Optimal. Leaf size=4 \[ \tan ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0173531, antiderivative size = 4, normalized size of antiderivative = 1., 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} \[ \tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 203
Rubi steps
\begin{align*} \int \frac{e^x}{1+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0027281, size = 4, normalized size = 1. \[ \tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 4, normalized size = 1. \begin{align*} \arctan \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80007, size = 4, normalized size = 1. \begin{align*} \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50125, size = 18, normalized size = 4.5 \begin{align*} \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.104927, size = 15, normalized size = 3.75 \begin{align*} \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + e^{x} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22691, size = 4, normalized size = 1. \begin{align*} \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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