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 = 0.03, 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} \[ \frac {4}{7} \left (e^x+1\right )^{7/4}-\frac {4}{3} \left (e^x+1\right )^{3/4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.39, size = 14, normalized size = 0.52 \[ \frac {4}{21} \, {\left (3 \, e^{x} - 4\right )} {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]
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 \[ \frac {4}{7} \, {\left (e^{x} + 1\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]
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 \[ -\frac {4 \left ({\mathrm e}^{x}+1\right )^{\frac {3}{4}}}{3}+\frac {4 \left ({\mathrm e}^{x}+1\right )^{\frac {7}{4}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 17, normalized size = 0.63 \[ \frac {4}{7} \, {\left (e^{x} + 1\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 14, normalized size = 0.52 \[ \frac {4\,{\left ({\mathrm {e}}^x+1\right )}^{3/4}\,\left (3\,{\mathrm {e}}^x-4\right )}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.25, size = 22, normalized size = 0.81 \[ \frac {4 \left (e^{x} + 1\right )^{\frac {7}{4}}}{7} - \frac {4 \left (e^{x} + 1\right )^{\frac {3}{4}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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