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

Optimal. Leaf size=63 \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right ),\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]

[Out]

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

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Rubi [A]  time = 0.0077939, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1097} \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + x^2 + 3*x^4],x]

[Out]

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

Rule 1097

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2+x^2+3 x^4}} \, dx &=\frac{\sqrt{1+x^2} \sqrt{-2+3 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{-2+3 x^2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{-2+x^2+3 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0298895, size = 48, normalized size = 0.76 \[ \frac{\sqrt{\left (\frac{2}{3}-x^2\right ) \left (x^2+1\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3 x^4+x^2-2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[-2 + x^2 + 3*x^4],x]

[Out]

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

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Maple [C]  time = 0.046, size = 43, normalized size = 0.7 \begin{align*}{-{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{6} \right ) \sqrt{{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\frac{1}{\sqrt{3\,{x}^{4}+{x}^{2}-2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(3*x^4 + x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)