3.904 \(\int \frac{1}{\sqrt{2+e x} \sqrt{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e} \]

[Out]

-(ArcTanh[Sqrt[2 - e*x]/2]/(Sqrt[3]*e))

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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]

-(ArcTanh[Sqrt[2 - e*x]/2]/(Sqrt[3]*e))

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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]

-sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(3*e)

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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]

(Sqrt[-2 + e*x]*Sqrt[2 + e*x]*ArcTan[Sqrt[-2 + e*x]/2])/(e*Sqrt[12 - 3*e^2*x^2])

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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]

-(-e^2*x^2+4)^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))/(e*x+2)^(1/2)/(-3*e*x+
6)^(1/2)/e

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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]

-1/3*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2))/e

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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]

1/6*sqrt(3)*log(-(sqrt(3)*(e^2*x^2 - 4*e*x - 12) + 4*sqrt(-3*e^2*x^2 + 12)*sqrt(
e*x + 2))/(e^2*x^2 + 4*e*x + 4))/e

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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]

sqrt(3)*Integral(1/(sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4)), x)/3

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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]

integrate(1/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)), x)