Integrand size = 535, antiderivative size = 27 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {x}{(3+x)^2}+\frac {x^2}{\left (x+\log \left (4-e^3+x+\log (x)\right )\right )^2} \] Output:
x/(3+x)^2+x^2/(x+ln(ln(x)-exp(3)+4+x))^2
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=x \left (\frac {1}{(3+x)^2}+\frac {x}{\left (x+\log \left (4-e^3+x+\log (x)\right )\right )^2}\right ) \] Input:
Integrate[(-54*x - 108*x^2 - 60*x^3 - 21*x^4 - 3*x^5 + E^3*(-3*x^3 + x^4) + (3*x^3 - x^4)*Log[x] + (216*x + 306*x^2 + 123*x^3 + 23*x^4 + 2*x^5 + E^3 *(-54*x - 63*x^2 - 15*x^3 - 2*x^4) + (54*x + 63*x^2 + 15*x^3 + 2*x^4)*Log[ x])*Log[4 - E^3 + x + Log[x]] + (36*x - 3*x^2 - 3*x^3 + E^3*(-9*x + 3*x^2) + (9*x - 3*x^2)*Log[x])*Log[4 - E^3 + x + Log[x]]^2 + (12 + E^3*(-3 + x) - x - x^2 + (3 - x)*Log[x])*Log[4 - E^3 + x + Log[x]]^3)/(108*x^3 + 135*x^ 4 + 63*x^5 + 13*x^6 + x^7 + E^3*(-27*x^3 - 27*x^4 - 9*x^5 - x^6) + (27*x^3 + 27*x^4 + 9*x^5 + x^6)*Log[x] + (324*x^2 + 405*x^3 + 189*x^4 + 39*x^5 + 3*x^6 + E^3*(-81*x^2 - 81*x^3 - 27*x^4 - 3*x^5) + (81*x^2 + 81*x^3 + 27*x^ 4 + 3*x^5)*Log[x])*Log[4 - E^3 + x + Log[x]] + (324*x + 405*x^2 + 189*x^3 + 39*x^4 + 3*x^5 + E^3*(-81*x - 81*x^2 - 27*x^3 - 3*x^4) + (81*x + 81*x^2 + 27*x^3 + 3*x^4)*Log[x])*Log[4 - E^3 + x + Log[x]]^2 + (108 + 135*x + 63* x^2 + 13*x^3 + x^4 + E^3*(-27 - 27*x - 9*x^2 - x^3) + (27 + 27*x + 9*x^2 + x^3)*Log[x])*Log[4 - E^3 + x + Log[x]]^3),x]
Output:
x*((3 + x)^(-2) + x/(x + Log[4 - E^3 + x + Log[x]])^2)
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.
| \(\displaystyle \int \frac {-3 x^5-21 x^4-60 x^3-108 x^2+\left (-x^2-x+e^3 (x-3)+(3-x) \log (x)+12\right ) \log ^3\left (x+\log (x)-e^3+4\right )+e^3 \left (x^4-3 x^3\right )+\left (3 x^3-x^4\right ) \log (x)+\left (-3 x^3-3 x^2+e^3 \left (3 x^2-9 x\right )+\left (9 x-3 x^2\right ) \log (x)+36 x\right ) \log ^2\left (x+\log (x)-e^3+4\right )+\left (2 x^5+23 x^4+123 x^3+306 x^2+e^3 \left (-2 x^4-15 x^3-63 x^2-54 x\right )+\left (2 x^4+15 x^3+63 x^2+54 x\right ) \log (x)+216 x\right ) \log \left (x+\log (x)-e^3+4\right )-54 x}{x^7+13 x^6+63 x^5+135 x^4+108 x^3+\left (x^4+13 x^3+63 x^2+e^3 \left (-x^3-9 x^2-27 x-27\right )+\left (x^3+9 x^2+27 x+27\right ) \log (x)+135 x+108\right ) \log ^3\left (x+\log (x)-e^3+4\right )+e^3 \left (-x^6-9 x^5-27 x^4-27 x^3\right )+\left (x^6+9 x^5+27 x^4+27 x^3\right ) \log (x)+\left (3 x^5+39 x^4+189 x^3+405 x^2+e^3 \left (-3 x^4-27 x^3-81 x^2-81 x\right )+\left (3 x^4+27 x^3+81 x^2+81 x\right ) \log (x)+324 x\right ) \log ^2\left (x+\log (x)-e^3+4\right )+\left (3 x^6+39 x^5+189 x^4+405 x^3+324 x^2+e^3 \left (-3 x^5-27 x^4-81 x^3-81 x^2\right )+\left (3 x^5+27 x^4+81 x^3+81 x^2\right ) \log (x)\right ) \log \left (x+\log (x)-e^3+4\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left ((x-3) x^3\right )+\left (2 x^3+15 x^2+63 x+54\right ) x \log \left (x+\log (x)-e^3+4\right )-(x-3) \log ^3\left (x+\log (x)-e^3+4\right )-3 (x-3) x \log ^2\left (x+\log (x)-e^3+4\right )\right )-\left (-x+e^3-4\right ) x \left (2 x^3+15 x^2+63 x+54\right ) \log \left (x+\log (x)-e^3+4\right )-x \left (3 x^4-\left (e^3-21\right ) x^3+3 \left (20+e^3\right ) x^2+108 x+54\right )+\left (-x+e^3-4\right ) (x-3) \log ^3\left (x+\log (x)-e^3+4\right )+3 \left (-x+e^3-4\right ) (x-3) x \log ^2\left (x+\log (x)-e^3+4\right )}{(x+3)^3 \left (x+\log (x)+4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x \left (-x^2-5 \left (1-\frac {e^3}{5}\right ) x-x \log (x)-1\right )}{\left (x+\log (x)+4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}+\frac {3-x}{(x+3)^3}+\frac {2 x}{\left (x+\log \left (x+\log (x)-e^3+4\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {x^3}{\left (-x-\log (x)-4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}dx+2 \int \frac {x^2 \log (x)}{\left (-x-\log (x)-4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}dx-2 \left (5-e^3\right ) \int \frac {x^2}{\left (x+\log (x)+4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}dx+2 \int \frac {x}{\left (-x-\log (x)-4 \left (1-\frac {e^3}{4}\right )\right ) \left (x+\log \left (x+\log (x)-e^3+4\right )\right )^3}dx+2 \int \frac {x}{\left (x+\log \left (x+\log (x)-e^3+4\right )\right )^2}dx+\frac {x}{(x+3)^2}\) |
Input:
Int[(-54*x - 108*x^2 - 60*x^3 - 21*x^4 - 3*x^5 + E^3*(-3*x^3 + x^4) + (3*x ^3 - x^4)*Log[x] + (216*x + 306*x^2 + 123*x^3 + 23*x^4 + 2*x^5 + E^3*(-54* x - 63*x^2 - 15*x^3 - 2*x^4) + (54*x + 63*x^2 + 15*x^3 + 2*x^4)*Log[x])*Lo g[4 - E^3 + x + Log[x]] + (36*x - 3*x^2 - 3*x^3 + E^3*(-9*x + 3*x^2) + (9* x - 3*x^2)*Log[x])*Log[4 - E^3 + x + Log[x]]^2 + (12 + E^3*(-3 + x) - x - x^2 + (3 - x)*Log[x])*Log[4 - E^3 + x + Log[x]]^3)/(108*x^3 + 135*x^4 + 63 *x^5 + 13*x^6 + x^7 + E^3*(-27*x^3 - 27*x^4 - 9*x^5 - x^6) + (27*x^3 + 27* x^4 + 9*x^5 + x^6)*Log[x] + (324*x^2 + 405*x^3 + 189*x^4 + 39*x^5 + 3*x^6 + E^3*(-81*x^2 - 81*x^3 - 27*x^4 - 3*x^5) + (81*x^2 + 81*x^3 + 27*x^4 + 3* x^5)*Log[x])*Log[4 - E^3 + x + Log[x]] + (324*x + 405*x^2 + 189*x^3 + 39*x ^4 + 3*x^5 + E^3*(-81*x - 81*x^2 - 27*x^3 - 3*x^4) + (81*x + 81*x^2 + 27*x ^3 + 3*x^4)*Log[x])*Log[4 - E^3 + x + Log[x]]^2 + (108 + 135*x + 63*x^2 + 13*x^3 + x^4 + E^3*(-27 - 27*x - 9*x^2 - x^3) + (27 + 27*x + 9*x^2 + x^3)* Log[x])*Log[4 - E^3 + x + Log[x]]^3),x]
Output:
$Aborted
Time = 8.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(\frac {x}{\left (3+x \right )^{2}}+\frac {x^{2}}{{\left (x +\ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )\right )}^{2}}\) | \(27\) |
| risch | \(\frac {x}{x^{2}+6 x +9}+\frac {x^{2}}{{\left (x +\ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )\right )}^{2}}\) | \(32\) |
| parallelrisch | \(-\frac {-45 x^{2}-36 x^{3}-5 x^{4}+9 \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )^{2}+18 x \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )+\ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )^{2} x^{2}+2 \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right ) x^{3}}{6 \left (x^{4}+2 \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right ) x^{3}+\ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )^{2} x^{2}+6 x^{3}+12 x^{2} \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )+6 x \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )^{2}+9 x^{2}+18 x \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )+9 \ln \left (\ln \left (x \right )-{\mathrm e}^{3}+4+x \right )^{2}\right )}\) | \(181\) |
Input:
int((((-x+3)*ln(x)+exp(3)*(-3+x)-x^2-x+12)*ln(ln(x)-exp(3)+4+x)^3+((-3*x^2 +9*x)*ln(x)+(3*x^2-9*x)*exp(3)-3*x^3-3*x^2+36*x)*ln(ln(x)-exp(3)+4+x)^2+(( 2*x^4+15*x^3+63*x^2+54*x)*ln(x)+(-2*x^4-15*x^3-63*x^2-54*x)*exp(3)+2*x^5+2 3*x^4+123*x^3+306*x^2+216*x)*ln(ln(x)-exp(3)+4+x)+(-x^4+3*x^3)*ln(x)+(x^4- 3*x^3)*exp(3)-3*x^5-21*x^4-60*x^3-108*x^2-54*x)/(((x^3+9*x^2+27*x+27)*ln(x )+(-x^3-9*x^2-27*x-27)*exp(3)+x^4+13*x^3+63*x^2+135*x+108)*ln(ln(x)-exp(3) +4+x)^3+((3*x^4+27*x^3+81*x^2+81*x)*ln(x)+(-3*x^4-27*x^3-81*x^2-81*x)*exp( 3)+3*x^5+39*x^4+189*x^3+405*x^2+324*x)*ln(ln(x)-exp(3)+4+x)^2+((3*x^5+27*x ^4+81*x^3+81*x^2)*ln(x)+(-3*x^5-27*x^4-81*x^3-81*x^2)*exp(3)+3*x^6+39*x^5+ 189*x^4+405*x^3+324*x^2)*ln(ln(x)-exp(3)+4+x)+(x^6+9*x^5+27*x^4+27*x^3)*ln (x)+(-x^6-9*x^5-27*x^4-27*x^3)*exp(3)+x^7+13*x^6+63*x^5+135*x^4+108*x^3),x ,method=_RETURNVERBOSE)
Output:
x/(3+x)^2+x^2/(x+ln(ln(x)-exp(3)+4+x))^2
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {x^{4} + 7 \, x^{3} + 2 \, x^{2} \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + x \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2}}{x^{4} + 6 \, x^{3} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2} + 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \left (x - e^{3} + \log \left (x\right ) + 4\right )} \] Input:
integrate((((3-x)*log(x)+exp(3)*(-3+x)-x^2-x+12)*log(log(x)-exp(3)+4+x)^3+ ((-3*x^2+9*x)*log(x)+(3*x^2-9*x)*exp(3)-3*x^3-3*x^2+36*x)*log(log(x)-exp(3 )+4+x)^2+((2*x^4+15*x^3+63*x^2+54*x)*log(x)+(-2*x^4-15*x^3-63*x^2-54*x)*ex p(3)+2*x^5+23*x^4+123*x^3+306*x^2+216*x)*log(log(x)-exp(3)+4+x)+(-x^4+3*x^ 3)*log(x)+(x^4-3*x^3)*exp(3)-3*x^5-21*x^4-60*x^3-108*x^2-54*x)/(((x^3+9*x^ 2+27*x+27)*log(x)+(-x^3-9*x^2-27*x-27)*exp(3)+x^4+13*x^3+63*x^2+135*x+108) *log(log(x)-exp(3)+4+x)^3+((3*x^4+27*x^3+81*x^2+81*x)*log(x)+(-3*x^4-27*x^ 3-81*x^2-81*x)*exp(3)+3*x^5+39*x^4+189*x^3+405*x^2+324*x)*log(log(x)-exp(3 )+4+x)^2+((3*x^5+27*x^4+81*x^3+81*x^2)*log(x)+(-3*x^5-27*x^4-81*x^3-81*x^2 )*exp(3)+3*x^6+39*x^5+189*x^4+405*x^3+324*x^2)*log(log(x)-exp(3)+4+x)+(x^6 +9*x^5+27*x^4+27*x^3)*log(x)+(-x^6-9*x^5-27*x^4-27*x^3)*exp(3)+x^7+13*x^6+ 63*x^5+135*x^4+108*x^3),x, algorithm="fricas")
Output:
(x^4 + 7*x^3 + 2*x^2*log(x - e^3 + log(x) + 4) + x*log(x - e^3 + log(x) + 4)^2 + 9*x^2)/(x^4 + 6*x^3 + (x^2 + 6*x + 9)*log(x - e^3 + log(x) + 4)^2 + 9*x^2 + 2*(x^3 + 6*x^2 + 9*x)*log(x - e^3 + log(x) + 4))
Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {x^{2}}{x^{2} + 2 x \log {\left (x + \log {\left (x \right )} - e^{3} + 4 \right )} + \log {\left (x + \log {\left (x \right )} - e^{3} + 4 \right )}^{2}} + \frac {x}{x^{2} + 6 x + 9} \] Input:
integrate((((3-x)*ln(x)+exp(3)*(-3+x)-x**2-x+12)*ln(ln(x)-exp(3)+4+x)**3+( (-3*x**2+9*x)*ln(x)+(3*x**2-9*x)*exp(3)-3*x**3-3*x**2+36*x)*ln(ln(x)-exp(3 )+4+x)**2+((2*x**4+15*x**3+63*x**2+54*x)*ln(x)+(-2*x**4-15*x**3-63*x**2-54 *x)*exp(3)+2*x**5+23*x**4+123*x**3+306*x**2+216*x)*ln(ln(x)-exp(3)+4+x)+(- x**4+3*x**3)*ln(x)+(x**4-3*x**3)*exp(3)-3*x**5-21*x**4-60*x**3-108*x**2-54 *x)/(((x**3+9*x**2+27*x+27)*ln(x)+(-x**3-9*x**2-27*x-27)*exp(3)+x**4+13*x* *3+63*x**2+135*x+108)*ln(ln(x)-exp(3)+4+x)**3+((3*x**4+27*x**3+81*x**2+81* x)*ln(x)+(-3*x**4-27*x**3-81*x**2-81*x)*exp(3)+3*x**5+39*x**4+189*x**3+405 *x**2+324*x)*ln(ln(x)-exp(3)+4+x)**2+((3*x**5+27*x**4+81*x**3+81*x**2)*ln( x)+(-3*x**5-27*x**4-81*x**3-81*x**2)*exp(3)+3*x**6+39*x**5+189*x**4+405*x* *3+324*x**2)*ln(ln(x)-exp(3)+4+x)+(x**6+9*x**5+27*x**4+27*x**3)*ln(x)+(-x* *6-9*x**5-27*x**4-27*x**3)*exp(3)+x**7+13*x**6+63*x**5+135*x**4+108*x**3), x)
Output:
x**2/(x**2 + 2*x*log(x + log(x) - exp(3) + 4) + log(x + log(x) - exp(3) + 4)**2) + x/(x**2 + 6*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {x^{4} + 7 \, x^{3} + 2 \, x^{2} \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + x \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2}}{x^{4} + 6 \, x^{3} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2} + 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \left (x - e^{3} + \log \left (x\right ) + 4\right )} \] Input:
integrate((((3-x)*log(x)+exp(3)*(-3+x)-x^2-x+12)*log(log(x)-exp(3)+4+x)^3+ ((-3*x^2+9*x)*log(x)+(3*x^2-9*x)*exp(3)-3*x^3-3*x^2+36*x)*log(log(x)-exp(3 )+4+x)^2+((2*x^4+15*x^3+63*x^2+54*x)*log(x)+(-2*x^4-15*x^3-63*x^2-54*x)*ex p(3)+2*x^5+23*x^4+123*x^3+306*x^2+216*x)*log(log(x)-exp(3)+4+x)+(-x^4+3*x^ 3)*log(x)+(x^4-3*x^3)*exp(3)-3*x^5-21*x^4-60*x^3-108*x^2-54*x)/(((x^3+9*x^ 2+27*x+27)*log(x)+(-x^3-9*x^2-27*x-27)*exp(3)+x^4+13*x^3+63*x^2+135*x+108) *log(log(x)-exp(3)+4+x)^3+((3*x^4+27*x^3+81*x^2+81*x)*log(x)+(-3*x^4-27*x^ 3-81*x^2-81*x)*exp(3)+3*x^5+39*x^4+189*x^3+405*x^2+324*x)*log(log(x)-exp(3 )+4+x)^2+((3*x^5+27*x^4+81*x^3+81*x^2)*log(x)+(-3*x^5-27*x^4-81*x^3-81*x^2 )*exp(3)+3*x^6+39*x^5+189*x^4+405*x^3+324*x^2)*log(log(x)-exp(3)+4+x)+(x^6 +9*x^5+27*x^4+27*x^3)*log(x)+(-x^6-9*x^5-27*x^4-27*x^3)*exp(3)+x^7+13*x^6+ 63*x^5+135*x^4+108*x^3),x, algorithm="maxima")
Output:
(x^4 + 7*x^3 + 2*x^2*log(x - e^3 + log(x) + 4) + x*log(x - e^3 + log(x) + 4)^2 + 9*x^2)/(x^4 + 6*x^3 + (x^2 + 6*x + 9)*log(x - e^3 + log(x) + 4)^2 + 9*x^2 + 2*(x^3 + 6*x^2 + 9*x)*log(x - e^3 + log(x) + 4))
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (26) = 52\).
Time = 1.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.48 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {x^{4} + 7 \, x^{3} + 2 \, x^{2} \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + x \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2}}{x^{4} + 2 \, x^{3} \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + x^{2} \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 6 \, x^{3} + 12 \, x^{2} \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + 6 \, x \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2} + 9 \, x^{2} + 18 \, x \log \left (x - e^{3} + \log \left (x\right ) + 4\right ) + 9 \, \log \left (x - e^{3} + \log \left (x\right ) + 4\right )^{2}} \] Input:
integrate((((3-x)*log(x)+exp(3)*(-3+x)-x^2-x+12)*log(log(x)-exp(3)+4+x)^3+ ((-3*x^2+9*x)*log(x)+(3*x^2-9*x)*exp(3)-3*x^3-3*x^2+36*x)*log(log(x)-exp(3 )+4+x)^2+((2*x^4+15*x^3+63*x^2+54*x)*log(x)+(-2*x^4-15*x^3-63*x^2-54*x)*ex p(3)+2*x^5+23*x^4+123*x^3+306*x^2+216*x)*log(log(x)-exp(3)+4+x)+(-x^4+3*x^ 3)*log(x)+(x^4-3*x^3)*exp(3)-3*x^5-21*x^4-60*x^3-108*x^2-54*x)/(((x^3+9*x^ 2+27*x+27)*log(x)+(-x^3-9*x^2-27*x-27)*exp(3)+x^4+13*x^3+63*x^2+135*x+108) *log(log(x)-exp(3)+4+x)^3+((3*x^4+27*x^3+81*x^2+81*x)*log(x)+(-3*x^4-27*x^ 3-81*x^2-81*x)*exp(3)+3*x^5+39*x^4+189*x^3+405*x^2+324*x)*log(log(x)-exp(3 )+4+x)^2+((3*x^5+27*x^4+81*x^3+81*x^2)*log(x)+(-3*x^5-27*x^4-81*x^3-81*x^2 )*exp(3)+3*x^6+39*x^5+189*x^4+405*x^3+324*x^2)*log(log(x)-exp(3)+4+x)+(x^6 +9*x^5+27*x^4+27*x^3)*log(x)+(-x^6-9*x^5-27*x^4-27*x^3)*exp(3)+x^7+13*x^6+ 63*x^5+135*x^4+108*x^3),x, algorithm="giac")
Output:
(x^4 + 7*x^3 + 2*x^2*log(x - e^3 + log(x) + 4) + x*log(x - e^3 + log(x) + 4)^2 + 9*x^2)/(x^4 + 2*x^3*log(x - e^3 + log(x) + 4) + x^2*log(x - e^3 + l og(x) + 4)^2 + 6*x^3 + 12*x^2*log(x - e^3 + log(x) + 4) + 6*x*log(x - e^3 + log(x) + 4)^2 + 9*x^2 + 18*x*log(x - e^3 + log(x) + 4) + 9*log(x - e^3 + log(x) + 4)^2)
Time = 5.60 (sec) , antiderivative size = 1690, normalized size of antiderivative = 62.59 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\text {Too large to display} \] Input:
int(-(54*x - log(x)*(3*x^3 - x^4) + log(x - exp(3) + log(x) + 4)^3*(x + lo g(x)*(x - 3) - exp(3)*(x - 3) + x^2 - 12) - log(x - exp(3) + log(x) + 4)*( 216*x + log(x)*(54*x + 63*x^2 + 15*x^3 + 2*x^4) - exp(3)*(54*x + 63*x^2 + 15*x^3 + 2*x^4) + 306*x^2 + 123*x^3 + 23*x^4 + 2*x^5) + log(x - exp(3) + l og(x) + 4)^2*(exp(3)*(9*x - 3*x^2) - 36*x - log(x)*(9*x - 3*x^2) + 3*x^2 + 3*x^3) + exp(3)*(3*x^3 - x^4) + 108*x^2 + 60*x^3 + 21*x^4 + 3*x^5)/(log(x )*(27*x^3 + 27*x^4 + 9*x^5 + x^6) + log(x - exp(3) + log(x) + 4)*(log(x)*( 81*x^2 + 81*x^3 + 27*x^4 + 3*x^5) + 324*x^2 + 405*x^3 + 189*x^4 + 39*x^5 + 3*x^6 - exp(3)*(81*x^2 + 81*x^3 + 27*x^4 + 3*x^5)) + log(x - exp(3) + log (x) + 4)^2*(324*x + log(x)*(81*x + 81*x^2 + 27*x^3 + 3*x^4) - exp(3)*(81*x + 81*x^2 + 27*x^3 + 3*x^4) + 405*x^2 + 189*x^3 + 39*x^4 + 3*x^5) - exp(3) *(27*x^3 + 27*x^4 + 9*x^5 + x^6) + log(x - exp(3) + log(x) + 4)^3*(135*x - exp(3)*(27*x + 9*x^2 + x^3 + 27) + log(x)*(27*x + 9*x^2 + x^3 + 27) + 63* x^2 + 13*x^3 + x^4 + 108) + 108*x^3 + 135*x^4 + 63*x^5 + 13*x^6 + x^7),x)
Output:
((217*x + 30*exp(3) - 4*exp(6) - 94*x*exp(3) + 11*x*exp(6) - 225*x^2*exp(3 ) - 182*x^3*exp(3) - 20*x^4*exp(3) + 28*x^2*exp(6) + 14*x^5*exp(3) + 29*x^ 3*exp(6) + 4*x^6*exp(3) + 3*x^4*exp(6) - x^2*exp(9) - 2*x^3*exp(9) + 533*x ^2 + 411*x^3 + 40*x^4 - 66*x^5 - 25*x^6 - 2*x^7 - 51)/(6*(x + x^2 - 1)^3) + (log(x)*(94*x + 8*exp(3) - 22*x*exp(3) - 56*x^2*exp(3) - 58*x^3*exp(3) - 6*x^4*exp(3) + 3*x^2*exp(6) + 6*x^3*exp(6) + 225*x^2 + 182*x^3 + 20*x^4 - 14*x^5 - 4*x^6 - 30))/(6*(x + x^2 - 1)^3) + (log(x)^2*(11*x - 3*x^2*exp(3 ) - 6*x^3*exp(3) + 28*x^2 + 29*x^3 + 3*x^4 - 4))/(6*(x + x^2 - 1)^3) + (x^ 2*log(x)^3*(2*x + 1))/(6*(x + x^2 - 1)^3))/(log(x)^2 + (5*x - x*exp(3) + x ^2 + 1)^2/x^2 + (2*log(x)*(5*x - x*exp(3) + x^2 + 1))/x) - (log(x)^3/(3*(x + x^2 - 1)) + (174*x - 38*exp(3) + 4*exp(6) - 77*x*exp(3) + 13*x*exp(6) - x*exp(9) - 56*x^2*exp(3) - 30*x^3*exp(3) - 4*x^4*exp(3) + 6*x^2*exp(6) + 3*x^3*exp(6) + 151*x^2 + 83*x^3 + 18*x^4 + x^5 + 89)/(3*x*(x + x^2 - 1)) + (log(x)^2*(13*x - 3*x*exp(3) + 6*x^2 + 3*x^3 + 4))/(3*x*(x + x^2 - 1)) + (log(x)*(77*x - 8*exp(3) - 26*x*exp(3) + 3*x*exp(6) - 12*x^2*exp(3) - 6*x^ 3*exp(3) + 56*x^2 + 30*x^3 + 4*x^4 + 38))/(3*x*(x + x^2 - 1)))/(log(x)^3 + (5*x - x*exp(3) + x^2 + 1)^3/x^3 + (3*log(x)^2*(5*x - x*exp(3) + x^2 + 1) )/x + (3*log(x)*(5*x - x*exp(3) + x^2 + 1)^2)/x^2) - ((x*(188*x + 8*exp(3) - 34*x*exp(3) - x*exp(6) - 500*x^2*exp(3) - 1221*x^3*exp(3) - 872*x^4*exp (3) + 57*x^2*exp(6) - 392*x^5*exp(3) + 173*x^3*exp(6) - 38*x^6*exp(3) +...
Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 7.78 \[ \int \frac {-54 x-108 x^2-60 x^3-21 x^4-3 x^5+e^3 \left (-3 x^3+x^4\right )+\left (3 x^3-x^4\right ) \log (x)+\left (216 x+306 x^2+123 x^3+23 x^4+2 x^5+e^3 \left (-54 x-63 x^2-15 x^3-2 x^4\right )+\left (54 x+63 x^2+15 x^3+2 x^4\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (36 x-3 x^2-3 x^3+e^3 \left (-9 x+3 x^2\right )+\left (9 x-3 x^2\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (12+e^3 (-3+x)-x-x^2+(3-x) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )}{108 x^3+135 x^4+63 x^5+13 x^6+x^7+e^3 \left (-27 x^3-27 x^4-9 x^5-x^6\right )+\left (27 x^3+27 x^4+9 x^5+x^6\right ) \log (x)+\left (324 x^2+405 x^3+189 x^4+39 x^5+3 x^6+e^3 \left (-81 x^2-81 x^3-27 x^4-3 x^5\right )+\left (81 x^2+81 x^3+27 x^4+3 x^5\right ) \log (x)\right ) \log \left (4-e^3+x+\log (x)\right )+\left (324 x+405 x^2+189 x^3+39 x^4+3 x^5+e^3 \left (-81 x-81 x^2-27 x^3-3 x^4\right )+\left (81 x+81 x^2+27 x^3+3 x^4\right ) \log (x)\right ) \log ^2\left (4-e^3+x+\log (x)\right )+\left (108+135 x+63 x^2+13 x^3+x^4+e^3 \left (-27-27 x-9 x^2-x^3\right )+\left (27+27 x+9 x^2+x^3\right ) \log (x)\right ) \log ^3\left (4-e^3+x+\log (x)\right )} \, dx=\frac {-\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2} x^{2}-5 \mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2} x -9 \mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x^{3}-10 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x^{2}-18 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x +x^{3}}{\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2} x^{2}+6 \mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2} x +9 \mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x^{3}+12 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x^{2}+18 \,\mathrm {log}\left (\mathrm {log}\left (x \right )-e^{3}+x +4\right ) x +x^{4}+6 x^{3}+9 x^{2}} \] Input:
int((((3-x)*log(x)+exp(3)*(-3+x)-x^2-x+12)*log(log(x)-exp(3)+4+x)^3+((-3*x ^2+9*x)*log(x)+(3*x^2-9*x)*exp(3)-3*x^3-3*x^2+36*x)*log(log(x)-exp(3)+4+x) ^2+((2*x^4+15*x^3+63*x^2+54*x)*log(x)+(-2*x^4-15*x^3-63*x^2-54*x)*exp(3)+2 *x^5+23*x^4+123*x^3+306*x^2+216*x)*log(log(x)-exp(3)+4+x)+(-x^4+3*x^3)*log (x)+(x^4-3*x^3)*exp(3)-3*x^5-21*x^4-60*x^3-108*x^2-54*x)/(((x^3+9*x^2+27*x +27)*log(x)+(-x^3-9*x^2-27*x-27)*exp(3)+x^4+13*x^3+63*x^2+135*x+108)*log(l og(x)-exp(3)+4+x)^3+((3*x^4+27*x^3+81*x^2+81*x)*log(x)+(-3*x^4-27*x^3-81*x ^2-81*x)*exp(3)+3*x^5+39*x^4+189*x^3+405*x^2+324*x)*log(log(x)-exp(3)+4+x) ^2+((3*x^5+27*x^4+81*x^3+81*x^2)*log(x)+(-3*x^5-27*x^4-81*x^3-81*x^2)*exp( 3)+3*x^6+39*x^5+189*x^4+405*x^3+324*x^2)*log(log(x)-exp(3)+4+x)+(x^6+9*x^5 +27*x^4+27*x^3)*log(x)+(-x^6-9*x^5-27*x^4-27*x^3)*exp(3)+x^7+13*x^6+63*x^5 +135*x^4+108*x^3),x)
Output:
( - log(log(x) - e**3 + x + 4)**2*x**2 - 5*log(log(x) - e**3 + x + 4)**2*x - 9*log(log(x) - e**3 + x + 4)**2 - 2*log(log(x) - e**3 + x + 4)*x**3 - 1 0*log(log(x) - e**3 + x + 4)*x**2 - 18*log(log(x) - e**3 + x + 4)*x + x**3 )/(log(log(x) - e**3 + x + 4)**2*x**2 + 6*log(log(x) - e**3 + x + 4)**2*x + 9*log(log(x) - e**3 + x + 4)**2 + 2*log(log(x) - e**3 + x + 4)*x**3 + 12 *log(log(x) - e**3 + x + 4)*x**2 + 18*log(log(x) - e**3 + x + 4)*x + x**4 + 6*x**3 + 9*x**2)