Optimal. Leaf size=22 \[ \frac {1+\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x} \]
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Rubi [F] time = 31.31, 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+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\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 {2 x^2+4 x^3+2 x \left (1+x+x^2\right ) \log \left (1+x+x^2\right )-\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2 \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx\\ &=\int \left (\frac {2 x^2+4 x^3+2 x \log \left (1+x+x^2\right )+2 x^2 \log \left (1+x+x^2\right )+2 x^3 \log \left (1+x+x^2\right )-e^3 \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-e^3 x \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-e^3 x^2 \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x^2 \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x^3 \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}{x^2 \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}-\frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2}\right ) \, dx\\ &=\int \frac {2 x^2+4 x^3+2 x \log \left (1+x+x^2\right )+2 x^2 \log \left (1+x+x^2\right )+2 x^3 \log \left (1+x+x^2\right )-e^3 \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-e^3 x \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-e^3 x^2 \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x^2 \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )-2 x^3 \log \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}{x^2 \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\int \frac {2 x^2 (1+2 x)-2 x \left (1+x+x^2\right ) \log \left (1+x+x^2\right ) \left (-1+\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )-e^3 \left (1+x+x^2\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}{x^2 \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\int \left (-\frac {1}{x^2}+\frac {2 \left (x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )\right )}{x \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \frac {x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )}{x \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \frac {x (1+2 x)+\left (1+x+x^2\right ) \log \left (1+x+x^2\right )}{x \left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \left (\frac {x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )}{x \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}-\frac {(1+x) \left (x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \frac {x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )}{x \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-2 \int \frac {(1+x) \left (x+2 x^2+\log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+x^2 \log \left (1+x+x^2\right )\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \frac {x (1+2 x)+\left (1+x+x^2\right ) \log \left (1+x+x^2\right )}{x \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-2 \int \frac {(1+x) \left (x (1+2 x)+\left (1+x+x^2\right ) \log \left (1+x+x^2\right )\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \left (\frac {1}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {2 x}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {\log \left (1+x+x^2\right )}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {\log \left (1+x+x^2\right )}{x \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {x \log \left (1+x+x^2\right )}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}\right ) \, dx-2 \int \left (\frac {x}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {3 x^2}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {2 x^3}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {\log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {2 x \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {2 x^2 \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}+\frac {x^3 \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{x}+2 \int \frac {1}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-2 \int \frac {x}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx+2 \int \frac {\log \left (1+x+x^2\right )}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx+2 \int \frac {\log \left (1+x+x^2\right )}{x \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx+2 \int \frac {x \log \left (1+x+x^2\right )}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-2 \int \frac {\log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-2 \int \frac {x^3 \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx+4 \int \frac {x}{\left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-4 \int \frac {x^3}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-4 \int \frac {x \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-4 \int \frac {x^2 \log \left (1+x+x^2\right )}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-6 \int \frac {x^2}{\left (1+x+x^2\right ) \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx-\int \frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 24, normalized size = 1.09 \begin {gather*} \frac {1}{x}+\frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 21, normalized size = 0.95 \begin {gather*} \frac {\log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 1}{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*} \int \frac {4 \, x^{3} - {\left ({\left (x^{2} + x + 1\right )} e^{3} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right ) \log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 2 \, x^{2} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right ) - {\left ({\left (x^{2} + x + 1\right )} e^{3} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )}{{\left ({\left (x^{4} + x^{3} + x^{2}\right )} e^{3} + 2 \, {\left (x^{5} + x^{4} + x^{3}\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.09
method | result | size |
risch | \(\frac {\ln \left (\ln \left (2 x \ln \left (x^{2}+x +1\right )+{\mathrm e}^{3}\right )\right )}{x}+\frac {1}{x}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 21, normalized size = 0.95 \begin {gather*} \frac {\log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 21, normalized size = 0.95 \begin {gather*} \frac {\ln \left (\ln \left ({\mathrm {e}}^3+2\,x\,\ln \left (x^2+x+1\right )\right )\right )+1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.43, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log {\left (\log {\left (2 x \log {\left (x^{2} + x + 1 \right )} + e^{3} \right )} \right )}}{x} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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