Integrand size = 74, antiderivative size = 21 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\log \left (-x+\frac {-4+x^4}{5 \log \left (\log \left (x^2\right )\right )}\right ) \]
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=-\log \left (\log \left (\log \left (x^2\right )\right )\right )+\log \left (4-x^4+5 x \log \left (\log \left (x^2\right )\right )\right ) \]
Integrate[(-8 + 2*x^4 - 4*x^4*Log[x^2]*Log[Log[x^2]] + 5*x*Log[x^2]*Log[Lo g[x^2]]^2)/((4*x - x^5)*Log[x^2]*Log[Log[x^2]] + 5*x^2*Log[x^2]*Log[Log[x^ 2]]^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.
\(\displaystyle \int \frac {2 x^4+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )-8}{5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^4+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )-8}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \left (-x^4+5 x \log \left (\log \left (x^2\right )\right )+4\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\frac {4 \log \left (x^2\right )+3 x^4 \log \left (x^2\right )-10 x}{x \log \left (x^2\right ) \left (x^4-5 x \log \left (\log \left (x^2\right )\right )-4\right )}+\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{x \left (x^4-5 \log \left (\log \left (x^2\right )\right ) x-4\right )}dx-10 \int \frac {1}{\log \left (x^2\right ) \left (x^4-5 \log \left (\log \left (x^2\right )\right ) x-4\right )}dx+3 \int \frac {x^3}{x^4-5 \log \left (\log \left (x^2\right )\right ) x-4}dx-\log \left (\log \left (\log \left (x^2\right )\right )\right )+\log (x)\) |
Int[(-8 + 2*x^4 - 4*x^4*Log[x^2]*Log[Log[x^2]] + 5*x*Log[x^2]*Log[Log[x^2] ]^2)/((4*x - x^5)*Log[x^2]*Log[Log[x^2]] + 5*x^2*Log[x^2]*Log[Log[x^2]]^2) ,x]
3.13.13.3.1 Defintions of rubi rules used
Time = 1.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(-\ln \left (\ln \left (\ln \left (x^{2}\right )\right )\right )+\ln \left (x^{4}-5 x \ln \left (\ln \left (x^{2}\right )\right )-4\right )\) | \(24\) |
int((5*x*ln(x^2)*ln(ln(x^2))^2-4*x^4*ln(x^2)*ln(ln(x^2))+2*x^4-8)/(5*x^2*l n(x^2)*ln(ln(x^2))^2+(-x^5+4*x)*ln(x^2)*ln(ln(x^2))),x,method=_RETURNVERBO SE)
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\frac {1}{2} \, \log \left (x^{2}\right ) + \log \left (-\frac {x^{4} - 5 \, x \log \left (\log \left (x^{2}\right )\right ) - 4}{x}\right ) - \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \]
integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4 -8)/(5*x^2*log(x^2)*log(log(x^2))^2+(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} + \frac {8 - 2 x^{4}}{10 x} \right )} - \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )} \]
integrate((5*x*ln(x**2)*ln(ln(x**2))**2-4*x**4*ln(x**2)*ln(ln(x**2))+2*x** 4-8)/(5*x**2*ln(x**2)*ln(ln(x**2))**2+(-x**5+4*x)*ln(x**2)*ln(ln(x**2))),x )
Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\log \left (x\right ) + \log \left (-\frac {x^{4} - 5 \, x \log \left (2\right ) - 5 \, x \log \left (\log \left (x\right )\right ) - 4}{5 \, x}\right ) - \log \left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) \]
integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4 -8)/(5*x^2*log(x^2)*log(log(x^2))^2+(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\log \left (-x^{4} + 5 \, x \log \left (\log \left (x^{2}\right )\right ) + 4\right ) - \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \]
integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4 -8)/(5*x^2*log(x^2)*log(log(x^2))^2+(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm=\
Time = 12.63 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.71 \[ \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\ln \left (\frac {20\,x\,\ln \left (\ln \left (x^2\right )\right )-4\,x^4+16}{\ln \left (x^2\right )}\right )-\ln \left (\frac {\ln \left (\ln \left (x^2\right )\right )\,\left (4\,\ln \left (x^2\right )-10\,x+3\,x^4\,\ln \left (x^2\right )\right )}{\ln \left (x^2\right )}\right )+\ln \left (4\,\ln \left (x^2\right )-10\,x+3\,x^4\,\ln \left (x^2\right )\right ) \]
int((2*x^4 + 5*x*log(x^2)*log(log(x^2))^2 - 4*x^4*log(x^2)*log(log(x^2)) - 8)/(5*x^2*log(x^2)*log(log(x^2))^2 + log(x^2)*log(log(x^2))*(4*x - x^5)), x)