Optimal. Leaf size=20 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0343111, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\csc ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[2 - b/x^2]*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.13383, 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(1/x**2/(2-b/x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.054696, 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[1/(Sqrt[2 - b/x^2]*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 64, normalized size = 3.2 \[ -{\frac{1}{x}\sqrt{2\,{x}^{2}-b}\ln \left ( 2\,{\frac{\sqrt{-b}\sqrt{2\,{x}^{2}-b}-b}{x}} \right ){\frac{1}{\sqrt{{\frac{2\,{x}^{2}-b}{{x}^{2}}}}}}{\frac{1}{\sqrt{-b}}}} \]
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 = 0.251337, 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(1/(x^2*sqrt(-b/x^2 + 2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.6291, size = 49, normalized size = 2.45 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{\frac{b}{x^{2}}}\right |}{2} > 1 \\- \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b}}{2 x} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
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]