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

Optimal. Leaf size=148 \[ \frac{\sqrt{\frac{3-\left (2-\sqrt{10}\right ) x^2}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-3} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-3}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2^{3/4} \sqrt{3} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{2 x^4+4 x^2-3}} \]

[Out]

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

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Rubi [A]  time = 0.109734, 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 \[ \frac{\sqrt{\frac{3-\left (2-\sqrt{10}\right ) x^2}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-3} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-3}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2^{3/4} \sqrt{3} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{2 x^4+4 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.17556, size = 136, normalized size = 0.92 \[ \frac{\sqrt [4]{2} \sqrt{3} \cdot 5^{\frac{3}{4}} \sqrt{\frac{x^{2} \left (- 2 \sqrt{10} + 4\right ) - 6}{x^{2} \left (4 + 2 \sqrt{10}\right ) - 6}} \sqrt{x^{2} \left (4 + 2 \sqrt{10}\right ) - 6} F\left (\operatorname{asin}{\left (\frac{2 \sqrt [4]{10} x}{\sqrt{x^{2} \left (4 + 2 \sqrt{10}\right ) - 6}} \right )}\middle | \frac{\sqrt{10}}{10} + \frac{1}{2}\right )}{60 \sqrt{- \frac{1}{x^{2} \left (4 + 2 \sqrt{10}\right ) - 6}} \sqrt{2 x^{4} + 4 x^{2} - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*sqrt(3)*5**(3/4)*sqrt((x**2*(-2*sqrt(10) + 4) - 6)/(x**2*(4 + 2*sqrt(10
)) - 6))*sqrt(x**2*(4 + 2*sqrt(10)) - 6)*elliptic_f(asin(2*10**(1/4)*x/sqrt(x**2
*(4 + 2*sqrt(10)) - 6)), sqrt(10)/10 + 1/2)/(60*sqrt(-1/(x**2*(4 + 2*sqrt(10)) -
6))*sqrt(2*x**4 + 4*x**2 - 3))

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Mathematica [C]  time = 0.116333, size = 83, normalized size = 0.56 \[ -\frac{i \sqrt{-2 x^4-4 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{2+\sqrt{10}}} x\right )|-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right )}{\sqrt{\sqrt{10}-2} \sqrt{2 x^4+4 x^2-3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.04, size = 84, normalized size = 0.6 \[ 3\,{\frac{\sqrt{1- \left ( 2/3-1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( 2/3+1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{6-3\,\sqrt{10}}x,i/3\sqrt{6}+i/3\sqrt{15} \right ) }{\sqrt{6-3\,\sqrt{10}}\sqrt{2\,{x}^{4}+4\,{x}^{2}-3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4+4*x^2-3)^(1/2),x)

[Out]

3/(6-3*10^(1/2))^(1/2)*(1-(2/3-1/3*10^(1/2))*x^2)^(1/2)*(1-(2/3+1/3*10^(1/2))*x^
2)^(1/2)/(2*x^4+4*x^2-3)^(1/2)*EllipticF(1/3*(6-3*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. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 4 \, x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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