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

Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right ),\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2 x^2+2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0159053, antiderivative size = 92, 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} \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 2*x^2 + 3*x^4],x]

[Out]

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

Mathematica [C]  time = 0.080682, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{3 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{3 x^2}{-1+i \sqrt{5}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{3}{-1-i \sqrt{5}}} x\right ),\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right )}{\sqrt{3} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{3 x^4+2 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.76, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{5}},1/3\,\sqrt{-6+3\,i\sqrt{5}} \right ) }{\sqrt{-2+2\,i\sqrt{5}}\sqrt{3\,{x}^{4}+2\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-2+2*I*5^(1/2))^(1/2)*(1-(-1/2+1/2*I*5^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*5^(1/2))*x^2)^(1/2)/(3*x^4+2*x^2+2)
^(1/2)*EllipticF(1/2*x*(-2+2*I*5^(1/2))^(1/2),1/3*(-6+3*I*5^(1/2))^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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