Optimal. Leaf size=34 \[ \frac {2 x}{x^2 \left (e^5-e^8 x^2\right )-\log \left ((x-\log (\log (x)))^2\right )} \]
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Rubi [F]
time = 2.16, 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 {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\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 {-4+2 \log (x) \left (x \left (2-e^5 x^2+3 e^8 x^4-\log \left ((x-\log (\log (x)))^2\right )\right )+\log (\log (x)) \left (e^5 x^2-3 e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )\right )}{\log (x) (x-\log (\log (x))) \left (e^5 x^2-e^8 x^4-\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx\\ &=\int \left (\frac {4 \left (-1+x \log (x)-e^5 x^3 \log (x)+2 e^8 x^5 \log (x)+e^5 x^2 \log (x) \log (\log (x))-2 e^8 x^4 \log (x) \log (\log (x))\right )}{\log (x) (x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}-\frac {2}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )} \, dx\right )+4 \int \frac {-1+x \log (x)-e^5 x^3 \log (x)+2 e^8 x^5 \log (x)+e^5 x^2 \log (x) \log (\log (x))-2 e^8 x^4 \log (x) \log (\log (x))}{\log (x) (x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )} \, dx\right )+4 \int \left (\frac {x}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}-\frac {e^5 x^3}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}+\frac {2 e^8 x^5}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}-\frac {1}{\log (x) (x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}+\frac {e^5 x^2 \log (\log (x))}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}-\frac {2 e^8 x^4 \log (\log (x))}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )} \, dx\right )+4 \int \frac {x}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx-4 \int \frac {1}{\log (x) (x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx-\left (4 e^5\right ) \int \frac {x^3}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx+\left (4 e^5\right ) \int \frac {x^2 \log (\log (x))}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx+\left (8 e^8\right ) \int \frac {x^5}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx-\left (8 e^8\right ) \int \frac {x^4 \log (\log (x))}{(x-\log (\log (x))) \left (-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 31, normalized size = 0.91 \begin {gather*} -\frac {2 x}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.42, size = 110, normalized size = 3.24
method | result | size |
risch | \(-\frac {4 x}{2 x^{4} {\mathrm e}^{8}-2 x^{2} {\mathrm e}^{5}-i \pi \mathrm {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right )^{2} \mathrm {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right ) \mathrm {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{3}+4 \ln \left (x -\ln \left (\ln \left (x \right )\right )\right )}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 29, normalized size = 0.85 \begin {gather*} -\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + 2 \, \log \left (-x + \log \left (\log \left (x\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 1.03 \begin {gather*} -\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + \log \left (x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 37, normalized size = 1.09 \begin {gather*} - \frac {2 x}{x^{4} e^{8} - x^{2} e^{5} + \log {\left (x^{2} - 2 x \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} \right )}^{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 61.91, size = 35, normalized size = 1.03 \begin {gather*} -\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + \log \left (x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\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 {\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )\,\left (2\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-2\,x\,\ln \left (x\right )\right )+\ln \left (x\right )\,\left (6\,{\mathrm {e}}^8\,x^5-2\,{\mathrm {e}}^5\,x^3+4\,x\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^5-6\,x^4\,{\mathrm {e}}^8\right )-4}{\left (\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-x\,\ln \left (x\right )\right )\,{\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )}^2+\left (\ln \left (x\right )\,\left (2\,x^3\,{\mathrm {e}}^5-2\,x^5\,{\mathrm {e}}^8\right )-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^5-2\,x^4\,{\mathrm {e}}^8\right )\right )\,\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^{16}\,x^9-2\,{\mathrm {e}}^{13}\,x^7+{\mathrm {e}}^{10}\,x^5\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left ({\mathrm {e}}^{16}\,x^8-2\,{\mathrm {e}}^{13}\,x^6+{\mathrm {e}}^{10}\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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