Integrand size = 100, antiderivative size = 23 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log (x+(5-x \log (x)) (5+2 x-\log (\log (2 x)))) \]
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(23)=46\).
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log \left (25+11 x-5 x (\log (x)-\log (2 x))-2 x^2 (\log (x)-\log (2 x))-5 x \log (2 x)-2 x^2 \log (2 x)-5 \log (\log (2 x))+x (\log (x)-\log (2 x)) \log (\log (2 x))+x \log (2 x) \log (\log (2 x))\right ) \]
Integrate[(-5 + x*Log[x] + (6*x - 2*x^2 + (-5*x - 4*x^2)*Log[x])*Log[2*x] + (x + x*Log[x])*Log[2*x]*Log[Log[2*x]])/((25*x + 11*x^2 + (-5*x^2 - 2*x^3 )*Log[x])*Log[2*x] + (-5*x + x^2*Log[x])*Log[2*x]*Log[Log[2*x]]),x]
Log[25 + 11*x - 5*x*(Log[x] - Log[2*x]) - 2*x^2*(Log[x] - Log[2*x]) - 5*x* Log[2*x] - 2*x^2*Log[2*x] - 5*Log[Log[2*x]] + x*(Log[x] - Log[2*x])*Log[Lo g[2*x]] + x*Log[2*x]*Log[Log[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.
\(\displaystyle \int \frac {\left (-2 x^2+\left (-4 x^2-5 x\right ) \log (x)+6 x\right ) \log (2 x)+x \log (x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))-5}{\left (x^2 \log (x)-5 x\right ) \log (\log (2 x)) \log (2 x)+\left (11 x^2+\left (-2 x^3-5 x^2\right ) \log (x)+25 x\right ) \log (2 x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-2 x^2+\left (-4 x^2-5 x\right ) \log (x)+6 x\right ) \log (2 x)+x \log (x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))-5}{x \log (2 x) \left (-2 x^2 \log (x)+11 x-5 x \log (x)+x \log (x) \log (\log (2 x))-5 \log (\log (2 x))+25\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^3 \log ^2(x) \log (2 x)-x^2 \log ^2(x)-20 x^2 \log (x) \log (2 x)+x^2 \log (2 x)+10 x \log (x)+55 x \log (2 x)-25}{x (x \log (x)-5) \log (2 x) \left (2 x^2 \log (x)-11 x+5 x \log (x)-x \log (x) \log (\log (2 x))+5 \log (\log (2 x))-25\right )}+\frac {\log (x)+1}{x \log (x)-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {x^2 \log ^2(x)}{(x \log (x)-5) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx-\int \frac {x \log ^2(x)}{(x \log (x)-5) \log (2 x) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx+55 \int \frac {1}{(x \log (x)-5) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx+\int \frac {x}{(x \log (x)-5) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx-20 \int \frac {x \log (x)}{(x \log (x)-5) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx-25 \int \frac {1}{x (x \log (x)-5) \log (2 x) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx+10 \int \frac {\log (x)}{(x \log (x)-5) \log (2 x) \left (2 \log (x) x^2+5 \log (x) x-\log (x) \log (\log (2 x)) x-11 x+5 \log (\log (2 x))-25\right )}dx+\log (5-x \log (x))\) |
Int[(-5 + x*Log[x] + (6*x - 2*x^2 + (-5*x - 4*x^2)*Log[x])*Log[2*x] + (x + x*Log[x])*Log[2*x]*Log[Log[2*x]])/((25*x + 11*x^2 + (-5*x^2 - 2*x^3)*Log[ x])*Log[2*x] + (-5*x + x^2*Log[x])*Log[2*x]*Log[Log[2*x]]),x]
3.1.70.3.1 Defintions of rubi rules used
Time = 2.58 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
parallelrisch | \(\ln \left (x^{2} \ln \left (x \right )-\frac {\ln \left (x \right ) \ln \left (\ln \left (2 x \right )\right ) x}{2}+\frac {5 x \ln \left (x \right )}{2}-\frac {11 x}{2}+\frac {5 \ln \left (\ln \left (2 x \right )\right )}{2}-\frac {25}{2}\right )\) | \(35\) |
default | \(\ln \left (2 x^{2} \ln \left (x \right )-\ln \left (x \right ) \ln \left (\ln \left (2\right )+\ln \left (x \right )\right ) x +5 x \ln \left (x \right )-11 x +5 \ln \left (\ln \left (2\right )+\ln \left (x \right )\right )-25\right )\) | \(38\) |
risch | \(\ln \left (x \right )+\ln \left (\ln \left (x \right )-\frac {5}{x}\right )+\ln \left (\ln \left (\ln \left (2\right )+\ln \left (x \right )\right )-\frac {2 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-11 x -25}{x \ln \left (x \right )-5}\right )\) | \(48\) |
int(((x*ln(x)+x)*ln(2*x)*ln(ln(2*x))+((-4*x^2-5*x)*ln(x)-2*x^2+6*x)*ln(2*x )+x*ln(x)-5)/((x^2*ln(x)-5*x)*ln(2*x)*ln(ln(2*x))+((-2*x^3-5*x^2)*ln(x)+11 *x^2+25*x)*ln(2*x)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log \left (x\right ) + \log \left (-\frac {{\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - {\left (x \log \left (x\right ) - 5\right )} \log \left (\log \left (2\right ) + \log \left (x\right )\right ) - 11 \, x - 25}{x \log \left (x\right ) - 5}\right ) + \log \left (\frac {x \log \left (x\right ) - 5}{x}\right ) \]
integrate(((x*log(x)+x)*log(2*x)*log(log(2*x))+((-4*x^2-5*x)*log(x)-2*x^2+ 6*x)*log(2*x)+x*log(x)-5)/((x^2*log(x)-5*x)*log(2*x)*log(log(2*x))+((-2*x^ 3-5*x^2)*log(x)+11*x^2+25*x)*log(2*x)),x, algorithm=\
log(x) + log(-((2*x^2 + 5*x)*log(x) - (x*log(x) - 5)*log(log(2) + log(x)) - 11*x - 25)/(x*log(x) - 5)) + log((x*log(x) - 5)/x)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.82 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (x \right )} - \frac {5}{x} \right )} + \log {\left (\log {\left (\log {\left (x \right )} + \log {\left (2 \right )} \right )} + \frac {- 2 x^{2} \log {\left (x \right )} - 5 x \log {\left (x \right )} + 11 x + 25}{x \log {\left (x \right )} - 5} \right )} \]
integrate(((x*ln(x)+x)*ln(2*x)*ln(ln(2*x))+((-4*x**2-5*x)*ln(x)-2*x**2+6*x )*ln(2*x)+x*ln(x)-5)/((x**2*ln(x)-5*x)*ln(2*x)*ln(ln(2*x))+((-2*x**3-5*x** 2)*ln(x)+11*x**2+25*x)*ln(2*x)),x)
log(x) + log(log(x) - 5/x) + log(log(log(x) + log(2)) + (-2*x**2*log(x) - 5*x*log(x) + 11*x + 25)/(x*log(x) - 5))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log \left (x\right ) + \log \left (-\frac {{\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - {\left (x \log \left (x\right ) - 5\right )} \log \left (\log \left (2\right ) + \log \left (x\right )\right ) - 11 \, x - 25}{x \log \left (x\right ) - 5}\right ) + \log \left (\frac {x \log \left (x\right ) - 5}{x}\right ) \]
integrate(((x*log(x)+x)*log(2*x)*log(log(2*x))+((-4*x^2-5*x)*log(x)-2*x^2+ 6*x)*log(2*x)+x*log(x)-5)/((x^2*log(x)-5*x)*log(2*x)*log(log(2*x))+((-2*x^ 3-5*x^2)*log(x)+11*x^2+25*x)*log(2*x)),x, algorithm=\
log(x) + log(-((2*x^2 + 5*x)*log(x) - (x*log(x) - 5)*log(log(2) + log(x)) - 11*x - 25)/(x*log(x) - 5)) + log((x*log(x) - 5)/x)
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\log \left (2 \, x^{2} \log \left (x\right ) - x \log \left (x\right ) \log \left (\log \left (2\right ) + \log \left (x\right )\right ) + 5 \, x \log \left (x\right ) - 11 \, x + 5 \, \log \left (\log \left (2\right ) + \log \left (x\right )\right ) - 25\right ) \]
integrate(((x*log(x)+x)*log(2*x)*log(log(2*x))+((-4*x^2-5*x)*log(x)-2*x^2+ 6*x)*log(2*x)+x*log(x)-5)/((x^2*log(x)-5*x)*log(2*x)*log(log(2*x))+((-2*x^ 3-5*x^2)*log(x)+11*x^2+25*x)*log(2*x)),x, algorithm=\
log(2*x^2*log(x) - x*log(x)*log(log(2) + log(x)) + 5*x*log(x) - 11*x + 5*l og(log(2) + log(x)) - 25)
Time = 13.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.48 \[ \int \frac {-5+x \log (x)+\left (6 x-2 x^2+\left (-5 x-4 x^2\right ) \log (x)\right ) \log (2 x)+(x+x \log (x)) \log (2 x) \log (\log (2 x))}{\left (25 x+11 x^2+\left (-5 x^2-2 x^3\right ) \log (x)\right ) \log (2 x)+\left (-5 x+x^2 \log (x)\right ) \log (2 x) \log (\log (2 x))} \, dx=\ln \left (\frac {x\,\ln \left (x\right )-5}{x}\right )+\ln \left (\frac {11\,x-5\,\ln \left (\ln \left (2\,x\right )\right )-2\,x^2\,\ln \left (x\right )-5\,x\,\ln \left (x\right )+x\,\ln \left (\ln \left (2\,x\right )\right )\,\ln \left (x\right )+25}{x\,\ln \left (x\right )-5}\right )+\ln \left (x\right ) \]
int((x*log(x) - log(2*x)*(log(x)*(5*x + 4*x^2) - 6*x + 2*x^2) + log(2*x)*l og(log(2*x))*(x + x*log(x)) - 5)/(log(2*x)*(25*x - log(x)*(5*x^2 + 2*x^3) + 11*x^2) - log(2*x)*log(log(2*x))*(5*x - x^2*log(x))),x)