Optimal. Leaf size=26 \[ 4-4 x-\frac {\log (5 x)}{x}+\log \left (\frac {e^3 x^2}{\log ^2(x)}\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6742, 14, 2302, 29, 2304} \begin {gather*} -4 x+2 \log (x)-2 \log (\log (x))-\frac {\log (5 x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 2302
Rule 2304
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-2 x-\log (x)+2 x \log (x)-4 x^2 \log (x)}{x^2 \log (x)}+\frac {\log (5 x)}{x^2}\right ) \, dx\\ &=\int \frac {-2 x-\log (x)+2 x \log (x)-4 x^2 \log (x)}{x^2 \log (x)} \, dx+\int \frac {\log (5 x)}{x^2} \, dx\\ &=-\frac {1}{x}-\frac {\log (5 x)}{x}+\int \left (\frac {-1+2 x-4 x^2}{x^2}-\frac {2}{x \log (x)}\right ) \, dx\\ &=-\frac {1}{x}-\frac {\log (5 x)}{x}-2 \int \frac {1}{x \log (x)} \, dx+\int \frac {-1+2 x-4 x^2}{x^2} \, dx\\ &=-\frac {1}{x}-\frac {\log (5 x)}{x}-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )+\int \left (-4-\frac {1}{x^2}+\frac {2}{x}\right ) \, dx\\ &=-4 x+2 \log (x)-\frac {\log (5 x)}{x}-2 \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.85 \begin {gather*} -4 x+2 \log (x)-\frac {\log (5 x)}{x}-2 \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 28, normalized size = 1.08 \begin {gather*} -\frac {4 \, x^{2} - {\left (2 \, x - 1\right )} \log \relax (x) + 2 \, x \log \left (\log \relax (x)\right ) + \log \relax (5)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 27, normalized size = 1.04 \begin {gather*} -4 \, x - \frac {\log \relax (5)}{x} - \frac {\log \relax (x)}{x} + 2 \, \log \relax (x) - 2 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 27, normalized size = 1.04
method | result | size |
norman | \(\frac {-4 x^{2}-\ln \left (5 x \right )}{x}+2 \ln \relax (x )-2 \ln \left (\ln \relax (x )\right )\) | \(27\) |
default | \(-4 x -\frac {\ln \relax (5)}{x}-\frac {\ln \relax (x )}{x}+2 \ln \relax (x )-2 \ln \left (\ln \relax (x )\right )\) | \(28\) |
risch | \(-\frac {\ln \relax (x )}{x}-\frac {-4 x \ln \relax (x )+8 x^{2}+2 \ln \relax (5)}{2 x}-2 \ln \left (\ln \relax (x )\right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 22, normalized size = 0.85 \begin {gather*} -4 \, x - \frac {\log \left (5 \, x\right )}{x} + 2 \, \log \relax (x) - 2 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 23, normalized size = 0.88 \begin {gather*} 2\,\ln \relax (x)-2\,\ln \left (\ln \relax (x)\right )-4\,x-\frac {\ln \relax (5)+\ln \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 24, normalized size = 0.92 \begin {gather*} - 4 x + 2 \log {\relax (x )} - 2 \log {\left (\log {\relax (x )} \right )} - \frac {\log {\relax (x )}}{x} - \frac {\log {\relax (5 )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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