Integrand size = 131, antiderivative size = 21 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \]
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \]
Integrate[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x] ] + (-12*x^3 + 2*x^3*Log[x] + (-60*x + 10*x*Log[x])*Log[6 - Log[x]])*Log[( x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Log[ 6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/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.
| \(\displaystyle \int \frac {6 x^3+x^3 (-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(10 x \log (x)-60 x) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )-5 x+(5 x \log (x)-30 x) \log (6-\log (x))}{\left (-6 x^2+x^2 \log (x)+(5 \log (x)-30) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-6 x^3+x^3 \log (x)-\left (-12 x^3+2 x^3 \log (x)+(10 x \log (x)-60 x) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )+5 x-(5 x \log (x)-30 x) \log (6-\log (x))}{(6-\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 x \log (x) \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {30 x \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {5 x}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {10 x \log (x) \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {60 x \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {x^3 \log (x)}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {6 x^3}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {2 x^3 \log (x)}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {12 x^3}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \int \frac {x}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx-30 \int \frac {x \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx+5 \int \frac {x \log (x) \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx-60 \int \frac {x \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx+10 \int \frac {x \log (x) \log (6-\log (x))}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx+6 \int \frac {x^3}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx-\int \frac {x^3 \log (x)}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx-12 \int \frac {x^3}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx+2 \int \frac {x^3 \log (x)}{(\log (x)-6) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}dx\) |
Int[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x]] + (- 12*x^3 + 2*x^3*Log[x] + (-60*x + 10*x*Log[x])*Log[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Log[6 - Lo g[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x]^2),x]
3.26.16.3.1 Defintions of rubi rules used
Time = 11.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19
| method | result | size |
| parallelrisch | \(\frac {x^{2}}{\ln \left (\frac {5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}}{x}\right )}\) | \(25\) |
| risch | \(-\frac {2 i x^{2}}{\pi \,\operatorname {csgn}\left (i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{3}+\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{2}+2 i \ln \left (x \right )-2 i \ln \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}\) | \(176\) |
int((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x^3*ln(x)-12*x^3)*ln((5*ln(-ln(x)+6 )+x^2)/x)+(5*x*ln(x)-30*x)*ln(-ln(x)+6)-x^3*ln(x)+6*x^3-5*x)/((5*ln(x)-30) *ln(-ln(x)+6)+x^2*ln(x)-6*x^2)/ln((5*ln(-ln(x)+6)+x^2)/x)^2,x,method=_RETU RNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )}{x}\right )} \]
integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5* log(-log(x)+6)+x^2)/x)+(5*x*log(x)-30*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5 *x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6)+ x^2)/x)^2,x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {x^{2} + 5 \log {\left (6 - \log {\left (x \right )} \right )}}{x} \right )}} \]
integrate((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x**3*ln(x)-12*x**3)*ln((5*ln( -ln(x)+6)+x**2)/x)+(5*x*ln(x)-30*x)*ln(-ln(x)+6)-x**3*ln(x)+6*x**3-5*x)/(( 5*ln(x)-30)*ln(-ln(x)+6)+x**2*ln(x)-6*x**2)/ln((5*ln(-ln(x)+6)+x**2)/x)**2 ,x)
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )\right ) - \log \left (x\right )} \]
integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5* log(-log(x)+6)+x^2)/x)+(5*x*log(x)-30*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5 *x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6)+ x^2)/x)^2,x, algorithm=\
Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )\right ) - \log \left (x\right )} \]
integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5* log(-log(x)+6)+x^2)/x)+(5*x*log(x)-30*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5 *x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6)+ x^2)/x)^2,x, algorithm=\
Timed out. \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=-\int \frac {5\,x+x^3\,\ln \left (x\right )+\ln \left (6-\ln \left (x\right )\right )\,\left (30\,x-5\,x\,\ln \left (x\right )\right )+\ln \left (\frac {5\,\ln \left (6-\ln \left (x\right )\right )+x^2}{x}\right )\,\left (\ln \left (6-\ln \left (x\right )\right )\,\left (60\,x-10\,x\,\ln \left (x\right )\right )-2\,x^3\,\ln \left (x\right )+12\,x^3\right )-6\,x^3}{{\ln \left (\frac {5\,\ln \left (6-\ln \left (x\right )\right )+x^2}{x}\right )}^2\,\left (\ln \left (6-\ln \left (x\right )\right )\,\left (5\,\ln \left (x\right )-30\right )+x^2\,\ln \left (x\right )-6\,x^2\right )} \,d x \]
int(-(5*x + x^3*log(x) + log(6 - log(x))*(30*x - 5*x*log(x)) + log((5*log( 6 - log(x)) + x^2)/x)*(log(6 - log(x))*(60*x - 10*x*log(x)) - 2*x^3*log(x) + 12*x^3) - 6*x^3)/(log((5*log(6 - log(x)) + x^2)/x)^2*(log(6 - log(x))*( 5*log(x) - 30) + x^2*log(x) - 6*x^2)),x)