Integrand size = 84, antiderivative size = 20 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {1+\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \]
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\[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx \\ & = \int \left (\frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2}\right ) \, dx \\ & = \int \frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx \\ & = \int \left (\frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )}-\frac {1}{x^2 \log (x)}-\frac {36}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}+\frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx \\ & = -\left (36 \int \frac {1}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx\right )+\int \frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )} \, dx-\int \frac {1}{x^2 \log (x)} \, dx+\int \frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx \\ & = \frac {1}{x}+6 \int \frac {1}{-2+3 x^2} \, dx-36 \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {1}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {3}{6-2 \log (x)+3 x^2 \log (x)}-\frac {2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx-\text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {1}{x}-\sqrt {6} \text {arctanh}\left (\sqrt {\frac {3}{2}} x\right )-\operatorname {ExpIntegralEi}(-\log (x))-2 \int \frac {1}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+3 \int \frac {1}{6-2 \log (x)+3 x^2 \log (x)} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {1}{x}+\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \]
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Time = 4.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50
method | result | size |
parallelrisch | \(\frac {24+24 \ln \left (\frac {3 x^{2} \ln \left (x \right )-2 \ln \left (x \right )+6}{3 \ln \left (x \right )}\right )}{24 x}\) | \(30\) |
risch | \(\frac {\ln \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{x}-\frac {-i \pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+2 \ln \left (\ln \left (x \right )\right )-2}{2 x}\) | \(181\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {\log \left (\frac {{\left (3 \, x^{2} - 2\right )} \log \left (x\right ) + 6}{3 \, \log \left (x\right )}\right ) + 1}{x} \]
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Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {\log {\left (\frac {\frac {\left (3 x^{2} - 2\right ) \log {\left (x \right )}}{3} + 2}{\log {\left (x \right )}} \right )}}{x} + \frac {1}{x} \]
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Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=-\frac {\log \left (3\right ) - \log \left ({\left (3 \, x^{2} - 2\right )} \log \left (x\right ) + 6\right ) + \log \left (\log \left (x\right )\right ) - 1}{x} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {\log \left (3 \, x^{2} \log \left (x\right ) - 2 \, \log \left (x\right ) + 6\right )}{x} - \frac {\log \left (3 \, \log \left (x\right )\right )}{x} + \frac {1}{x} \]
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Time = 17.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx=\frac {\ln \left (\frac {\frac {\ln \left (x\right )\,\left (3\,x^2-2\right )}{3}+2}{\ln \left (x\right )}\right )+1}{x} \]
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