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

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 = 0.0293956, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\text{csch}^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.80731, 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((2+b/x**2)**(1/2)/(2*x**2+b),x)

[Out]

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

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Mathematica [B]  time = 0.0424659, size = 54, normalized size = 2.7 \[ \frac{x \sqrt{\frac{b}{x^2}+2} \left (\log (x)-\log \left (\sqrt{b} \sqrt{b+2 x^2}+b\right )\right )}{\sqrt{b} \sqrt{b+2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 50, normalized size = 2.5 \[ -{x\sqrt{{\frac{2\,{x}^{2}+b}{{x}^{2}}}}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{2\,{x}^{2}+b}+b}{x}} \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{2\,{x}^{2}+b}}}} \]

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 = 0.287161, 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(sqrt(b/x^2 + 2)/(2*x^2 + b),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 [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 = 0.263193, size = 59, normalized size = 2.95 \[ \frac{\arctan \left (\frac{\sqrt{2 \, x^{2} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x\right )}{\sqrt{-b}} - \frac{\arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ){\rm sign}\left (x\right )}{\sqrt{-b}} \]

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))*sig
n(x)/sqrt(-b)