Optimal. Leaf size=19 \[ 2+3 x-\frac {1}{\log (x (\log (4)+x \log (x)))} \]
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Rubi [A]
time = 0.58, 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 = {2641, 6823,
6874, 6818} \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 2641
Rule 6818
Rule 6823
Rule 6874
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 = 0.08, 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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Maple [A]
time = 0.39, size = 22, normalized size = 1.16
method | result | size |
default | \(3 x -\frac {1}{\ln \left (x^{2} \ln \left (x \right )+2 x \ln \left (2\right )\right )}\) | \(22\) |
risch | \(3 x -\frac {2 i}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right ) \mathrm {csgn}\left (i x \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )\right )^{3}+2 i \ln \left (x \right )+2 i \ln \left (\frac {x \ln \left (x \right )}{2}+\ln \left (2\right )\right )}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 36, normalized size = 1.89 \begin {gather*} \frac {3 \, x \log \left (x \log \left (x\right ) + 2 \, \log \left (2\right )\right ) + 3 \, x \log \left (x\right ) - 1}{\log \left (x \log \left (x\right ) + 2 \, \log \left (2\right )\right ) + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 1.79 \begin {gather*} \frac {3 \, x \log \left (x^{2} \log \left (x\right ) + 2 \, x \log \left (2\right )\right ) - 1}{\log \left (x^{2} \log \left (x\right ) + 2 \, x \log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 19, normalized size = 1.00 \begin {gather*} 3 x - \frac {1}{\log {\left (x^{2} \log {\left (x \right )} + 2 x \log {\left (2 \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 21, normalized size = 1.11 \begin {gather*} 3 \, x - \frac {1}{\log \left (x \log \left (x\right ) + 2 \, \log \left (2\right )\right ) + \log \left (x\right )} \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 \left (x\right )+2\,x\,\ln \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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