Optimal. Leaf size=10 \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]
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Rubi [A] time = 0.0208617, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2249, 203} \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{5 x}}{1+e^{10 x}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^{5 x}\right )\\ &=\frac{1}{5} \tan ^{-1}\left (e^{5 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0031279, size = 10, normalized size = 1. \[ \frac{1}{5} \tan ^{-1}\left (e^{5 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 8, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( \left ({{\rm e}^{x}} \right ) ^{5} \right ) }{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45586, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.85661, size = 28, normalized size = 2.8 \begin{align*} \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.112689, size = 17, normalized size = 1.7 \begin{align*} \operatorname{RootSum}{\left (100 z^{2} + 1, \left ( i \mapsto i \log{\left (10 i + e^{5 x} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.73921, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{5} \, \arctan \left (e^{\left (5 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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