Optimal. Leaf size=29 \[ e^{-x} \left (-x+\left (-e^5+\log (x)\right ) \left (-9+x-\frac {4}{\log (\log (2))}\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(29)=58\).
time = 0.33, antiderivative size = 88, normalized size of antiderivative = 3.03, number of steps
used = 14, number of rules used = 7, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {12, 6874,
2207, 2225, 2634, 2230, 2209} \begin {gather*} \left (1+e^5\right ) \left (-e^{-x}\right ) x+e^{-x}-\left (1+e^5\right ) e^{-x}+e^{-x} \log (x)+\frac {e^{-x} \log (x) (x \log (\log (2))-2 (2+5 \log (\log (2))))}{\log (\log (2))}+\frac {2 e^{5-x} (2+5 \log (\log (2)))}{\log (\log (2))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-x} \left (-4-4 e^5 x+4 x \log (x)+\left (-9+x^2+e^5 \left (-10 x+x^2\right )+\left (10 x-x^2\right ) \log (x)\right ) \log (\log (2))\right )}{x} \, dx}{\log (\log (2))}\\ &=\frac {\int \left (-e^{-x} \log (x) (-4-10 \log (\log (2))+x \log (\log (2)))+\frac {e^{-x} \left (-4-9 \log (\log (2))+\left (1+e^5\right ) x^2 \log (\log (2))-2 e^5 x (2+5 \log (\log (2)))\right )}{x}\right ) \, dx}{\log (\log (2))}\\ &=-\frac {\int e^{-x} \log (x) (-4-10 \log (\log (2))+x \log (\log (2))) \, dx}{\log (\log (2))}+\frac {\int \frac {e^{-x} \left (-4-9 \log (\log (2))+\left (1+e^5\right ) x^2 \log (\log (2))-2 e^5 x (2+5 \log (\log (2)))\right )}{x} \, dx}{\log (\log (2))}\\ &=e^{-x} \log (x)+\frac {e^{-x} \log (x) (x \log (\log (2))-2 (2+5 \log (\log (2))))}{\log (\log (2))}+\frac {\int \frac {e^{-x} (4+9 \log (\log (2))-x \log (\log (2)))}{x} \, dx}{\log (\log (2))}+\frac {\int \left (\frac {e^{-x} (-4-9 \log (\log (2)))}{x}+e^{-x} \left (1+e^5\right ) x \log (\log (2))-2 e^{5-x} (2+5 \log (\log (2)))\right ) \, dx}{\log (\log (2))}\\ &=e^{-x} \log (x)+\frac {e^{-x} \log (x) (x \log (\log (2))-2 (2+5 \log (\log (2))))}{\log (\log (2))}+\left (1+e^5\right ) \int e^{-x} x \, dx+\frac {\int \left (-e^{-x} \log (\log (2))+\frac {e^{-x} (4+9 \log (\log (2)))}{x}\right ) \, dx}{\log (\log (2))}+\frac {(-4-9 \log (\log (2))) \int \frac {e^{-x}}{x} \, dx}{\log (\log (2))}-\frac {(2 (2+5 \log (\log (2)))) \int e^{5-x} \, dx}{\log (\log (2))}\\ &=-e^{-x} \left (1+e^5\right ) x+e^{-x} \log (x)+\frac {2 e^{5-x} (2+5 \log (\log (2)))}{\log (\log (2))}-\frac {\text {Ei}(-x) (4+9 \log (\log (2)))}{\log (\log (2))}+\frac {e^{-x} \log (x) (x \log (\log (2))-2 (2+5 \log (\log (2))))}{\log (\log (2))}+\left (1+e^5\right ) \int e^{-x} \, dx+\frac {(4+9 \log (\log (2))) \int \frac {e^{-x}}{x} \, dx}{\log (\log (2))}-\int e^{-x} \, dx\\ &=e^{-x}-e^{-x} \left (1+e^5\right )-e^{-x} \left (1+e^5\right ) x+e^{-x} \log (x)+\frac {2 e^{5-x} (2+5 \log (\log (2)))}{\log (\log (2))}+\frac {e^{-x} \log (x) (x \log (\log (2))-2 (2+5 \log (\log (2))))}{\log (\log (2))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 50, normalized size = 1.72 \begin {gather*} \frac {e^{-x} \left (-x \log (\log (2))+e^5 (4+9 \log (\log (2))-x \log (\log (2)))+\log (x) (-4-9 \log (\log (2))+x \log (\log (2)))\right )}{\log (\log (2))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 50, normalized size = 1.72
method | result | size |
norman | \(\left (x \ln \left (x \right )+\left (-{\mathrm e}^{5}-1\right ) x +\frac {{\mathrm e}^{5} \left (9 \ln \left (\ln \left (2\right )\right )+4\right )}{\ln \left (\ln \left (2\right )\right )}-\frac {\left (9 \ln \left (\ln \left (2\right )\right )+4\right ) \ln \left (x \right )}{\ln \left (\ln \left (2\right )\right )}\right ) {\mathrm e}^{-x}\) | \(50\) |
risch | \(\frac {\left (x \ln \left (\ln \left (2\right )\right )-9 \ln \left (\ln \left (2\right )\right )-4\right ) {\mathrm e}^{-x} \ln \left (x \right )}{\ln \left (\ln \left (2\right )\right )}-\frac {\left ({\mathrm e}^{5} \ln \left (\ln \left (2\right )\right ) x -9 \,{\mathrm e}^{5} \ln \left (\ln \left (2\right )\right )+x \ln \left (\ln \left (2\right )\right )-4 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-x}}{\ln \left (\ln \left (2\right )\right )}\) | \(61\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (26) = 52\).
time = 0.31, size = 54, normalized size = 1.86 \begin {gather*} -\frac {4 \, e^{\left (-x\right )} \log \left (x\right ) - {\left ({\left (x - 9\right )} e^{\left (-x\right )} \log \left (x\right ) - {\left ({\left (x - 9\right )} e^{5} + x\right )} e^{\left (-x\right )}\right )} \log \left (\log \left (2\right )\right ) - 4 \, e^{\left (-x + 5\right )}}{\log \left (\log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (20) = 40\).
time = 0.23, size = 65, normalized size = 2.24 \begin {gather*} \frac {\left (x \log {\left (x \right )} \log {\left (\log {\left (2 \right )} \right )} - x \log {\left (\log {\left (2 \right )} \right )} - x e^{5} \log {\left (\log {\left (2 \right )} \right )} - 4 \log {\left (x \right )} - 9 \log {\left (x \right )} \log {\left (\log {\left (2 \right )} \right )} + 9 e^{5} \log {\left (\log {\left (2 \right )} \right )} + 4 e^{5}\right ) e^{- x}}{\log {\left (\log {\left (2 \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs.
\(2 (26) = 52\).
time = 0.41, size = 78, normalized size = 2.69 \begin {gather*} \frac {x e^{\left (-x\right )} \log \left (x\right ) \log \left (\log \left (2\right )\right ) - x e^{\left (-x\right )} \log \left (\log \left (2\right )\right ) - x e^{\left (-x + 5\right )} \log \left (\log \left (2\right )\right ) - 9 \, e^{\left (-x\right )} \log \left (x\right ) \log \left (\log \left (2\right )\right ) - 4 \, e^{\left (-x\right )} \log \left (x\right ) + 9 \, e^{\left (-x + 5\right )} \log \left (\log \left (2\right )\right ) + 4 \, e^{\left (-x + 5\right )}}{\log \left (\log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 69, normalized size = 2.38 \begin {gather*} \frac {{\mathrm {e}}^{5-x}\,\left (9\,\ln \left (\ln \left (2\right )\right )+4\right )-{\mathrm {e}}^{-x}\,\ln \left (x\right )\,\left (9\,\ln \left (\ln \left (2\right )\right )+4\right )}{\ln \left (\ln \left (2\right )\right )}-\frac {x\,\left ({\mathrm {e}}^{-x}\,\ln \left (\ln \left (2\right )\right )\,\left ({\mathrm {e}}^5+1\right )-{\mathrm {e}}^{-x}\,\ln \left (\ln \left (2\right )\right )\,\ln \left (x\right )\right )}{\ln \left (\ln \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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