Optimal. Leaf size=10 \[ \frac{1}{2} \tan ^{-1}\left (e^{x^2}\right ) \]
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Rubi [A] time = 0.126492, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6715, 2249, 203} \[ \frac{1}{2} \tan ^{-1}\left (e^{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6715
Rule 2249
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{x^2} x}{1+e^{2 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^x}{1+e^{2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^{x^2}\right )\\ &=\frac{1}{2} \tan ^{-1}\left (e^{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0158131, size = 10, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (e^{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 8, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ({{\rm e}^{{x}^{2}}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49493, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{2} \, \arctan \left (e^{\left (x^{2}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.814956, size = 28, normalized size = 2.8 \begin{align*} \frac{1}{2} \, \arctan \left (e^{\left (x^{2}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.129271, size = 17, normalized size = 1.7 \begin{align*} \operatorname{RootSum}{\left (16 z^{2} + 1, \left ( i \mapsto i \log{\left (4 i + e^{x^{2}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25936, size = 9, normalized size = 0.9 \begin{align*} \frac{1}{2} \, \arctan \left (e^{\left (x^{2}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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