Optimal. Leaf size=20 \[ \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 2249, 203} \[ \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 2249
Rubi steps
\begin {align*} \int \frac {2 e^x}{2+3 e^{2 x}} \, dx &=2 \int \frac {e^x}{2+3 e^{2 x}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,e^x\right )\\ &=\sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 19, normalized size = 0.95 \[ \frac {1}{3} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 13, normalized size = 0.65 \[ \frac {1}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.70 \[ \frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, {\mathrm e}^{x}}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 13, normalized size = 0.65 \[ \frac {1}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 13, normalized size = 0.65 \[ \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,{\mathrm {e}}^x}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 15, normalized size = 0.75 \[ \operatorname {RootSum} {\left (6 z^{2} + 1, \left (i \mapsto i \log {\left (2 i + e^{x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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