Optimal. Leaf size=73 \[ \frac{8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac{216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]
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Rubi [A] time = 0.0474549, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2248, 43} \[ \frac{8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac{216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(3-x)^{3/4}} \, dx,x,e^{x/2}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{27}{(3-x)^{3/4}}-27 \sqrt [4]{3-x}+9 (3-x)^{5/4}-(3-x)^{9/4}\right ) \, dx,x,e^{x/2}\right )\\ &=-216 \sqrt [4]{3-e^{x/2}}+\frac{216}{5} \left (3-e^{x/2}\right )^{5/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac{8}{13} \left (3-e^{x/2}\right )^{13/4}\\ \end{align*}
Mathematica [A] time = 0.0195386, size = 44, normalized size = 0.6 \[ -\frac{8}{65} \sqrt [4]{3-e^{x/2}} \left (96 e^{x/2}+20 e^x+5 e^{3 x/2}+1152\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 37, normalized size = 0.5 \begin{align*}{\frac{8}{65} \left ( 5\,{{\rm e}^{3/2\,x}}+20\,{{\rm e}^{x}}+96\,{{\rm e}^{x/2}}+1152 \right ) \left ( -3+{{\rm e}^{{\frac{x}{2}}}} \right ) \left ( 3-{{\rm e}^{{\frac{x}{2}}}} \right ) ^{-{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931849, size = 66, normalized size = 0.9 \begin{align*} \frac{8}{13} \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{13}{4}} - 8 \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{9}{4}} + \frac{216}{5} \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{5}{4}} - 216 \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83639, size = 101, normalized size = 1.38 \begin{align*} -\frac{8}{65} \,{\left (5 \, e^{\left (\frac{3}{2} \, x\right )} + 96 \, e^{\left (\frac{1}{2} \, x\right )} + 20 \, e^{x} + 1152\right )}{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{2 x}}{\left (3 - e^{\frac{x}{2}}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14824, size = 88, normalized size = 1.21 \begin{align*} -\frac{8}{13} \,{\left (e^{\left (\frac{1}{2} \, x\right )} - 3\right )}^{3}{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{1}{4}} - 8 \,{\left (e^{\left (\frac{1}{2} \, x\right )} - 3\right )}^{2}{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{1}{4}} + \frac{216}{5} \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{5}{4}} - 216 \,{\left (-e^{\left (\frac{1}{2} \, x\right )} + 3\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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