Optimal. Leaf size=21 \[ \log \left (-x+\frac {-4+x^4}{5 \log \left (\log \left (x^2\right )\right )}\right ) \]
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Rubi [F]
time = 1.32, 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^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (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 {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \left (4-x^4+5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\frac {-10 x+4 \log \left (x^2\right )+3 x^4 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx\\ &=\log (x)-2 \int \frac {1}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\int \frac {-10 x+4 \log \left (x^2\right )+3 x^4 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\log (x)+\int \left (\frac {4}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}+\frac {3 x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )}-\frac {10}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\log \left (x^2\right )\right )\\ &=\log (x)+3 \int \frac {x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )} \, dx+4 \int \frac {1}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-10 \int \frac {1}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\log \left (x^2\right )\right )\right )\\ &=\log (x)-\log \left (\log \left (\log \left (x^2\right )\right )\right )+3 \int \frac {x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )} \, dx+4 \int \frac {1}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-10 \int \frac {1}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 25, normalized size = 1.19 \begin {gather*} -\log \left (\log \left (\log \left (x^2\right )\right )\right )+\log \left (4-x^4+5 x \log \left (\log \left (x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {5 x \ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )^{2}-4 x^{4} \ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )+2 x^{4}-8}{5 x^{2} \ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )^{2}+\left (-x^{5}+4 x \right ) \ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 34, normalized size = 1.62 \begin {gather*} \log \left (x\right ) + \log \left (-\frac {x^{4} - 5 \, x \log \left (2\right ) - 5 \, x \log \left (\log \left (x\right )\right ) - 4}{5 \, x}\right ) - \log \left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 34, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, \log \left (x^{2}\right ) + \log \left (-\frac {x^{4} - 5 \, x \log \left (\log \left (x^{2}\right )\right ) - 4}{x}\right ) - \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 29, normalized size = 1.38 \begin {gather*} \log {\left (x \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} + \frac {8 - 2 x^{4}}{10 x} \right )} - \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 25, normalized size = 1.19 \begin {gather*} \log \left (-x^{4} + 5 \, x \log \left (\log \left (x^{2}\right )\right ) + 4\right ) - \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.73, size = 78, normalized size = 3.71 \begin {gather*} \ln \left (\frac {20\,x\,\ln \left (\ln \left (x^2\right )\right )-4\,x^4+16}{\ln \left (x^2\right )}\right )-\ln \left (\frac {\ln \left (\ln \left (x^2\right )\right )\,\left (4\,\ln \left (x^2\right )-10\,x+3\,x^4\,\ln \left (x^2\right )\right )}{\ln \left (x^2\right )}\right )+\ln \left (4\,\ln \left (x^2\right )-10\,x+3\,x^4\,\ln \left (x^2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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