Optimal. Leaf size=26 \[ -e^{-x} x+e^x x-e^{-x}-e^x \]
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Rubi [A] time = 0.0174877, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {14, 2176, 2194} \[ -e^{-x} x+e^x x-e^{-x}-e^x \]
Antiderivative was successfully verified.
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Rule 14
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \left (e^{-x}+e^x\right ) x \, dx &=\int \left (e^{-x} x+e^x x\right ) \, dx\\ &=\int e^{-x} x \, dx+\int e^x x \, dx\\ &=-e^{-x} x+e^x x+\int e^{-x} \, dx-\int e^x \, dx\\ &=-e^{-x}-e^x-e^{-x} x+e^x x\\ \end{align*}
Mathematica [A] time = 0.0208862, size = 20, normalized size = 0.77 \[ e^{-x} \left (e^{2 x} (x-1)-x-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 23, normalized size = 0.9 \begin{align*} - \left ({{\rm e}^{x}} \right ) ^{-1}-{{\rm e}^{x}}-{\frac{x}{{{\rm e}^{x}}}}+{{\rm e}^{x}}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965757, size = 22, normalized size = 0.85 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} +{\left (x - 1\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.847601, size = 46, normalized size = 1.77 \begin{align*}{\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - x - 1\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09574, size = 14, normalized size = 0.54 \begin{align*} \left (- x - 1\right ) e^{- x} + \left (x - 1\right ) e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21823, size = 22, normalized size = 0.85 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} +{\left (x - 1\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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