3.79 \(\int \frac{1}{\sqrt{3+8 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=110 \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right ),-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]

[Out]

(Sqrt[(3 + (4 - Sqrt[10])*x^2)/(3 + (4 + Sqrt[10])*x^2)]*(3 + (4 + Sqrt[10])*x^2)*EllipticF[ArcTan[Sqrt[(4 + S
qrt[10])/3]*x], (-2*(5 - 2*Sqrt[10]))/3])/(Sqrt[3*(4 + Sqrt[10])]*Sqrt[3 + 8*x^2 + 2*x^4])

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Rubi [A]  time = 0.0757186, 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} \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

(Sqrt[(3 + (4 - Sqrt[10])*x^2)/(3 + (4 + Sqrt[10])*x^2)]*(3 + (4 + Sqrt[10])*x^2)*EllipticF[ArcTan[Sqrt[(4 + S
qrt[10])/3]*x], (-2*(5 - 2*Sqrt[10]))/3])/(Sqrt[3*(4 + Sqrt[10])]*Sqrt[3 + 8*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+8 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3+\left (4-\sqrt{10}\right ) x^2}{3+\left (4+\sqrt{10}\right ) x^2}} \left (3+\left (4+\sqrt{10}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{3+8 x^2+2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0794726, size = 98, normalized size = 0.89 \[ -\frac{i \sqrt{\frac{-2 x^2+\sqrt{10}-4}{\sqrt{10}-4}} \sqrt{2 x^2+\sqrt{10}+4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{4+\sqrt{10}}} x\right ),\frac{13}{3}+\frac{4 \sqrt{10}}{3}\right )}{\sqrt{4 x^4+16 x^2+6}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

((-I)*Sqrt[(-4 + Sqrt[10] - 2*x^2)/(-4 + Sqrt[10])]*Sqrt[4 + Sqrt[10] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(4 +
 Sqrt[10])]*x], 13/3 + (4*Sqrt[10])/3])/Sqrt[6 + 16*x^2 + 4*x^4]

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Maple [A]  time = 0.231, size = 82, normalized size = 0.8 \begin{align*} 3\,{\frac{\sqrt{1- \left ( -4/3+1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -4/3-1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-12+3\,\sqrt{10}},2/3\,\sqrt{6}+1/3\,\sqrt{15} \right ) }{\sqrt{-12+3\,\sqrt{10}}\sqrt{2\,{x}^{4}+8\,{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^4+8*x^2+3)^(1/2),x)

[Out]

3/(-12+3*10^(1/2))^(1/2)*(1-(-4/3+1/3*10^(1/2))*x^2)^(1/2)*(1-(-4/3-1/3*10^(1/2))*x^2)^(1/2)/(2*x^4+8*x^2+3)^(
1/2)*EllipticF(1/3*x*(-12+3*10^(1/2))^(1/2),2/3*6^(1/2)+1/3*15^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+8*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 8*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} + 8 \, x^{2} + 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+8*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 + 8*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} + 8 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**4+8*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 8*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} + 8 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+8*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 8*x^2 + 3), x)