Optimal. Leaf size=27 \[ \frac {2}{9} x \left (-\frac {1}{3}+x-\frac {x^2}{\log \left (\frac {e^x}{x}\right )}\right )^2 \]
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Rubi [F]
time = 0.42, 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 {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{\log ^3\left (\frac {e^x}{x}\right )} \, dx\\ &=\frac {1}{81} \int \left (2 \left (1-12 x+27 x^2\right )-\frac {36 (-1+x) x^4}{\log ^3\left (\frac {e^x}{x}\right )}+\frac {6 x^2 \left (2-8 x+21 x^2\right )}{\log ^2\left (\frac {e^x}{x}\right )}-\frac {36 x^2 (-1+4 x)}{\log \left (\frac {e^x}{x}\right )}\right ) \, dx\\ &=\frac {2}{81} \int \left (1-12 x+27 x^2\right ) \, dx+\frac {2}{27} \int \frac {x^2 \left (2-8 x+21 x^2\right )}{\log ^2\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {(-1+x) x^4}{\log ^3\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {x^2 (-1+4 x)}{\log \left (\frac {e^x}{x}\right )} \, dx\\ &=\frac {2 x}{81}-\frac {4 x^2}{27}+\frac {2 x^3}{9}+\frac {2}{27} \int \left (\frac {2 x^2}{\log ^2\left (\frac {e^x}{x}\right )}-\frac {8 x^3}{\log ^2\left (\frac {e^x}{x}\right )}+\frac {21 x^4}{\log ^2\left (\frac {e^x}{x}\right )}\right ) \, dx-\frac {4}{9} \int \left (-\frac {x^4}{\log ^3\left (\frac {e^x}{x}\right )}+\frac {x^5}{\log ^3\left (\frac {e^x}{x}\right )}\right ) \, dx-\frac {4}{9} \int \left (-\frac {x^2}{\log \left (\frac {e^x}{x}\right )}+\frac {4 x^3}{\log \left (\frac {e^x}{x}\right )}\right ) \, dx\\ &=\frac {2 x}{81}-\frac {4 x^2}{27}+\frac {2 x^3}{9}+\frac {4}{27} \int \frac {x^2}{\log ^2\left (\frac {e^x}{x}\right )} \, dx+\frac {4}{9} \int \frac {x^4}{\log ^3\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {x^5}{\log ^3\left (\frac {e^x}{x}\right )} \, dx+\frac {4}{9} \int \frac {x^2}{\log \left (\frac {e^x}{x}\right )} \, dx-\frac {16}{27} \int \frac {x^3}{\log ^2\left (\frac {e^x}{x}\right )} \, dx+\frac {14}{9} \int \frac {x^4}{\log ^2\left (\frac {e^x}{x}\right )} \, dx-\frac {16}{9} \int \frac {x^3}{\log \left (\frac {e^x}{x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs.
\(2(22)=44\).
time = 1.02, size = 197, normalized size = 7.30
method | result | size |
default | \(\frac {\frac {2 x \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \left (x \right )\right )^{2}}{81}+\frac {2 \left (9 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \left (x \right )\right )^{2}-6 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+6 x -6 \ln \left (x \right )+1\right ) x^{3}}{81}+\frac {2 \left (-6 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \left (x \right )\right )^{2}+2 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-2 x +2 \ln \left (x \right )\right ) x^{2}}{81}+\frac {2 x \ln \left (x \right )^{2}}{81}+\frac {2 \left (6-18 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+18 x -18 \ln \left (x \right )\right ) \ln \left (x \right ) x^{3}}{81}-\frac {4 \ln \left (x \right ) x \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \left (x \right )\right )}{81}+\frac {2 \left (12 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-12 x +12 \ln \left (x \right )-2\right ) \ln \left (x \right ) x^{2}}{81}-\frac {4 x^{2} \ln \left (x \right )^{2}}{27}+\frac {2 x^{3} \ln \left (x \right )^{2}}{9}}{\ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}}\) | \(197\) |
risch | \(\frac {2 x^{3}}{9}-\frac {4 x^{2}}{27}+\frac {2 x}{81}+\frac {8 x^{3} \left (3 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-3 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-3 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+3 i \pi x \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+3 x^{2}+6 x \ln \left (x \right )-6 x \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{x}\right )\right )}{27 \left (2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x}\right )+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )\right )^{2}}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (24) = 48\).
time = 0.31, size = 57, normalized size = 2.11 \begin {gather*} \frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 2 \, x^{4} - 2 \, {\left (3 \, x^{4} - x^{3}\right )} \log \left (x\right )\right )}}{27 \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (24) = 48\).
time = 0.38, size = 59, normalized size = 2.19 \begin {gather*} \frac {2 \, {\left (9 \, x^{5} + {\left (9 \, x^{3} - 6 \, x^{2} + x\right )} \log \left (\frac {e^{x}}{x}\right )^{2} - 6 \, {\left (3 \, x^{4} - x^{3}\right )} \log \left (\frac {e^{x}}{x}\right )\right )}}{81 \, \log \left (\frac {e^{x}}{x}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (20) = 40\).
time = 0.06, size = 49, normalized size = 1.81 \begin {gather*} \frac {2 x^{3}}{9} - \frac {4 x^{2}}{27} + \frac {2 x}{81} + \frac {6 x^{5} + \left (- 12 x^{4} + 4 x^{3}\right ) \log {\left (\frac {e^{x}}{x} \right )}}{27 \log {\left (\frac {e^{x}}{x} \right )}^{2}} \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. 56 vs.
\(2 (24) = 48\).
time = 0.39, size = 56, normalized size = 2.07 \begin {gather*} \frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 6 \, x^{4} \log \left (x\right ) - 2 \, x^{4} + 2 \, x^{3} \log \left (x\right )\right )}}{27 \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 303, normalized size = 11.22 \begin {gather*} \frac {245\,x}{81}+\frac {2\,\ln \left (x\right )}{3}+\frac {\frac {x\,\left (9\,x^4+8\,x^3-2\,x^2\right )}{27\,\left (x-1\right )}+\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (-12\,x^3+7\,x^2+12\,x-2\right )}{9\,{\left (x-1\right )}^3}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x^2-6\,x+1\right )}{3\,{\left (x-1\right )}^3}}{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}+\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (\frac {24\,x^5-92\,x^4+128\,x^3-\frac {230\,x^2}{3}+18\,x-\frac {2}{3}}{x^3-3\,x^2+3\,x-1}-\frac {\frac {80\,x^5}{3}-96\,x^4+128\,x^3-\frac {224\,x^2}{3}+16\,x}{x^3-3\,x^2+3\,x-1}\right )+\frac {\frac {38\,x^2}{9}-9\,x+\frac {38}{9}}{x^3-3\,x^2+3\,x-1}+\frac {\frac {2\,x^5}{9}-\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (21\,x^2-8\,x+2\right )}{27\,\left (x-1\right )}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x-1\right )}{9\,\left (x-1\right )}}{{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2}+\frac {65\,x^2}{27}+\frac {2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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