Integrand size = 178, antiderivative size = 32 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\frac {3}{x \log \left (\log \left (x \log \left (e^{2 x-2 x^2}\right )+\frac {4 \log (4)}{\log (x)}\right )\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(32)=64\).
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=-\frac {3 \left (\log (256)-2 (-1+x) x^2 \log (x)\right ) \left (\log (16)+x^2 (-2+3 x) \log ^2(x)\right )}{x \left (-2 \log (4)+(-1+x) x^2 \log (x)\right ) \left (\log (256)+2 x^2 (-2+3 x) \log ^2(x)\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \]
Integrate[(-6*Log[4] + (6*x^2 - 9*x^3)*Log[x]^2 + (6*Log[4]*Log[x] + (3*x^ 2 - 3*x^3)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[L og[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]])/((-2*x^2*Log[4]*Log[x] + (-x^4 + x^5)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log [Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]]^2),x]
(-3*(Log[256] - 2*(-1 + x)*x^2*Log[x])*(Log[16] + x^2*(-2 + 3*x)*Log[x]^2) )/(x*(-2*Log[4] + (-1 + x)*x^2*Log[x])*(Log[256] + 2*x^2*(-2 + 3*x)*Log[x] ^2)*Log[Log[-2*(-1 + x)*x^2 + Log[256]/Log[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.
| \(\displaystyle \int \frac {\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (\left (3 x^2-3 x^3\right ) \log ^2(x)+6 \log (4) \log (x)\right ) \log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right ) \log \left (\log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right )\right )-6 \log (4)}{\left (\left (x^5-x^4\right ) \log ^2(x)-2 x^2 \log (4) \log (x)\right ) \log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (6 x^2-9 x^3\right ) \log ^2(x)\right )-\left (\left (3 x^2-3 x^3\right ) \log ^2(x)+6 \log (4) \log (x)\right ) \log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right ) \log \left (\log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right )\right )+6 \log (4)}{x^2 \log (x) \left (x^3 (-\log (x))+x^2 \log (x)+2 \log (4)\right ) \log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {\left (2 x^2-2 x^3\right ) \log (x)+4 \log (4)}{\log (x)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3}{x^2 \log \left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}-\frac {3 \left (3 x^3 \log ^2(x)-2 x^2 \log ^2(x)+2 \log (4)\right )}{x^2 \log (x) \left (x^3 \log (x)-x^2 \log (x)-2 \log (4)\right ) \log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right ) \log ^2\left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {1}{x^2 \log \left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}dx-6 \int \frac {\log (x)}{\left (-\log (x) x^3+\log (x) x^2+\log (16)\right ) \log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right ) \log ^2\left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}dx-3 \log (16) \int \frac {1}{x^2 \log (x) \left (\log (x) x^3-\log (x) x^2-2 \log (4)\right ) \log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right ) \log ^2\left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}dx-9 \int \frac {x \log (x)}{\left (\log (x) x^3-\log (x) x^2-2 \log (4)\right ) \log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right ) \log ^2\left (\log \left (\frac {\log (256)}{\log (x)}-2 (x-1) x^2\right )\right )}dx\) |
Int[(-6*Log[4] + (6*x^2 - 9*x^3)*Log[x]^2 + (6*Log[4]*Log[x] + (3*x^2 - 3* x^3)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4* Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]])/((-2*x^2*Log[4]*Log[x] + (-x^4 + x^5)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[( 4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]]^2),x]
3.10.55.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 24.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.72
\[\frac {3}{x \ln \left (3 \ln \left (2\right )-\ln \left (\ln \left (x \right )\right )+\ln \left (-\frac {x^{3} \ln \left (x \right )}{4}+\frac {x^{2} \ln \left (x \right )}{4}+\ln \left (2\right )\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \left (x \right )}{4}+\frac {x^{2} \ln \left (x \right )}{4}+\ln \left (2\right )\right )}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \left (x \right )}{4}+\frac {x^{2} \ln \left (x \right )}{4}+\ln \left (2\right )\right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {x^{3} \ln \left (x \right )}{4}+\frac {x^{2} \ln \left (x \right )}{4}+\ln \left (2\right )\right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i \left (-\frac {x^{3} \ln \left (x \right )}{4}+\frac {x^{2} \ln \left (x \right )}{4}+\ln \left (2\right )\right )\right )\right )}{2}\right )}\]
int((((-3*x^3+3*x^2)*ln(x)^2+12*ln(2)*ln(x))*ln(((-2*x^3+2*x^2)*ln(x)+8*ln (2))/ln(x))*ln(ln(((-2*x^3+2*x^2)*ln(x)+8*ln(2))/ln(x)))+(-9*x^3+6*x^2)*ln (x)^2-12*ln(2))/((x^5-x^4)*ln(x)^2-4*x^2*ln(2)*ln(x))/ln(((-2*x^3+2*x^2)*l n(x)+8*ln(2))/ln(x))/ln(ln(((-2*x^3+2*x^2)*ln(x)+8*ln(2))/ln(x)))^2,x)
3/x/ln(3*ln(2)-ln(ln(x))+ln(-1/4*x^3*ln(x)+1/4*x^2*ln(x)+ln(2))-1/2*I*Pi*c sgn(I/ln(x)*(-1/4*x^3*ln(x)+1/4*x^2*ln(x)+ln(2)))*(-csgn(I/ln(x)*(-1/4*x^3 *ln(x)+1/4*x^2*ln(x)+ln(2)))+csgn(I/ln(x)))*(-csgn(I/ln(x)*(-1/4*x^3*ln(x) +1/4*x^2*ln(x)+ln(2)))+csgn(I*(-1/4*x^3*ln(x)+1/4*x^2*ln(x)+ln(2)))))
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\frac {3}{x \log \left (\log \left (-\frac {2 \, {\left ({\left (x^{3} - x^{2}\right )} \log \left (x\right ) - 4 \, \log \left (2\right )\right )}}{\log \left (x\right )}\right )\right )} \]
integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)* log(x)+8*log(2))/log(x))*log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))) +(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log(x ))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*lo g(x)+8*log(2))/log(x)))^2,x, algorithm=\
Time = 1.97 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\frac {3}{x \log {\left (\log {\left (\frac {\left (- 2 x^{3} + 2 x^{2}\right ) \log {\left (x \right )} + 8 \log {\left (2 \right )}}{\log {\left (x \right )}} \right )} \right )}} \]
integrate((((-3*x**3+3*x**2)*ln(x)**2+12*ln(2)*ln(x))*ln(((-2*x**3+2*x**2) *ln(x)+8*ln(2))/ln(x))*ln(ln(((-2*x**3+2*x**2)*ln(x)+8*ln(2))/ln(x)))+(-9* x**3+6*x**2)*ln(x)**2-12*ln(2))/((x**5-x**4)*ln(x)**2-4*x**2*ln(2)*ln(x))/ ln(((-2*x**3+2*x**2)*ln(x)+8*ln(2))/ln(x))/ln(ln(((-2*x**3+2*x**2)*ln(x)+8 *ln(2))/ln(x)))**2,x)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\frac {3}{x \log \left (i \, \pi + \log \left (2\right ) + \log \left ({\left (x^{3} - x^{2}\right )} \log \left (x\right ) - 4 \, \log \left (2\right )\right ) - \log \left (\log \left (x\right )\right )\right )} \]
integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)* log(x)+8*log(2))/log(x))*log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))) +(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log(x ))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*lo g(x)+8*log(2))/log(x)))^2,x, algorithm=\
Time = 0.63 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\frac {3}{x \log \left (\log \left (2\right ) + \log \left (-x^{3} \log \left (x\right ) + x^{2} \log \left (x\right ) + 4 \, \log \left (2\right )\right ) - \log \left (\log \left (x\right )\right )\right )} \]
integrate((((-3*x^3+3*x^2)*log(x)^2+12*log(2)*log(x))*log(((-2*x^3+2*x^2)* log(x)+8*log(2))/log(x))*log(log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))) +(-9*x^3+6*x^2)*log(x)^2-12*log(2))/((x^5-x^4)*log(x)^2-4*x^2*log(2)*log(x ))/log(((-2*x^3+2*x^2)*log(x)+8*log(2))/log(x))/log(log(((-2*x^3+2*x^2)*lo g(x)+8*log(2))/log(x)))^2,x, algorithm=\
Timed out. \[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx=\text {Hanged} \]
int(-(log(x)^2*(6*x^2 - 9*x^3) - 12*log(2) + log((8*log(2) + log(x)*(2*x^2 - 2*x^3))/log(x))*log(log((8*log(2) + log(x)*(2*x^2 - 2*x^3))/log(x)))*(l og(x)^2*(3*x^2 - 3*x^3) + 12*log(2)*log(x)))/(log((8*log(2) + log(x)*(2*x^ 2 - 2*x^3))/log(x))*log(log((8*log(2) + log(x)*(2*x^2 - 2*x^3))/log(x)))^2 *(log(x)^2*(x^4 - x^5) + 4*x^2*log(2)*log(x))),x)