3.4.3 \(\int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+(-3+\sqrt {3}) x^2}} \, dx\) [303]

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

[Out]

-1/6*(x^2*(9-3*3^(1/2)))^(1/2)/x/(9-3*3^(1/2))^(1/2)*EllipticF(1/3*(9-x^2*(9-3*3^(1/2)))^(1/2),1/2*(2+2*3^(1/2
))^(1/2))*(3+3^(1/2))^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {431} \begin {gather*} -\frac {1}{6} \sqrt {3+\sqrt {3}} F\left (\text {ArcCos}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \end {gather*}

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]

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

Rule 431

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

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Mathematica [A]
time = 2.06, size = 81, normalized size = 1.72 \begin {gather*} \frac {\sqrt {-3+3 \sqrt {3}-2 \sqrt {3} x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt [4]{3}}\right )|2-\sqrt {3}\right )}{3^{3/4} \sqrt {6-6 \sqrt {3}+4 \sqrt {3} x^2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(64)=128\).
time = 0.43, size = 207, normalized size = 4.40

method	result	size
default	\(\frac {\sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {2}\, \sqrt {-\left (4 x^{2} \sqrt {3}-6 x^{2}-3 \sqrt {3}+3\right ) \left (\sqrt {3}-1\right )}\, \sqrt {-\left (3-3 \sqrt {3}+2 x^{2} \sqrt {3}\right ) \left (\sqrt {3}-1\right )}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {3}\, \sqrt {\left (2 \sqrt {3}-3\right ) \left (\sqrt {3}-1\right )}}{3 \sqrt {3}-3}, \frac {\sqrt {\left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}}{\sqrt {3}-1}\right ) \left (-3+\sqrt {3}\right )}{18 \left (\sqrt {3}-1\right )^{2} \left (2 x^{4} \sqrt {3}-2 x^{4}-6 x^{2} \sqrt {3}+6 x^{2}+3 \sqrt {3}-3\right ) \sqrt {\left (2 \sqrt {3}-3\right ) \left (\sqrt {3}-1\right )}}\)	\(207\)
elliptic	\(\frac {\sqrt {\left (x^{2} \sqrt {3}-3 x^{2}+3\right ) \left (3-3 \sqrt {3}+2 x^{2} \sqrt {3}\right )}\, \sqrt {6}\, \sqrt {9-\frac {6 \left (2 \sqrt {3}-3\right ) x^{2}}{\sqrt {3}-1}}\, \sqrt {9-\frac {6 \sqrt {3}\, x^{2}}{\sqrt {3}-1}}\, \EllipticF \left (\frac {x \sqrt {6}\, \sqrt {\frac {2 \sqrt {3}-3}{\sqrt {3}-1}}}{3}, \frac {\sqrt {-9-\frac {6 \left (18 \sqrt {3}-18\right ) \sqrt {3}}{\left (\sqrt {3}-1\right ) \left (6-6 \sqrt {3}\right )}}}{3}\right )}{18 \sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {\frac {2 \sqrt {3}-3}{\sqrt {3}-1}}\, \sqrt {18 x^{2} \sqrt {3}-18 x^{2}+6 x^{4}-6 x^{4} \sqrt {3}+9-9 \sqrt {3}}}\)	\(223\)

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*x^2*3^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/18*(x^2*3^(1/2)-3*x^2+3)^(1/2)*(3-3*3^(1/2)+2*x^2*3^(1/2))^(1/2)*2^(1/2)/(3^(1/2)-1)^2*(-(4*x^2*3^(1/2)-6*x^
2-3*3^(1/2)+3)*(3^(1/2)-1))^(1/2)*(-(3-3*3^(1/2)+2*x^2*3^(1/2))*(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*x^2*3^(1/2)+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 3 x^{2} + \sqrt {3} x^{2} + 3} \sqrt {2 \sqrt {3} x^{2} - 3 \sqrt {3} + 3}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {\left (\sqrt {3}-3\right )\,x^2+3}\,\sqrt {2\,\sqrt {3}\,x^2-3\,\sqrt {3}+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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