csc ^{-1}(x)\right |-1\right )}{\sqrt{2-2 x^2} \sqrt{-x^2-1}} \]

[Out]

-((Sqrt[1 - x^(-4)]*x^2*EllipticF[ArcCsc[x], -1])/(Sqrt[2 - 2*x^2]*Sqrt[-1 - x^2
]))

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Rubi [A]  time = 0.0399208, antiderivative size = 65, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{2 \sqrt{-x^2-1} \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.64475, size = 42, normalized size = 1. \[ - \frac{\sqrt{- 2 x^{2} + 2} \sqrt{- x^{2} - 1} \sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{- 2 x^{4} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x**2 + 2)*sqrt(-x**2 - 1)*sqrt(-x**4 + 1)*elliptic_f(asin(x), -1)/(-2*x
**4 + 2)

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Mathematica [A]  time = 0.0477581, size = 30, normalized size = 0.71 \[ \frac{\sqrt{x^2+1} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 30, normalized size = 0.7 \[{{\frac{i}{2}}{\it EllipticF} \left ( ix,i \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 35.2287, size = 73, normalized size = 1.74 \[ \frac{\sqrt{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} - \frac{\sqrt{2}{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)),
x**(-4))/(16*pi**(3/2)) - sqrt(2)*meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0
, 1/2, 0), (-1/4, 1/4, 1/4)), exp_polar(-2*I*pi)/x**4)/(16*pi**(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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