3.532 \(\int \frac{e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt{-2+e^{3 x/4}+e^{3 x/2}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{2-5 e^{3 x/4}}{4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

[Out]

(2*ArcTanh[(2 - 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/3

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Rubi [A]  time = 0.166763, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{2-5 e^{3 x/4}}{4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]),x]

[Out]

(2*ArcTanh[(2 - 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/3

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Rubi in Sympy [A]  time = 24.2406, size = 36, normalized size = 0.9 \[ - \frac{2 \operatorname{atanh}{\left (\frac{5 e^{\frac{3 x}{4}} - 2}{4 \sqrt{e^{\frac{3 x}{4}} + e^{\frac{3 x}{2}} - 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))**(1/2),x)

[Out]

-2*atanh((5*exp(3*x/4) - 2)/(4*sqrt(exp(3*x/4) + exp(3*x/2) - 2)))/3

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Mathematica [A]  time = 0.040818, size = 55, normalized size = 1.38 \[ \frac{2}{3} \log \left (2-e^{3 x/4}\right )-\frac{2}{3} \log \left (-4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}-5 e^{3 x/4}+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]),x]

[Out]

(2*Log[2 - E^((3*x)/4)])/3 - (2*Log[2 - 5*E^((3*x)/4) - 4*Sqrt[-2 + E^((3*x)/4)
+ E^((3*x)/2)]])/3

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{1{{\rm e}^{{\frac{3\,x}{4}}}} \left ( -2+{{\rm e}^{{\frac{3\,x}{4}}}} \right ) ^{-1}{\frac{1}{\sqrt{-2+{{\rm e}^{{\frac{3\,x}{4}}}}+{{\rm e}^{{\frac{3\,x}{2}}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)

[Out]

int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)

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Maxima [A]  time = 1.6532, size = 53, normalized size = 1.32 \[ -\frac{2}{3} \, \log \left (\frac{4 \, \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac{3}{4} \, x\right )} - 2 \right |}} + \frac{8}{{\left | e^{\left (\frac{3}{4} \, x\right )} - 2 \right |}} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(3/4*x)/(sqrt(e^(3/2*x) + e^(3/4*x) - 2)*(e^(3/4*x) - 2)),x, algorithm="maxima")

[Out]

-2/3*log(4*sqrt(e^(3/2*x) + e^(3/4*x) - 2)/abs(e^(3/4*x) - 2) + 8/abs(e^(3/4*x)
- 2) + 5)

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Fricas [A]  time = 0.22534, size = 62, normalized size = 1.55 \[ -\frac{2}{3} \, \log \left (\sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} + 4\right ) + \frac{2}{3} \, \log \left (\sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(3/4*x)/(sqrt(e^(3/2*x) + e^(3/4*x) - 2)*(e^(3/4*x) - 2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\frac{3 x}{4}}}{\left (e^{\frac{3 x}{4}} - 2\right ) \sqrt{e^{\frac{3 x}{4}} + e^{\frac{3 x}{2}} - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.244508, size = 65, normalized size = 1.62 \[ -\frac{2}{3} \,{\rm ln}\left ({\left | \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} + 4 \right |}\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(3/4*x)/(sqrt(e^(3/2*x) + e^(3/4*x) - 2)*(e^(3/4*x) - 2)),x, algorithm="giac")

[Out]

-2/3*ln(abs(sqrt(e^(3/2*x) + e^(3/4*x) - 2) - e^(3/4*x) + 4)) + 2/3*ln(abs(sqrt(
e^(3/2*x) + e^(3/4*x) - 2) - e^(3/4*x)))