3.1007 \(\int \frac{x^2}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{2} F\left (\left .\cos ^{-1}(x)\right |2\right )-\frac{1}{2} E\left (\left .\cos ^{-1}(x)\right |2\right ) \]

[Out]

-EllipticE[ArcCos[x], 2]/2 - EllipticF[ArcCos[x], 2]/2

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Rubi [A]  time = 0.101875, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{1}{2} F\left (\left .\cos ^{-1}(x)\right |2\right )-\frac{1}{2} E\left (\left .\cos ^{-1}(x)\right |2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-EllipticE[ArcCos[x], 2]/2 - EllipticF[ArcCos[x], 2]/2

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Rubi in Sympy [A]  time = 18.7968, size = 14, normalized size = 0.82 \[ - \frac{E\left (\operatorname{acos}{\left (x \right )}\middle | 2\right )}{2} - \frac{F\left (\operatorname{acos}{\left (x \right )}\middle | 2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-elliptic_e(acos(x), 2)/2 - elliptic_f(acos(x), 2)/2

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Mathematica [B]  time = 0.0518936, size = 37, normalized size = 2.18 \[ \frac{\sqrt{1-2 x^2} \left (F\left (\left .\sin ^{-1}(x)\right |2\right )-E\left (\left .\sin ^{-1}(x)\right |2\right )\right )}{2 \sqrt{2 x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 34, normalized size = 2. \[{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) -{\it EllipticE} \left ( x,\sqrt{2} \right ) }{2}\sqrt{-2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(EllipticF(x,2^(1/2))-EllipticE(x,2^(1/2)))*(-2*x^2+1)^(1/2)/(2*x^2-1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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