Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.057053, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^2]*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.13101, size = 20, normalized size = 0.83 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.0293485, size = 50, normalized size = 2.08 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b}}\right )}{\sqrt{a} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^2]*x),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.008, size = 46, normalized size = 1.9 \[{\frac{1}{x}\sqrt{a{x}^{2}+b}\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(1/2)/x,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/(sqrt(a + b/x^2)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.241666, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.25846, size = 17, normalized size = 0.71 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x),x, algorithm="giac")
[Out]