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 = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.60 \[ -\frac {8}{65} \sqrt [4]{3-e^{x/2}} \left (96 e^{x/2}+20 e^x+5 e^{3 x/2}+1152\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.25, size = 30, normalized size = 0.41 \[ -\frac {8}{65} \, {\left (5 \, e^{\left (\frac {3}{2} \, x\right )} + 96 \, e^{\left (\frac {1}{2} \, x\right )} + 20 \, e^{x} + 1152\right )} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 65, normalized size = 0.89 \[ -\frac {8}{13} \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{3} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} - 8 \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{2} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.51
method | result | size |
risch | \(\frac {8 \left (5 \,{\mathrm e}^{\frac {3 x}{2}}+20 \,{\mathrm e}^{x}+96 \,{\mathrm e}^{\frac {x}{2}}+1152\right ) \left (-3+{\mathrm e}^{\frac {x}{2}}\right )}{65 \left (3-{\mathrm e}^{\frac {x}{2}}\right )^{\frac {3}{4}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 49, normalized size = 0.67 \[ \frac {8}{13} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {13}{4}} - 8 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {9}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 30, normalized size = 0.41 \[ -{\left (3-{\mathrm {e}}^{x/2}\right )}^{1/4}\,\left (\frac {768\,{\mathrm {e}}^{x/2}}{65}+\frac {8\,{\mathrm {e}}^{\frac {3\,x}{2}}}{13}+\frac {32\,{\mathrm {e}}^x}{13}+\frac {9216}{65}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{2 x}}{\left (3 - e^{\frac {x}{2}}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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