Optimal. Leaf size=22 \[ \log \left (\frac {x}{\log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}\right ) \]
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Rubi [F]
time = 0.68, 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+\left (-4 x-2 x^2\right ) \log (5)+\left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}{\left (\left (-x+4 x^2+x^3\right ) \log (5)+x \log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\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 {1-\frac {2 \left (1+x^2 \log (5)+x \log (25)\right )}{\left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}}{x} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2 \left (1+x^2 \log (5)+x \log (25)\right )}{x \left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}\right ) \, dx\\ &=\log (x)-2 \int \frac {1+x^2 \log (5)+x \log (25)}{x \left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )} \, dx\\ &=\log (x)-2 \int \left (\frac {1}{x \left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}+\frac {x \log (5)}{\left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}+\frac {\log (25)}{\left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}\right ) \, dx\\ &=\log (x)-2 \int \frac {1}{x \left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )} \, dx-(2 \log (5)) \int \frac {x}{\left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )} \, dx-(2 \log (25)) \int \frac {1}{\left (-\log (5)+4 x \log (5)+x^2 \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2+\left (-4 x-2 x^2\right ) \log (5)+\left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )}{\left (\left (-x+4 x^2+x^3\right ) \log (5)+x \log \left (x^2\right )\right ) \log \left (\left (-1+4 x+x^2\right ) \log (5)+\log \left (x^2\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (\ln \left (x^{2}\right )+\left (x^{2}+4 x -1\right ) \ln \left (5\right )\right ) \ln \left (\ln \left (x^{2}\right )+\left (x^{2}+4 x -1\right ) \ln \left (5\right )\right )+\left (-2 x^{2}-4 x \right ) \ln \left (5\right )-2}{\left (x \ln \left (x^{2}\right )+\left (x^{3}+4 x^{2}-x \right ) \ln \left (5\right )\right ) \ln \left (\ln \left (x^{2}\right )+\left (x^{2}+4 x -1\right ) \ln \left (5\right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 27, normalized size = 1.23 \begin {gather*} \log \left (x\right ) - \log \left (\log \left (x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right ) - \log \left (5\right ) + 2 \, \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 = 0.35, size = 27, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, \log \left (x^{2}\right ) - \log \left (\log \left ({\left (x^{2} + 4 \, x - 1\right )} \log \left (5\right ) + \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 = 0.23, size = 22, normalized size = 1.00 \begin {gather*} \log {\left (x \right )} - \log {\left (\log {\left (\left (x^{2} + 4 x - 1\right ) \log {\left (5 \right )} + \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 = 0.44, size = 27, normalized size = 1.23 \begin {gather*} \log \left (x\right ) - \log \left (\log \left (x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right ) - \log \left (5\right ) + \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 = 1.72, size = 23, normalized size = 1.05 \begin {gather*} \ln \left (x\right )-\ln \left (\ln \left (\ln \left (x^2\right )+\ln \left (5\right )\,\left (x^2+4\,x-1\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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