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

Optimal. Leaf size=141 \[ \frac{\sqrt{\frac{2-\left (2-\sqrt{10}\right ) x^2}{2-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-2}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{2-\left (2+\sqrt{10}\right ) x^2}} \sqrt{3 x^4+4 x^2-2}} \]

[Out]

(Sqrt[(2 - (2 - Sqrt[10])*x^2)/(2 - (2 + Sqrt[10])*x^2)]*Sqrt[-2 + (2 + Sqrt[10]
)*x^2]*EllipticF[ArcSin[(2^(3/4)*5^(1/4)*x)/Sqrt[-2 + (2 + Sqrt[10])*x^2]], (5 +
Sqrt[10])/10])/(2*10^(1/4)*Sqrt[(2 - (2 + Sqrt[10])*x^2)^(-1)]*Sqrt[-2 + 4*x^2
+ 3*x^4])

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Rubi [A]  time = 0.121361, antiderivative size = 141, 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 \[ \frac{\sqrt{\frac{2-\left (2-\sqrt{10}\right ) x^2}{2-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-2}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{2-\left (2+\sqrt{10}\right ) x^2}} \sqrt{3 x^4+4 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(2 - (2 - Sqrt[10])*x^2)/(2 - (2 + Sqrt[10])*x^2)]*Sqrt[-2 + (2 + Sqrt[10]
)*x^2]*EllipticF[ArcSin[(2^(3/4)*5^(1/4)*x)/Sqrt[-2 + (2 + Sqrt[10])*x^2]], (5 +
Sqrt[10])/10])/(2*10^(1/4)*Sqrt[(2 - (2 + Sqrt[10])*x^2)^(-1)]*Sqrt[-2 + 4*x^2
+ 3*x^4])

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Rubi in Sympy [A]  time = 4.16877, size = 126, normalized size = 0.89 \[ \frac{10^{\frac{3}{4}} \sqrt{\frac{x^{2} \left (- 2 \sqrt{10} + 4\right ) - 4}{x^{2} \left (4 + 2 \sqrt{10}\right ) - 4}} \sqrt{x^{2} \left (4 + 2 \sqrt{10}\right ) - 4} F\left (\operatorname{asin}{\left (\frac{2 \sqrt [4]{10} x}{\sqrt{x^{2} \left (4 + 2 \sqrt{10}\right ) - 4}} \right )}\middle | \frac{\sqrt{10}}{10} + \frac{1}{2}\right )}{40 \sqrt{- \frac{1}{x^{2} \left (4 + 2 \sqrt{10}\right ) - 4}} \sqrt{3 x^{4} + 4 x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10**(3/4)*sqrt((x**2*(-2*sqrt(10) + 4) - 4)/(x**2*(4 + 2*sqrt(10)) - 4))*sqrt(x*
*2*(4 + 2*sqrt(10)) - 4)*elliptic_f(asin(2*10**(1/4)*x/sqrt(x**2*(4 + 2*sqrt(10)
) - 4)), sqrt(10)/10 + 1/2)/(40*sqrt(-1/(x**2*(4 + 2*sqrt(10)) - 4))*sqrt(3*x**4
+ 4*x**2 - 2))

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Mathematica [C]  time = 0.114596, size = 81, normalized size = 0.57 \[ -\frac{i \sqrt{-3 x^4-4 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{-1+\sqrt{\frac{5}{2}}} x\right )|\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right )}{\sqrt{\sqrt{10}-2} \sqrt{3 x^4+4 x^2-2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*Sqrt[2 - 4*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/2]]*x], (-7 -
2*Sqrt[10])/3])/(Sqrt[-2 + Sqrt[10]]*Sqrt[-2 + 4*x^2 + 3*x^4])

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Maple [C]  time = 0.041, size = 84, normalized size = 0.6 \[ 2\,{\frac{\sqrt{1- \left ( -1/2\,\sqrt{10}+1 \right ){x}^{2}}\sqrt{1- \left ( 1+1/2\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{4-2\,\sqrt{10}}x,i/3\sqrt{6}+i/3\sqrt{15} \right ) }{\sqrt{4-2\,\sqrt{10}}\sqrt{3\,{x}^{4}+4\,{x}^{2}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+4*x^2-2)^(1/2),x)

[Out]

2/(4-2*10^(1/2))^(1/2)*(1-(-1/2*10^(1/2)+1)*x^2)^(1/2)*(1-(1+1/2*10^(1/2))*x^2)^
(1/2)/(3*x^4+4*x^2-2)^(1/2)*EllipticF(1/2*(4-2*10^(1/2))^(1/2)*x,1/3*I*6^(1/2)+1
/3*I*15^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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