Integrand size = 145, antiderivative size = 26 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 x}{\log \left (\log \left (x+\left (-5+x^2\right )^2+4 (2 x+\log (4 x))\right )\right )} \]
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 x}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \]
Integrate[(-8 - 18*x + 40*x^2 - 8*x^4 + (50 + 18*x - 20*x^2 + 2*x^4 + 8*Lo g[4*x])*Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^ 2 + x^4 + 4*Log[4*x]]])/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9 *x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4* 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.
\(\displaystyle \int \frac {-8 x^4+40 x^2+\left (2 x^4-20 x^2+18 x+8 \log (4 x)+50\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )-18 x-8}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2}{\log \left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}-\frac {2 \left (4 x^4-20 x^2+9 x+4\right )}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 \int \frac {1}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}dx-18 \int \frac {x}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}dx+40 \int \frac {x^2}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}dx-8 \int \frac {x^4}{\left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right ) \log ^2\left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}dx+2 \int \frac {1}{\log \left (\log \left (x^4-10 x^2+9 x+4 \log (4 x)+25\right )\right )}dx\) |
Int[(-8 - 18*x + 40*x^2 - 8*x^4 + (50 + 18*x - 20*x^2 + 2*x^4 + 8*Log[4*x] )*Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^ 4 + 4*Log[4*x]]])/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9*x - 1 0*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2 ),x]
3.24.48.3.1 Defintions of rubi rules used
Time = 23.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {2 x}{\ln \left (\ln \left (4 \ln \left (4 x \right )+x^{4}-10 x^{2}+9 x +25\right )\right )}\) | \(27\) |
parallelrisch | \(\frac {2 x}{\ln \left (\ln \left (4 \ln \left (4 x \right )+x^{4}-10 x^{2}+9 x +25\right )\right )}\) | \(27\) |
int(((8*ln(4*x)+2*x^4-20*x^2+18*x+50)*ln(4*ln(4*x)+x^4-10*x^2+9*x+25)*ln(l n(4*ln(4*x)+x^4-10*x^2+9*x+25))-8*x^4+40*x^2-18*x-8)/(4*ln(4*x)+x^4-10*x^2 +9*x+25)/ln(4*ln(4*x)+x^4-10*x^2+9*x+25)/ln(ln(4*ln(4*x)+x^4-10*x^2+9*x+25 ))^2,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 4 \, \log \left (4 \, x\right ) + 25\right )\right )} \]
integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x +25)*log(log(4*log(4*x)+x^4-10*x^2+9*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4* x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*x) +x^4-10*x^2+9*x+25))^2,x, algorithm=\
Time = 1.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 x}{\log {\left (\log {\left (x^{4} - 10 x^{2} + 9 x + 4 \log {\left (4 x \right )} + 25 \right )} \right )}} \]
integrate(((8*ln(4*x)+2*x**4-20*x**2+18*x+50)*ln(4*ln(4*x)+x**4-10*x**2+9* x+25)*ln(ln(4*ln(4*x)+x**4-10*x**2+9*x+25))-8*x**4+40*x**2-18*x-8)/(4*ln(4 *x)+x**4-10*x**2+9*x+25)/ln(4*ln(4*x)+x**4-10*x**2+9*x+25)/ln(ln(4*ln(4*x) +x**4-10*x**2+9*x+25))**2,x)
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 8 \, \log \left (2\right ) + 4 \, \log \left (x\right ) + 25\right )\right )} \]
integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x +25)*log(log(4*log(4*x)+x^4-10*x^2+9*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4* x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*x) +x^4-10*x^2+9*x+25))^2,x, algorithm=\
Time = 0.74 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 4 \, \log \left (4 \, x\right ) + 25\right )\right )} \]
integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x +25)*log(log(4*log(4*x)+x^4-10*x^2+9*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4* x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*x) +x^4-10*x^2+9*x+25))^2,x, algorithm=\
Time = 10.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx=\frac {2\,x}{\ln \left (\ln \left (9\,x+4\,\ln \left (4\,x\right )-10\,x^2+x^4+25\right )\right )} \]
int(-(18*x - 40*x^2 + 8*x^4 - log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25)*lo g(log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25))*(18*x + 8*log(4*x) - 20*x^2 + 2*x^4 + 50) + 8)/(log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25)*log(log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25))^2*(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25) ),x)