Optimal. Leaf size=18 \[ 2+\frac {1}{3+\frac {3 \log (x) \log (\log (x))}{2+x}} \]
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Rubi [F] time = 1.57, 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 {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \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 {-2-x-(2+x-x \log (x)) \log (\log (x))}{3 x (2+x+\log (x) \log (\log (x)))^2} \, dx\\ &=\frac {1}{3} \int \frac {-2-x-(2+x-x \log (x)) \log (\log (x))}{x (2+x+\log (x) \log (\log (x)))^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {(2+x) (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}+\frac {-2-x+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {(2+x) (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )+\frac {1}{3} \int \frac {-2-x+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {-2-x+\log (x)+x \log (x)}{\log (x) (2+x+\log (x) \log (\log (x)))^2}+\frac {2 (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {1}{2+x+\log (x) \log (\log (x))}-\frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))}-\frac {2}{x \log (x) (2+x+\log (x) \log (\log (x)))}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {-2-x+\log (x)+x \log (x)}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )+\frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \frac {-2-x+\log (x)+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx\\ &=\frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {1}{3} \int \left (\frac {1}{(2+x+\log (x) \log (\log (x)))^2}+\frac {x}{(2+x+\log (x) \log (\log (x)))^2}-\frac {2}{\log (x) (2+x+\log (x) \log (\log (x)))^2}-\frac {x}{\log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \left (\frac {1}{(2+x+\log (x) \log (\log (x)))^2}+\frac {1}{x (2+x+\log (x) \log (\log (x)))^2}-\frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))^2}-\frac {2}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{(2+x+\log (x) \log (\log (x)))^2} \, dx\right )-\frac {1}{3} \int \frac {x}{(2+x+\log (x) \log (\log (x)))^2} \, dx+\frac {1}{3} \int \frac {x}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx+\frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \frac {1}{(2+x+\log (x) \log (\log (x)))^2} \, dx-\frac {2}{3} \int \frac {1}{x (2+x+\log (x) \log (\log (x)))^2} \, dx+2 \left (\frac {2}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx+\frac {4}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 20, normalized size = 1.11 \begin {gather*} -\frac {-2-x}{3 (2+x+\log (x) \log (\log (x)))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 16, normalized size = 0.89 \begin {gather*} \frac {x + 2}{3 \, {\left (\log \relax (x) \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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giac [A] time = 0.97, size = 16, normalized size = 0.89 \begin {gather*} \frac {x + 2}{3 \, {\left (\log \relax (x) \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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maple [A] time = 0.03, size = 17, normalized size = 0.94
method | result | size |
risch | \(\frac {2+x}{3 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+3 x +6}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 16, normalized size = 0.89 \begin {gather*} \frac {x + 2}{3 \, {\left (\log \relax (x) \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 = 6.12, size = 63, normalized size = 3.50 \begin {gather*} \frac {\frac {4\,x}{3}-\ln \relax (x)\,\left (\frac {x^3}{3}+x^2+\frac {2\,x}{3}\right )+\frac {4\,x^2}{3}+\frac {x^3}{3}}{\left (x+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)+2\right )\,\left (2\,x-x^2\,\ln \relax (x)-x\,\ln \relax (x)+x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 17, normalized size = 0.94 \begin {gather*} \frac {x + 2}{3 x + 3 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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