3.78 \(\int \frac{1}{\sqrt{3+9 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=110 \[ \frac{\sqrt{\frac{\left (9-\sqrt{57}\right ) x^2+6}{\left (9+\sqrt{57}\right ) x^2+6}} \left (\left (9+\sqrt{57}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right ),\frac{1}{4} \left (3 \sqrt{57}-19\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{2 x^4+9 x^2+3}} \]

[Out]

(Sqrt[(6 + (9 - Sqrt[57])*x^2)/(6 + (9 + Sqrt[57])*x^2)]*(6 + (9 + Sqrt[57])*x^2)*EllipticF[ArcTan[Sqrt[(9 + S
qrt[57])/6]*x], (-19 + 3*Sqrt[57])/4])/(Sqrt[6*(9 + Sqrt[57])]*Sqrt[3 + 9*x^2 + 2*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.091197, antiderivative size = 110, 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 = {1099} \[ \frac{\sqrt{\frac{\left (9-\sqrt{57}\right ) x^2+6}{\left (9+\sqrt{57}\right ) x^2+6}} \left (\left (9+\sqrt{57}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right )|\frac{1}{4} \left (-19+3 \sqrt{57}\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{2 x^4+9 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 9*x^2 + 2*x^4],x]

[Out]

(Sqrt[(6 + (9 - Sqrt[57])*x^2)/(6 + (9 + Sqrt[57])*x^2)]*(6 + (9 + Sqrt[57])*x^2)*EllipticF[ArcTan[Sqrt[(9 + S
qrt[57])/6]*x], (-19 + 3*Sqrt[57])/4])/(Sqrt[6*(9 + Sqrt[57])]*Sqrt[3 + 9*x^2 + 2*x^4])

Rule 1099

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3+9 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{6+\left (9-\sqrt{57}\right ) x^2}{6+\left (9+\sqrt{57}\right ) x^2}} \left (6+\left (9+\sqrt{57}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right )|\frac{1}{4} \left (-19+3 \sqrt{57}\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{3+9 x^2+2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0721278, size = 97, normalized size = 0.88 \[ -\frac{i \sqrt{\frac{-4 x^2+\sqrt{57}-9}{\sqrt{57}-9}} \sqrt{4 x^2+\sqrt{57}+9} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{9+\sqrt{57}}}\right ),\frac{23}{4}+\frac{3 \sqrt{57}}{4}\right )}{2 \sqrt{2 x^4+9 x^2+3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[3 + 9*x^2 + 2*x^4],x]

[Out]

((-I/2)*Sqrt[(-9 + Sqrt[57] - 4*x^2)/(-9 + Sqrt[57])]*Sqrt[9 + Sqrt[57] + 4*x^2]*EllipticF[I*ArcSinh[(2*x)/Sqr
t[9 + Sqrt[57]]], 23/4 + (3*Sqrt[57])/4])/Sqrt[3 + 9*x^2 + 2*x^4]

________________________________________________________________________________________

Maple [A]  time = 0.236, size = 82, normalized size = 0.8 \begin{align*} 6\,{\frac{\sqrt{1- \left ( -3/2+1/6\,\sqrt{57} \right ){x}^{2}}\sqrt{1- \left ( -3/2-1/6\,\sqrt{57} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-54+6\,\sqrt{57}},3/4\,\sqrt{6}+1/4\,\sqrt{38} \right ) }{\sqrt{-54+6\,\sqrt{57}}\sqrt{2\,{x}^{4}+9\,{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^4+9*x^2+3)^(1/2),x)

[Out]

6/(-54+6*57^(1/2))^(1/2)*(1-(-3/2+1/6*57^(1/2))*x^2)^(1/2)*(1-(-3/2-1/6*57^(1/2))*x^2)^(1/2)/(2*x^4+9*x^2+3)^(
1/2)*EllipticF(1/6*x*(-54+6*57^(1/2))^(1/2),3/4*6^(1/2)+1/4*38^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{4} + 9 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+9*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 9*x^2 + 3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 9 \, x^{2} + 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+9*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 + 9*x^2 + 3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 x^{4} + 9 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**4+9*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 9*x**2 + 3), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{4} + 9 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+9*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 9*x^2 + 3), x)