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