Optimal. Leaf size=19 \[ 2+3 x-\frac {1}{\log (x (\log (4)+x \log (x)))} \]
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Rubi [A] time = 1.14, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2561, 6691, 6742, 6686} \begin {gather*} 3 x-\frac {1}{\log \left (x^2 \log (x)+x \log (4)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2561
Rule 6686
Rule 6691
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x+\log (4)+2 x \log (x)+\left (3 x \log (4)+3 x^2 \log (x)\right ) \log ^2\left (x \log (4)+x^2 \log (x)\right )}{x^2 \left (\frac {\log (4)}{x}+\log (x)\right ) \log ^2\left (x \log (4)+x^2 \log (x)\right )} \, dx\\ &=\int \frac {x+\log (4)+2 x \log (x)+\left (3 x \log (4)+3 x^2 \log (x)\right ) \log ^2\left (x \log (4)+x^2 \log (x)\right )}{x (\log (4)+x \log (x)) \log ^2\left (x \log (4)+x^2 \log (x)\right )} \, dx\\ &=\int \left (3+\frac {x+\log (4)+2 x \log (x)}{x (\log (4)+x \log (x)) \log ^2\left (x \log (4)+x^2 \log (x)\right )}\right ) \, dx\\ &=3 x+\int \frac {x+\log (4)+2 x \log (x)}{x (\log (4)+x \log (x)) \log ^2\left (x \log (4)+x^2 \log (x)\right )} \, dx\\ &=3 x-\frac {1}{\log \left (x \log (4)+x^2 \log (x)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 18, normalized size = 0.95 \begin {gather*} 3 x-\frac {1}{\log (x (\log (4)+x \log (x)))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 34, normalized size = 1.79 \begin {gather*} \frac {3 \, x \log \left (x^{2} \log \relax (x) + 2 \, x \log \relax (2)\right ) - 1}{\log \left (x^{2} \log \relax (x) + 2 \, x \log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 21, normalized size = 1.11 \begin {gather*} 3 \, x - \frac {1}{\log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 130, normalized size = 6.84
method | result | size |
risch | \(3 x -\frac {2 i}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )\right )^{3}+2 i \ln \relax (x )+2 i \ln \left (\frac {x \ln \relax (x )}{2}+\ln \relax (2)\right )}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 36, normalized size = 1.89 \begin {gather*} \frac {3 \, x \log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + 3 \, x \log \relax (x) - 1}{\log \left (x \log \relax (x) + 2 \, \log \relax (2)\right ) + \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 21, normalized size = 1.11 \begin {gather*} 3\,x-\frac {1}{\ln \left (x^2\,\ln \relax (x)+2\,x\,\ln \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} 3 x - \frac {1}{\log {\left (x^{2} \log {\relax (x )} + 2 x \log {\relax (2 )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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