3.777 \(\int \frac{\sqrt{2-\frac{b}{x^2}}}{-b+2 x^2} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

[Out]

-(ArcCsc[(Sqrt[2]*x)/Sqrt[b]]/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]

-(ArcCsc[(Sqrt[2]*x)/Sqrt[b]]/Sqrt[b])

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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]

-asin(sqrt(2)*sqrt(b)/(2*x))/sqrt(b)

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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]

[Out]

((-I)*Sqrt[2 - b/x^2]*x*Log[(2*((-I)*Sqrt[b] + Sqrt[-b + 2*x^2]))/x])/(Sqrt[b]*S
qrt[-b + 2*x^2])

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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]

-((2*x^2-b)/x^2)^(1/2)*x/(2*x^2-b)^(1/2)/(-b)^(1/2)*ln(2*((-b)^(1/2)*(2*x^2-b)^(
1/2)-b)/x)

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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]

Exception raised: ValueError

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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]

[-1/2*sqrt(-b)*log(-(b*x*sqrt((2*x^2 - b)/x^2) + (x^2 - b)*sqrt(-b))/x^2)/b, -ar
ctan(sqrt(b)/(x*sqrt((2*x^2 - b)/x^2)))/sqrt(b)]

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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]

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

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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]

arctan(sqrt(2*x^2 - b)/sqrt(b))*sign(x)/sqrt(b) - arctan(sqrt(-b)/sqrt(b))*sign(
x)/sqrt(b)