Integrand size = 69, antiderivative size = 22 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=5 \left (-x+x \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )\right ) \]
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-5 x+5 x \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right ) \]
Integrate[(10*Log[x^4/18] - 5*Log[x^2]*Log[Log[x^2]] + (20*Log[x^2] + 5*Lo g[x^2]*Log[x^4/18])*Log[Log[x^2]]*Log[Log[Log[x^2]]])/(Log[x^2]*Log[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.
\(\displaystyle \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (\frac {x^4}{18}\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (5 \left (\log \left (\frac {x^4}{18}\right )+4\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )-\frac {5 \left (\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )-2 \log \left (\frac {x^4}{18}\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 20 \int \log \left (\log \left (\log \left (x^2\right )\right )\right )dx+10 \int \frac {\log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}dx+5 \int \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )dx-5 x\) |
Int[(10*Log[x^4/18] - 5*Log[x^2]*Log[Log[x^2]] + (20*Log[x^2] + 5*Log[x^2] *Log[x^4/18])*Log[Log[x^2]]*Log[Log[Log[x^2]]])/(Log[x^2]*Log[Log[x^2]]),x ]
3.9.83.3.1 Defintions of rubi rules used
Time = 0.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
method | result | size |
parallelrisch | \(5 x \ln \left (\frac {x^{4}}{18}\right ) \ln \left (\ln \left (\ln \left (x^{2}\right )\right )\right )-5 x\) | \(20\) |
risch | \(\left (20 x \ln \left (x \right )-\frac {5 i \pi x \operatorname {csgn}\left (i x^{3}\right )^{3}}{2}+\frac {5 i \pi x \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+\frac {5 i \pi x \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}}{2}-\frac {5 i \pi x \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (i x \right )}{2}+5 i x \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-\frac {5 i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {5 i \pi x \operatorname {csgn}\left (i x^{4}\right )^{3}}{2}-\frac {5 i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}}{2}+\frac {5 i \pi x \operatorname {csgn}\left (i x^{4}\right )^{2} \operatorname {csgn}\left (i x \right )}{2}+\frac {5 i \pi x \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )}{2}-\frac {5 i \pi x \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )}{2}-5 x \ln \left (2\right )-10 x \ln \left (3\right )\right ) \ln \left (\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )\right )-5 x\) | \(265\) |
int(((5*ln(x^2)*ln(1/18*x^4)+20*ln(x^2))*ln(ln(x^2))*ln(ln(ln(x^2)))-5*ln( x^2)*ln(ln(x^2))+10*ln(1/18*x^4))/ln(x^2)/ln(ln(x^2)),x,method=_RETURNVERB OSE)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-5 \, {\left (x \log \left (18\right ) - 2 \, x \log \left (x^{2}\right )\right )} \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 5 \, x \]
integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(lo g(x^2)))-5*log(x^2)*log(log(x^2))+10*log(1/18*x^4))/log(x^2)/log(log(x^2)) ,x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=- 5 x + \left (10 x \log {\left (x^{2} \right )} - 5 x \log {\left (18 \right )}\right ) \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )} \]
integrate(((5*ln(x**2)*ln(1/18*x**4)+20*ln(x**2))*ln(ln(x**2))*ln(ln(ln(x* *2)))-5*ln(x**2)*ln(ln(x**2))+10*ln(1/18*x**4))/ln(x**2)/ln(ln(x**2)),x)
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-5 \, {\left (x {\left (2 \, \log \left (3\right ) + \log \left (2\right )\right )} - 4 \, x \log \left (x\right )\right )} \log \left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) - 5 \, x \]
integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(lo g(x^2)))-5*log(x^2)*log(log(x^2))+10*log(1/18*x^4))/log(x^2)/log(log(x^2)) ,x, algorithm=\
Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-5 \, {\left (x \log \left (18\right ) - 2 \, x \log \left (x^{2}\right )\right )} \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 5 \, x \]
integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(lo g(x^2)))-5*log(x^2)*log(log(x^2))+10*log(1/18*x^4))/log(x^2)/log(log(x^2)) ,x, algorithm=\
Time = 9.51 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-5\,x-\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )\,\left (20\,x+x\,\left (5\,\ln \left (18\right )-20\right )-10\,x\,\ln \left (x^2\right )\right ) \]