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

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

[Out]

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

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Rubi [A]  time = 0.0288026, 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, Rules used = {1098} \[ \frac{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2} \sqrt{\frac{\left (2+\sqrt{10}\right ) x^2+2}{\left (2-\sqrt{10}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2}}\right )|\frac{1}{10} \left (5-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{\left (2-\sqrt{10}\right ) x^2+2}} \sqrt{3 x^4-4 x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 4*x^2 + 3*x^4],x]

[Out]

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

Rule 1098

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)/(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 + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-4 x^2+3 x^4}} \, dx &=\frac{\sqrt{-2-\left (2-\sqrt{10}\right ) x^2} \sqrt{\frac{2+\left (2+\sqrt{10}\right ) x^2}{2+\left (2-\sqrt{10}\right ) x^2}} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-2-\left (2-\sqrt{10}\right ) x^2}}\right )|\frac{1}{10} \left (5-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{2+\left (2-\sqrt{10}\right ) x^2}} \sqrt{-2-4 x^2+3 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0638357, size = 81, normalized size = 0.55 \[ -\frac{i \sqrt{-3 x^4+4 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{2}}} x\right ),\frac{1}{3} \left (2 \sqrt{10}-7\right )\right )}{\sqrt{2+\sqrt{10}} \sqrt{3 x^4-4 x^2-2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[2 + 4*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/2]]*x], (-7 + 2*Sqrt[10])/3])/(Sqrt[2 + Sqrt
[10]]*Sqrt[-2 - 4*x^2 + 3*x^4])

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Maple [C]  time = 0.182, size = 84, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1-1/2\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -1+1/2\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{-4-2\,\sqrt{10}}x,i/3\sqrt{15}-i/3\sqrt{6} \right ) }{\sqrt{-4-2\,\sqrt{10}}\sqrt{3\,{x}^{4}-4\,{x}^{2}-2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-4-2*10^(1/2))^(1/2)*(1-(-1-1/2*10^(1/2))*x^2)^(1/2)*(1-(-1+1/2*10^(1/2))*x^2)^(1/2)/(3*x^4-4*x^2-2)^(1/2)*
EllipticF(1/2*(-4-2*10^(1/2))^(1/2)*x,1/3*I*15^(1/2)-1/3*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} - 4 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(3*x^4 - 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} - 4 \, x^{2} - 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(3*x^4 - 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} - 4 x^{2} - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(3*x**4 - 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} - 4 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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