Optimal. Leaf size=31 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0393342, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(-1),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.44295, size = 26, normalized size = 0.84 \[ \frac{x}{a} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0157899, size = 31, normalized size = 1. \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(-1),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.002, size = 27, normalized size = 0.9 \[{\frac{x}{a}}-{\frac{b}{a}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^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/(a + b/x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.237185, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 2 \, x}{2 \, a}, -\frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) - x}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a + b/x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.18514, size = 56, normalized size = 1.81 \[ \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} + \frac{x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.225645, size = 35, normalized size = 1.13 \[ -\frac{b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a + b/x^2),x, algorithm="giac")
[Out]