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

Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{24} \left (12+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-x^2+3}} \]

[Out]

((3 + Sqrt[6]*x^2)*Sqrt[(3 - x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(2/3)^(1/4)*x], (12 + Sqrt[6
])/24])/(2*6^(1/4)*Sqrt[3 - x^2 + 2*x^4])

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Rubi [A]  time = 0.0096487, antiderivative size = 90, 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, Rules used = {1103} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{24} \left (12+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - x^2 + 2*x^4],x]

[Out]

((3 + Sqrt[6]*x^2)*Sqrt[(3 - x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(2/3)^(1/4)*x], (12 + Sqrt[6
])/24])/(2*6^(1/4)*Sqrt[3 - x^2 + 2*x^4])

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-x^2+2 x^4}} \, dx &=\frac{\left (3+\sqrt{6} x^2\right ) \sqrt{\frac{3-x^2+2 x^4}{\left (3+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{24} \left (12+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3-x^2+2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0682245, size = 142, normalized size = 1.58 \[ -\frac{i \sqrt{1-\frac{4 x^2}{1-i \sqrt{23}}} \sqrt{1-\frac{4 x^2}{1+i \sqrt{23}}} \text{EllipticF}\left (i \sinh ^{-1}\left (2 \sqrt{-\frac{1}{1-i \sqrt{23}}} x\right ),\frac{1-i \sqrt{23}}{1+i \sqrt{23}}\right )}{2 \sqrt{-\frac{1}{1-i \sqrt{23}}} \sqrt{2 x^4-x^2+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - x^2 + 2*x^4],x]

[Out]

((-I/2)*Sqrt[1 - (4*x^2)/(1 - I*Sqrt[23])]*Sqrt[1 - (4*x^2)/(1 + I*Sqrt[23])]*EllipticF[I*ArcSinh[2*Sqrt[-(1 -
I*Sqrt[23])^(-1)]*x], (1 - I*Sqrt[23])/(1 + I*Sqrt[23])])/(Sqrt[-(1 - I*Sqrt[23])^(-1)]*Sqrt[3 - x^2 + 2*x^4]
)

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Maple [C]  time = 0.737, size = 87, normalized size = 1. \begin{align*} 6\,{\frac{\sqrt{1- \left ( 1/6+i/6\sqrt{23} \right ){x}^{2}}\sqrt{1- \left ( 1/6-i/6\sqrt{23} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{6+6\,i\sqrt{23}},1/6\,\sqrt{-33-3\,i\sqrt{23}} \right ) }{\sqrt{6+6\,i\sqrt{23}}\sqrt{2\,{x}^{4}-{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/(6+6*I*23^(1/2))^(1/2)*(1-(1/6+1/6*I*23^(1/2))*x^2)^(1/2)*(1-(1/6-1/6*I*23^(1/2))*x^2)^(1/2)/(2*x^4-x^2+3)^(
1/2)*EllipticF(1/6*x*(6+6*I*23^(1/2))^(1/2),1/6*(-33-3*I*23^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 - x^2 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(2*x^4 - x^2 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 - x^2 + 3), x)