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

Optimal. Leaf size=49 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0313948, antiderivative size = 49, 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 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.4532, size = 48, normalized size = 0.98 \[ \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{- \frac{- x^{2} - 2}{x^{2} + 1}} \sqrt{- x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0537498, size = 53, normalized size = 1.08 \[ -\frac{i \sqrt{x^2+1} \sqrt{x^2+2} F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\left (x^2+1\right ) \left (x^2+2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.028, size = 33, normalized size = 0.7 \[{{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ( ix,{\frac{\sqrt{2}}{2}} \right ) \sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{x^{2} + 2} \sqrt{-x^{2} - 1}}{x^{4} + 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(x^2 + 2)*sqrt(-x^2 - 1)/(x^4 + 3*x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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