Optimal. Leaf size=12 \[ \frac {1}{2} \tan ^{-1}\left (\frac {e^x}{2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2249, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {e^x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{4+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{4+x^2} \, dx,x,e^x\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {e^x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\frac {e^x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 7, normalized size = 0.58 \[ \frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 7, normalized size = 0.58 \[ \frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 8, normalized size = 0.67 \[ \frac {\arctan \left (\frac {{\mathrm e}^{x}}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 7, normalized size = 0.58 \[ \frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 7, normalized size = 0.58 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 15, normalized size = 1.25 \[ \operatorname {RootSum} {\left (16 z^{2} + 1, \left (i \mapsto i \log {\left (8 i + e^{x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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