Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0305389, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 - b/x^2]/(-b + 2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 3.35642, size = 20, normalized size = 1. \[ - \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-b/x**2)**(1/2)/(2*x**2-b),x)
[Out]
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Mathematica [C] time = 0.0451672, size = 64, normalized size = 3.2 \[ -\frac{i x \sqrt{2-\frac{b}{x^2}} \log \left (\frac{2 \left (\sqrt{2 x^2-b}-i \sqrt{b}\right )}{x}\right )}{\sqrt{b} \sqrt{2 x^2-b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 - b/x^2]/(-b + 2*x^2),x]
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Maple [B] time = 0.013, size = 62, normalized size = 3.1 \[ -{x\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{-b}}}{\frac{1}{\sqrt{2\,{x}^{2}-b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-b/x^2)^(1/2)/(2*x^2-b),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b/x^2 + 2)/(2*x^2 - b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287642, size = 1, normalized size = 0.05 \[ \left [-\frac{\sqrt{-b} \log \left (-\frac{b x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}} +{\left (x^{2} - b\right )} \sqrt{-b}}{x^{2}}\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{b}}{x \sqrt{\frac{2 \, x^{2} - b}{x^{2}}}}\right )}{\sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b/x^2 + 2)/(2*x^2 - b),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \frac{b}{x^{2}} + 2}}{- b + 2 x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2-b/x**2)**(1/2)/(2*x**2-b),x)
[Out]
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GIAC/XCAS [A] time = 0.266131, size = 54, normalized size = 2.7 \[ \frac{\arctan \left (\frac{\sqrt{2 \, x^{2} - b}}{\sqrt{b}}\right ){\rm sign}\left (x\right )}{\sqrt{b}} - \frac{\arctan \left (\frac{\sqrt{-b}}{\sqrt{b}}\right ){\rm sign}\left (x\right )}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b/x^2 + 2)/(2*x^2 - b),x, algorithm="giac")
[Out]