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

Optimal. Leaf size=25 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

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Rubi [A]  time = 0.043342, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

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Rubi in Sympy [A]  time = 11.2578, size = 19, normalized size = 0.76 \[ - \frac{\operatorname{asin}{\left (x \right )}}{2} - \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{- x^{2} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asin(x)/2 - atanh(x/sqrt(-x**2 + 1))/2

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Mathematica [A]  time = 0.0128124, size = 25, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

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Maple [B]  time = 0.104, size = 187, normalized size = 7.5 \[{\frac{\sqrt{2}}{2} \left ({\frac{1}{4}\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\, \left ( x-1/2\,\sqrt{2} \right ) \sqrt{2}+2}}-{\frac{\sqrt{2}\arcsin \left ( x \right ) }{4}}-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\sqrt{2} \left ( - \left ( x-{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}+1 \right ){\frac{1}{\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\, \left ( x-1/2\,\sqrt{2} \right ) \sqrt{2}+2}}}} \right ) } \right ) }-{\frac{\sqrt{2}}{2} \left ({\frac{1}{4}\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\, \left ( x+1/2\,\sqrt{2} \right ) \sqrt{2}+2}}+{\frac{\sqrt{2}\arcsin \left ( x \right ) }{4}}-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\sqrt{2} \left ( \left ( x+{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}+1 \right ){\frac{1}{\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\, \left ( x+1/2\,\sqrt{2} \right ) \sqrt{2}+2}}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(2*x^2-1),x)

[Out]

1/2*2^(1/2)*(1/4*(-4*(x-1/2*2^(1/2))^2-4*(x-1/2*2^(1/2))*2^(1/2)+2)^(1/2)-1/4*2^
(1/2)*arcsin(x)-1/4*2^(1/2)*arctanh((-(x-1/2*2^(1/2))*2^(1/2)+1)*2^(1/2)/(-4*(x-
1/2*2^(1/2))^2-4*(x-1/2*2^(1/2))*2^(1/2)+2)^(1/2)))-1/2*2^(1/2)*(1/4*(-4*(x+1/2*
2^(1/2))^2+4*(x+1/2*2^(1/2))*2^(1/2)+2)^(1/2)+1/4*2^(1/2)*arcsin(x)-1/4*2^(1/2)*
arctanh(((x+1/2*2^(1/2))*2^(1/2)+1)*2^(1/2)/(-4*(x+1/2*2^(1/2))^2+4*(x+1/2*2^(1/
2))*2^(1/2)+2)^(1/2)))

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Maxima [A]  time = 1.49764, size = 143, normalized size = 5.72 \[ -\frac{1}{8} \, \sqrt{2}{\left (2 \, \sqrt{2} \arcsin \left (x\right ) - \sqrt{2} \log \left (\frac{1}{4} \, \sqrt{2} + \frac{\sqrt{2} \sqrt{-x^{2} + 1}}{{\left | \left (2 \, \sqrt{2}\right ) + 4 \, x \right |}} + \frac{1}{{\left | \left (2 \, \sqrt{2}\right ) + 4 \, x \right |}}\right ) + \sqrt{2} \log \left (-\frac{1}{4} \, \sqrt{2} + \frac{\sqrt{2} \sqrt{-x^{2} + 1}}{{\left | 4 \, x - 2 \, \sqrt{2} \right |}} + \frac{1}{{\left | 4 \, x - 2 \, \sqrt{2} \right |}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*(2*sqrt(2)*arcsin(x) - sqrt(2)*log(1/4*sqrt(2) + sqrt(2)*sqrt(-x^2
+ 1)/abs((2*sqrt(2)) + 4*x) + 1/abs((2*sqrt(2)) + 4*x)) + sqrt(2)*log(-1/4*sqrt(
2) + sqrt(2)*sqrt(-x^2 + 1)/abs(4*x - 2*sqrt(2)) + 1/abs(4*x - 2*sqrt(2))))

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Fricas [A]  time = 0.210409, size = 100, normalized size = 4. \[ \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 1}{\left (x + 1\right )} - x - 1}{x^{2}}\right ) - \frac{1}{4} \, \log \left (-\frac{x^{2} - \sqrt{-x^{2} + 1}{\left (x - 1\right )} + x - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239943, size = 159, normalized size = 6.36 \[ -\frac{1}{4} \, \pi{\rm sign}\left (x\right ) - \frac{1}{2} \, \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*pi*sign(x) - 1/2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2
+ 1) - 1)) - 1/4*ln(abs(-x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x + 2)) +
1/4*ln(abs(-x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x - 2))