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 = 0.0259199, antiderivative size = 27, normalized size of antiderivative = 1., 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 2248
Rule 43
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*}
Mathematica [A] time = 0.0101943, 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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Maple [A] time = 0.02, size = 18, normalized size = 0.7 \begin{align*} -{\frac{3}{2} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{2}{3}}}}+{\frac{3}{5} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991878, size = 23, normalized size = 0.85 \begin{align*} \frac{3}{5} \,{\left (e^{x} + 1\right )}^{\frac{5}{3}} - \frac{3}{2} \,{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.78088, size = 46, normalized size = 1.7 \begin{align*} \frac{3}{10} \,{\left (2 \, e^{x} - 3\right )}{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.89628, size = 22, normalized size = 0.81 \begin{align*} \frac{3 \left (e^{x} + 1\right )^{\frac{5}{3}}}{5} - \frac{3 \left (e^{x} + 1\right )^{\frac{2}{3}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19501, size = 23, normalized size = 0.85 \begin{align*} \frac{3}{5} \,{\left (e^{x} + 1\right )}^{\frac{5}{3}} - \frac{3}{2} \,{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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