Optimal. Leaf size=19 \[ 2^{-4 e^{-x}} \sqrt [5]{\log (x)}+\log (x) \]
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Rubi [A]
time = 0.66, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6874, 2326}
\begin {gather*} \log (x)+2^{-4 e^{-x}} \sqrt [5]{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {e^{-x} \left (5 e^x \log (x)+e^{\frac {1}{5} e^{-x} \left (-20 \log (2)+e^x \log (\log (x))\right )} \left (e^x+20 x \log (2) \log (x)\right )\right )}{x \log (x)} \, dx\\ &=\frac {1}{5} \int \left (\frac {5}{x}+\frac {2^{-4 e^{-x}} e^{-x} \left (e^x+20 x \log (2) \log (x)\right )}{x \log ^{\frac {4}{5}}(x)}\right ) \, dx\\ &=\log (x)+\frac {1}{5} \int \frac {2^{-4 e^{-x}} e^{-x} \left (e^x+20 x \log (2) \log (x)\right )}{x \log ^{\frac {4}{5}}(x)} \, dx\\ &=2^{-4 e^{-x}} \sqrt [5]{\log (x)}+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.29, size = 33, normalized size = 1.74 \begin {gather*} \frac {1}{5} \left (\frac {2^{1-4 e^{-x}} \log (32) \sqrt [5]{\log (x)}}{\log (4)}+5 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 15, normalized size = 0.79
method | result | size |
risch | \(\ln \left (x \right )+\ln \left (x \right )^{\frac {1}{5}} \left (\frac {1}{16}\right )^{{\mathrm e}^{-x}}\) | \(15\) |
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 = 0.31, size = 21, normalized size = 1.11 \begin {gather*} e^{\left (\frac {1}{5} \, {\left (e^{x} \log \left (\log \left (x\right )\right ) - 20 \, \log \left (2\right )\right )} e^{\left (-x\right )}\right )} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 22, normalized size = 1.16 \begin {gather*} e^{\left (\frac {e^{x} \log {\left (\log {\left (x \right )} \right )}}{5} - 4 \log {\left (2 \right )}\right ) e^{- x}} + \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.50, size = 18, normalized size = 0.95 \begin {gather*} \ln \left (x\right )+\frac {{\ln \left (x\right )}^{1/5}}{2^{4\,{\mathrm {e}}^{-x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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