Optimal. Leaf size=21 \[ \left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \]
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Rubi [C] time = 0.39, antiderivative size = 178, normalized size of antiderivative = 8.48, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 6719, 2310, 2181, 2366, 6557} \begin {gather*} -\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \log ^{\frac {1}{\log (5)}-1}(5) x^{\frac {1}{\log (5)}} (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )+\left (\frac {139 \log (5)}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (1+\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \log ^{\frac {1}{\log (5)}-1}(5) x^{\frac {1}{\log (5)}} \log ^{1-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2181
Rule 2310
Rule 2366
Rule 6557
Rule 6719
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \int \frac {(1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (x)} \, dx}{\log (5)}\\ &=\frac {\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int x^{-1-\frac {1}{\log (5)}} (1-\log (x)) \log ^{-1+\frac {1}{\log (5)}}(x) \, dx}{\log (5)}\\ &=-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\frac {\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \sqrt [\log (5)]{\log (5)}}{x} \, dx}{\log (5)}\\ &=-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )}{x} \, dx\\ &=-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\right ) \operatorname {Subst}\left (\int \Gamma \left (\frac {1}{\log (5)},\frac {x}{\log (5)}\right ) \, dx,x,\log (x)\right )\\ &=-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) \log ^{1-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}+x^{\frac {1}{\log (5)}} \Gamma \left (1+\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \left (\frac {139 \log (5)}{12}\right )^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}-\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \log ^{-1+\frac {1}{\log (5)}}(5) (1-\log (x)) \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 21, normalized size = 1.00 \begin {gather*} \left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 12, normalized size = 0.57 \begin {gather*} \left (-\frac {139 \, \log \relax (x)}{12 \, x}\right )^{\left (\frac {1}{\log \relax (5)}\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*} \int -\frac {\left (-\frac {139 \, \log \relax (x)}{12 \, x}\right )^{\left (\frac {1}{\log \relax (5)}\right )} {\left (\log \relax (x) - 1\right )}}{x \log \relax (5) \log \relax (x)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.60, size = 15, normalized size = 0.71
method | result | size |
norman | \({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \relax (x )}{12 x}\right )}{\ln \relax (5)}}\) | \(15\) |
risch | \(x^{-\frac {1}{\ln \relax (5)}} \ln \relax (x )^{\frac {1}{\ln \relax (5)}} \left (\frac {1}{4}\right )^{\frac {1}{\ln \relax (5)}} \left (\frac {1}{3}\right )^{\frac {1}{\ln \relax (5)}} 139^{\frac {1}{\ln \relax (5)}} {\mathrm e}^{-\frac {i \pi \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right )+2 \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-2\right )}{2 \ln \relax (5)}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 45, normalized size = 2.14 \begin {gather*} \frac {139^{\left (\frac {1}{\log \relax (5)}\right )} e^{\left (-\frac {\log \relax (x)}{\log \relax (5)} + \frac {\log \left (-\log \relax (x)\right )}{\log \relax (5)}\right )}}{3^{\left (\frac {1}{\log \relax (5)}\right )} 2^{\frac {2}{\log \relax (5)}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 12, normalized size = 0.57 \begin {gather*} {\left (-\frac {139\,\ln \relax (x)}{12\,x}\right )}^{\frac {1}{\ln \relax (5)}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.75, size = 24, normalized size = 1.14 \begin {gather*} \frac {139^{\frac {1}{\log {\relax (5 )}}} \left (- \frac {\log {\relax (x )}}{x}\right )^{\frac {1}{\log {\relax (5 )}}}}{12^{\frac {1}{\log {\relax (5 )}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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