Optimal. Leaf size=25 \[ \log \left (-\frac {x}{5-x}+(-3+x) x+\frac {\log ^2(x)}{x}\right ) \]
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Rubi [F]
time = 11.59, 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 {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {80 x^2-80 x^3+23 x^4-2 x^5-\left (50-20 x+2 x^2\right ) \log (x)-\left (-25+10 x-x^2\right ) \log ^2(x)}{(5-x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\int \frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\int \left (\frac {160 x^2-136 x^3+36 x^4-3 x^5-50 \log (x)+20 x \log (x)-2 x^2 \log (x)}{5 x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{5 (-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\frac {1}{5} \int \frac {160 x^2-136 x^3+36 x^4-3 x^5-50 \log (x)+20 x \log (x)-2 x^2 \log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {1}{5} \int \frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\frac {1}{5} \int \frac {x^2 \left (160-136 x+36 x^2-3 x^3\right )-2 (-5+x)^2 \log (x)}{(-4+x)^2 x^3+(-5+x) x \log ^2(x)} \, dx+\frac {1}{5} \int \left (-\frac {160 x^2}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {136 x^3}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {36 x^4}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {3 x^5}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {50 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {20 x \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {2 x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\frac {1}{5} \int \left (\frac {160 x}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {136 x^2}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}+\frac {36 x^3}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {3 x^4}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}+\frac {20 \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {50 \log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {2 x \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}\right ) \, dx+\frac {2}{5} \int \frac {x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {3}{5} \int \frac {x^5}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-4 \int \frac {x \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-\frac {36}{5} \int \frac {x^4}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+10 \int \frac {\log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {136}{5} \int \frac {x^3}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-32 \int \frac {x^2}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\frac {2}{5} \int \frac {x^2 \log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {2}{5} \int \frac {x \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx+\frac {3}{5} \int \frac {x^5}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {3}{5} \int \frac {x^4}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-4 \int \frac {x \log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+4 \int \frac {\log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-\frac {36}{5} \int \frac {x^4}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+\frac {36}{5} \int \frac {x^3}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx+10 \int \frac {\log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-10 \int \frac {\log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {136}{5} \int \frac {x^3}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {136}{5} \int \frac {x^2}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-32 \int \frac {x^2}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+32 \int \frac {x}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 40, normalized size = 1.60 \begin {gather*} -\log (5-x)-\log (x)+\log \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 39, normalized size = 1.56
method | result | size |
risch | \(-\ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}+\frac {x^{2} \left (x^{2}-8 x +16\right )}{x -5}\right )\) | \(29\) |
default | \(-\ln \left (x \right )+\ln \left (x^{4}+x \ln \left (x \right )^{2}-8 x^{3}-5 \ln \left (x \right )^{2}+16 x^{2}\right )-\ln \left (x -5\right )\) | \(39\) |
norman | \(-\ln \left (x \right )+\ln \left (x^{4}+x \ln \left (x \right )^{2}-8 x^{3}-5 \ln \left (x \right )^{2}+16 x^{2}\right )-\ln \left (x -5\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 34, normalized size = 1.36 \begin {gather*} -\log \left (x\right ) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \left (x\right )^{2} + 16 \, x^{2}}{x - 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 34, normalized size = 1.36 \begin {gather*} -\log \left (x\right ) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \left (x\right )^{2} + 16 \, x^{2}}{x - 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 26, normalized size = 1.04 \begin {gather*} - \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} + \frac {x^{4} - 8 x^{3} + 16 x^{2}}{x - 5} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 38, normalized size = 1.52 \begin {gather*} \log \left (x^{4} - 8 \, x^{3} + x \log \left (x\right )^{2} + 16 \, x^{2} - 5 \, \log \left (x\right )^{2}\right ) - \log \left (x - 5\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (x\right )\,\left (2\,x^2-20\,x+50\right )-{\ln \left (x\right )}^2\,\left (x^2-10\,x+25\right )-80\,x^2+80\,x^3-23\,x^4+2\,x^5}{56\,x^4-80\,x^3-13\,x^5+x^6+{\ln \left (x\right )}^2\,\left (x^3-10\,x^2+25\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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