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

Optimal. Leaf size=153 \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+6}{\left (3-\sqrt{33}\right ) x^2+6}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2\ 3^{3/4} \sqrt [4]{11} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+6}} \sqrt{2 x^4-3 x^2-3}} \]

[Out]

(Sqrt[-6 - (3 - Sqrt[33])*x^2]*Sqrt[(6 + (3 + Sqrt[33])*x^2)/(6 + (3 - Sqrt[33])
*x^2)]*EllipticF[ArcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-6 - (3 - Sqrt[33])*x^2]], (11
- Sqrt[33])/22])/(2*3^(3/4)*11^(1/4)*Sqrt[(6 + (3 - Sqrt[33])*x^2)^(-1)]*Sqrt[-
3 - 3*x^2 + 2*x^4])

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Rubi [A]  time = 0.116393, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+6}{\left (3-\sqrt{33}\right ) x^2+6}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2\ 3^{3/4} \sqrt [4]{11} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+6}} \sqrt{2 x^4-3 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-6 - (3 - Sqrt[33])*x^2]*Sqrt[(6 + (3 + Sqrt[33])*x^2)/(6 + (3 - Sqrt[33])
*x^2)]*EllipticF[ArcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-6 - (3 - Sqrt[33])*x^2]], (11
- Sqrt[33])/22])/(2*3^(3/4)*11^(1/4)*Sqrt[(6 + (3 - Sqrt[33])*x^2)^(-1)]*Sqrt[-
3 - 3*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 3.90481, size = 128, normalized size = 0.84 \[ \frac{11^{\frac{3}{4}} \sqrt [4]{3} \sqrt{\frac{x^{2} \left (- \sqrt{33} - 3\right ) - 6}{x^{2} \left (-3 + \sqrt{33}\right ) - 6}} \sqrt{x^{2} \left (-3 + \sqrt{33}\right ) - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{x^{2} \left (-3 + \sqrt{33}\right ) - 6}} \right )}\middle | - \frac{\sqrt{33}}{22} + \frac{1}{2}\right )}{66 \sqrt{- \frac{1}{x^{2} \left (-3 + \sqrt{33}\right ) - 6}} \sqrt{2 x^{4} - 3 x^{2} - 3}} \]

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]

11**(3/4)*3**(1/4)*sqrt((x**2*(-sqrt(33) - 3) - 6)/(x**2*(-3 + sqrt(33)) - 6))*s
qrt(x**2*(-3 + sqrt(33)) - 6)*elliptic_f(asin(sqrt(2)*33**(1/4)*x/sqrt(x**2*(-3
+ sqrt(33)) - 6)), -sqrt(33)/22 + 1/2)/(66*sqrt(-1/(x**2*(-3 + sqrt(33)) - 6))*s
qrt(2*x**4 - 3*x**2 - 3))

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Mathematica [C]  time = 0.0979657, size = 78, normalized size = 0.51 \[ -\frac{i \sqrt{-4 x^4+6 x^2+6} F\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right )|\frac{1}{4} \left (-7+\sqrt{33}\right )\right )}{\sqrt{3+\sqrt{33}} \sqrt{2 x^4-3 x^2-3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*Sqrt[6 + 6*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-3 + Sqrt[33]]], (-
7 + Sqrt[33])/4])/(Sqrt[3 + Sqrt[33]]*Sqrt[-3 - 3*x^2 + 2*x^4])

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Maple [C]  time = 0.041, size = 84, normalized size = 0.6 \[ 6\,{\frac{\sqrt{1- \left ( -1/2-1/6\,\sqrt{33} \right ){x}^{2}}\sqrt{1- \left ( -1/2+1/6\,\sqrt{33} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,\sqrt{-18-6\,\sqrt{33}}x,i/4\sqrt{22}-i/4\sqrt{6} \right ) }{\sqrt{-18-6\,\sqrt{33}}\sqrt{2\,{x}^{4}-3\,{x}^{2}-3}}} \]

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