Optimal. Leaf size=24 \[ \left (\frac {4}{3} \left (-5+e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \]
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Rubi [F] time = 2.22, 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 {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} (1-\log (x)) \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}-\frac {2 \left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )+\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} (1-\log (x)) \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)} \, dx\\ &=-\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )+\int \left (-\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3}+\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )-\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3} \, dx+\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.76, size = 24, normalized size = 1.00 \begin {gather*} \left (\frac {4}{3} \left (-5+e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 15, normalized size = 0.62 \begin {gather*} \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \relax (x)}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\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 (2 \, \log \relax (x) \log \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \relax (x)}{3 \, x}\right ) + \log \relax (x) - 1\right )} \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \relax (x)}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\right )}}{x^{3} \log \relax (x)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 82, normalized size = 3.42
method | result | size |
risch | \(x^{-\frac {1}{x^{2}}} \ln \relax (x )^{\frac {1}{x^{2}}} \left (\frac {4}{3}\right )^{\frac {1}{x^{2}}} \left ({\mathrm e}^{2}-5\right )^{\frac {1}{x^{2}}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right )}{2 x^{2}}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 39, normalized size = 1.62 \begin {gather*} e^{\left (-\frac {\log \relax (3)}{x^{2}} + \frac {2 \, \log \relax (2)}{x^{2}} - \frac {\log \relax (x)}{x^{2}} + \frac {\log \left (e^{2} - 5\right )}{x^{2}} + \frac {\log \left (\log \relax (x)\right )}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 20, normalized size = 0.83 \begin {gather*} {\left (-\frac {20\,\ln \relax (x)-4\,{\mathrm {e}}^2\,\ln \relax (x)}{3\,x}\right )}^{\frac {1}{x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 20, normalized size = 0.83 \begin {gather*} e^{\frac {\log {\left (\frac {\left (- \frac {20}{3} + \frac {4 e^{2}}{3}\right ) \log {\relax (x )}}{x} \right )}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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