Optimal. Leaf size=32 \[ -\frac{x}{2 \left (e^{2 x}+1\right )}+\frac{x}{2}-\frac{1}{4} \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.0550637, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2283, 2191, 2282, 36, 29, 31} \[ -\frac{x}{2 \left (e^{2 x}+1\right )}+\frac{x}{2}-\frac{1}{4} \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2283
Rule 2191
Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{\left (e^{-x}+e^x\right )^2} \, dx &=\int \frac{e^{2 x} x}{\left (1+e^{2 x}\right )^2} \, dx\\ &=-\frac{x}{2 \left (1+e^{2 x}\right )}+\frac{1}{2} \int \frac{1}{1+e^{2 x}} \, dx\\ &=-\frac{x}{2 \left (1+e^{2 x}\right )}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,e^{2 x}\right )\\ &=-\frac{x}{2 \left (1+e^{2 x}\right )}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{2 x}\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^{2 x}\right )\\ &=\frac{x}{2}-\frac{x}{2 \left (1+e^{2 x}\right )}-\frac{1}{4} \log \left (1+e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0350698, size = 31, normalized size = 0.97 \[ \frac{e^{2 x} x}{2 e^{2 x}+2}-\frac{1}{4} \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 26, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( \left ({{\rm e}^{x}} \right ) ^{2}+1 \right ) }{4}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}x}{2\, \left ({{\rm e}^{x}} \right ) ^{2}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43541, size = 34, normalized size = 1.06 \begin{align*} \frac{x e^{\left (2 \, x\right )}}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} - \frac{1}{4} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80123, size = 89, normalized size = 2.78 \begin{align*} \frac{2 \, x e^{\left (2 \, x\right )} -{\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{4 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098359, size = 24, normalized size = 0.75 \begin{align*} - \frac{x}{2} + \frac{x}{2 + 2 e^{- 2 x}} - \frac{\log{\left (1 + e^{- 2 x} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1543, size = 54, normalized size = 1.69 \begin{align*} \frac{2 \, x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right )}{4 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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