Optimal. Leaf size=21 \[ \frac {1}{5} e^{-136-x} \left (\log (4)+\frac {1}{2} x \log (x)\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(21)=42\).
time = 0.10, antiderivative size = 60, normalized size of antiderivative = 2.86, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 6874, 2225,
2207, 2634} \begin {gather*} \frac {e^{-x-136}}{10}+\frac {1}{10} e^{-x-136} \log (x)-\frac {1}{10} e^{-x-136} (1-x) \log (x)-\frac {1}{10} e^{-x-136} (1-\log (16)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int e^{-136-x} (1-2 \log (4)+(1-x) \log (x)) \, dx\\ &=\frac {1}{10} \int \left (e^{-136-x} (1-2 \log (4))-e^{-136-x} (-1+x) \log (x)\right ) \, dx\\ &=-\left (\frac {1}{10} \int e^{-136-x} (-1+x) \log (x) \, dx\right )+\frac {1}{10} (1-2 \log (4)) \int e^{-136-x} \, dx\\ &=-\frac {1}{10} e^{-136-x} (1-\log (16))+\frac {1}{10} e^{-136-x} \log (x)-\frac {1}{10} e^{-136-x} (1-x) \log (x)-\frac {1}{10} \int e^{-136-x} \, dx\\ &=\frac {e^{-136-x}}{10}-\frac {1}{10} e^{-136-x} (1-\log (16))+\frac {1}{10} e^{-136-x} \log (x)-\frac {1}{10} e^{-136-x} (1-x) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.86 \begin {gather*} \frac {1}{10} e^{-136-x} (\log (16)+x \log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 18, normalized size = 0.86
method | result | size |
norman | \(\left (\frac {x \ln \left (x \right )}{10}+\frac {2 \ln \left (2\right )}{5}\right ) {\mathrm e}^{-x -136}\) | \(18\) |
risch | \(\frac {x \,{\mathrm e}^{-x -136} \ln \left (x \right )}{10}+\frac {2 \ln \left (2\right ) {\mathrm e}^{-x -136}}{5}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 22, normalized size = 1.05 \begin {gather*} \frac {1}{10} \, x e^{\left (-x - 136\right )} \log \left (x\right ) + \frac {2}{5} \, e^{\left (-x - 136\right )} \log \left (2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 17, normalized size = 0.81 \begin {gather*} \frac {\left (x \log {\left (x \right )} + 4 \log {\left (2 \right )}\right ) e^{- x - 136}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{10} \, {\left (x e^{\left (-x\right )} \log \left (x\right ) + 4 \, e^{\left (-x\right )} \log \left (2\right )\right )} e^{\left (-136\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.41, size = 17, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{-x-136}\,\left (\frac {\ln \left (16\right )}{10}+\frac {x\,\ln \left (x\right )}{10}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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