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

Optimal. Leaf size=150 \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+3}{\left (1-\sqrt{7}\right ) x^2+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+3}} \sqrt{2 x^4-2 x^2-3}} \]

[Out]

(Sqrt[-3 - (1 - Sqrt[7])*x^2]*Sqrt[(3 + (1 + Sqrt[7])*x^2)/(3 + (1 - Sqrt[7])*x^
2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-3 - (1 - Sqrt[7])*x^2]], (7 - Sqr
t[7])/14])/(Sqrt[6]*7^(1/4)*Sqrt[(3 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-3 - 2*x^2 +
2*x^4])

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Rubi [A]  time = 0.0930872, antiderivative size = 150, 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 \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+3}{\left (1-\sqrt{7}\right ) x^2+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+3}} \sqrt{2 x^4-2 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-3 - (1 - Sqrt[7])*x^2]*Sqrt[(3 + (1 + Sqrt[7])*x^2)/(3 + (1 - Sqrt[7])*x^
2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-3 - (1 - Sqrt[7])*x^2]], (7 - Sqr
t[7])/14])/(Sqrt[6]*7^(1/4)*Sqrt[(3 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-3 - 2*x^2 +
2*x^4])

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Rubi in Sympy [A]  time = 4.01815, size = 133, normalized size = 0.89 \[ \frac{\sqrt{6} \cdot 7^{\frac{3}{4}} \sqrt{\frac{x^{2} \left (- 2 \sqrt{7} - 2\right ) - 6}{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 6}} \sqrt{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 6} F\left (\operatorname{asin}{\left (\frac{2 \sqrt [4]{7} x}{\sqrt{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 6}} \right )}\middle | - \frac{\sqrt{7}}{14} + \frac{1}{2}\right )}{84 \sqrt{- \frac{1}{x^{2} \left (-2 + 2 \sqrt{7}\right ) - 6}} \sqrt{2 x^{4} - 2 x^{2} - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*7**(3/4)*sqrt((x**2*(-2*sqrt(7) - 2) - 6)/(x**2*(-2 + 2*sqrt(7)) - 6))*s
qrt(x**2*(-2 + 2*sqrt(7)) - 6)*elliptic_f(asin(2*7**(1/4)*x/sqrt(x**2*(-2 + 2*sq
rt(7)) - 6)), -sqrt(7)/14 + 1/2)/(84*sqrt(-1/(x**2*(-2 + 2*sqrt(7)) - 6))*sqrt(2
*x**4 - 2*x**2 - 3))

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Mathematica [C]  time = 0.0861445, size = 81, normalized size = 0.54 \[ -\frac{i \sqrt{-2 x^4+2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}} \sqrt{2 x^4-2 x^2-3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.041, size = 84, normalized size = 0.6 \[ 3\,{\frac{\sqrt{1- \left ( -1/3-1/3\,\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -1/3+1/3\,\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{-3-3\,\sqrt{7}}x,i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{-3-3\,\sqrt{7}}\sqrt{2\,{x}^{4}-2\,{x}^{2}-3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4-2*x^2-3)^(1/2),x)

[Out]

3/(-3-3*7^(1/2))^(1/2)*(1-(-1/3-1/3*7^(1/2))*x^2)^(1/2)*(1-(-1/3+1/3*7^(1/2))*x^
2)^(1/2)/(2*x^4-2*x^2-3)^(1/2)*EllipticF(1/3*(-3-3*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. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 2 \, x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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