Optimal. Leaf size=26 \[ -e^{-x} x+e^x x-e^{-x}-e^x \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.40, size = 18, normalized size = 0.69 \[ {\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - x - 1\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 16, normalized size = 0.62 \[ -{\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.88 \[ x \,{\mathrm e}^{x}-x \,{\mathrm e}^{-x}-{\mathrm e}^{x}-{\mathrm e}^{-x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 16, normalized size = 0.62 \[ -{\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 10, normalized size = 0.38 \[ 2\,x\,\mathrm {sinh}\relax (x)-2\,\mathrm {cosh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 14, normalized size = 0.54 \[ \left (- x - 1\right ) e^{- x} + \left (x - 1\right ) e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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