Optimal. Leaf size=23 \[ \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x} \]
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Rubi [F] time = 1.70, 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 {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\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 {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{x^2 \left (-4-x+x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx\\ &=\int \frac {-8+2 x-6 x^2-\left (4+x-x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right ) \log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2 \left (4+x-x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\\ &=\int \left (\frac {2 \left (4-x+3 x^2\right )}{x^2 \left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}-\frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {4-x+3 x^2}{x^2 \left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=2 \int \left (-\frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}+\frac {1}{2 x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}+\frac {9-x}{2 \left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\right )+\int \frac {1}{x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx+\int \frac {9-x}{\left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\right )+\int \left (\frac {9}{\left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}-\frac {x}{\left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}\right ) \, dx+\int \frac {1}{x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\right )+9 \int \frac {1}{\left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx+\int \frac {1}{x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {x}{\left (-4-x+x^2\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\right )+9 \int \left (-\frac {2}{\sqrt {17} \left (1+\sqrt {17}-2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}-\frac {2}{\sqrt {17} \left (-1+\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}\right ) \, dx-\int \left (\frac {1+\frac {1}{\sqrt {17}}}{\left (-1-\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}+\frac {1-\frac {1}{\sqrt {17}}}{\left (-1+\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )}\right ) \, dx+\int \frac {1}{x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {1}{x^2 \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx\right )-\frac {18 \int \frac {1}{\left (1+\sqrt {17}-2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx}{\sqrt {17}}-\frac {18 \int \frac {1}{\left (-1+\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx}{\sqrt {17}}-\frac {1}{17} \left (17-\sqrt {17}\right ) \int \frac {1}{\left (-1+\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\frac {1}{17} \left (17+\sqrt {17}\right ) \int \frac {1}{\left (-1-\sqrt {17}+2 x\right ) \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx+\int \frac {1}{x \log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )} \, dx-\int \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 49, normalized size = 2.13 \begin {gather*} \frac {\log \left (\log \left (\frac {x^{8} - 4 \, x^{7} - 10 \, x^{6} + 44 \, x^{5} + 49 \, x^{4} - 176 \, x^{3} - 160 \, x^{2} + 256 \, x + 256}{128 \, x^{2}}\right )\right )}{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 {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{2}+x +4\right ) \ln \left (\frac {x^{8}-4 x^{7}-10 x^{6}+44 x^{5}+49 x^{4}-176 x^{3}-160 x^{2}+256 x +256}{128 x^{2}}\right ) \ln \left (\ln \left (\frac {x^{8}-4 x^{7}-10 x^{6}+44 x^{5}+49 x^{4}-176 x^{3}-160 x^{2}+256 x +256}{128 x^{2}}\right )\right )+6 x^{2}-2 x +8}{\left (x^{4}-x^{3}-4 x^{2}\right ) \ln \left (\frac {x^{8}-4 x^{7}-10 x^{6}+44 x^{5}+49 x^{4}-176 x^{3}-160 x^{2}+256 x +256}{128 x^{2}}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left (-7 \, \log \relax (2) + 4 \, \log \left (x^{2} - x - 4\right ) - 2 \, \log \relax (x)\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 49, normalized size = 2.13 \begin {gather*} \frac {\ln \left (\ln \left (\frac {x^8-4\,x^7-10\,x^6+44\,x^5+49\,x^4-176\,x^3-160\,x^2+256\,x+256}{128\,x^2}\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 56, normalized size = 2.43 \begin {gather*} \frac {\log {\left (\log {\left (\frac {\frac {x^{8}}{128} - \frac {x^{7}}{32} - \frac {5 x^{6}}{64} + \frac {11 x^{5}}{32} + \frac {49 x^{4}}{128} - \frac {11 x^{3}}{8} - \frac {5 x^{2}}{4} + 2 x + 2}{x^{2}} \right )} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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