Optimal. Leaf size=24 \[ \frac {1}{5} e^{x/4} (1-2 x) \log \left (\frac {1}{5} x \log (5)\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps
used = 14, number of rules used = 7, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 14, 2225,
2209, 2634, 2207, 2230} \begin {gather*} \frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{20} \int \frac {e^{x/4} (4-8 x)+e^{x/4} \left (-7 x-2 x^2\right ) \log \left (\frac {1}{5} x \log (5)\right )}{x} \, dx\\ &=\frac {1}{20} \int \left (-8 e^{x/4}+\frac {4 e^{x/4}}{x}-7 e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-2 e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )\right ) \, dx\\ &=-\left (\frac {1}{10} \int e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right ) \, dx\right )+\frac {1}{5} \int \frac {e^{x/4}}{x} \, dx-\frac {7}{20} \int e^{x/4} \log \left (\frac {1}{5} x \log (5)\right ) \, dx-\frac {2}{5} \int e^{x/4} \, dx\\ &=-\frac {8 e^{x/4}}{5}+\frac {\text {Ei}\left (\frac {x}{4}\right )}{5}+\frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )+\frac {1}{10} \int \frac {4 e^{x/4} (-4+x)}{x} \, dx+\frac {7}{20} \int \frac {4 e^{x/4}}{x} \, dx\\ &=-\frac {8 e^{x/4}}{5}+\frac {\text {Ei}\left (\frac {x}{4}\right )}{5}+\frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )+\frac {2}{5} \int \frac {e^{x/4} (-4+x)}{x} \, dx+\frac {7}{5} \int \frac {e^{x/4}}{x} \, dx\\ &=-\frac {8 e^{x/4}}{5}+\frac {8 \text {Ei}\left (\frac {x}{4}\right )}{5}+\frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )+\frac {2}{5} \int \left (e^{x/4}-\frac {4 e^{x/4}}{x}\right ) \, dx\\ &=-\frac {8 e^{x/4}}{5}+\frac {8 \text {Ei}\left (\frac {x}{4}\right )}{5}+\frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )+\frac {2}{5} \int e^{x/4} \, dx-\frac {8}{5} \int \frac {e^{x/4}}{x} \, dx\\ &=\frac {1}{5} e^{x/4} \log \left (\frac {1}{5} x \log (5)\right )-\frac {2}{5} e^{x/4} x \log \left (\frac {1}{5} x \log (5)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 24, normalized size = 1.00 \begin {gather*} -\frac {1}{5} e^{x/4} (-1+2 x) \log \left (\frac {1}{5} x \log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 18, normalized size = 0.75
method | result | size |
risch | \(-\frac {{\mathrm e}^{\frac {x}{4}} \left (2 x -1\right ) \ln \left (\frac {x \ln \left (5\right )}{5}\right )}{5}\) | \(18\) |
norman | \(\frac {{\mathrm e}^{\frac {x}{4}} \ln \left (\frac {x \ln \left (5\right )}{5}\right )}{5}-\frac {2 x \,{\mathrm e}^{\frac {x}{4}} \ln \left (\frac {x \ln \left (5\right )}{5}\right )}{5}\) | \(27\) |
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 = 0.40, size = 17, normalized size = 0.71 \begin {gather*} -\frac {1}{5} \, {\left (2 \, x - 1\right )} e^{\left (\frac {1}{4} \, x\right )} \log \left (\frac {1}{5} \, x \log \left (5\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 26, normalized size = 1.08 \begin {gather*} \frac {\left (- 2 x \log {\left (\frac {x \log {\left (5 \right )}}{5} \right )} + \log {\left (\frac {x \log {\left (5 \right )}}{5} \right )}\right ) e^{\frac {x}{4}}}{5} \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. 41 vs.
\(2 (17) = 34\).
time = 0.41, size = 41, normalized size = 1.71 \begin {gather*} \frac {2}{5} \, x e^{\left (\frac {1}{4} \, x\right )} \log \left (5\right ) - \frac {2}{5} \, x e^{\left (\frac {1}{4} \, x\right )} \log \left (x \log \left (5\right )\right ) - \frac {1}{5} \, e^{\left (\frac {1}{4} \, x\right )} \log \left (5\right ) + \frac {1}{5} \, e^{\left (\frac {1}{4} \, x\right )} \log \left (x \log \left (5\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.41, size = 21, normalized size = 0.88 \begin {gather*} -\frac {{\mathrm {e}}^{x/4}\,\left (2\,x-1\right )\,\left (\ln \left (\ln \left (5\right )\right )-\ln \left (5\right )+\ln \left (x\right )\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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