Optimal. Leaf size=27 \[ \frac {2}{x}+x-\log \left (-4+\frac {1}{5} e^{\frac {x}{\log (\log (x))}}+\log (x)\right ) \]
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Rubi [F] time = 4.40, 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 (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\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 \frac {-\left (\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))\right )-e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{x^2 \left (20-e^{\frac {x}{\log (\log (x))}}-5 \log (x)\right ) \log (x) \log ^2(\log (x))} \, dx\\ &=\int \left (\frac {5 \left (4 x-x \log (x)-4 x \log (x) \log (\log (x))+x \log ^2(x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{x \log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}+\frac {x^2-x^2 \log (x) \log (\log (x))-2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log ^2(\log (x))}{x^2 \log (x) \log ^2(\log (x))}\right ) \, dx\\ &=5 \int \frac {4 x-x \log (x)-4 x \log (x) \log (\log (x))+x \log ^2(x) \log (\log (x))-\log (x) \log ^2(\log (x))}{x \log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx+\int \frac {x^2-x^2 \log (x) \log (\log (x))-2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log ^2(\log (x))}{x^2 \log (x) \log ^2(\log (x))} \, dx\\ &=5 \int \left (-\frac {1}{x \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right )}-\frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}+\frac {4}{\log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}-\frac {4}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))}+\frac {\log (x)}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))}\right ) \, dx+\int \left (1-\frac {2}{x^2}+\frac {1}{\log (x) \log ^2(\log (x))}-\frac {1}{\log (\log (x))}\right ) \, dx\\ &=\frac {2}{x}+x-5 \int \frac {1}{x \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right )} \, dx-5 \int \frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx+5 \int \frac {\log (x)}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))} \, dx+20 \int \frac {1}{\log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx-20 \int \frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))} \, dx+\int \frac {1}{\log (x) \log ^2(\log (x))} \, dx-\int \frac {1}{\log (\log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 27, normalized size = 1.00 \begin {gather*} \frac {2}{x}+x-\log \left (20-e^{\frac {x}{\log (\log (x))}}-5 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^{2} - x \log \left (e^{\frac {x}{\log \left (\log \relax (x)\right )}} + 5 \, \log \relax (x) - 20\right ) + 2}{x} \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*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 1.04
method | result | size |
risch | \(\frac {x^{2}+2}{x}-\ln \left (5 \ln \relax (x )+{\mathrm e}^{\frac {x}{\ln \left (\ln \relax (x )\right )}}-20\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^{2} + 2}{x} - \log \left (e^{\frac {x}{\log \left (\log \relax (x)\right )}} + 5 \, \log \relax (x) - 20\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.20, size = 24, normalized size = 0.89 \begin {gather*} x-\ln \left (5\,\ln \relax (x)+{\mathrm {e}}^{\frac {x}{\ln \left (\ln \relax (x)\right )}}-20\right )+\frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 20, normalized size = 0.74 \begin {gather*} x - \log {\left (e^{\frac {x}{\log {\left (\log {\relax (x )} \right )}}} + 5 \log {\relax (x )} - 20 \right )} + \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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