Integrand size = 84, antiderivative size = 20 \begin {dmath*} \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} \end {dmath*}
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \begin {dmath*} \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} \end {dmath*}
Integrate[(-6 - 6*Log[x] + (2 + 3*x^2)*Log[x]^2 + (-6*Log[x] + (2 - 3*x^2) *Log[x]^2)*Log[(6 + (-2 + 3*x^2)*Log[x])/(3*Log[x])])/(6*x^2*Log[x] + (-2* x^2 + 3*x^4)*Log[x]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (3 x^2+2\right ) \log ^2(x)+\left (\left (2-3 x^2\right ) \log ^2(x)-6 \log (x)\right ) \log \left (\frac {\left (3 x^2-2\right ) \log (x)+6}{3 \log (x)}\right )-6 \log (x)-6}{6 x^2 \log (x)+\left (3 x^4-2 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (3 x^2+2\right ) \log ^2(x)+\left (\left (2-3 x^2\right ) \log ^2(x)-6 \log (x)\right ) \log \left (\frac {\left (3 x^2-2\right ) \log (x)+6}{3 \log (x)}\right )-6 \log (x)-6}{x^2 \log (x) \left (3 x^2 \log (x)-2 \log (x)+6\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^2 \log ^2(x)+2 \log ^2(x)-6 \log (x)-6}{x^2 \log (x) \left (3 x^2 \log (x)-2 \log (x)+6\right )}-\frac {\log \left (x^2+\frac {2}{\log (x)}-\frac {2}{3}\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {1}{3 \log (x) x^2-2 \log (x)+6}dx-2 \int \frac {1}{x^2 \left (3 \log (x) x^2-2 \log (x)+6\right )}dx+9 \sqrt {2} \int \frac {1}{\left (\sqrt {2}-\sqrt {3} x\right ) \left (3 \log (x) x^2-2 \log (x)+6\right )}dx+9 \sqrt {2} \int \frac {1}{\left (\sqrt {3} x+\sqrt {2}\right ) \left (3 \log (x) x^2-2 \log (x)+6\right )}dx-\int \frac {\log \left (x^2+\frac {2}{\log (x)}-\frac {2}{3}\right )}{x^2}dx-\sqrt {6} \text {arctanh}\left (\sqrt {\frac {3}{2}} x\right )-\operatorname {ExpIntegralEi}(-\log (x))+\frac {1}{x}\) |
Int[(-6 - 6*Log[x] + (2 + 3*x^2)*Log[x]^2 + (-6*Log[x] + (2 - 3*x^2)*Log[x ]^2)*Log[(6 + (-2 + 3*x^2)*Log[x])/(3*Log[x])])/(6*x^2*Log[x] + (-2*x^2 + 3*x^4)*Log[x]^2),x]
3.6.31.3.1 Defintions of rubi rules used
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 | \(\frac {\ln \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{x}-\frac {-i \pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )-\frac {2 \ln \left (x \right )}{3}+2\right )}{\ln \left (x \right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+2 \ln \left (\ln \left (x \right )\right )-2}{2 x}\) | \(181\) |
int((((-3*x^2+2)*ln(x)^2-6*ln(x))*ln(1/3*((3*x^2-2)*ln(x)+6)/ln(x))+(3*x^2 +2)*ln(x)^2-6*ln(x)-6)/((3*x^4-2*x^2)*ln(x)^2+6*x^2*ln(x)),x,method=_RETUR NVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \begin {dmath*} \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} \end {dmath*}
integrate((((-3*x^2+2)*log(x)^2-6*log(x))*log(1/3*((3*x^2-2)*log(x)+6)/log (x))+(3*x^2+2)*log(x)^2-6*log(x)-6)/((3*x^4-2*x^2)*log(x)^2+6*x^2*log(x)), x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \begin {dmath*} \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} \end {dmath*}
integrate((((-3*x**2+2)*ln(x)**2-6*ln(x))*ln(1/3*((3*x**2-2)*ln(x)+6)/ln(x ))+(3*x**2+2)*ln(x)**2-6*ln(x)-6)/((3*x**4-2*x**2)*ln(x)**2+6*x**2*ln(x)), x)
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \begin {dmath*} \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} \end {dmath*}
integrate((((-3*x^2+2)*log(x)^2-6*log(x))*log(1/3*((3*x^2-2)*log(x)+6)/log (x))+(3*x^2+2)*log(x)^2-6*log(x)-6)/((3*x^4-2*x^2)*log(x)^2+6*x^2*log(x)), x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \begin {dmath*} \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} \end {dmath*}
integrate((((-3*x^2+2)*log(x)^2-6*log(x))*log(1/3*((3*x^2-2)*log(x)+6)/log (x))+(3*x^2+2)*log(x)^2-6*log(x)-6)/((3*x^4-2*x^2)*log(x)^2+6*x^2*log(x)), x, algorithm=\
Time = 17.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \begin {dmath*} \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} \end {dmath*}