Integrand size = 340, antiderivative size = 24 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {(-2+8 x)^2}{6561 x^2 (-3+x+\log (x+\log (x)))^4} \] Output:
1/81*(8*x-2)^2/x^2/(x+ln(x+ln(x))-3)^2/(9*x+9*ln(x+ln(x))-27)^2
Time = 3.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {4 (-1+4 x)^2}{6561 x^2 (-3+x+\log (x+\log (x)))^4} \] Input:
Integrate[(-16 + 136*x - 248*x^2 - 96*x^3 - 256*x^4 + (24 - 120*x + 160*x^ 2 - 256*x^3)*Log[x] + (-8*x + 32*x^2 + (-8 + 32*x)*Log[x])*Log[x + Log[x]] )/(-1594323*x^4 + 2657205*x^5 - 1771470*x^6 + 590490*x^7 - 98415*x^8 + 656 1*x^9 + (-1594323*x^3 + 2657205*x^4 - 1771470*x^5 + 590490*x^6 - 98415*x^7 + 6561*x^8)*Log[x] + (2657205*x^4 - 3542940*x^5 + 1771470*x^6 - 393660*x^ 7 + 32805*x^8 + (2657205*x^3 - 3542940*x^4 + 1771470*x^5 - 393660*x^6 + 32 805*x^7)*Log[x])*Log[x + Log[x]] + (-1771470*x^4 + 1771470*x^5 - 590490*x^ 6 + 65610*x^7 + (-1771470*x^3 + 1771470*x^4 - 590490*x^5 + 65610*x^6)*Log[ x])*Log[x + Log[x]]^2 + (590490*x^4 - 393660*x^5 + 65610*x^6 + (590490*x^3 - 393660*x^4 + 65610*x^5)*Log[x])*Log[x + Log[x]]^3 + (-98415*x^4 + 32805 *x^5 + (-98415*x^3 + 32805*x^4)*Log[x])*Log[x + Log[x]]^4 + (6561*x^4 + 65 61*x^3*Log[x])*Log[x + Log[x]]^5),x]
Output:
(4*(-1 + 4*x)^2)/(6561*x^2*(-3 + x + Log[x + Log[x]])^4)
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.
| \(\displaystyle \int \frac {-256 x^4-96 x^3-248 x^2+\left (32 x^2-8 x+(32 x-8) \log (x)\right ) \log (x+\log (x))+\left (-256 x^3+160 x^2-120 x+24\right ) \log (x)+136 x-16}{6561 x^9-98415 x^8+590490 x^7-1771470 x^6+2657205 x^5-1594323 x^4+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))+\left (32805 x^5-98415 x^4+\left (32805 x^4-98415 x^3\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (65610 x^6-393660 x^5+590490 x^4+\left (65610 x^5-393660 x^4+590490 x^3\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (65610 x^7-590490 x^6+1771470 x^5-1771470 x^4+\left (65610 x^6-590490 x^5+1771470 x^4-1771470 x^3\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (6561 x^8-98415 x^7+590490 x^6-1771470 x^5+2657205 x^4-1594323 x^3\right ) \log (x)+\left (32805 x^8-393660 x^7+1771470 x^6-3542940 x^5+2657205 x^4+\left (32805 x^7-393660 x^6+1771470 x^5-3542940 x^4+2657205 x^3\right ) \log (x)\right ) \log (x+\log (x))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 (1-4 x) \left (-8 x^3-5 x^2-\log (x) \left (8 x^2-3 x-\log (x+\log (x))+3\right )-9 x+x \log (x+\log (x))+2\right )}{6561 x^3 (x+\log (x)) (-x-\log (x+\log (x))+3)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8 \int \frac {(1-4 x) \left (-8 x^3-5 x^2+\log (x+\log (x)) x-9 x-\log (x) \left (8 x^2-3 x-\log (x+\log (x))+3\right )+2\right )}{x^3 (x+\log (x)) (-x-\log (x+\log (x))+3)^5}dx}{6561}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {8 \int \left (\frac {4 x-1}{x^3 (x+\log (x+\log (x))-3)^4}-\frac {2 (4 x-1)^2 \left (x^2+\log (x) x+x+1\right )}{x^3 (x+\log (x)) (x+\log (x+\log (x))-3)^5}\right )dx}{6561}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \left (-2 \int \frac {1}{x^3 (x+\log (x)) (x+\log (x+\log (x))-3)^5}dx-\int \frac {1}{x^3 (x+\log (x+\log (x))-3)^4}dx+14 \int \frac {1}{x^2 (x+\log (x)) (x+\log (x+\log (x))-3)^5}dx-2 \int \frac {\log (x)}{x^2 (x+\log (x)) (x+\log (x+\log (x))-3)^5}dx+4 \int \frac {1}{x^2 (x+\log (x+\log (x))-3)^4}dx+16 \int \frac {1}{(x+\log (x)) (x+\log (x+\log (x))-3)^5}dx+14 \int \frac {1}{x (x+\log (x)) (x+\log (x+\log (x))-3)^5}dx+16 \int \frac {\log (x)}{x (x+\log (x)) (x+\log (x+\log (x))-3)^5}dx+\frac {8}{(-x-\log (x+\log (x))+3)^4}\right )}{6561}\) |
Input:
Int[(-16 + 136*x - 248*x^2 - 96*x^3 - 256*x^4 + (24 - 120*x + 160*x^2 - 25 6*x^3)*Log[x] + (-8*x + 32*x^2 + (-8 + 32*x)*Log[x])*Log[x + Log[x]])/(-15 94323*x^4 + 2657205*x^5 - 1771470*x^6 + 590490*x^7 - 98415*x^8 + 6561*x^9 + (-1594323*x^3 + 2657205*x^4 - 1771470*x^5 + 590490*x^6 - 98415*x^7 + 656 1*x^8)*Log[x] + (2657205*x^4 - 3542940*x^5 + 1771470*x^6 - 393660*x^7 + 32 805*x^8 + (2657205*x^3 - 3542940*x^4 + 1771470*x^5 - 393660*x^6 + 32805*x^ 7)*Log[x])*Log[x + Log[x]] + (-1771470*x^4 + 1771470*x^5 - 590490*x^6 + 65 610*x^7 + (-1771470*x^3 + 1771470*x^4 - 590490*x^5 + 65610*x^6)*Log[x])*Lo g[x + Log[x]]^2 + (590490*x^4 - 393660*x^5 + 65610*x^6 + (590490*x^3 - 393 660*x^4 + 65610*x^5)*Log[x])*Log[x + Log[x]]^3 + (-98415*x^4 + 32805*x^5 + (-98415*x^3 + 32805*x^4)*Log[x])*Log[x + Log[x]]^4 + (6561*x^4 + 6561*x^3 *Log[x])*Log[x + Log[x]]^5),x]
Output:
$Aborted
Time = 8.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(\frac {\frac {64}{6561} x^{2}-\frac {32}{6561} x +\frac {4}{6561}}{x^{2} \left (x +\ln \left (x +\ln \left (x \right )\right )-3\right )^{4}}\) | \(26\) |
| risch | \(\frac {\frac {64}{6561} x^{2}-\frac {32}{6561} x +\frac {4}{6561}}{x^{2} \left (x +\ln \left (x +\ln \left (x \right )\right )-3\right )^{4}}\) | \(26\) |
| parallelrisch | \(\frac {34560 x^{2}-17280 x +2160}{3542940 x^{2} \left (x^{4}+4 \ln \left (x +\ln \left (x \right )\right ) x^{3}+6 \ln \left (x +\ln \left (x \right )\right )^{2} x^{2}+4 \ln \left (x +\ln \left (x \right )\right )^{3} x +\ln \left (x +\ln \left (x \right )\right )^{4}-12 x^{3}-36 \ln \left (x +\ln \left (x \right )\right ) x^{2}-36 \ln \left (x +\ln \left (x \right )\right )^{2} x -12 \ln \left (x +\ln \left (x \right )\right )^{3}+54 x^{2}+108 \ln \left (x +\ln \left (x \right )\right ) x +54 \ln \left (x +\ln \left (x \right )\right )^{2}-108 x -108 \ln \left (x +\ln \left (x \right )\right )+81\right )}\) | \(128\) |
Input:
int((((32*x-8)*ln(x)+32*x^2-8*x)*ln(x+ln(x))+(-256*x^3+160*x^2-120*x+24)*l n(x)-256*x^4-96*x^3-248*x^2+136*x-16)/((6561*x^3*ln(x)+6561*x^4)*ln(x+ln(x ))^5+((32805*x^4-98415*x^3)*ln(x)+32805*x^5-98415*x^4)*ln(x+ln(x))^4+((656 10*x^5-393660*x^4+590490*x^3)*ln(x)+65610*x^6-393660*x^5+590490*x^4)*ln(x+ ln(x))^3+((65610*x^6-590490*x^5+1771470*x^4-1771470*x^3)*ln(x)+65610*x^7-5 90490*x^6+1771470*x^5-1771470*x^4)*ln(x+ln(x))^2+((32805*x^7-393660*x^6+17 71470*x^5-3542940*x^4+2657205*x^3)*ln(x)+32805*x^8-393660*x^7+1771470*x^6- 3542940*x^5+2657205*x^4)*ln(x+ln(x))+(6561*x^8-98415*x^7+590490*x^6-177147 0*x^5+2657205*x^4-1594323*x^3)*ln(x)+6561*x^9-98415*x^8+590490*x^7-1771470 *x^6+2657205*x^5-1594323*x^4),x,method=_RETURNVERBOSE)
Output:
4/6561*(16*x^2-8*x+1)/x^2/(x+ln(x+ln(x))-3)^4
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.83 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {4 \, {\left (16 \, x^{2} - 8 \, x + 1\right )}}{6561 \, {\left (x^{6} + x^{2} \log \left (x + \log \left (x\right )\right )^{4} - 12 \, x^{5} + 54 \, x^{4} + 4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )^{3} - 108 \, x^{3} + 6 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )^{2} + 81 \, x^{2} + 4 \, {\left (x^{5} - 9 \, x^{4} + 27 \, x^{3} - 27 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )\right )}} \] Input:
integrate((((32*x-8)*log(x)+32*x^2-8*x)*log(x+log(x))+(-256*x^3+160*x^2-12 0*x+24)*log(x)-256*x^4-96*x^3-248*x^2+136*x-16)/((6561*x^3*log(x)+6561*x^4 )*log(x+log(x))^5+((32805*x^4-98415*x^3)*log(x)+32805*x^5-98415*x^4)*log(x +log(x))^4+((65610*x^5-393660*x^4+590490*x^3)*log(x)+65610*x^6-393660*x^5+ 590490*x^4)*log(x+log(x))^3+((65610*x^6-590490*x^5+1771470*x^4-1771470*x^3 )*log(x)+65610*x^7-590490*x^6+1771470*x^5-1771470*x^4)*log(x+log(x))^2+((3 2805*x^7-393660*x^6+1771470*x^5-3542940*x^4+2657205*x^3)*log(x)+32805*x^8- 393660*x^7+1771470*x^6-3542940*x^5+2657205*x^4)*log(x+log(x))+(6561*x^8-98 415*x^7+590490*x^6-1771470*x^5+2657205*x^4-1594323*x^3)*log(x)+6561*x^9-98 415*x^8+590490*x^7-1771470*x^6+2657205*x^5-1594323*x^4),x, algorithm="fric as")
Output:
4/6561*(16*x^2 - 8*x + 1)/(x^6 + x^2*log(x + log(x))^4 - 12*x^5 + 54*x^4 + 4*(x^3 - 3*x^2)*log(x + log(x))^3 - 108*x^3 + 6*(x^4 - 6*x^3 + 9*x^2)*log (x + log(x))^2 + 81*x^2 + 4*(x^5 - 9*x^4 + 27*x^3 - 27*x^2)*log(x + log(x) ))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.88 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {64 x^{2} - 32 x + 4}{6561 x^{6} - 78732 x^{5} + 354294 x^{4} - 708588 x^{3} + 6561 x^{2} \log {\left (x + \log {\left (x \right )} \right )}^{4} + 531441 x^{2} + \left (26244 x^{3} - 78732 x^{2}\right ) \log {\left (x + \log {\left (x \right )} \right )}^{3} + \left (39366 x^{4} - 236196 x^{3} + 354294 x^{2}\right ) \log {\left (x + \log {\left (x \right )} \right )}^{2} + \left (26244 x^{5} - 236196 x^{4} + 708588 x^{3} - 708588 x^{2}\right ) \log {\left (x + \log {\left (x \right )} \right )}} \] Input:
integrate((((32*x-8)*ln(x)+32*x**2-8*x)*ln(x+ln(x))+(-256*x**3+160*x**2-12 0*x+24)*ln(x)-256*x**4-96*x**3-248*x**2+136*x-16)/((6561*x**3*ln(x)+6561*x **4)*ln(x+ln(x))**5+((32805*x**4-98415*x**3)*ln(x)+32805*x**5-98415*x**4)* ln(x+ln(x))**4+((65610*x**5-393660*x**4+590490*x**3)*ln(x)+65610*x**6-3936 60*x**5+590490*x**4)*ln(x+ln(x))**3+((65610*x**6-590490*x**5+1771470*x**4- 1771470*x**3)*ln(x)+65610*x**7-590490*x**6+1771470*x**5-1771470*x**4)*ln(x +ln(x))**2+((32805*x**7-393660*x**6+1771470*x**5-3542940*x**4+2657205*x**3 )*ln(x)+32805*x**8-393660*x**7+1771470*x**6-3542940*x**5+2657205*x**4)*ln( x+ln(x))+(6561*x**8-98415*x**7+590490*x**6-1771470*x**5+2657205*x**4-15943 23*x**3)*ln(x)+6561*x**9-98415*x**8+590490*x**7-1771470*x**6+2657205*x**5- 1594323*x**4),x)
Output:
(64*x**2 - 32*x + 4)/(6561*x**6 - 78732*x**5 + 354294*x**4 - 708588*x**3 + 6561*x**2*log(x + log(x))**4 + 531441*x**2 + (26244*x**3 - 78732*x**2)*lo g(x + log(x))**3 + (39366*x**4 - 236196*x**3 + 354294*x**2)*log(x + log(x) )**2 + (26244*x**5 - 236196*x**4 + 708588*x**3 - 708588*x**2)*log(x + log( x)))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (22) = 44\).
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.83 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {4 \, {\left (16 \, x^{2} - 8 \, x + 1\right )}}{6561 \, {\left (x^{6} + x^{2} \log \left (x + \log \left (x\right )\right )^{4} - 12 \, x^{5} + 54 \, x^{4} + 4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )^{3} - 108 \, x^{3} + 6 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )^{2} + 81 \, x^{2} + 4 \, {\left (x^{5} - 9 \, x^{4} + 27 \, x^{3} - 27 \, x^{2}\right )} \log \left (x + \log \left (x\right )\right )\right )}} \] Input:
integrate((((32*x-8)*log(x)+32*x^2-8*x)*log(x+log(x))+(-256*x^3+160*x^2-12 0*x+24)*log(x)-256*x^4-96*x^3-248*x^2+136*x-16)/((6561*x^3*log(x)+6561*x^4 )*log(x+log(x))^5+((32805*x^4-98415*x^3)*log(x)+32805*x^5-98415*x^4)*log(x +log(x))^4+((65610*x^5-393660*x^4+590490*x^3)*log(x)+65610*x^6-393660*x^5+ 590490*x^4)*log(x+log(x))^3+((65610*x^6-590490*x^5+1771470*x^4-1771470*x^3 )*log(x)+65610*x^7-590490*x^6+1771470*x^5-1771470*x^4)*log(x+log(x))^2+((3 2805*x^7-393660*x^6+1771470*x^5-3542940*x^4+2657205*x^3)*log(x)+32805*x^8- 393660*x^7+1771470*x^6-3542940*x^5+2657205*x^4)*log(x+log(x))+(6561*x^8-98 415*x^7+590490*x^6-1771470*x^5+2657205*x^4-1594323*x^3)*log(x)+6561*x^9-98 415*x^8+590490*x^7-1771470*x^6+2657205*x^5-1594323*x^4),x, algorithm="maxi ma")
Output:
4/6561*(16*x^2 - 8*x + 1)/(x^6 + x^2*log(x + log(x))^4 - 12*x^5 + 54*x^4 + 4*(x^3 - 3*x^2)*log(x + log(x))^3 - 108*x^3 + 6*(x^4 - 6*x^3 + 9*x^2)*log (x + log(x))^2 + 81*x^2 + 4*(x^5 - 9*x^4 + 27*x^3 - 27*x^2)*log(x + log(x) ))
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (22) = 44\).
Time = 1.09 (sec) , antiderivative size = 432, normalized size of antiderivative = 18.00 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx =\text {Too large to display} \] Input:
integrate((((32*x-8)*log(x)+32*x^2-8*x)*log(x+log(x))+(-256*x^3+160*x^2-12 0*x+24)*log(x)-256*x^4-96*x^3-248*x^2+136*x-16)/((6561*x^3*log(x)+6561*x^4 )*log(x+log(x))^5+((32805*x^4-98415*x^3)*log(x)+32805*x^5-98415*x^4)*log(x +log(x))^4+((65610*x^5-393660*x^4+590490*x^3)*log(x)+65610*x^6-393660*x^5+ 590490*x^4)*log(x+log(x))^3+((65610*x^6-590490*x^5+1771470*x^4-1771470*x^3 )*log(x)+65610*x^7-590490*x^6+1771470*x^5-1771470*x^4)*log(x+log(x))^2+((3 2805*x^7-393660*x^6+1771470*x^5-3542940*x^4+2657205*x^3)*log(x)+32805*x^8- 393660*x^7+1771470*x^6-3542940*x^5+2657205*x^4)*log(x+log(x))+(6561*x^8-98 415*x^7+590490*x^6-1771470*x^5+2657205*x^4-1594323*x^3)*log(x)+6561*x^9-98 415*x^8+590490*x^7-1771470*x^6+2657205*x^5-1594323*x^4),x, algorithm="giac ")
Output:
4/6561*(16*x^4 + 16*x^3*log(x) + 8*x^3 - 8*x^2*log(x) + 9*x^2 + x*log(x) - 7*x + 1)/(x^8 + 4*x^7*log(x + log(x)) + 6*x^6*log(x + log(x))^2 + 4*x^5*l og(x + log(x))^3 + x^4*log(x + log(x))^4 + x^7*log(x) + 4*x^6*log(x + log( x))*log(x) + 6*x^5*log(x + log(x))^2*log(x) + 4*x^4*log(x + log(x))^3*log( x) + x^3*log(x + log(x))^4*log(x) - 11*x^7 - 32*x^6*log(x + log(x)) - 30*x ^5*log(x + log(x))^2 - 8*x^4*log(x + log(x))^3 + x^3*log(x + log(x))^4 - 1 2*x^6*log(x) - 36*x^5*log(x + log(x))*log(x) - 36*x^4*log(x + log(x))^2*lo g(x) - 12*x^3*log(x + log(x))^3*log(x) + 43*x^6 + 76*x^5*log(x + log(x)) + 24*x^4*log(x + log(x))^2 - 8*x^3*log(x + log(x))^3 + x^2*log(x + log(x))^ 4 + 54*x^5*log(x) + 108*x^4*log(x + log(x))*log(x) + 54*x^3*log(x + log(x) )^2*log(x) - 66*x^5 - 36*x^4*log(x + log(x)) + 18*x^3*log(x + log(x))^2 - 12*x^2*log(x + log(x))^3 - 108*x^4*log(x) - 108*x^3*log(x + log(x))*log(x) + 27*x^4 + 54*x^2*log(x + log(x))^2 + 81*x^3*log(x) - 27*x^3 - 108*x^2*lo g(x + log(x)) + 81*x^2)
Timed out. \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\int -\frac {248\,x^2-\ln \left (x+\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (32\,x-8\right )-8\,x+32\,x^2\right )-136\,x+96\,x^3+256\,x^4+\ln \left (x\right )\,\left (256\,x^3-160\,x^2+120\,x-24\right )+16}{{\ln \left (x+\ln \left (x\right )\right )}^3\,\left (\ln \left (x\right )\,\left (65610\,x^5-393660\,x^4+590490\,x^3\right )+590490\,x^4-393660\,x^5+65610\,x^6\right )+{\ln \left (x+\ln \left (x\right )\right )}^5\,\left (6561\,x^3\,\ln \left (x\right )+6561\,x^4\right )-{\ln \left (x+\ln \left (x\right )\right )}^2\,\left (\ln \left (x\right )\,\left (-65610\,x^6+590490\,x^5-1771470\,x^4+1771470\,x^3\right )+1771470\,x^4-1771470\,x^5+590490\,x^6-65610\,x^7\right )+\ln \left (x+\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (32805\,x^7-393660\,x^6+1771470\,x^5-3542940\,x^4+2657205\,x^3\right )+2657205\,x^4-3542940\,x^5+1771470\,x^6-393660\,x^7+32805\,x^8\right )-{\ln \left (x+\ln \left (x\right )\right )}^4\,\left (\ln \left (x\right )\,\left (98415\,x^3-32805\,x^4\right )+98415\,x^4-32805\,x^5\right )-1594323\,x^4+2657205\,x^5-1771470\,x^6+590490\,x^7-98415\,x^8+6561\,x^9-\ln \left (x\right )\,\left (-6561\,x^8+98415\,x^7-590490\,x^6+1771470\,x^5-2657205\,x^4+1594323\,x^3\right )} \,d x \] Input:
int(-(248*x^2 - log(x + log(x))*(log(x)*(32*x - 8) - 8*x + 32*x^2) - 136*x + 96*x^3 + 256*x^4 + log(x)*(120*x - 160*x^2 + 256*x^3 - 24) + 16)/(log(x + log(x))^3*(log(x)*(590490*x^3 - 393660*x^4 + 65610*x^5) + 590490*x^4 - 393660*x^5 + 65610*x^6) + log(x + log(x))^5*(6561*x^3*log(x) + 6561*x^4) - log(x + log(x))^2*(log(x)*(1771470*x^3 - 1771470*x^4 + 590490*x^5 - 65610 *x^6) + 1771470*x^4 - 1771470*x^5 + 590490*x^6 - 65610*x^7) + log(x + log( x))*(log(x)*(2657205*x^3 - 3542940*x^4 + 1771470*x^5 - 393660*x^6 + 32805* x^7) + 2657205*x^4 - 3542940*x^5 + 1771470*x^6 - 393660*x^7 + 32805*x^8) - log(x + log(x))^4*(log(x)*(98415*x^3 - 32805*x^4) + 98415*x^4 - 32805*x^5 ) - 1594323*x^4 + 2657205*x^5 - 1771470*x^6 + 590490*x^7 - 98415*x^8 + 656 1*x^9 - log(x)*(1594323*x^3 - 2657205*x^4 + 1771470*x^5 - 590490*x^6 + 984 15*x^7 - 6561*x^8)),x)
Output:
int(-(248*x^2 - log(x + log(x))*(log(x)*(32*x - 8) - 8*x + 32*x^2) - 136*x + 96*x^3 + 256*x^4 + log(x)*(120*x - 160*x^2 + 256*x^3 - 24) + 16)/(log(x + log(x))^3*(log(x)*(590490*x^3 - 393660*x^4 + 65610*x^5) + 590490*x^4 - 393660*x^5 + 65610*x^6) + log(x + log(x))^5*(6561*x^3*log(x) + 6561*x^4) - log(x + log(x))^2*(log(x)*(1771470*x^3 - 1771470*x^4 + 590490*x^5 - 65610 *x^6) + 1771470*x^4 - 1771470*x^5 + 590490*x^6 - 65610*x^7) + log(x + log( x))*(log(x)*(2657205*x^3 - 3542940*x^4 + 1771470*x^5 - 393660*x^6 + 32805* x^7) + 2657205*x^4 - 3542940*x^5 + 1771470*x^6 - 393660*x^7 + 32805*x^8) - log(x + log(x))^4*(log(x)*(98415*x^3 - 32805*x^4) + 98415*x^4 - 32805*x^5 ) - 1594323*x^4 + 2657205*x^5 - 1771470*x^6 + 590490*x^7 - 98415*x^8 + 656 1*x^9 - log(x)*(1594323*x^3 - 2657205*x^4 + 1771470*x^5 - 590490*x^6 + 984 15*x^7 - 6561*x^8)), x)
Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 5.29 \[ \int \frac {-16+136 x-248 x^2-96 x^3-256 x^4+\left (24-120 x+160 x^2-256 x^3\right ) \log (x)+\left (-8 x+32 x^2+(-8+32 x) \log (x)\right ) \log (x+\log (x))}{-1594323 x^4+2657205 x^5-1771470 x^6+590490 x^7-98415 x^8+6561 x^9+\left (-1594323 x^3+2657205 x^4-1771470 x^5+590490 x^6-98415 x^7+6561 x^8\right ) \log (x)+\left (2657205 x^4-3542940 x^5+1771470 x^6-393660 x^7+32805 x^8+\left (2657205 x^3-3542940 x^4+1771470 x^5-393660 x^6+32805 x^7\right ) \log (x)\right ) \log (x+\log (x))+\left (-1771470 x^4+1771470 x^5-590490 x^6+65610 x^7+\left (-1771470 x^3+1771470 x^4-590490 x^5+65610 x^6\right ) \log (x)\right ) \log ^2(x+\log (x))+\left (590490 x^4-393660 x^5+65610 x^6+\left (590490 x^3-393660 x^4+65610 x^5\right ) \log (x)\right ) \log ^3(x+\log (x))+\left (-98415 x^4+32805 x^5+\left (-98415 x^3+32805 x^4\right ) \log (x)\right ) \log ^4(x+\log (x))+\left (6561 x^4+6561 x^3 \log (x)\right ) \log ^5(x+\log (x))} \, dx=\frac {\frac {64}{6561} x^{2}-\frac {32}{6561} x +\frac {4}{6561}}{x^{2} \left (\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{4}+4 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{3} x -12 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{3}+6 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2} x^{2}-36 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2} x +54 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2}+4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right ) x^{3}-36 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right ) x^{2}+108 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right ) x -108 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )+x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )} \] Input:
int((((32*x-8)*log(x)+32*x^2-8*x)*log(x+log(x))+(-256*x^3+160*x^2-120*x+24 )*log(x)-256*x^4-96*x^3-248*x^2+136*x-16)/((6561*x^3*log(x)+6561*x^4)*log( x+log(x))^5+((32805*x^4-98415*x^3)*log(x)+32805*x^5-98415*x^4)*log(x+log(x ))^4+((65610*x^5-393660*x^4+590490*x^3)*log(x)+65610*x^6-393660*x^5+590490 *x^4)*log(x+log(x))^3+((65610*x^6-590490*x^5+1771470*x^4-1771470*x^3)*log( x)+65610*x^7-590490*x^6+1771470*x^5-1771470*x^4)*log(x+log(x))^2+((32805*x ^7-393660*x^6+1771470*x^5-3542940*x^4+2657205*x^3)*log(x)+32805*x^8-393660 *x^7+1771470*x^6-3542940*x^5+2657205*x^4)*log(x+log(x))+(6561*x^8-98415*x^ 7+590490*x^6-1771470*x^5+2657205*x^4-1594323*x^3)*log(x)+6561*x^9-98415*x^ 8+590490*x^7-1771470*x^6+2657205*x^5-1594323*x^4),x)
Output:
(4*(16*x**2 - 8*x + 1))/(6561*x**2*(log(log(x) + x)**4 + 4*log(log(x) + x) **3*x - 12*log(log(x) + x)**3 + 6*log(log(x) + x)**2*x**2 - 36*log(log(x) + x)**2*x + 54*log(log(x) + x)**2 + 4*log(log(x) + x)*x**3 - 36*log(log(x) + x)*x**2 + 108*log(log(x) + x)*x - 108*log(log(x) + x) + x**4 - 12*x**3 + 54*x**2 - 108*x + 81))