Optimal. Leaf size=29 \[ \frac {2 e^{e^8} x}{\left (5-e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \]
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Rubi [F]
time = 5.51, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{e^8} \left (20-4 e^x\right )+e^{e^8} \left (-20+4 e^x\right ) \log (x)+e^{e^8} \left (10+e^x (-2+2 x)\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}{\left (25-10 e^x+e^{2 x}\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^8} \left (20-4 e^x\right )+e^{e^8} \left (-20+4 e^x\right ) \log (x)+e^{e^8} \left (10+e^x (-2+2 x)\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}{\left (5-e^x\right )^2 \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\\ &=\int \left (\frac {10 e^{e^8} x}{\left (-5+e^x\right )^2 \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}+\frac {2 e^{e^8} \left (-2+2 \log (x)-\log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )+x \log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )\right )}{\left (-5+e^x\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}\right ) \, dx\\ &=\left (2 e^{e^8}\right ) \int \frac {-2+2 \log (x)-\log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )+x \log (x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}{\left (-5+e^x\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx+\left (10 e^{e^8}\right ) \int \frac {x}{\left (-5+e^x\right )^2 \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\\ &=\left (2 e^{e^8}\right ) \int \frac {2-\log (x) \left (2+(-1+x) \log \left (\frac {x}{\log (x)}\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )\right )}{\left (5-e^x\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx+\left (10 e^{e^8}\right ) \int \frac {x}{\left (-5+e^x\right )^2 \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\\ &=\left (2 e^{e^8}\right ) \int \left (\frac {2}{\left (-5+e^x\right ) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}-\frac {2}{\left (-5+e^x\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}-\frac {1}{\left (-5+e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}+\frac {x}{\left (-5+e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )}\right ) \, dx+\left (10 e^{e^8}\right ) \int \frac {x}{\left (-5+e^x\right )^2 \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\\ &=-\left (\left (2 e^{e^8}\right ) \int \frac {1}{\left (-5+e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\right )+\left (2 e^{e^8}\right ) \int \frac {x}{\left (-5+e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx+\left (4 e^{e^8}\right ) \int \frac {1}{\left (-5+e^x\right ) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx-\left (4 e^{e^8}\right ) \int \frac {1}{\left (-5+e^x\right ) \log (x) \log \left (\frac {x}{\log (x)}\right ) \log ^2\left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx+\left (10 e^{e^8}\right ) \int \frac {x}{\left (-5+e^x\right )^2 \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.21, size = 27, normalized size = 0.93 \begin {gather*} -\frac {2 e^{e^8} x}{\left (-5+e^x\right ) \log \left (\log ^2\left (\frac {x}{\log (x)}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.25, size = 710, normalized size = 24.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(710\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 23, normalized size = 0.79 \begin {gather*} -\frac {x e^{\left (e^{8}\right )}}{{\left (e^{x} - 5\right )} \log \left (\log \left (x\right ) - \log \left (\log \left (x\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 24, normalized size = 0.83 \begin {gather*} -\frac {2 \, x e^{\left (e^{8}\right )}}{{\left (e^{x} - 5\right )} \log \left (\log \left (\frac {x}{\log \left (x\right )}\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 34, normalized size = 1.17 \begin {gather*} - \frac {2 x e^{e^{8}}}{e^{x} \log {\left (\log {\left (\frac {x}{\log {\left (x \right )}} \right )}^{2} \right )} - 5 \log {\left (\log {\left (\frac {x}{\log {\left (x \right )}} \right )}^{2} \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. 50 vs.
\(2 (24) = 48\).
time = 37.57, size = 50, normalized size = 1.72 \begin {gather*} -\frac {2 \, x e^{\left (e^{8}\right )}}{e^{x} \log \left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right ) - 5 \, \log \left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.32, size = 24, normalized size = 0.83 \begin {gather*} -\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^8}}{\ln \left ({\ln \left (\frac {x}{\ln \left (x\right )}\right )}^2\right )\,\left ({\mathrm {e}}^x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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