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

Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]

[Out]

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

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Rubi [A]  time = 0.0438645, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.31563, size = 31, normalized size = 0.97 \[ \frac{\sqrt{2} \sqrt{- x^{2} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0295661, size = 32, normalized size = 1. \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.035, size = 34, normalized size = 1.1 \[{-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},i\sqrt{2} \right ) \sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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(1/(sqrt(x^2 + 2)*sqrt(x^2 - 1)), x)

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

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 - 1)*(x + 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)