Integrand size = 78, antiderivative size = 30 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=5-\frac {6 \left (x+2 \left (-5+x+\log (5)+\frac {x}{-\frac {3}{x}+\log \left (x^2\right )}\right )\right )}{x} \]
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\[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=\int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {12 \left (3 x^2+2 x^3+9 (-5+\log (5))-6 x (-5+\log (5)) \log \left (x^2\right )+x^2 (-5+\log (5)) \log ^2\left (x^2\right )\right )}{x^2 \left (3-x \log \left (x^2\right )\right )^2} \, dx \\ & = 12 \int \frac {3 x^2+2 x^3+9 (-5+\log (5))-6 x (-5+\log (5)) \log \left (x^2\right )+x^2 (-5+\log (5)) \log ^2\left (x^2\right )}{x^2 \left (3-x \log \left (x^2\right )\right )^2} \, dx \\ & = 12 \int \left (\frac {-5+\log (5)}{x^2}+\frac {3+2 x}{\left (-3+x \log \left (x^2\right )\right )^2}\right ) \, dx \\ & = \frac {12 (5-\log (5))}{x}+12 \int \frac {3+2 x}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx \\ & = \frac {12 (5-\log (5))}{x}+12 \int \left (\frac {3}{\left (-3+x \log \left (x^2\right )\right )^2}+\frac {2 x}{\left (-3+x \log \left (x^2\right )\right )^2}\right ) \, dx \\ & = \frac {12 (5-\log (5))}{x}+24 \int \frac {x}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx+36 \int \frac {1}{\left (-3+x \log \left (x^2\right )\right )^2} \, dx \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=12 \left (\frac {5-\log (5)}{x}-\frac {x}{-3+x \log \left (x^2\right )}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {60}{x}-\frac {12 \ln \left (5\right )}{x}-\frac {12 x}{x \ln \left (x^{2}\right )-3}\) | \(27\) |
default | \(-\frac {12 \ln \left (5\right )}{x}+\frac {-180+60 x \ln \left (x^{2}\right )-12 x^{2}}{x \left (x \ln \left (x^{2}\right )-3\right )}\) | \(37\) |
norman | \(\frac {\left (60-12 \ln \left (5\right )\right ) x \ln \left (x^{2}\right )-12 x^{2}-180+36 \ln \left (5\right )}{x \left (x \ln \left (x^{2}\right )-3\right )}\) | \(38\) |
parallelrisch | \(-\frac {360+24 x \ln \left (x^{2}\right ) \ln \left (5\right )+24 x^{2}-120 x \ln \left (x^{2}\right )-72 \ln \left (5\right )}{2 x \left (x \ln \left (x^{2}\right )-3\right )}\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=-\frac {12 \, {\left (x^{2} + {\left (x \log \left (5\right ) - 5 \, x\right )} \log \left (x^{2}\right ) - 3 \, \log \left (5\right ) + 15\right )}}{x^{2} \log \left (x^{2}\right ) - 3 \, x} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=- \frac {12 x}{x \log {\left (x^{2} \right )} - 3} - \frac {-60 + 12 \log {\left (5 \right )}}{x} \]
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Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=-\frac {12 \, {\left (2 \, x {\left (\log \left (5\right ) - 5\right )} \log \left (x\right ) + x^{2} - 3 \, \log \left (5\right ) + 15\right )}}{2 \, x^{2} \log \left (x\right ) - 3 \, x} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=-\frac {12 \, x}{x \log \left (x^{2}\right ) - 3} - \frac {12 \, {\left (\log \left (5\right ) - 5\right )}}{x} \]
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Time = 8.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-540+36 x^2+24 x^3+108 \log (5)+(360 x-72 x \log (5)) \log \left (x^2\right )+\left (-60 x^2+12 x^2 \log (5)\right ) \log ^2\left (x^2\right )}{9 x^2-6 x^3 \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )} \, dx=-\frac {12\,x^3-x\,\left (36\,\ln \left (5\right )-180\right )+x^2\,\ln \left (x^2\right )\,\left (12\,\ln \left (5\right )-60\right )}{x^3\,\ln \left (x^2\right )-3\,x^2} \]
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