3.846 \(\int \frac{1}{\sqrt{\left (1-x^2\right ) \left (3+x^2\right )}} \, dx\)

Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

[Out]

EllipticF[ArcSin[x], -1/3]/Sqrt[3]

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Rubi [A]  time = 0.0383122, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(1 - x^2)*(3 + x^2)],x]

[Out]

EllipticF[ArcSin[x], -1/3]/Sqrt[3]

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Rubi in Sympy [A]  time = 3.23253, size = 14, normalized size = 1.17 \[ \frac{\sqrt{3} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((-x**2+1)*(x**2+3))**(1/2),x)

[Out]

sqrt(3)*elliptic_f(asin(x), -1/3)/3

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Mathematica [C]  time = 0.0249868, size = 18, normalized size = 1.5 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(1 - x^2)*(3 + x^2)],x]

[Out]

(-I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3]

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Maple [B]  time = 0.01, size = 43, normalized size = 3.6 \[{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-x^2+1)*(x^2+3))^(1/2),x)

[Out]

1/3*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/2)*EllipticF(x,1/3*I*3^(1/2
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="fricas")

[Out]

integral(1/sqrt(-x^4 - 2*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (- x^{2} + 1\right ) \left (x^{2} + 3\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x**2+1)*(x**2+3))**(1/2),x)

[Out]

Integral(1/sqrt((-x**2 + 1)*(x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="giac")

[Out]

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