Optimal. Leaf size=20 \[ \log (x) \left (x^4+\log \left (2+\frac {1}{4+\frac {x}{3}}+x\right )\right ) \]
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Rubi [A]
time = 0.62, antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps
used = 28, number of rules used = 7, integrand size = 103, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6820, 2404,
2341, 2354, 2438, 14, 2604} \begin {gather*} x^4 \log (x)+\log (x) \log \left (\frac {x^2+14 x+27}{x+12}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2341
Rule 2354
Rule 2404
Rule 2438
Rule 2604
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\left (141+24 x+x^2+1296 x^3+780 x^4+104 x^5+4 x^6\right ) \log (x)}{324+195 x+26 x^2+x^3}+\frac {x^4+\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x}\right ) \, dx\\ &=\int \frac {\left (141+24 x+x^2+1296 x^3+780 x^4+104 x^5+4 x^6\right ) \log (x)}{324+195 x+26 x^2+x^3} \, dx+\int \frac {x^4+\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x} \, dx\\ &=\int \left (4 x^3 \log (x)+\frac {\left (141+24 x+x^2\right ) \log (x)}{324+195 x+26 x^2+x^3}\right ) \, dx+\int \left (x^3+\frac {\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x}\right ) \, dx\\ &=\frac {x^4}{4}+4 \int x^3 \log (x) \, dx+\int \frac {\left (141+24 x+x^2\right ) \log (x)}{324+195 x+26 x^2+x^3} \, dx+\int \frac {\log \left (\frac {27+14 x+x^2}{12+x}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-\int \frac {(12+x) \left (\frac {14+2 x}{12+x}-\frac {27+14 x+x^2}{(12+x)^2}\right ) \log (x)}{27+14 x+x^2} \, dx+\int \left (\frac {\log (x)}{-12-x}+\frac {2 (7+x) \log (x)}{27+14 x+x^2}\right ) \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )+2 \int \frac {(7+x) \log (x)}{27+14 x+x^2} \, dx+\int \frac {\log (x)}{-12-x} \, dx-\int \left (\frac {\log (x)}{-12-x}+\frac {2 (7+x) \log (x)}{27+14 x+x^2}\right ) \, dx\\ &=x^4 \log (x)-\log \left (1+\frac {x}{12}\right ) \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-2 \int \frac {(7+x) \log (x)}{27+14 x+x^2} \, dx+2 \int \left (\frac {\log (x)}{14-2 \sqrt {22}+2 x}+\frac {\log (x)}{14+2 \sqrt {22}+2 x}\right ) \, dx+\int \frac {\log \left (1+\frac {x}{12}\right )}{x} \, dx-\int \frac {\log (x)}{-12-x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-\text {Li}_2\left (-\frac {x}{12}\right )+2 \int \frac {\log (x)}{14-2 \sqrt {22}+2 x} \, dx+2 \int \frac {\log (x)}{14+2 \sqrt {22}+2 x} \, dx-2 \int \left (\frac {\log (x)}{14-2 \sqrt {22}+2 x}+\frac {\log (x)}{14+2 \sqrt {22}+2 x}\right ) \, dx-\int \frac {\log \left (1+\frac {x}{12}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (1+\frac {x}{7-\sqrt {22}}\right )+\log (x) \log \left (1+\frac {x}{7+\sqrt {22}}\right )+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )-2 \int \frac {\log (x)}{14-2 \sqrt {22}+2 x} \, dx-2 \int \frac {\log (x)}{14+2 \sqrt {22}+2 x} \, dx-\int \frac {\log \left (1+\frac {2 x}{14-2 \sqrt {22}}\right )}{x} \, dx-\int \frac {\log \left (1+\frac {2 x}{14+2 \sqrt {22}}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )+\text {Li}_2\left (-\frac {x}{7-\sqrt {22}}\right )+\text {Li}_2\left (-\frac {x}{7+\sqrt {22}}\right )+\int \frac {\log \left (1+\frac {2 x}{14-2 \sqrt {22}}\right )}{x} \, dx+\int \frac {\log \left (1+\frac {2 x}{14+2 \sqrt {22}}\right )}{x} \, dx\\ &=x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 25, normalized size = 1.25 \begin {gather*} x^4 \log (x)+\log (x) \log \left (\frac {27+14 x+x^2}{12+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 26, normalized size = 1.30
method | result | size |
default | \(\ln \left (x \right ) \ln \left (\frac {x^{2}+14 x +27}{x +12}\right )+x^{4} \ln \left (x \right )\) | \(26\) |
risch | \(\ln \left (x \right ) \ln \left (x^{2}+14 x +27\right )-\ln \left (x \right ) \ln \left (x +12\right )+x^{4} \ln \left (x \right )-\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i}{x +12}\right ) \mathrm {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i}{x +12}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i \left (x^{2}+14 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{2}}{2}-\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+14 x +27\right )}{x +12}\right )^{3}}{2}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 27, normalized size = 1.35 \begin {gather*} x^{4} \log \left (x\right ) + \log \left (x^{2} + 14 \, x + 27\right ) \log \left (x\right ) - \log \left (x + 12\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 25, normalized size = 1.25 \begin {gather*} x^{4} \log \left (x\right ) + \log \left (x\right ) \log \left (\frac {x^{2} + 14 \, x + 27}{x + 12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 22, normalized size = 1.10 \begin {gather*} x^{4} \log {\left (x \right )} + \log {\left (x \right )} \log {\left (\frac {x^{2} + 14 x + 27}{x + 12} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 27, normalized size = 1.35 \begin {gather*} x^{4} \log \left (x\right ) + \log \left (x^{2} + 14 \, x + 27\right ) \log \left (x\right ) - \log \left (x + 12\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 22, normalized size = 1.10 \begin {gather*} \ln \left (x\right )\,\left (\ln \left (\frac {x^2+14\,x+27}{x+12}\right )+x^4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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