Integrand size = 484, antiderivative size = 33 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=-1-x+\frac {x}{\left (e^{e^5}+\frac {x}{2}\right ) (x-\log (\log (5+x+\log (3))))} \] Output:
-1+x/(exp(exp(5))+1/2*x)/(x-ln(ln(ln(3)+5+x)))-x
\[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=\int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx \] Input:
Integrate[(4*E^E^5*x + 2*x^2 + (-10*x^2 - 2*x^3 - 5*x^4 - x^5 + (-2*x^2 - x^4)*Log[3] + E^(2*E^5)*(-20*x^2 - 4*x^3 - 4*x^2*Log[3]) + E^E^5*(-20*x^3 - 4*x^4 - 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (10*x^3 + 2*x^4 + 2*x^3*Log [3] + E^(2*E^5)*(40*x + 8*x^2 + 8*x*Log[3]) + E^E^5*(-20 - 4*x + 40*x^2 + 8*x^3 + (-4 + 8*x^2)*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (-5*x^2 - x^3 + E^(2*E^5)*(-20 - 4*x - 4*Log[3]) - x^2*Log[3] + E^E^5*( -20*x - 4*x^2 - 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^ 2)/((5*x^4 + x^5 + x^4*Log[3] + E^(2*E^5)*(20*x^2 + 4*x^3 + 4*x^2*Log[3]) + E^E^5*(20*x^3 + 4*x^4 + 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (-10*x^3 - 2*x^4 - 2*x^3*Log[3] + E^(2*E^5)*(-40*x - 8*x^2 - 8*x*Log[3]) + E^E^5*(-40 *x^2 - 8*x^3 - 8*x^2*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (5*x^2 + x^3 + x^2*Log[3] + E^(2*E^5)*(20 + 4*x + 4*Log[3]) + E^E^5*(20 *x + 4*x^2 + 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^2), x]
Output:
Integrate[(4*E^E^5*x + 2*x^2 + (-10*x^2 - 2*x^3 - 5*x^4 - x^5 + (-2*x^2 - x^4)*Log[3] + E^(2*E^5)*(-20*x^2 - 4*x^3 - 4*x^2*Log[3]) + E^E^5*(-20*x^3 - 4*x^4 - 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (10*x^3 + 2*x^4 + 2*x^3*Log [3] + E^(2*E^5)*(40*x + 8*x^2 + 8*x*Log[3]) + E^E^5*(-20 - 4*x + 40*x^2 + 8*x^3 + (-4 + 8*x^2)*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (-5*x^2 - x^3 + E^(2*E^5)*(-20 - 4*x - 4*Log[3]) - x^2*Log[3] + E^E^5*( -20*x - 4*x^2 - 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^ 2)/((5*x^4 + x^5 + x^4*Log[3] + E^(2*E^5)*(20*x^2 + 4*x^3 + 4*x^2*Log[3]) + E^E^5*(20*x^3 + 4*x^4 + 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (-10*x^3 - 2*x^4 - 2*x^3*Log[3] + E^(2*E^5)*(-40*x - 8*x^2 - 8*x*Log[3]) + E^E^5*(-40 *x^2 - 8*x^3 - 8*x^2*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (5*x^2 + x^3 + x^2*Log[3] + E^(2*E^5)*(20 + 4*x + 4*Log[3]) + E^E^5*(20 *x + 4*x^2 + 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^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.
| \(\displaystyle \int \frac {2 x^2+\left (-x^3-5 x^2-x^2 \log (3)+e^{e^5} \left (-4 x^2-20 x-4 x \log (3)\right )+e^{2 e^5} (-4 x-20-4 \log (3))\right ) \log (x+5+\log (3)) \log ^2(\log (x+5+\log (3)))+\left (2 x^4+10 x^3+2 x^3 \log (3)+e^{2 e^5} \left (8 x^2+40 x+8 x \log (3)\right )+e^{e^5} \left (8 x^3+40 x^2+\left (8 x^2-4\right ) \log (3)-4 x-20\right )\right ) \log (x+5+\log (3)) \log (\log (x+5+\log (3)))+\left (-x^5-5 x^4-2 x^3-10 x^2+e^{e^5} \left (-4 x^4-20 x^3-4 x^3 \log (3)\right )+\left (-x^4-2 x^2\right ) \log (3)+e^{2 e^5} \left (-4 x^3-20 x^2-4 x^2 \log (3)\right )\right ) \log (x+5+\log (3))+4 e^{e^5} x}{\left (x^3+5 x^2+x^2 \log (3)+e^{e^5} \left (4 x^2+20 x+4 x \log (3)\right )+e^{2 e^5} (4 x+20+4 \log (3))\right ) \log (x+5+\log (3)) \log ^2(\log (x+5+\log (3)))+\left (-2 x^4-10 x^3-2 x^3 \log (3)+e^{2 e^5} \left (-8 x^2-40 x-8 x \log (3)\right )+e^{e^5} \left (-8 x^3-40 x^2-8 x^2 \log (3)\right )\right ) \log (x+5+\log (3)) \log (\log (x+5+\log (3)))+\left (x^5+5 x^4+x^4 \log (3)+e^{e^5} \left (4 x^4+20 x^3+4 x^3 \log (3)\right )+e^{2 e^5} \left (4 x^3+20 x^2+4 x^2 \log (3)\right )\right ) \log (x+5+\log (3))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \left (x+2 e^{e^5}\right )-\log (x+5+\log (3)) \left (x^2 \left (x^3+x^2 (5+\log (3))+2 x+e^{e^5} x (4 x+20+\log (81))+e^{2 e^5} (4 x+20+\log (81))+10+\log (9)\right )-\left (x^3 (2 x+10+\log (9))+4 e^{e^5} \left (2 x^2-1\right ) (x+5+\log (3))+8 e^{2 e^5} x (x+5+\log (3))\right ) \log (\log (x+5+\log (3)))+\left (x+2 e^{e^5}\right )^2 (x+5+\log (3)) \log ^2(\log (x+5+\log (3)))\right )}{\left (x+2 e^{e^5}\right )^2 (x+5+\log (3)) \log (x+5+\log (3)) (x-\log (\log (x+5+\log (3))))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x \left (-2 x^2 \log (x+5+\log (3))+2 x-10 x \left (1+\frac {1}{5} \left (2 e^{e^5}+\log (3)\right )\right ) \log (x+5+\log (3))-20 e^{e^5} \left (1+\frac {\log (3)}{5}\right ) \log (x+5+\log (3))+4 e^{e^5}\right )}{\left (x+2 e^{e^5}\right )^2 (x+5+\log (3)) \log (x+5+\log (3)) (x-\log (\log (x+5+\log (3))))^2}+\frac {4 e^{e^5}}{\left (x+2 e^{e^5}\right )^2 (x-\log (\log (x+5+\log (3))))}-1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 e^{e^5} \int \frac {1}{(x-\log (\log (x+\log (3)+5)))^2}dx}{5-2 e^{e^5}+\log (3)}+\frac {4 e^{e^5} (5+\log (3)) \int \frac {1}{\left (x+2 e^{e^5}\right ) (x-\log (\log (x+\log (3)+5)))^2}dx}{5-2 e^{e^5}+\log (3)}-\frac {8 e^{2 e^5} \int \frac {1}{\left (x+2 e^{e^5}\right ) (x-\log (\log (x+\log (3)+5)))^2}dx}{5-2 e^{e^5}+\log (3)}-\frac {4 e^{e^5} \int \frac {1}{\left (x+2 e^{e^5}\right ) \log (x+\log (3)+5) (x-\log (\log (x+\log (3)+5)))^2}dx}{5-2 e^{e^5}+\log (3)}+4 e^{e^5} \int \frac {1}{\left (x+2 e^{e^5}\right )^2 (x-\log (\log (x+\log (3)+5)))}dx-x+\frac {2 (5+\log (3))}{\left (5-2 e^{e^5}+\log (3)\right ) (x-\log (\log (x+5+\log (3))))}\) |
Input:
Int[(4*E^E^5*x + 2*x^2 + (-10*x^2 - 2*x^3 - 5*x^4 - x^5 + (-2*x^2 - x^4)*L og[3] + E^(2*E^5)*(-20*x^2 - 4*x^3 - 4*x^2*Log[3]) + E^E^5*(-20*x^3 - 4*x^ 4 - 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (10*x^3 + 2*x^4 + 2*x^3*Log[3] + E^(2*E^5)*(40*x + 8*x^2 + 8*x*Log[3]) + E^E^5*(-20 - 4*x + 40*x^2 + 8*x^3 + (-4 + 8*x^2)*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (-5 *x^2 - x^3 + E^(2*E^5)*(-20 - 4*x - 4*Log[3]) - x^2*Log[3] + E^E^5*(-20*x - 4*x^2 - 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^2)/((5 *x^4 + x^5 + x^4*Log[3] + E^(2*E^5)*(20*x^2 + 4*x^3 + 4*x^2*Log[3]) + E^E^ 5*(20*x^3 + 4*x^4 + 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (-10*x^3 - 2*x^4 - 2*x^3*Log[3] + E^(2*E^5)*(-40*x - 8*x^2 - 8*x*Log[3]) + E^E^5*(-40*x^2 - 8*x^3 - 8*x^2*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (5* x^2 + x^3 + x^2*Log[3] + E^(2*E^5)*(20 + 4*x + 4*Log[3]) + E^E^5*(20*x + 4 *x^2 + 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^2),x]
Output:
$Aborted
Time = 6.54 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(-x +\frac {2 x}{\left (2 \,{\mathrm e}^{{\mathrm e}^{5}}+x \right ) \left (x -\ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )\right )}\) | \(30\) |
| default | \(-\ln \left (3\right )-5-x +\frac {2 \ln \left (\ln \left (3\right )+5+x \right ) x \left (\ln \left (3\right )+5+x \right )}{\left (\left (\ln \left (3\right )+5+x \right ) \ln \left (\ln \left (3\right )+5+x \right )-1\right ) \left (-x -2 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) \left (-x +\ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )\right )}-\frac {2 x}{\left (\left (\ln \left (3\right )+5+x \right ) \ln \left (\ln \left (3\right )+5+x \right )-1\right ) \left (-x -2 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) \left (-x +\ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )\right )}\) | \(107\) |
| parallelrisch | \(-\frac {-2 x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )-x^{2} \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )-20 x \,{\mathrm e}^{{\mathrm e}^{5}}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} x +20 \,{\mathrm e}^{{\mathrm e}^{5}} \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )+10 x \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )-2 x^{2} \ln \left (3\right )+x^{3}-10 x^{2}-4 \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{5}} x +4 \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{5}} \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )+2 \ln \left (3\right ) \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right ) x}{2 x \,{\mathrm e}^{{\mathrm e}^{5}}-2 \,{\mathrm e}^{{\mathrm e}^{5}} \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )+x^{2}-x \ln \left (\ln \left (\ln \left (3\right )+5+x \right )\right )}\) | \(152\) |
Input:
int((((-4*ln(3)-4*x-20)*exp(exp(5))^2+(-4*x*ln(3)-4*x^2-20*x)*exp(exp(5))- x^2*ln(3)-x^3-5*x^2)*ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))^2+((8*x*ln(3)+8*x^2+4 0*x)*exp(exp(5))^2+((8*x^2-4)*ln(3)+8*x^3+40*x^2-4*x-20)*exp(exp(5))+2*x^3 *ln(3)+2*x^4+10*x^3)*ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))+((-4*x^2*ln(3)-4*x^3- 20*x^2)*exp(exp(5))^2+(-4*x^3*ln(3)-4*x^4-20*x^3)*exp(exp(5))+(-x^4-2*x^2) *ln(3)-x^5-5*x^4-2*x^3-10*x^2)*ln(ln(3)+5+x)+4*x*exp(exp(5))+2*x^2)/(((4*l n(3)+20+4*x)*exp(exp(5))^2+(4*x*ln(3)+4*x^2+20*x)*exp(exp(5))+x^2*ln(3)+x^ 3+5*x^2)*ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))^2+((-8*x*ln(3)-8*x^2-40*x)*exp(ex p(5))^2+(-8*x^2*ln(3)-8*x^3-40*x^2)*exp(exp(5))-2*x^3*ln(3)-2*x^4-10*x^3)* ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))+((4*x^2*ln(3)+4*x^3+20*x^2)*exp(exp(5))^2+ (4*x^3*ln(3)+4*x^4+20*x^3)*exp(exp(5))+x^4*ln(3)+x^5+5*x^4)*ln(ln(3)+5+x)) ,x,method=_RETURNVERBOSE)
Output:
-x+2*x/(2*exp(exp(5))+x)/(x-ln(ln(ln(3)+5+x)))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=-\frac {x^{3} + 2 \, x^{2} e^{\left (e^{5}\right )} - {\left (x^{2} + 2 \, x e^{\left (e^{5}\right )}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right ) - 2 \, x}{x^{2} + 2 \, x e^{\left (e^{5}\right )} - {\left (x + 2 \, e^{\left (e^{5}\right )}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right )} \] Input:
integrate((((-4*log(3)-4*x-20)*exp(exp(5))^2+(-4*x*log(3)-4*x^2-20*x)*exp( exp(5))-x^2*log(3)-x^3-5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x))^2+((8*x *log(3)+8*x^2+40*x)*exp(exp(5))^2+((8*x^2-4)*log(3)+8*x^3+40*x^2-4*x-20)*e xp(exp(5))+2*x^3*log(3)+2*x^4+10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((-4*x^2*log(3)-4*x^3-20*x^2)*exp(exp(5))^2+(-4*x^3*log(3)-4*x^4-20*x^3)* exp(exp(5))+(-x^4-2*x^2)*log(3)-x^5-5*x^4-2*x^3-10*x^2)*log(log(3)+5+x)+4* x*exp(exp(5))+2*x^2)/(((4*log(3)+20+4*x)*exp(exp(5))^2+(4*x*log(3)+4*x^2+2 0*x)*exp(exp(5))+x^2*log(3)+x^3+5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x) )^2+((-8*x*log(3)-8*x^2-40*x)*exp(exp(5))^2+(-8*x^2*log(3)-8*x^3-40*x^2)*e xp(exp(5))-2*x^3*log(3)-2*x^4-10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((4*x^2*log(3)+4*x^3+20*x^2)*exp(exp(5))^2+(4*x^3*log(3)+4*x^4+20*x^3)*ex p(exp(5))+x^4*log(3)+x^5+5*x^4)*log(log(3)+5+x)),x, algorithm="fricas")
Output:
-(x^3 + 2*x^2*e^(e^5) - (x^2 + 2*x*e^(e^5))*log(log(x + log(3) + 5)) - 2*x )/(x^2 + 2*x*e^(e^5) - (x + 2*e^(e^5))*log(log(x + log(3) + 5)))
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=- x - \frac {2 x}{- x^{2} - 2 x e^{e^{5}} + \left (x + 2 e^{e^{5}}\right ) \log {\left (\log {\left (x + \log {\left (3 \right )} + 5 \right )} \right )}} \] Input:
integrate((((-4*ln(3)-4*x-20)*exp(exp(5))**2+(-4*x*ln(3)-4*x**2-20*x)*exp( exp(5))-x**2*ln(3)-x**3-5*x**2)*ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))**2+((8*x*l n(3)+8*x**2+40*x)*exp(exp(5))**2+((8*x**2-4)*ln(3)+8*x**3+40*x**2-4*x-20)* exp(exp(5))+2*x**3*ln(3)+2*x**4+10*x**3)*ln(ln(3)+5+x)*ln(ln(ln(3)+5+x))+( (-4*x**2*ln(3)-4*x**3-20*x**2)*exp(exp(5))**2+(-4*x**3*ln(3)-4*x**4-20*x** 3)*exp(exp(5))+(-x**4-2*x**2)*ln(3)-x**5-5*x**4-2*x**3-10*x**2)*ln(ln(3)+5 +x)+4*x*exp(exp(5))+2*x**2)/(((4*ln(3)+20+4*x)*exp(exp(5))**2+(4*x*ln(3)+4 *x**2+20*x)*exp(exp(5))+x**2*ln(3)+x**3+5*x**2)*ln(ln(3)+5+x)*ln(ln(ln(3)+ 5+x))**2+((-8*x*ln(3)-8*x**2-40*x)*exp(exp(5))**2+(-8*x**2*ln(3)-8*x**3-40 *x**2)*exp(exp(5))-2*x**3*ln(3)-2*x**4-10*x**3)*ln(ln(3)+5+x)*ln(ln(ln(3)+ 5+x))+((4*x**2*ln(3)+4*x**3+20*x**2)*exp(exp(5))**2+(4*x**3*ln(3)+4*x**4+2 0*x**3)*exp(exp(5))+x**4*ln(3)+x**5+5*x**4)*ln(ln(3)+5+x)),x)
Output:
-x - 2*x/(-x**2 - 2*x*exp(exp(5)) + (x + 2*exp(exp(5)))*log(log(x + log(3) + 5)))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=-\frac {x^{3} + 2 \, x^{2} e^{\left (e^{5}\right )} - {\left (x^{2} + 2 \, x e^{\left (e^{5}\right )}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right ) - 2 \, x}{x^{2} + 2 \, x e^{\left (e^{5}\right )} - {\left (x + 2 \, e^{\left (e^{5}\right )}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right )} \] Input:
integrate((((-4*log(3)-4*x-20)*exp(exp(5))^2+(-4*x*log(3)-4*x^2-20*x)*exp( exp(5))-x^2*log(3)-x^3-5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x))^2+((8*x *log(3)+8*x^2+40*x)*exp(exp(5))^2+((8*x^2-4)*log(3)+8*x^3+40*x^2-4*x-20)*e xp(exp(5))+2*x^3*log(3)+2*x^4+10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((-4*x^2*log(3)-4*x^3-20*x^2)*exp(exp(5))^2+(-4*x^3*log(3)-4*x^4-20*x^3)* exp(exp(5))+(-x^4-2*x^2)*log(3)-x^5-5*x^4-2*x^3-10*x^2)*log(log(3)+5+x)+4* x*exp(exp(5))+2*x^2)/(((4*log(3)+20+4*x)*exp(exp(5))^2+(4*x*log(3)+4*x^2+2 0*x)*exp(exp(5))+x^2*log(3)+x^3+5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x) )^2+((-8*x*log(3)-8*x^2-40*x)*exp(exp(5))^2+(-8*x^2*log(3)-8*x^3-40*x^2)*e xp(exp(5))-2*x^3*log(3)-2*x^4-10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((4*x^2*log(3)+4*x^3+20*x^2)*exp(exp(5))^2+(4*x^3*log(3)+4*x^4+20*x^3)*ex p(exp(5))+x^4*log(3)+x^5+5*x^4)*log(log(3)+5+x)),x, algorithm="maxima")
Output:
-(x^3 + 2*x^2*e^(e^5) - (x^2 + 2*x*e^(e^5))*log(log(x + log(3) + 5)) - 2*x )/(x^2 + 2*x*e^(e^5) - (x + 2*e^(e^5))*log(log(x + log(3) + 5)))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
Time = 0.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.30 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=-\frac {x^{3} + 2 \, x^{2} e^{\left (e^{5}\right )} - x^{2} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right ) - 2 \, x e^{\left (e^{5}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right ) - 2 \, x}{x^{2} + 2 \, x e^{\left (e^{5}\right )} - x \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right ) - 2 \, e^{\left (e^{5}\right )} \log \left (\log \left (x + \log \left (3\right ) + 5\right )\right )} \] Input:
integrate((((-4*log(3)-4*x-20)*exp(exp(5))^2+(-4*x*log(3)-4*x^2-20*x)*exp( exp(5))-x^2*log(3)-x^3-5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x))^2+((8*x *log(3)+8*x^2+40*x)*exp(exp(5))^2+((8*x^2-4)*log(3)+8*x^3+40*x^2-4*x-20)*e xp(exp(5))+2*x^3*log(3)+2*x^4+10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((-4*x^2*log(3)-4*x^3-20*x^2)*exp(exp(5))^2+(-4*x^3*log(3)-4*x^4-20*x^3)* exp(exp(5))+(-x^4-2*x^2)*log(3)-x^5-5*x^4-2*x^3-10*x^2)*log(log(3)+5+x)+4* x*exp(exp(5))+2*x^2)/(((4*log(3)+20+4*x)*exp(exp(5))^2+(4*x*log(3)+4*x^2+2 0*x)*exp(exp(5))+x^2*log(3)+x^3+5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x) )^2+((-8*x*log(3)-8*x^2-40*x)*exp(exp(5))^2+(-8*x^2*log(3)-8*x^3-40*x^2)*e xp(exp(5))-2*x^3*log(3)-2*x^4-10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x)) +((4*x^2*log(3)+4*x^3+20*x^2)*exp(exp(5))^2+(4*x^3*log(3)+4*x^4+20*x^3)*ex p(exp(5))+x^4*log(3)+x^5+5*x^4)*log(log(3)+5+x)),x, algorithm="giac")
Output:
-(x^3 + 2*x^2*e^(e^5) - x^2*log(log(x + log(3) + 5)) - 2*x*e^(e^5)*log(log (x + log(3) + 5)) - 2*x)/(x^2 + 2*x*e^(e^5) - x*log(log(x + log(3) + 5)) - 2*e^(e^5)*log(log(x + log(3) + 5)))
Timed out. \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=\text {Hanged} \] Input:
int((4*x*exp(exp(5)) - log(x + log(3) + 5)*(log(3)*(2*x^2 + x^4) + exp(exp (5))*(4*x^3*log(3) + 20*x^3 + 4*x^4) + 10*x^2 + 2*x^3 + 5*x^4 + x^5 + exp( 2*exp(5))*(4*x^2*log(3) + 20*x^2 + 4*x^3)) + 2*x^2 - log(log(x + log(3) + 5))^2*log(x + log(3) + 5)*(exp(2*exp(5))*(4*x + 4*log(3) + 20) + x^2*log(3 ) + 5*x^2 + x^3 + exp(exp(5))*(20*x + 4*x*log(3) + 4*x^2)) + log(log(x + l og(3) + 5))*log(x + log(3) + 5)*(exp(exp(5))*(log(3)*(8*x^2 - 4) - 4*x + 4 0*x^2 + 8*x^3 - 20) + exp(2*exp(5))*(40*x + 8*x*log(3) + 8*x^2) + 2*x^3*lo g(3) + 10*x^3 + 2*x^4))/(log(x + log(3) + 5)*(exp(exp(5))*(4*x^3*log(3) + 20*x^3 + 4*x^4) + x^4*log(3) + 5*x^4 + x^5 + exp(2*exp(5))*(4*x^2*log(3) + 20*x^2 + 4*x^3)) + log(log(x + log(3) + 5))^2*log(x + log(3) + 5)*(exp(2* exp(5))*(4*x + 4*log(3) + 20) + x^2*log(3) + 5*x^2 + x^3 + exp(exp(5))*(20 *x + 4*x*log(3) + 4*x^2)) - log(log(x + log(3) + 5))*log(x + log(3) + 5)*( exp(2*exp(5))*(40*x + 8*x*log(3) + 8*x^2) + exp(exp(5))*(8*x^2*log(3) + 40 *x^2 + 8*x^3) + 2*x^3*log(3) + 10*x^3 + 2*x^4)),x)
Output:
\text{Hanged}
Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36 \[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx=\frac {x \left (-2 e^{e^{5}} \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right )+x +5\right )\right )+2 e^{e^{5}} x -\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right )+x +5\right )\right ) x +x^{2}-2\right )}{2 e^{e^{5}} \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right )+x +5\right )\right )-2 e^{e^{5}} x +\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right )+x +5\right )\right ) x -x^{2}} \] Input:
int((((-4*log(3)-4*x-20)*exp(exp(5))^2+(-4*x*log(3)-4*x^2-20*x)*exp(exp(5) )-x^2*log(3)-x^3-5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x))^2+((8*x*log(3 )+8*x^2+40*x)*exp(exp(5))^2+((8*x^2-4)*log(3)+8*x^3+40*x^2-4*x-20)*exp(exp (5))+2*x^3*log(3)+2*x^4+10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x))+((-4* x^2*log(3)-4*x^3-20*x^2)*exp(exp(5))^2+(-4*x^3*log(3)-4*x^4-20*x^3)*exp(ex p(5))+(-x^4-2*x^2)*log(3)-x^5-5*x^4-2*x^3-10*x^2)*log(log(3)+5+x)+4*x*exp( exp(5))+2*x^2)/(((4*log(3)+20+4*x)*exp(exp(5))^2+(4*x*log(3)+4*x^2+20*x)*e xp(exp(5))+x^2*log(3)+x^3+5*x^2)*log(log(3)+5+x)*log(log(log(3)+5+x))^2+(( -8*x*log(3)-8*x^2-40*x)*exp(exp(5))^2+(-8*x^2*log(3)-8*x^3-40*x^2)*exp(exp (5))-2*x^3*log(3)-2*x^4-10*x^3)*log(log(3)+5+x)*log(log(log(3)+5+x))+((4*x ^2*log(3)+4*x^3+20*x^2)*exp(exp(5))^2+(4*x^3*log(3)+4*x^4+20*x^3)*exp(exp( 5))+x^4*log(3)+x^5+5*x^4)*log(log(3)+5+x)),x)
Output:
(x*( - 2*e**(e**5)*log(log(log(3) + x + 5)) + 2*e**(e**5)*x - log(log(log( 3) + x + 5))*x + x**2 - 2))/(2*e**(e**5)*log(log(log(3) + x + 5)) - 2*e**( e**5)*x + log(log(log(3) + x + 5))*x - x**2)