Optimal. Leaf size=33 \[ \frac {\left (x-(x-\log (2))^2 \log ^2(2)\right ) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{10 x} \]
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Rubi [F] time = 1.99, 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 {x-x^2 \log ^2(2)+2 x \log ^3(2)-\log ^4(2)+\left (\left (-x^2-x^3\right ) \log ^2(2)+(1+x) \log ^4(2)\right ) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right ) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{\left (10 x^2+10 x^3\right ) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\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 {-x^2 \log ^2(2)-\log ^4(2)+x \left (1+2 \log ^3(2)\right )+\left (\left (-x^2-x^3\right ) \log ^2(2)+(1+x) \log ^4(2)\right ) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right ) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{\left (10 x^2+10 x^3\right ) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx\\ &=\int \frac {-x^2 \log ^2(2)-\log ^4(2)+x \left (1+2 \log ^3(2)\right )+\left (\left (-x^2-x^3\right ) \log ^2(2)+(1+x) \log ^4(2)\right ) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right ) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{x^2 (10+10 x) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx\\ &=\int \left (\frac {-x^2 \log ^2(2)-\log ^4(2)+x \left (1+2 \log ^3(2)\right )}{10 x^2 (1+x) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )}-\frac {(x-\log (2)) \log ^2(2) (x+\log (2)) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{10 x^2}\right ) \, dx\\ &=\frac {1}{10} \int \frac {-x^2 \log ^2(2)-\log ^4(2)+x \left (1+2 \log ^3(2)\right )}{x^2 (1+x) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx-\frac {1}{10} \log ^2(2) \int \frac {(x-\log (2)) (x+\log (2)) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{10} \int \left (-\frac {\log ^4(2)}{x^2 \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )}+\frac {-1-\log ^2(2)-2 \log ^3(2)-\log ^4(2)}{(1+x) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )}+\frac {1+2 \log ^3(2)+\log ^4(2)}{x \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )}\right ) \, dx-\frac {1}{10} \log ^2(2) \int \left (\log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )-\frac {\log ^2(2) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{x^2}\right ) \, dx\\ &=-\left (\frac {1}{10} \log ^2(2) \int \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right ) \, dx\right )-\frac {1}{10} \log ^4(2) \int \frac {1}{x^2 \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx+\frac {1}{10} \log ^4(2) \int \frac {\log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{x^2} \, dx+\frac {1}{10} \left (-1-\log ^2(2)-2 \log ^3(2)-\log ^4(2)\right ) \int \frac {1}{(1+x) \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx+\frac {1}{10} \left (1+2 \log ^3(2)+\log ^4(2)\right ) \int \frac {1}{x \log \left (\frac {x}{1+x}\right ) \log \left (\log \left (\frac {x}{1+x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 41, normalized size = 1.24 \begin {gather*} -\frac {\left (x^2 \log ^2(2)+\log ^4(2)-x \left (1+2 \log ^3(2)\right )\right ) \log \left (\log \left (\log \left (\frac {x}{1+x}\right )\right )\right )}{10 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 38, normalized size = 1.15 \begin {gather*} -\frac {{\left (x^{2} \log \relax (2)^{2} - 2 \, x \log \relax (2)^{3} + \log \relax (2)^{4} - x\right )} \log \left (\log \left (\log \left (\frac {x}{x + 1}\right )\right )\right )}{10 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 49, normalized size = 1.48 \begin {gather*} \frac {1}{10} \, {\left (2 \, \log \relax (2)^{3} + 1\right )} \log \left (\log \left (-\log \left (x + 1\right ) + \log \relax (x)\right )\right ) - \frac {1}{10} \, {\left (x \log \relax (2)^{2} + \frac {\log \relax (2)^{4}}{x}\right )} \log \left (\log \left (\log \left (\frac {x}{x + 1}\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x +1\right ) \ln \relax (2)^{4}+\left (-x^{3}-x^{2}\right ) \ln \relax (2)^{2}\right ) \ln \left (\frac {x}{x +1}\right ) \ln \left (\ln \left (\frac {x}{x +1}\right )\right ) \ln \left (\ln \left (\ln \left (\frac {x}{x +1}\right )\right )\right )-\ln \relax (2)^{4}+2 x \ln \relax (2)^{3}-x^{2} \ln \relax (2)^{2}+x}{\left (10 x^{3}+10 x^{2}\right ) \ln \left (\frac {x}{x +1}\right ) \ln \left (\ln \left (\frac {x}{x +1}\right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 40, normalized size = 1.21 \begin {gather*} -\frac {{\left (x^{2} \log \relax (2)^{2} + \log \relax (2)^{4} - {\left (2 \, \log \relax (2)^{3} + 1\right )} x\right )} \log \left (\log \left (-\log \left (x + 1\right ) + \log \relax (x)\right )\right )}{10 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 39, normalized size = 1.18 \begin {gather*} \frac {\ln \left (\ln \left (\ln \left (\frac {x}{x+1}\right )\right )\right )\,\left (x-x^2\,{\ln \relax (2)}^2+2\,x\,{\ln \relax (2)}^3-{\ln \relax (2)}^4\right )}{10\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 48, normalized size = 1.45 \begin {gather*} \frac {\left (2 \log {\relax (2 )}^{3} + 1\right ) \log {\left (\log {\left (\log {\left (\frac {x}{x + 1} \right )} \right )} \right )}}{10} + \frac {\left (- x^{2} \log {\relax (2 )}^{2} - \log {\relax (2 )}^{4}\right ) \log {\left (\log {\left (\log {\left (\frac {x}{x + 1} \right )} \right )} \right )}}{10 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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