Optimal. Leaf size=17 \[ 2+\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \]
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Rubi [F] time = 0.42, 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 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-1+\log (x)}{\log ^2(x)}-\frac {2}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )}-\frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \frac {-1+\log (x)}{\log ^2(x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=\frac {x}{\log (x)}+\text {li}(x)-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=\frac {x}{\log (x)}-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} \frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 44, normalized size = 2.59 \begin {gather*} \frac {x \log \left (\frac {5}{4}\right )^{2} + 2 \, x \log \left (\frac {5}{4}\right ) \log \relax (x) + x \log \relax (x)^{2} + 1}{\log \left (\frac {5}{4}\right )^{2} \log \relax (x) + 2 \, \log \left (\frac {5}{4}\right ) \log \relax (x)^{2} + \log \relax (x)^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 167, normalized size = 9.82 \begin {gather*} \frac {x \log \relax (5)^{2} - 4 \, x \log \relax (5) \log \relax (2) + 4 \, x \log \relax (2)^{2} + 1}{\log \relax (5)^{2} \log \relax (x) - 4 \, \log \relax (5) \log \relax (2) \log \relax (x) + 4 \, \log \relax (2)^{2} \log \relax (x)} - \frac {2 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)}{\log \relax (5)^{4} - 8 \, \log \relax (5)^{3} \log \relax (2) + 24 \, \log \relax (5)^{2} \log \relax (2)^{2} - 32 \, \log \relax (5) \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + 2 \, \log \relax (5)^{3} \log \relax (x) - 12 \, \log \relax (5)^{2} \log \relax (2) \log \relax (x) + 24 \, \log \relax (5) \log \relax (2)^{2} \log \relax (x) - 16 \, \log \relax (2)^{3} \log \relax (x) + \log \relax (5)^{2} \log \relax (x)^{2} - 4 \, \log \relax (5) \log \relax (2) \log \relax (x)^{2} + 4 \, \log \relax (2)^{2} \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 65, normalized size = 3.82
method | result | size |
risch | \(\frac {4-16 x \ln \relax (2) \ln \relax (x )-16 x \ln \relax (2) \ln \relax (5)+16 x \ln \relax (2)^{2}+4 x \ln \relax (x )^{2}+4 x \ln \relax (5)^{2}+8 x \ln \relax (5) \ln \relax (x )}{\left (2 \ln \relax (5)-4 \ln \relax (2)+2 \ln \relax (x )\right )^{2} \ln \relax (x )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.64, size = 186, normalized size = 10.94 \begin {gather*} -\frac {3 \, \log \relax (5) - 6 \, \log \relax (2) + 2 \, \log \relax (x)}{\log \relax (5)^{4} - 8 \, \log \relax (5)^{3} \log \relax (2) + 24 \, \log \relax (5)^{2} \log \relax (2)^{2} - 32 \, \log \relax (5) \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + {\left (\log \relax (5)^{2} - 4 \, \log \relax (5) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (\log \relax (5)^{3} - 6 \, \log \relax (5)^{2} \log \relax (2) + 12 \, \log \relax (5) \log \relax (2)^{2} - 8 \, \log \relax (2)^{3}\right )} \log \relax (x)} + \frac {\log \relax (5) - 2 \, \log \relax (2) + 2 \, \log \relax (x)}{{\left (\log \relax (5)^{2} - 4 \, \log \relax (5) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2} + {\left (\log \relax (5)^{3} - 6 \, \log \relax (5)^{2} \log \relax (2) + 12 \, \log \relax (5) \log \relax (2)^{2} - 8 \, \log \relax (2)^{3}\right )} \log \relax (x)} + {\rm Ei}\left (\log \relax (x)\right ) - \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 21, normalized size = 1.24 \begin {gather*} \frac {x\,{\ln \left (\frac {5\,x}{4}\right )}^2+1}{{\ln \left (\frac {5\,x}{4}\right )}^2\,\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.43, size = 94, normalized size = 5.53 \begin {gather*} \frac {x \log {\relax (x )}^{2} - 4 x \log {\relax (2 )} \log {\relax (5 )} + 4 x \log {\relax (2 )}^{2} + x \log {\relax (5 )}^{2} + \left (- 4 x \log {\relax (2 )} + 2 x \log {\relax (5 )}\right ) \log {\relax (x )} + 1}{\log {\relax (x )}^{3} + \left (- 4 \log {\relax (2 )} + 2 \log {\relax (5 )}\right ) \log {\relax (x )}^{2} + \left (- 4 \log {\relax (2 )} \log {\relax (5 )} + 4 \log {\relax (2 )}^{2} + \log {\relax (5 )}^{2}\right ) \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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