Optimal. Leaf size=22 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right ) \]
[Out]
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Rubi [A] time = 0.0994283, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x^(-1)]/(1 - x^2),x]
[Out]
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Rubi in Sympy [A] time = 6.16368, size = 20, normalized size = 0.91 \[ \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{1 + \frac{1}{x}}}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+1/x)**(1/2)/(-x**2+1),x)
[Out]
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Mathematica [A] time = 0.0327234, size = 38, normalized size = 1.73 \[ \frac{\log \left (\left (2 \sqrt{2} \sqrt{\frac{1}{x}+1}+3\right ) x+1\right )-\log (1-x)}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + x^(-1)]/(1 - x^2),x]
[Out]
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Maple [B] time = 0.019, size = 41, normalized size = 1.9 \[{\frac{\sqrt{2}x}{2}\sqrt{{\frac{1+x}{x}}}{\it Artanh} \left ({\frac{ \left ( 3\,x+1 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+1/x)^(1/2)/(-x^2+1),x)
[Out]
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Maxima [A] time = 0.777649, size = 54, normalized size = 2.45 \[ -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{\frac{x + 1}{x}}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{\frac{x + 1}{x}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(1/x + 1)/(x^2 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27427, size = 45, normalized size = 2.05 \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} x \sqrt{\frac{x + 1}{x}} + 3 \, x + 1}{x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(1/x + 1)/(x^2 - 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{1 + \frac{1}{x}}}{x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+1/x)**(1/2)/(-x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.290333, size = 99, normalized size = 4.5 \[ \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right ){\rm sign}\left (x\right ) - \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + x} + 2 \right |}}\right ){\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(1/x + 1)/(x^2 - 1),x, algorithm="giac")
[Out]