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

Optimal. Leaf size=47 \[ -\frac{1}{6} \sqrt{3+\sqrt{3}} \text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right ),\frac{1}{2} \left (1+\sqrt{3}\right )\right ) \]

[Out]

-(Sqrt[3 + Sqrt[3]]*EllipticF[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sqrt[3])/2])/6

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Rubi [A]  time = 0.0616624, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {420} \[ -\frac{1}{6} \sqrt{3+\sqrt{3}} F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]),x]

[Out]

-(Sqrt[3 + Sqrt[3]]*EllipticF[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sqrt[3])/2])/6

Rule 420

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> -Simp[EllipticF[ArcCos[Rt[-(d/c), 2]
*x], (b*c)/(b*c - a*d)]/(Sqrt[c]*Rt[-(d/c), 2]*Sqrt[a - (b*c)/d]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &
& GtQ[c, 0] && GtQ[a - (b*c)/d, 0]

Rubi steps

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

Mathematica [A]  time = 0.147288, size = 81, normalized size = 1.72 \[ \frac{\sqrt{-2 \sqrt{3} x^2+3 \sqrt{3}-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1+\sqrt{3}} x}{\sqrt [4]{3}}\right ),2-\sqrt{3}\right )}{3^{3/4} \sqrt{4 \sqrt{3} x^2-6 \sqrt{3}+6}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.181, size = 207, normalized size = 4.4 \begin{align*}{\frac{\sqrt{2} \left ( -3+\sqrt{3} \right ) }{18\, \left ( \sqrt{3}-1 \right ) ^{2} \left ( 2\,{x}^{4}\sqrt{3}-2\,{x}^{4}-6\,\sqrt{3}{x}^{2}+6\,{x}^{2}+3\,\sqrt{3}-3 \right ) \sqrt{ \left ( 2\,\sqrt{3}-3 \right ) \left ( \sqrt{3}-1 \right ) }}\sqrt{\sqrt{3}{x}^{2}-3\,{x}^{2}+3}\sqrt{3-3\,\sqrt{3}+2\,\sqrt{3}{x}^{2}}\sqrt{- \left ( 4\,\sqrt{3}{x}^{2}-6\,{x}^{2}-3\,\sqrt{3}+3 \right ) \left ( \sqrt{3}-1 \right ) }\sqrt{- \left ( 3-3\,\sqrt{3}+2\,\sqrt{3}{x}^{2} \right ) \left ( \sqrt{3}-1 \right ) }{\it EllipticF} \left ({\frac{x\sqrt{2}\sqrt{3}\sqrt{ \left ( 2\,\sqrt{3}-3 \right ) \left ( \sqrt{3}-1 \right ) }}{3\,\sqrt{3}-3}},{\frac{\sqrt{ \left ( \sqrt{3}-1 \right ) \left ( 1+\sqrt{3} \right ) }}{\sqrt{3}-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(3^(1/2)*x^2-3*x^2+3)^(1/2)*(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)*2^(1/2)/(3^(1/2)-1)^2*(-(4*3^(1/2)*x^2-6*x^
2-3*3^(1/2)+3)*(3^(1/2)-1))^(1/2)*(-(3-3*3^(1/2)+2*3^(1/2)*x^2)*(3^(1/2)-1))^(1/2)*EllipticF(1/3*x*2^(1/2)*3^(
1/2)/(3^(1/2)-1)*((2*3^(1/2)-3)*(3^(1/2)-1))^(1/2),1/(3^(1/2)-1)*((3^(1/2)-1)*(1+3^(1/2)))^(1/2))*(-3+3^(1/2))
/(2*x^4*3^(1/2)-2*x^4-6*3^(1/2)*x^2+6*x^2+3*3^(1/2)-3)/((2*3^(1/2)-3)*(3^(1/2)-1))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{\sqrt{3} x^{2} - 3 \, x^{2} + 3} \sqrt{\sqrt{3}{\left (2 \, x^{2} - 3\right )} + 3}{\left (\sqrt{3} + 1\right )}}{6 \,{\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/6*sqrt(sqrt(3)*x^2 - 3*x^2 + 3)*sqrt(sqrt(3)*(2*x^2 - 3) + 3)*(sqrt(3) + 1)/(2*x^4 - 6*x^2 + 3), x
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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