Optimal. Leaf size=34 \[ \frac {5-x}{2 x \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \]
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Rubi [F] time = 4.09, 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 {5-x+\left (5-51 x+20 x^2-2 x^3\right ) \log (x) \log (\log (x))+5 \log (x) \log (\log (x)) \log \left (\frac {1}{5} x \log (\log (x))\right )}{\left (50 x^4-20 x^5+2 x^6\right ) \log (x) \log (\log (x))+\left (-20 x^3+4 x^4\right ) \log (x) \log (\log (x)) \log \left (\frac {1}{5} x \log (\log (x))\right )+2 x^2 \log (x) \log (\log (x)) \log ^2\left (\frac {1}{5} x \log (\log (x))\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 {5-x+\log (x) \log (\log (x)) \left (5-51 x+20 x^2-2 x^3+5 \log \left (\frac {1}{5} x \log (\log (x))\right )\right )}{2 x^2 \log (x) \log (\log (x)) \left ((-5+x) x+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {5-x+\log (x) \log (\log (x)) \left (5-51 x+20 x^2-2 x^3+5 \log \left (\frac {1}{5} x \log (\log (x))\right )\right )}{x^2 \log (x) \log (\log (x)) \left ((-5+x) x+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {(-5+x) \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {5}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {(-5+x) \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\right )+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {5 \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}\right ) \, dx\right )+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\right )+\frac {5}{2} \int \frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\frac {1}{2 \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx+\frac {5}{2} \int \left (\frac {2}{\left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}-\frac {5}{x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}\right ) \, dx\\ &=-\frac {1}{2 \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx+5 \int \frac {1}{\left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx-\frac {25}{2} \int \frac {1}{x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.20, size = 28, normalized size = 0.82 \begin {gather*} \frac {-5+x}{2 x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 25, normalized size = 0.74 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} + x \log \left (\frac {1}{5} \, x \log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 31, normalized size = 0.91 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} - x \log \relax (5) + x \log \relax (x) + x \log \left (\log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 111, normalized size = 3.26
method | result | size |
risch | \(\frac {x -5}{x \left (-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \left (\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{3}+2 x^{2}-2 \ln \relax (5)-10 x +2 \ln \relax (x )+2 \ln \left (\ln \left (\ln \relax (x )\right )\right )\right )}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 31, normalized size = 0.91 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} - x \log \relax (5) + x \log \relax (x) + x \log \left (\log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {x-5\,\ln \left (\ln \relax (x)\right )\,\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )\,\ln \relax (x)+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (2\,x^3-20\,x^2+51\,x-5\right )-5}{\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (2\,x^6-20\,x^5+50\,x^4\right )+2\,x^2\,\ln \left (\ln \relax (x)\right )\,{\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )}^2\,\ln \relax (x)-\ln \left (\ln \relax (x)\right )\,\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )\,\ln \relax (x)\,\left (20\,x^3-4\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 26, normalized size = 0.76 \begin {gather*} \frac {x - 5}{2 x^{3} - 10 x^{2} + 2 x \log {\left (\frac {x \log {\left (\log {\relax (x )} \right )}}{5} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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