Integrand size = 106, antiderivative size = 22 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log \left (x+\frac {3-2 x+\log (2)}{-2+\log (\log (x))}\right )} \]
Time = 0.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log \left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \]
Integrate[(15 - 10*x + 5*Log[2] - 40*x*Log[x] + 30*x*Log[x]*Log[Log[x]] - 5*x*Log[x]*Log[Log[x]]^2)/(((-6*x + 8*x^2 - 2*x*Log[2])*Log[x] + (3*x - 6* x^2 + x*Log[2])*Log[x]*Log[Log[x]] + x^2*Log[x]*Log[Log[x]]^2)*Log[(3 - 4* x + Log[2] + x*Log[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.
| \(\displaystyle \int \frac {-10 x-5 x \log (x) \log ^2(\log (x))+30 x \log (x) \log (\log (x))-40 x \log (x)+15+5 \log (2)}{\left (x^2 \log (x) \log ^2(\log (x))+\left (-6 x^2+3 x+x \log (2)\right ) \log (x) \log (\log (x))+\left (8 x^2-6 x-2 x \log (2)\right ) \log (x)\right ) \log ^2\left (\frac {-4 x+x \log (\log (x))+3+\log (2)}{\log (\log (x))-2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {5 \left (-2 x-x \log (x) \log ^2(\log (x))+6 x \log (x) \log (\log (x))-8 x \log (x)+3 \left (1+\frac {\log (2)}{3}\right )\right )}{x \log (x) (2-\log (\log (x))) \left (4 x+x (-\log (\log (x)))-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {-4 x+x \log (\log (x))+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {-x \log (x) \log ^2(\log (x))+6 x \log (x) \log (\log (x))-2 x-8 x \log (x)+\log (2)+3}{x \log (x) (2-\log (\log (x))) (\log (\log (x)) x-4 x+\log (2)+3) \log ^2\left (-\frac {\log (\log (x)) x-4 x+\log (2)+3}{2-\log (\log (x))}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {-x \log (x) \log ^2(\log (x))+6 x \log (x) \log (\log (x))-2 x-8 x \log (x)+\log (2)+3}{x \log (x) (2-\log (\log (x))) (\log (\log (x)) x-4 x+\log (2)+3) \log ^2\left (-\frac {\log (\log (x)) x-4 x+\log (2)+3}{2-\log (\log (x))}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -5 \int \frac {x \log (x) \log ^2(\log (x))-6 x \log (x) \log (\log (x))+2 x+8 x \log (x)-3 \left (1+\frac {\log (2)}{3}\right )}{x \log (x) (2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (-\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{2-\log (\log (x))}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {\log ^2(\log (x))}{(2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}+\frac {6 \log (\log (x))}{(\log (\log (x))-2) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}+\frac {-3-\log (2)}{x \log (x) (2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}+\frac {2}{\log (x) (2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}+\frac {8}{(2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \left (8 \int \frac {1}{(2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx+2 \int \frac {1}{\log (x) (2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx-(3+\log (2)) \int \frac {1}{x \log (x) (2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx+6 \int \frac {\log (\log (x))}{(\log (\log (x))-2) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx+\int \frac {\log ^2(\log (x))}{(2-\log (\log (x))) \left (-\log (\log (x)) x+4 x-3 \left (1+\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {\log (\log (x)) x-4 x+3 \left (1+\frac {\log (2)}{3}\right )}{\log (\log (x))-2}\right )}dx\right )\) |
Int[(15 - 10*x + 5*Log[2] - 40*x*Log[x] + 30*x*Log[x]*Log[Log[x]] - 5*x*Lo g[x]*Log[Log[x]]^2)/(((-6*x + 8*x^2 - 2*x*Log[2])*Log[x] + (3*x - 6*x^2 + x*Log[2])*Log[x]*Log[Log[x]] + x^2*Log[x]*Log[Log[x]]^2)*Log[(3 - 4*x + Lo g[2] + x*Log[Log[x]])/(-2 + Log[Log[x]])]^2),x]
3.21.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 231.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
| method | result | size |
| parallelrisch | \(\frac {5}{\ln \left (\frac {x \ln \left (\ln \left (x \right )\right )+\ln \left (2\right )+3-4 x}{\ln \left (\ln \left (x \right )\right )-2}\right )}\) | \(26\) |
| risch | \(\frac {10 i}{\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )-2}\right ) \operatorname {csgn}\left (i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )}{\ln \left (\ln \left (x \right )\right )-2}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )-2}\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )}{\ln \left (\ln \left (x \right )\right )-2}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )}{\ln \left (\ln \left (x \right )\right )-2}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )}{\ln \left (\ln \left (x \right )\right )-2}\right )^{3}-2 i \ln \left (\ln \left (\ln \left (x \right )\right )-2\right )+2 i \ln \left (\left (\ln \left (\ln \left (x \right )\right )-4\right ) x +3+\ln \left (2\right )\right )}\) | \(187\) |
| default | \(\text {Expression too large to display}\) | \(459\) |
| parts | \(\text {Expression too large to display}\) | \(459\) |
int((-5*x*ln(x)*ln(ln(x))^2+30*x*ln(x)*ln(ln(x))-40*x*ln(x)+5*ln(2)+15-10* x)/(x^2*ln(x)*ln(ln(x))^2+(x*ln(2)-6*x^2+3*x)*ln(x)*ln(ln(x))+(-2*x*ln(2)+ 8*x^2-6*x)*ln(x))/ln((x*ln(ln(x))+ln(2)+3-4*x)/(ln(ln(x))-2))^2,x,method=_ RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log \left (\frac {x \log \left (\log \left (x\right )\right ) - 4 \, x + \log \left (2\right ) + 3}{\log \left (\log \left (x\right )\right ) - 2}\right )} \]
integrate((-5*x*log(x)*log(log(x))^2+30*x*log(x)*log(log(x))-40*x*log(x)+5 *log(2)+15-10*x)/(x^2*log(x)*log(log(x))^2+(x*log(2)-6*x^2+3*x)*log(x)*log (log(x))+(-2*x*log(2)+8*x^2-6*x)*log(x))/log((x*log(log(x))+log(2)+3-4*x)/ (log(log(x))-2))^2,x, algorithm=\
Time = 1.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log {\left (\frac {x \log {\left (\log {\left (x \right )} \right )} - 4 x + \log {\left (2 \right )} + 3}{\log {\left (\log {\left (x \right )} \right )} - 2} \right )}} \]
integrate((-5*x*ln(x)*ln(ln(x))**2+30*x*ln(x)*ln(ln(x))-40*x*ln(x)+5*ln(2) +15-10*x)/(x**2*ln(x)*ln(ln(x))**2+(x*ln(2)-6*x**2+3*x)*ln(x)*ln(ln(x))+(- 2*x*ln(2)+8*x**2-6*x)*ln(x))/ln((x*ln(ln(x))+ln(2)+3-4*x)/(ln(ln(x))-2))** 2,x)
Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log \left (x \log \left (\log \left (x\right )\right ) - 4 \, x + \log \left (2\right ) + 3\right ) - \log \left (\log \left (\log \left (x\right )\right ) - 2\right )} \]
integrate((-5*x*log(x)*log(log(x))^2+30*x*log(x)*log(log(x))-40*x*log(x)+5 *log(2)+15-10*x)/(x^2*log(x)*log(log(x))^2+(x*log(2)-6*x^2+3*x)*log(x)*log (log(x))+(-2*x*log(2)+8*x^2-6*x)*log(x))/log((x*log(log(x))+log(2)+3-4*x)/ (log(log(x))-2))^2,x, algorithm=\
Time = 0.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {5}{\log \left (x \log \left (\log \left (x\right )\right ) - 4 \, x + \log \left (2\right ) + 3\right ) - \log \left (\log \left (\log \left (x\right )\right ) - 2\right )} \]
integrate((-5*x*log(x)*log(log(x))^2+30*x*log(x)*log(log(x))-40*x*log(x)+5 *log(2)+15-10*x)/(x^2*log(x)*log(log(x))^2+(x*log(2)-6*x^2+3*x)*log(x)*log (log(x))+(-2*x*log(2)+8*x^2-6*x)*log(x))/log((x*log(log(x))+log(2)+3-4*x)/ (log(log(x))-2))^2,x, algorithm=\
Time = 18.90 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.41 \[ \int \frac {15-10 x+5 \log (2)-40 x \log (x)+30 x \log (x) \log (\log (x))-5 x \log (x) \log ^2(\log (x))}{\left (\left (-6 x+8 x^2-2 x \log (2)\right ) \log (x)+\left (3 x-6 x^2+x \log (2)\right ) \log (x) \log (\log (x))+x^2 \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {3-4 x+\log (2)+x \log (\log (x))}{-2+\log (\log (x))}\right )} \, dx=\frac {\left (\ln \left (\ln \left (x\right )\right )-2\right )\,\left (\ln \left (2\right )-4\,x+x\,\ln \left (\ln \left (x\right )\right )+3\right )\,\left (5\,x\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2-30\,x\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )+10\,x-\ln \left (32\right )+40\,x\,\ln \left (x\right )-15\right )}{\ln \left (\frac {\ln \left (2\right )-4\,x+x\,\ln \left (\ln \left (x\right )\right )+3}{\ln \left (\ln \left (x\right )\right )-2}\right )\,\left (x\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2-6\,x\,\ln \left (x\right )\,\ln \left (\ln \left (x\right )\right )+2\,x-\ln \left (2\right )+8\,x\,\ln \left (x\right )-3\right )\,\left (8\,x+3\,\ln \left (\ln \left (x\right )\right )-\ln \left (4\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (2\right )-6\,x\,\ln \left (\ln \left (x\right )\right )+x\,{\ln \left (\ln \left (x\right )\right )}^2-6\right )} \]
int(-(10*x - 5*log(2) + 40*x*log(x) - 30*x*log(log(x))*log(x) + 5*x*log(lo g(x))^2*log(x) - 15)/(log((log(2) - 4*x + x*log(log(x)) + 3)/(log(log(x)) - 2))^2*(log(log(x))*log(x)*(3*x + x*log(2) - 6*x^2) - log(x)*(6*x + 2*x*l og(2) - 8*x^2) + x^2*log(log(x))^2*log(x))),x)
((log(log(x)) - 2)*(log(2) - 4*x + x*log(log(x)) + 3)*(10*x - log(32) + 40 *x*log(x) - 30*x*log(log(x))*log(x) + 5*x*log(log(x))^2*log(x) - 15))/(log ((log(2) - 4*x + x*log(log(x)) + 3)/(log(log(x)) - 2))*(2*x - log(2) + 8*x *log(x) - 6*x*log(log(x))*log(x) + x*log(log(x))^2*log(x) - 3)*(8*x + 3*lo g(log(x)) - log(4) + log(log(x))*log(2) - 6*x*log(log(x)) + x*log(log(x))^ 2 - 6))