Optimal. Leaf size=27 \[ \frac{4}{7} \left (e^x+1\right )^{7/4}-\frac{4}{3} \left (e^x+1\right )^{3/4} \]
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Rubi [A] time = 0.0258179, 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} \[ \frac{4}{7} \left (e^x+1\right )^{7/4}-\frac{4}{3} \left (e^x+1\right )^{3/4} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{\sqrt [4]{1+e^x}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1+x}} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt [4]{1+x}}+(1+x)^{3/4}\right ) \, dx,x,e^x\right )\\ &=-\frac{4}{3} \left (1+e^x\right )^{3/4}+\frac{4}{7} \left (1+e^x\right )^{7/4}\\ \end{align*}
Mathematica [A] time = 0.0099306, size = 20, normalized size = 0.74 \[ \frac{4}{21} \left (e^x+1\right )^{3/4} \left (3 e^x-4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 18, normalized size = 0.7 \begin{align*} -{\frac{4}{3} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{3}{4}}}}+{\frac{4}{7} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{7}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959522, size = 23, normalized size = 0.85 \begin{align*} \frac{4}{7} \,{\left (e^{x} + 1\right )}^{\frac{7}{4}} - \frac{4}{3} \,{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.771953, size = 46, normalized size = 1.7 \begin{align*} \frac{4}{21} \,{\left (3 \, e^{x} - 4\right )}{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.62772, size = 22, normalized size = 0.81 \begin{align*} \frac{4 \left (e^{x} + 1\right )^{\frac{7}{4}}}{7} - \frac{4 \left (e^{x} + 1\right )^{\frac{3}{4}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23547, size = 23, normalized size = 0.85 \begin{align*} \frac{4}{7} \,{\left (e^{x} + 1\right )}^{\frac{7}{4}} - \frac{4}{3} \,{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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