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