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

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

[Out]

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

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Rubi [A]  time = 0.18325, 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 \[ \frac{\sqrt{\frac{\left (9-\sqrt{57}\right ) x^2+6}{\left (9+\sqrt{57}\right ) x^2+6}} \left (\left (9+\sqrt{57}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right )|\frac{1}{4} \left (-19+3 \sqrt{57}\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{2 x^4+9 x^2+3}} \]

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

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Rubi in Sympy [A]  time = 4.98422, size = 102, normalized size = 0.93 \[ \frac{\sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{57} + 9\right ) + 6}{x^{2} \left (\sqrt{57} + 9\right ) + 6}} \left (x^{2} \left (\sqrt{57} + 9\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{57} + 9}}{6} \right )}\middle | - \frac{19}{4} + \frac{3 \sqrt{57}}{4}\right )}{6 \sqrt{\sqrt{57} + 9} \sqrt{2 x^{4} + 9 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4+9*x**2+3)**(1/2),x)

[Out]

sqrt(6)*sqrt((x**2*(-sqrt(57) + 9) + 6)/(x**2*(sqrt(57) + 9) + 6))*(x**2*(sqrt(5
7) + 9) + 6)*elliptic_f(atan(sqrt(6)*x*sqrt(sqrt(57) + 9)/6), -19/4 + 3*sqrt(57)
/4)/(6*sqrt(sqrt(57) + 9)*sqrt(2*x**4 + 9*x**2 + 3))

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Mathematica [C]  time = 0.127315, size = 97, normalized size = 0.88 \[ -\frac{i \sqrt{\frac{-4 x^2+\sqrt{57}-9}{\sqrt{57}-9}} \sqrt{4 x^2+\sqrt{57}+9} F\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{9+\sqrt{57}}}\right )|\frac{23}{4}+\frac{3 \sqrt{57}}{4}\right )}{2 \sqrt{2 x^4+9 x^2+3}} \]

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)/Sqrt[9 + Sqrt[57]]], 23/4 + (3*Sqrt[57])/4])/Sqrt[3 +
9*x^2 + 2*x^4]

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

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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 9 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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