3.31 \(\int \frac{1}{\sqrt{3+3 x^2-2 x^4}} \, dx\)

Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{33}-3}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{33}}}\right )|\frac{1}{4} \left (-7-\sqrt{33}\right )\right ) \]

[Out]

Sqrt[2/(-3 + Sqrt[33])]*EllipticF[ArcSin[(2*x)/Sqrt[3 + Sqrt[33]]], (-7 - Sqrt[3
3])/4]

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Rubi [A]  time = 0.179597, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \sqrt{\frac{2}{\sqrt{33}-3}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{33}}}\right )|\frac{1}{4} \left (-7-\sqrt{33}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 + 3*x^2 - 2*x^4],x]

[Out]

Sqrt[2/(-3 + Sqrt[33])]*EllipticF[ArcSin[(2*x)/Sqrt[3 + Sqrt[33]]], (-7 - Sqrt[3
3])/4]

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Rubi in Sympy [A]  time = 12.7166, size = 56, normalized size = 1.24 \[ - \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{-3 + \sqrt{33}}}{6} \right )}\middle | - \frac{7}{4} - \frac{\sqrt{33}}{4}\right )}{\sqrt{3 + \sqrt{33}} \left (- \sqrt{33} + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-2*x**4+3*x**2+3)**(1/2),x)

[Out]

-4*sqrt(3)*elliptic_f(asin(sqrt(6)*x*sqrt(-3 + sqrt(33))/6), -7/4 - sqrt(33)/4)/
(sqrt(3 + sqrt(33))*(-sqrt(33) + 3))

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Mathematica [C]  time = 0.0889822, size = 50, normalized size = 1.11 \[ -i \sqrt{\frac{2}{3+\sqrt{33}}} F\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right )|\frac{1}{4} \left (-7+\sqrt{33}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[3 + 3*x^2 - 2*x^4],x]

[Out]

(-I)*Sqrt[2/(3 + Sqrt[33])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-3 + Sqrt[33]]], (-7
+ Sqrt[33])/4]

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Maple [B]  time = 0.101, size = 84, normalized size = 1.9 \[ 6\,{\frac{\sqrt{1- \left ( -1/2+1/6\,\sqrt{33} \right ){x}^{2}}\sqrt{1- \left ( -1/2-1/6\,\sqrt{33} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-18+6\,\sqrt{33}},i/4\sqrt{6}+i/4\sqrt{22} \right ) }{\sqrt{-18+6\,\sqrt{33}}\sqrt{-2\,{x}^{4}+3\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4+3*x^2+3)^(1/2),x)

[Out]

6/(-18+6*33^(1/2))^(1/2)*(1-(-1/2+1/6*33^(1/2))*x^2)^(1/2)*(1-(-1/2-1/6*33^(1/2)
)*x^2)^(1/2)/(-2*x^4+3*x^2+3)^(1/2)*EllipticF(1/6*x*(-18+6*33^(1/2))^(1/2),1/4*I
*6^(1/2)+1/4*I*22^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + 3 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + 3*x^2 + 3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + 3*x^2 + 3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 2 x^{4} + 3 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-2*x**4+3*x**2+3)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + 3 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-2*x^4 + 3*x^2 + 3),x, algorithm="giac")

[Out]

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