Integrand size = 153, antiderivative size = 30 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x \left (-4+x-\frac {5-\frac {3 x^2}{(6-\log (\log (x)))^2}}{-1-x}\right ) \]
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=-4 x+x^2-\frac {5}{1+x}-\frac {3 x^3}{(1+x) (-6+\log (\log (x)))^2} \]
Integrate[(6*x^2 + 6*x^3 + (-216 + 1296*x + 54*x^2 - 396*x^3)*Log[x] + (10 8 - 648*x - 9*x^2 + 210*x^3)*Log[x]*Log[Log[x]] + (-18 + 108*x - 36*x^3)*L og[x]*Log[Log[x]]^2 + (1 - 6*x + 2*x^3)*Log[x]*Log[Log[x]]^3)/((-216 - 432 *x - 216*x^2)*Log[x] + (108 + 216*x + 108*x^2)*Log[x]*Log[Log[x]] + (-18 - 36*x - 18*x^2)*Log[x]*Log[Log[x]]^2 + (1 + 2*x + x^2)*Log[x]*Log[Log[x]]^ 3),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.
| \(\displaystyle \int \frac {6 x^3+\left (2 x^3-6 x+1\right ) \log (x) \log ^3(\log (x))+\left (-36 x^3+108 x-18\right ) \log (x) \log ^2(\log (x))+6 x^2+\left (-396 x^3+54 x^2+1296 x-216\right ) \log (x)+\left (210 x^3-9 x^2-648 x+108\right ) \log (x) \log (\log (x))}{\left (x^2+2 x+1\right ) \log (x) \log ^3(\log (x))+\left (-18 x^2-36 x-18\right ) \log (x) \log ^2(\log (x))+\left (108 x^2+216 x+108\right ) \log (x) \log (\log (x))+\left (-216 x^2-432 x-216\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-6 (x+1) x^2-\log (x) (\log (\log (x))-6) \left (66 x^3+\left (2 x^3-6 x+1\right ) \log ^2(\log (x))-12 \left (2 x^3-6 x+1\right ) \log (\log (x))-9 x^2-216 x+36\right )}{(x+1)^2 \log (x) (6-\log (\log (x)))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^3-6 x+1}{(x+1)^2}-\frac {3 (2 x+3) x^2}{(x+1)^2 (\log (\log (x))-6)^2}+\frac {6 x^2}{(x+1) \log (x) (\log (\log (x))-6)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \int \frac {1}{\log (x) (\log (\log (x))-6)^3}dx+6 \int \frac {x}{\log (x) (\log (\log (x))-6)^3}dx+6 \int \frac {1}{(x+1) \log (x) (\log (\log (x))-6)^3}dx+3 \int \frac {1}{(\log (\log (x))-6)^2}dx-6 \int \frac {x}{(\log (\log (x))-6)^2}dx-3 \int \frac {1}{(x+1)^2 (\log (\log (x))-6)^2}dx+x^2-4 x-\frac {5}{x+1}\) |
Int[(6*x^2 + 6*x^3 + (-216 + 1296*x + 54*x^2 - 396*x^3)*Log[x] + (108 - 64 8*x - 9*x^2 + 210*x^3)*Log[x]*Log[Log[x]] + (-18 + 108*x - 36*x^3)*Log[x]* Log[Log[x]]^2 + (1 - 6*x + 2*x^3)*Log[x]*Log[Log[x]]^3)/((-216 - 432*x - 2 16*x^2)*Log[x] + (108 + 216*x + 108*x^2)*Log[x]*Log[Log[x]] + (-18 - 36*x - 18*x^2)*Log[x]*Log[Log[x]]^2 + (1 + 2*x + x^2)*Log[x]*Log[Log[x]]^3),x]
3.14.5.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(\frac {x^{3}-3 x^{2}-4 x -5}{1+x}-\frac {3 x^{3}}{\left (1+x \right ) \left (\ln \left (\ln \left (x \right )\right )-6\right )^{2}}\) | \(38\) |
| parallelrisch | \(-\frac {432+144 \ln \left (\ln \left (x \right )\right ) x^{3}+36 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-432 x^{2} \ln \left (\ln \left (x \right )\right )+12 \ln \left (\ln \left (x \right )\right )^{2}-144 \ln \left (\ln \left (x \right )\right )-396 x^{3}+1296 x^{2}-12 \ln \left (\ln \left (x \right )\right )^{2} x^{3}}{12 \left (\ln \left (\ln \left (x \right )\right )^{2}-12 \ln \left (\ln \left (x \right )\right )+36\right ) \left (1+x \right )}\) | \(82\) |
int(((2*x^3-6*x+1)*ln(x)*ln(ln(x))^3+(-36*x^3+108*x-18)*ln(x)*ln(ln(x))^2+ (210*x^3-9*x^2-648*x+108)*ln(x)*ln(ln(x))+(-396*x^3+54*x^2+1296*x-216)*ln( x)+6*x^3+6*x^2)/((x^2+2*x+1)*ln(x)*ln(ln(x))^3+(-18*x^2-36*x-18)*ln(x)*ln( ln(x))^2+(108*x^2+216*x+108)*ln(x)*ln(ln(x))+(-216*x^2-432*x-216)*ln(x)),x ,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {33 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} - 108 \, x^{2} - 12 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right ) - 144 \, x - 180}{{\left (x + 1\right )} \log \left (\log \left (x\right )\right )^{2} - 12 \, {\left (x + 1\right )} \log \left (\log \left (x\right )\right ) + 36 \, x + 36} \]
integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*lo g(log(x))^2+(210*x^3-9*x^2-648*x+108)*log(x)*log(log(x))+(-396*x^3+54*x^2+ 1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x^2 -36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-21 6*x^2-432*x-216)*log(x)),x, algorithm=\
(33*x^3 + (x^3 - 3*x^2 - 4*x - 5)*log(log(x))^2 - 108*x^2 - 12*(x^3 - 3*x^ 2 - 4*x - 5)*log(log(x)) - 144*x - 180)/((x + 1)*log(log(x))^2 - 12*(x + 1 )*log(log(x)) + 36*x + 36)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=- \frac {3 x^{3}}{36 x + \left (- 12 x - 12\right ) \log {\left (\log {\left (x \right )} \right )} + \left (x + 1\right ) \log {\left (\log {\left (x \right )} \right )}^{2} + 36} + x^{2} - 4 x - \frac {5}{x + 1} \]
integrate(((2*x**3-6*x+1)*ln(x)*ln(ln(x))**3+(-36*x**3+108*x-18)*ln(x)*ln( ln(x))**2+(210*x**3-9*x**2-648*x+108)*ln(x)*ln(ln(x))+(-396*x**3+54*x**2+1 296*x-216)*ln(x)+6*x**3+6*x**2)/((x**2+2*x+1)*ln(x)*ln(ln(x))**3+(-18*x**2 -36*x-18)*ln(x)*ln(ln(x))**2+(108*x**2+216*x+108)*ln(x)*ln(ln(x))+(-216*x* *2-432*x-216)*ln(x)),x)
-3*x**3/(36*x + (-12*x - 12)*log(log(x)) + (x + 1)*log(log(x))**2 + 36) + x**2 - 4*x - 5/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {33 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} - 108 \, x^{2} - 12 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right ) - 144 \, x - 180}{{\left (x + 1\right )} \log \left (\log \left (x\right )\right )^{2} - 12 \, {\left (x + 1\right )} \log \left (\log \left (x\right )\right ) + 36 \, x + 36} \]
integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*lo g(log(x))^2+(210*x^3-9*x^2-648*x+108)*log(x)*log(log(x))+(-396*x^3+54*x^2+ 1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x^2 -36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-21 6*x^2-432*x-216)*log(x)),x, algorithm=\
(33*x^3 + (x^3 - 3*x^2 - 4*x - 5)*log(log(x))^2 - 108*x^2 - 12*(x^3 - 3*x^ 2 - 4*x - 5)*log(log(x)) - 144*x - 180)/((x + 1)*log(log(x))^2 - 12*(x + 1 )*log(log(x)) + 36*x + 36)
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x^{2} - \frac {3 \, x^{3}}{x \log \left (\log \left (x\right )\right )^{2} - 12 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} + 36 \, x - 12 \, \log \left (\log \left (x\right )\right ) + 36} - 4 \, x - \frac {5}{x + 1} \]
integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*lo g(log(x))^2+(210*x^3-9*x^2-648*x+108)*log(x)*log(log(x))+(-396*x^3+54*x^2+ 1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x^2 -36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-21 6*x^2-432*x-216)*log(x)),x, algorithm=\
x^2 - 3*x^3/(x*log(log(x))^2 - 12*x*log(log(x)) + log(log(x))^2 + 36*x - 1 2*log(log(x)) + 36) - 4*x - 5/(x + 1)
Time = 8.49 (sec) , antiderivative size = 259, normalized size of antiderivative = 8.63 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x^2-\frac {5}{x+1}-\frac {\frac {3\,x\,\left (9\,x^2\,\ln \left (x\right )+6\,x^3\,\ln \left (x\right )+x^2+x^3\right )}{{\left (x+1\right )}^2}-\frac {3\,x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^3+3\,x^2\right )}{2\,{\left (x+1\right )}^2}}{{\ln \left (\ln \left (x\right )\right )}^2-12\,\ln \left (\ln \left (x\right )\right )+36}-\frac {\frac {3\,x\,\ln \left (x\right )\,\left (54\,x^2\,\ln \left (x\right )+66\,x^3\,\ln \left (x\right )+24\,x^4\,\ln \left (x\right )+21\,x^2+35\,x^3+14\,x^4\right )}{2\,{\left (x+1\right )}^3}-\frac {3\,x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (9\,x^2\,\ln \left (x\right )+11\,x^3\,\ln \left (x\right )+4\,x^4\,\ln \left (x\right )+3\,x^2+5\,x^3+2\,x^4\right )}{2\,{\left (x+1\right )}^3}}{\ln \left (\ln \left (x\right )\right )-6}-4\,x+{\ln \left (x\right )}^2\,\left (\frac {-6\,x^5-\frac {33\,x^4}{2}+\frac {81\,x^2}{2}+\frac {81\,x}{2}+\frac {27}{2}}{x^3+3\,x^2+3\,x+1}-\frac {27}{2}\right )-\frac {\ln \left (x\right )\,\left (3\,x^4+\frac {9\,x^3}{2}\right )}{x^2+2\,x+1} \]
int(-(6*x^2 + 6*x^3 + log(x)*(1296*x + 54*x^2 - 396*x^3 - 216) - log(log(x ))*log(x)*(648*x + 9*x^2 - 210*x^3 - 108) + log(log(x))^3*log(x)*(2*x^3 - 6*x + 1) - log(log(x))^2*log(x)*(36*x^3 - 108*x + 18))/(log(x)*(432*x + 21 6*x^2 + 216) - log(log(x))*log(x)*(216*x + 108*x^2 + 108) - log(log(x))^3* log(x)*(2*x + x^2 + 1) + log(log(x))^2*log(x)*(36*x + 18*x^2 + 18)),x)
x^2 - 5/(x + 1) - ((3*x*(9*x^2*log(x) + 6*x^3*log(x) + x^2 + x^3))/(x + 1) ^2 - (3*x*log(log(x))*log(x)*(3*x^2 + 2*x^3))/(2*(x + 1)^2))/(log(log(x))^ 2 - 12*log(log(x)) + 36) - ((3*x*log(x)*(54*x^2*log(x) + 66*x^3*log(x) + 2 4*x^4*log(x) + 21*x^2 + 35*x^3 + 14*x^4))/(2*(x + 1)^3) - (3*x*log(log(x)) *log(x)*(9*x^2*log(x) + 11*x^3*log(x) + 4*x^4*log(x) + 3*x^2 + 5*x^3 + 2*x ^4))/(2*(x + 1)^3))/(log(log(x)) - 6) - 4*x + log(x)^2*(((81*x)/2 + (81*x^ 2)/2 - (33*x^4)/2 - 6*x^5 + 27/2)/(3*x + 3*x^2 + x^3 + 1) - 27/2) - (log(x )*((9*x^3)/2 + 3*x^4))/(2*x + x^2 + 1)