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

Optimal. Leaf size=47 \[ \frac{\left (1-x^2\right ) \sqrt{1-\frac{1}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt{x^2-2}\right )}{x \sqrt{x^2-2}} \]

[Out]

((1 - x^2)*Sqrt[1 - (1 - x^2)^(-2)]*ArcTan[Sqrt[-2 + x^2]])/(x*Sqrt[-2 + x^2])

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Rubi [A]  time = 0.784073, antiderivative size = 73, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\left (1-x^2\right ) \sqrt{x^4-2 x^2} \sqrt{1-\frac{1}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt{x^2-2}\right )}{x \sqrt{x^2-2} \sqrt{\left (x^2-1\right )^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

((1 - x^2)*Sqrt[-2*x^2 + x^4]*Sqrt[1 - (1 - x^2)^(-2)]*ArcTan[Sqrt[-2 + x^2]])/(
x*Sqrt[-2 + x^2]*Sqrt[-1 + (-1 + x^2)^2])

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Rubi in Sympy [A]  time = 35.4131, size = 61, normalized size = 1.3 \[ \frac{\sqrt{1 - \frac{1}{\left (x^{2} - 1\right )^{2}}} \left (- x^{2} + 1\right ) \sqrt{x^{4} - 2 x^{2}} \operatorname{atan}{\left (\sqrt{x^{2} - 2} \right )}}{x \sqrt{x^{2} - 2} \sqrt{\left (x^{2} - 1\right )^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(1 - 1/(x**2 - 1)**2)*(-x**2 + 1)*sqrt(x**4 - 2*x**2)*atan(sqrt(x**2 - 2))/(
x*sqrt(x**2 - 2)*sqrt((x**2 - 1)**2 - 1))

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Mathematica [A]  time = 0.0401975, size = 91, normalized size = 1.94 \[ \frac{1}{2} \tan ^{-1}\left (\frac{(x-1) (x+1) (x+2) \sqrt{\frac{x^2 \left (x^2-2\right )}{\left (x^2-1\right )^2}}}{x \left (x^2-2\right )}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{(x-2) (x-1) (x+1) \sqrt{\frac{x^2 \left (x^2-2\right )}{\left (x^2-1\right )^2}}}{x \left (x^2-2\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[((-2 + x)*(-1 + x)*(1 + x)*Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2])/(x*(-2 +
x^2))]/2 + ArcTan[((-1 + x)*(1 + x)*(2 + x)*Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2]
)/(x*(-2 + x^2))]/2

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Maple [A]  time = 0.028, size = 63, normalized size = 1.3 \[{\frac{{x}^{2}-1}{2\,x}\sqrt{{\frac{{x}^{2} \left ({x}^{2}-2 \right ) }{ \left ({x}^{2}-1 \right ) ^{2}}}} \left ( \arctan \left ({(2+x){\frac{1}{\sqrt{{x}^{2}-2}}}} \right ) -\arctan \left ({(x-2){\frac{1}{\sqrt{{x}^{2}-2}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2*(x^2-2)/(x^2-1)^2)^(1/2)*(x^2-1)*(arctan((2+x)/(x^2-2)^(1/2))-arctan((x
-2)/(x^2-2)^(1/2)))/x/(x^2-2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279441, size = 107, normalized size = 2.28 \[ \arctan \left (\frac{x^{3} -{\left (x^{3} - x\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}} - 2 \, x}{x^{2} -{\left (x^{2} - 1\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan((x^3 - (x^3 - x)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1)) - 2*x)/(x^2 - (x^2
- 1)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265688, size = 24, normalized size = 0.51 \[ -\arctan \left (\sqrt{x^{2} - 2}\right ){\rm sign}\left (x^{3} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(sqrt(x^2 - 2))*sign(x^3 - x)