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

Optimal. Leaf size=148 \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-2} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+2}{\left (1-\sqrt{7}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-2}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{2 \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+2}} \sqrt{3 x^4-2 x^2-2}} \]

[Out]

(Sqrt[-2 - (1 - Sqrt[7])*x^2]*Sqrt[(2 + (1 + Sqrt[7])*x^2)/(2 + (1 - Sqrt[7])*x^
2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[7])*x^2]], (7 - Sqr
t[7])/14])/(2*7^(1/4)*Sqrt[(2 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-2 - 2*x^2 + 3*x^4
])

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Rubi [A]  time = 0.0942734, antiderivative size = 148, 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 \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-2} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+2}{\left (1-\sqrt{7}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-2}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{2 \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+2}} \sqrt{3 x^4-2 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-2 - (1 - Sqrt[7])*x^2]*Sqrt[(2 + (1 + Sqrt[7])*x^2)/(2 + (1 - Sqrt[7])*x^
2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[7])*x^2]], (7 - Sqr
t[7])/14])/(2*7^(1/4)*Sqrt[(2 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-2 - 2*x^2 + 3*x^4
])

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Rubi in Sympy [A]  time = 3.93774, size = 128, normalized size = 0.86 \[ \frac{7^{\frac{3}{4}} \sqrt{\frac{x^{2} \left (- 2 \sqrt{7} - 2\right ) - 4}{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 4}} \sqrt{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 4} F\left (\operatorname{asin}{\left (\frac{2 \sqrt [4]{7} x}{\sqrt{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 4}} \right )}\middle | - \frac{\sqrt{7}}{14} + \frac{1}{2}\right )}{28 \sqrt{- \frac{1}{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 4}} \sqrt{3 x^{4} - 2 x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7**(3/4)*sqrt((x**2*(-2*sqrt(7) - 2) - 4)/(x**2*(-2 + 2*sqrt(7)) - 4))*sqrt(x**2
*(-2 + 2*sqrt(7)) - 4)*elliptic_f(asin(2*7**(1/4)*x/sqrt(x**2*(-2 + 2*sqrt(7)) -
4)), -sqrt(7)/14 + 1/2)/(28*sqrt(-1/(x**2*(-2 + 2*sqrt(7)) - 4))*sqrt(3*x**4 -
2*x**2 - 2))

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Mathematica [C]  time = 0.0850432, size = 81, normalized size = 0.55 \[ -\frac{i \sqrt{-3 x^4+2 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}} \sqrt{3 x^4-2 x^2-2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*Sqrt[2 + 2*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4
+ Sqrt[7])/3])/(Sqrt[1 + Sqrt[7]]*Sqrt[-2 - 2*x^2 + 3*x^4])

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Maple [C]  time = 0.042, size = 84, normalized size = 0.6 \[ 2\,{\frac{\sqrt{1- \left ( -1/2-1/2\,\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -1/2+1/2\,\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{-2-2\,\sqrt{7}}x,i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{-2-2\,\sqrt{7}}\sqrt{3\,{x}^{4}-2\,{x}^{2}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4-2*x^2-2)^(1/2),x)

[Out]

2/(-2-2*7^(1/2))^(1/2)*(1-(-1/2-1/2*7^(1/2))*x^2)^(1/2)*(1-(-1/2+1/2*7^(1/2))*x^
2)^(1/2)/(3*x^4-2*x^2-2)^(1/2)*EllipticF(1/2*(-2-2*7^(1/2))^(1/2)*x,1/6*I*42^(1/
2)-1/6*I*6^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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