3.1926 \(\int \frac{1}{\sqrt{2+\frac{b}{x^2}} x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0344475, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\text{csch}^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 + b/x^2]*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.74194, size = 20, normalized size = 1. \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0469895, size = 56, normalized size = 2.8 \[ \frac{\sqrt{b+2 x^2} \left (\log (x)-\log \left (\sqrt{b} \sqrt{b+2 x^2}+b\right )\right )}{\sqrt{b} x \sqrt{\frac{b}{x^2}+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 52, normalized size = 2.6 \[ -{\frac{1}{x}\sqrt{2\,{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{2\,{x}^{2}+b}+b}{x}} \right ){\frac{1}{\sqrt{{\frac{2\,{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(2+b/x^2)^(1/2),x)

[Out]

-1/((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(1/(x^2*sqrt(b/x^2 + 2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246944, size = 1, normalized size = 0.05 \[ \left [\frac{\log \left (\frac{b x \sqrt{\frac{2 \, x^{2} + b}{x^{2}}} -{\left (x^{2} + b\right )} \sqrt{b}}{x^{2}}\right )}{2 \, \sqrt{b}}, \frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{2 \, x^{2} + b}{x^{2}}}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.33151, size = 20, normalized size = 1. \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{\frac{b}{x^{2}} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^2*sqrt(b/x^2 + 2)), x)