Optimal. Leaf size=27 \[ \frac {1}{4} \left (6-e^2-e^x \left (2+e^x-x\right )-2 x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.26, number of steps
used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2225, 2207}
\begin {gather*} -\frac {1}{4} e^x (1-x)-\frac {e^x}{4}-\frac {e^{2 x}}{4}-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (-2-2 e^{2 x}+e^x (-1+x)\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{4} \int e^x (-1+x) \, dx-\frac {1}{2} \int e^{2 x} \, dx\\ &=-\frac {e^{2 x}}{4}-\frac {1}{4} e^x (1-x)-\frac {x}{2}-\frac {\int e^x \, dx}{4}\\ &=-\frac {e^x}{4}-\frac {e^{2 x}}{4}-\frac {1}{4} e^x (1-x)-\frac {x}{2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{4} \left (-e^{2 x}+e^x (-2+x)-2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 20, normalized size = 0.74
method | result | size |
risch | \(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} \left (x -2\right )}{4}-\frac {x}{2}\) | \(18\) |
default | \(-\frac {x}{2}-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} x}{4}-\frac {{\mathrm e}^{x}}{2}\) | \(20\) |
norman | \(-\frac {x}{2}-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} x}{4}-\frac {{\mathrm e}^{x}}{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{4} \, {\left (x - 1\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} - \frac {1}{4} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{4} \, {\left (x - 2\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.70 \begin {gather*} - \frac {x}{2} + \frac {\left (4 x - 8\right ) e^{x}}{16} - \frac {e^{2 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{4} \, {\left (x - 2\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 19, normalized size = 0.70 \begin {gather*} \frac {x\,{\mathrm {e}}^x}{4}-\frac {{\mathrm {e}}^{2\,x}}{4}-\frac {{\mathrm {e}}^x}{2}-\frac {x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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