Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0673881, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.64317, size = 27, normalized size = 1.08 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+2)**(1/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0396641, size = 50, normalized size = 2. \[ \frac{\sqrt{e x-2} \sqrt{e x+2} \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )}{e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.016, size = 47, normalized size = 1.9 \[ -{\frac{1}{e}\sqrt{-{e}^{2}{x}^{2}+4}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ){\frac{1}{\sqrt{ex+2}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.802914, size = 24, normalized size = 0.96 \[ -\frac{i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.226715, size = 88, normalized size = 3.52 \[ \frac{\sqrt{3} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right )}{6 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{3} \int \frac{1}{\sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+2)**(1/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)),x, algorithm="giac")
[Out]