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

Optimal. Leaf size=44 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4-\sqrt{7}\right )\right )}{\sqrt{\sqrt{7}-1}} \]

[Out]

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

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Rubi [A]  time = 0.152414, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4-\sqrt{7}\right )\right )}{\sqrt{\sqrt{7}-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.2571, size = 71, normalized size = 1.61 \[ \frac{2 \sqrt{6} F\left (\operatorname{asin}{\left (\frac{\sqrt{3} x \sqrt{-1 + \sqrt{7}}}{3} \right )}\middle | - \frac{4}{3} - \frac{\sqrt{7}}{3}\right )}{\sqrt{-2 + 2 \sqrt{7}} \sqrt{-1 + \sqrt{7}} \sqrt{2 + 2 \sqrt{7}}} \]

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]

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

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Mathematica [C]  time = 0.0741948, size = 49, normalized size = 1.11 \[ -\frac{i F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.094, size = 84, normalized size = 1.9 \[ 3\,{\frac{\sqrt{1- \left ( -1/3+1/3\,\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -1/3-1/3\,\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-3+3\,\sqrt{7}},i/6\sqrt{6}+i/6\sqrt{42} \right ) }{\sqrt{-3+3\,\sqrt{7}}\sqrt{-2\,{x}^{4}+2\,{x}^{2}+3}}} \]

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