Optimal. Leaf size=39 \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0502188, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-9 + 4*x^2]/x^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.8086, size = 31, normalized size = 0.79 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{4 x^{2} - 9}}{3} \right )}}{3} - \frac{\sqrt{4 x^{2} - 9}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4*x**2-9)**(1/2)/x**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0159016, size = 37, normalized size = 0.95 \[ -\frac{\sqrt{4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{3}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-9 + 4*x^2]/x^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 41, normalized size = 1.1 \[{\frac{1}{18\,{x}^{2}} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{9}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{3}\arctan \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}-9}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4*x^2-9)^(1/2)/x^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.48651, size = 47, normalized size = 1.21 \[ -\frac{2}{9} \, \sqrt{4 \, x^{2} - 9} + \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} - \frac{2}{3} \, \arcsin \left (\frac{3}{2 \,{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.230625, size = 134, normalized size = 3.44 \[ \frac{48 \, x^{3} + 8 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) - 3 \,{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} - 9} - 108 \, x}{6 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.07295, size = 99, normalized size = 2.54 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} + \frac{i}{x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9 i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} - \frac{1}{x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x**2-9)**(1/2)/x**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.204703, size = 39, normalized size = 1. \[ -\frac{\sqrt{4 \, x^{2} - 9}}{2 \, x^{2}} + \frac{2}{3} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="giac")
[Out]