Optimal. Leaf size=18 \[ e^{-3+x} \left (\frac {7-x}{2}+\log (x)\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.56, number of steps
used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 14, 2225,
2209, 2207, 2634} \begin {gather*} -\frac {1}{2} e^{x-3} x+\frac {7 e^{x-3}}{2}+e^{x-3} \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2207
Rule 2209
Rule 2225
Rule 2634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^x \left (2+6 x-x^2\right )+2 e^x x \log (x)}{x} \, dx}{2 e^3}\\ &=\frac {\int \left (6 e^x+\frac {2 e^x}{x}-e^x x+2 e^x \log (x)\right ) \, dx}{2 e^3}\\ &=-\frac {\int e^x x \, dx}{2 e^3}+\frac {\int \frac {e^x}{x} \, dx}{e^3}+\frac {\int e^x \log (x) \, dx}{e^3}+\frac {3 \int e^x \, dx}{e^3}\\ &=3 e^{-3+x}-\frac {1}{2} e^{-3+x} x+\frac {\text {Ei}(x)}{e^3}+e^{-3+x} \log (x)+\frac {\int e^x \, dx}{2 e^3}-\frac {\int \frac {e^x}{x} \, dx}{e^3}\\ &=\frac {7 e^{-3+x}}{2}-\frac {1}{2} e^{-3+x} x+e^{-3+x} \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{2} e^{-3+x} (7-x+2 \log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 18, normalized size = 1.00
method | result | size |
risch | \(\ln \left (x \right ) {\mathrm e}^{x -3}-\frac {\left (x -7\right ) {\mathrm e}^{x -3}}{2}\) | \(18\) |
norman | \({\mathrm e}^{x} {\mathrm e}^{-3} \ln \left (x \right )+\frac {7 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{2}-\frac {{\mathrm e}^{-3} {\mathrm e}^{x} x}{2}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 1.17 \begin {gather*} -\frac {1}{2} \, {\left ({\left (x - 1\right )} e^{x} - 2 \, e^{x} \log \left (x\right ) - 6 \, e^{x}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 17, normalized size = 0.94 \begin {gather*} -\frac {1}{2} \, {\left ({\left (x - 7\right )} e^{x} - 2 \, e^{x} \log \left (x\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 15, normalized size = 0.83 \begin {gather*} \frac {\left (- x + 2 \log {\left (x \right )} + 7\right ) e^{x}}{2 e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 1.06 \begin {gather*} -\frac {1}{2} \, {\left (x e^{x} - 2 \, e^{x} \log \left (x\right ) - 7 \, e^{x}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.51, size = 12, normalized size = 0.67 \begin {gather*} {\mathrm {e}}^{x-3}\,\left (\ln \left (x\right )-\frac {x}{2}+\frac {7}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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