Optimal. Leaf size=27 \[ \frac {3}{5} \left (e^x+1\right )^{5/3}-\frac {3}{2} \left (e^x+1\right )^{2/3} \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac {3}{5} \left (e^x+1\right )^{5/3}-\frac {3}{2} \left (e^x+1\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{\sqrt [3]{1+e^x}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x}} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [3]{1+x}}+(1+x)^{2/3}\right ) \, dx,x,e^x\right )\\ &=-\frac {3}{2} \left (1+e^x\right )^{2/3}+\frac {3}{5} \left (1+e^x\right )^{5/3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.74 \[ \frac {3}{10} \left (e^x+1\right )^{2/3} \left (2 e^x-3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 14, normalized size = 0.52 \[ \frac {3}{10} \, {\left (2 \, e^{x} - 3\right )} {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 17, normalized size = 0.63 \[ \frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 18, normalized size = 0.67 \[ -\frac {3 \left ({\mathrm e}^{x}+1\right )^{\frac {2}{3}}}{2}+\frac {3 \left ({\mathrm e}^{x}+1\right )^{\frac {5}{3}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 17, normalized size = 0.63 \[ \frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 14, normalized size = 0.52 \[ \frac {3\,{\left ({\mathrm {e}}^x+1\right )}^{2/3}\,\left (2\,{\mathrm {e}}^x-3\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.99, size = 22, normalized size = 0.81 \[ \frac {3 \left (e^{x} + 1\right )^{\frac {5}{3}}}{5} - \frac {3 \left (e^{x} + 1\right )^{\frac {2}{3}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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