3.299 \(\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-
1 + x^2])

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Rubi [A]  time = 0.0994968, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-
1 + x^2])

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Rubi in Sympy [A]  time = 15.7275, size = 41, normalized size = 0.89 \[ \frac{\sqrt{- x^{2} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{- 4 \sqrt{3} + 7} \sqrt{x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2-1)**(1/2)/(7+x**2-4*3**(1/2))**(1/2),x)

[Out]

sqrt(-x**2 + 1)*elliptic_f(asin(x), -7 - 4*sqrt(3))/(sqrt(-4*sqrt(3) + 7)*sqrt(x
**2 - 1))

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Mathematica [A]  time = 0.107458, size = 48, normalized size = 1.04 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|\frac{1}{-7+4 \sqrt{3}}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], (-7 + 4*Sqrt[3])^(-1)])/(Sqrt[7 - 4*Sqrt[3]]
*Sqrt[-1 + x^2])

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Maple [B]  time = 0.245, size = 117, normalized size = 2.5 \[{\frac{-i \left ( -2+\sqrt{3} \right ) }{ \left ( 4\,\sqrt{3}-7 \right ) \left ( -{x}^{4}+4\,\sqrt{3}{x}^{2}-6\,{x}^{2}-4\,\sqrt{3}+7 \right ) }{\it EllipticF} \left ({\frac{ix}{-2+\sqrt{3}}},2\,i-i\sqrt{3} \right ) \sqrt{-{x}^{2}+1}\sqrt{- \left ( -{x}^{2}+4\,\sqrt{3}-7 \right ) \left ( -4\,\sqrt{3}+7 \right ) }\sqrt{{x}^{2}-1}\sqrt{7+{x}^{2}-4\,\sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x)

[Out]

-I*EllipticF(I*x/(-2+3^(1/2)),2*I-I*3^(1/2))*(-x^2+1)^(1/2)*(-(-x^2+4*3^(1/2)-7)
*(-4*3^(1/2)+7))^(1/2)/(4*3^(1/2)-7)*(-2+3^(1/2))*(x^2-1)^(1/2)*(7+x^2-4*3^(1/2)
)^(1/2)/(-x^4+4*3^(1/2)*x^2-6*x^2-4*3^(1/2)+7)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \sqrt{x^{2} - 4 \sqrt{3} + 7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2-1)**(1/2)/(7+x**2-4*3**(1/2))**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*sqrt(x**2 - 4*sqrt(3) + 7)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)