Integrand size = 16, antiderivative size = 9 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=x \log ^{-\log (\pi )}(x) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 9.67, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2336, 2212, 2408, 19, 6692} \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=(-\log (x))^{1+\log (\pi )} \log \left (\frac {x}{\pi }\right ) \log ^{-\log (e \pi )}(x) \Gamma (-\log (\pi ),-\log (x))+(-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x) \Gamma (1-\log (\pi ),-\log (x))+(-\log (x))^{\log (\pi )} \log ^{1-\log (\pi )}(x) \Gamma (-\log (\pi ),-\log (x)) \]
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Rule 19
Rule 2212
Rule 2336
Rule 2408
Rule 6692
Rubi steps \begin{align*} \text {integral}& = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\int \frac {\Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)}{x} \, dx \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\left ((-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)\right ) \int \frac {\Gamma (-\log (\pi ),-\log (x))}{x} \, dx \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right )+\left ((-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)\right ) \text {Subst}(\int \Gamma (-\log (\pi ),-x) \, dx,x,\log (x)) \\ & = \Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{1-\log (\pi )}(x)+\Gamma (1-\log (\pi ),-\log (x)) (-\log (x))^{\log (\pi )} \log ^{-\log (\pi )}(x)+\Gamma (-\log (\pi ),-\log (x)) (-\log (x))^{1+\log (\pi )} \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \\ \end{align*}
\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx \]
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\[\int \ln \left (\frac {x}{\pi }\right ) \ln \left (x \right )^{-\ln \left (\pi \,{\mathrm e}\right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (9) = 18\).
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 3.33 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\frac {x \log \left (\pi \right ) + x \log \left (\frac {x}{\pi }\right )}{{\left (\log \left (\pi \right ) + \log \left (\frac {x}{\pi }\right )\right )}^{\log \left (\pi \right ) + 1}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (7) = 14\).
Time = 18.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 6.22 \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=- \frac {\left (- \log {\left (x \right )}\right )^{1 + \log {\left (\pi \right )}} \log {\left (\pi \right )} \Gamma \left (- \log {\left (\pi \right )}, - \log {\left (x \right )}\right )}{\log {\left (x \right )}^{1 + \log {\left (\pi \right )}}} + \frac {\left (- \log {\left (x \right )}\right )^{\log {\left (\pi \right )}} \Gamma \left (1 - \log {\left (\pi \right )}, - \log {\left (x \right )}\right )}{\log {\left (x \right )}^{\log {\left (\pi \right )}}} \]
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\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int { \frac {\log \left (\frac {x}{\pi }\right )}{\log \left (x\right )^{\log \left (\pi e\right )}} \,d x } \]
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\[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int { \frac {\log \left (\frac {x}{\pi }\right )}{\log \left (x\right )^{\log \left (\pi e\right )}} \,d x } \]
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Timed out. \[ \int \log ^{-\log (e \pi )}(x) \log \left (\frac {x}{\pi }\right ) \, dx=\int \frac {\ln \left (\frac {x}{\Pi }\right )}{{\ln \left (x\right )}^{\ln \left (\Pi \,\mathrm {e}\right )}} \,d x \]
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