Integrand size = 120, antiderivative size = 33 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=4 \left (-2+\frac {x^4}{\log ^2\left (\frac {5 e^x}{3 \left (-1+e^{e^2}-x\right ) x}\right )}\right ) \] Output:
4*x^4/ln(5/3/x*exp(x)/(exp(exp(2))-x-1))^2-8
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 x^4}{\log ^2\left (-\frac {5 e^x}{3 x \left (1-e^{e^2}+x\right )}\right )} \] Input:
Integrate[(-8*x^3 - 8*x^4 + 8*x^5 + E^E^2*(8*x^3 - 8*x^4) + (-16*x^3 + 16* E^E^2*x^3 - 16*x^4)*Log[(5*E^x)/(-3*x + 3*E^E^2*x - 3*x^2)])/((-1 + E^E^2 - x)*Log[(5*E^x)/(-3*x + 3*E^E^2*x - 3*x^2)]^3),x]
Output:
(4*x^4)/Log[(-5*E^x)/(3*x*(1 - E^E^2 + 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.
| \(\displaystyle \int \frac {8 x^5-8 x^4-8 x^3+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^4+16 e^{e^2} x^3-16 x^3\right ) \log \left (\frac {5 e^x}{-3 x^2+3 e^{e^2} x-3 x}\right )}{\left (-x+e^{e^2}-1\right ) \log ^3\left (\frac {5 e^x}{-3 x^2+3 e^{e^2} x-3 x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-8 x^5+8 x^4+8 x^3-e^{e^2} \left (8 x^3-8 x^4\right )-\left (-16 x^4+16 e^{e^2} x^3-16 x^3\right ) \log \left (\frac {5 e^x}{-3 x^2+3 e^{e^2} x-3 x}\right )}{\left (x-e^{e^2}+1\right ) \log ^3\left (\frac {5 e^x}{\left (3 \left (e^{e^2}-1\right )-3 x\right ) x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {16 x^3 \log \left (-\frac {5 e^x}{3 x \left (x-e^{e^2}+1\right )}\right )}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}+\frac {8 \left (-x^2+\left (1+e^{e^2}\right ) x-e^{e^2}+1\right ) x^3}{\left (x-e^{e^2}+1\right ) \log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -8 \int \frac {x^4}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx+16 \int \frac {x^3}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx+16 \int \frac {x^3 \log \left (-\frac {5 e^x}{3 x \left (x-e^{e^2}+1\right )}\right )}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx-8 \left (1-e^{e^2}\right ) \int \frac {x^2}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx-8 \left (1-e^{e^2}\right )^3 \int \frac {1}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx-8 \left (1-e^{e^2}\right )^4 \int \frac {1}{\left (-x+e^{e^2}-1\right ) \log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx+8 \left (1-e^{e^2}\right )^2 \int \frac {x}{\log ^3\left (\frac {5 e^x}{3 \left (-x+e^{e^2}-1\right ) x}\right )}dx\) |
Input:
Int[(-8*x^3 - 8*x^4 + 8*x^5 + E^E^2*(8*x^3 - 8*x^4) + (-16*x^3 + 16*E^E^2* x^3 - 16*x^4)*Log[(5*E^x)/(-3*x + 3*E^E^2*x - 3*x^2)])/((-1 + E^E^2 - x)*L og[(5*E^x)/(-3*x + 3*E^E^2*x - 3*x^2)]^3),x]
Output:
$Aborted
Time = 2.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(\frac {4 x^{4}}{\ln \left (\frac {5 \,{\mathrm e}^{x}}{3 x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )^{2}}\) | \(26\) |
| default | \(\frac {16 \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right ) x^{4}}{\left (x \,{\mathrm e}^{{\mathrm e}^{2}}-x^{2}-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right ) \left (-\ln \left (5\right )+\ln \left (3\right )-\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )\right )}-\frac {4 x^{4} \left (4 \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{2}}-4 x \ln \left (3\right )-4 \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{2}}+4 \,{\mathrm e}^{{\mathrm e}^{2}} \ln \left (x \right )-5 x \,{\mathrm e}^{{\mathrm e}^{2}}+4 \,{\mathrm e}^{{\mathrm e}^{2}} \ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )-4 \left (\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-x +\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )\right ) {\mathrm e}^{{\mathrm e}^{2}}+4 x \ln \left (5\right )-4 x \ln \left (x \right )+5 x^{2}-4 x \ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )+4 x \left (\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-x +\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )\right )-4 \ln \left (3\right )+{\mathrm e}^{{\mathrm e}^{2}}+4 \ln \left (5\right )-x +4 \ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-1\right )}{{\left (-\ln \left (5\right )+\ln \left (3\right )-\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )\right )}^{2} \left (x \,{\mathrm e}^{{\mathrm e}^{2}}-x^{2}-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\) | \(296\) |
| parts | \(\frac {16 \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right ) x^{4}}{\left (x \,{\mathrm e}^{{\mathrm e}^{2}}-x^{2}-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right ) \left (-\ln \left (5\right )+\ln \left (3\right )-\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )\right )}-\frac {4 x^{4} \left (4 \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{2}}-4 x \ln \left (3\right )-4 \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{2}}+4 \,{\mathrm e}^{{\mathrm e}^{2}} \ln \left (x \right )-5 x \,{\mathrm e}^{{\mathrm e}^{2}}+4 \,{\mathrm e}^{{\mathrm e}^{2}} \ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )-4 \left (\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-x +\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )\right ) {\mathrm e}^{{\mathrm e}^{2}}+4 x \ln \left (5\right )-4 x \ln \left (x \right )+5 x^{2}-4 x \ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )+4 x \left (\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-x +\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )\right )-4 \ln \left (3\right )+{\mathrm e}^{{\mathrm e}^{2}}+4 \ln \left (5\right )-x +4 \ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )-1\right )}{{\left (-\ln \left (5\right )+\ln \left (3\right )-\ln \left (\frac {{\mathrm e}^{x}}{x \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )}\right )\right )}^{2} \left (x \,{\mathrm e}^{{\mathrm e}^{2}}-x^{2}-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\) | \(296\) |
| risch | \(\frac {16 x^{4}}{{\left (-2 \ln \left (5\right )+2 \ln \left (3\right )+2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x}\right )+2 \ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\right )-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-{\mathrm e}^{{\mathrm e}^{2}}+x +1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (-{\mathrm e}^{{\mathrm e}^{2}}+x +1\right )}\right )^{2}\right )}^{2}}\) | \(307\) |
Input:
int(((16*x^3*exp(exp(2))-16*x^4-16*x^3)*ln(5*exp(x)/(3*x*exp(exp(2))-3*x^2 -3*x))+(-8*x^4+8*x^3)*exp(exp(2))+8*x^5-8*x^4-8*x^3)/(exp(exp(2))-x-1)/ln( 5*exp(x)/(3*x*exp(exp(2))-3*x^2-3*x))^3,x,method=_RETURNVERBOSE)
Output:
4*x^4/ln(5/3/x*exp(x)/(exp(exp(2))-x-1))^2
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 \, x^{4}}{\log \left (-\frac {5 \, e^{x}}{3 \, {\left (x^{2} - x e^{\left (e^{2}\right )} + x\right )}}\right )^{2}} \] Input:
integrate(((16*x^3*exp(exp(2))-16*x^4-16*x^3)*log(5*exp(x)/(3*x*exp(exp(2) )-3*x^2-3*x))+(-8*x^4+8*x^3)*exp(exp(2))+8*x^5-8*x^4-8*x^3)/(exp(exp(2))-x -1)/log(5*exp(x)/(3*x*exp(exp(2))-3*x^2-3*x))^3,x, algorithm="fricas")
Output:
4*x^4/log(-5/3*e^x/(x^2 - x*e^(e^2) + x))^2
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 x^{4}}{\log {\left (\frac {5 e^{x}}{- 3 x^{2} - 3 x + 3 x e^{e^{2}}} \right )}^{2}} \] Input:
integrate(((16*x**3*exp(exp(2))-16*x**4-16*x**3)*ln(5*exp(x)/(3*x*exp(exp( 2))-3*x**2-3*x))+(-8*x**4+8*x**3)*exp(exp(2))+8*x**5-8*x**4-8*x**3)/(exp(e xp(2))-x-1)/ln(5*exp(x)/(3*x*exp(exp(2))-3*x**2-3*x))**3,x)
Output:
4*x**4/log(5*exp(x)/(-3*x**2 - 3*x + 3*x*exp(exp(2))))**2
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.58 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 \, x^{4}}{x^{2} + 2 \, x {\left (\log \left (5\right ) - \log \left (3\right )\right )} + \log \left (5\right )^{2} - 2 \, \log \left (5\right ) \log \left (3\right ) + \log \left (3\right )^{2} - 2 \, {\left (x + \log \left (5\right ) - \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (x + \log \left (5\right ) - \log \left (3\right ) - \log \left (x\right )\right )} \log \left (-x + e^{\left (e^{2}\right )} - 1\right ) + \log \left (-x + e^{\left (e^{2}\right )} - 1\right )^{2}} \] Input:
integrate(((16*x^3*exp(exp(2))-16*x^4-16*x^3)*log(5*exp(x)/(3*x*exp(exp(2) )-3*x^2-3*x))+(-8*x^4+8*x^3)*exp(exp(2))+8*x^5-8*x^4-8*x^3)/(exp(exp(2))-x -1)/log(5*exp(x)/(3*x*exp(exp(2))-3*x^2-3*x))^3,x, algorithm="maxima")
Output:
4*x^4/(x^2 + 2*x*(log(5) - log(3)) + log(5)^2 - 2*log(5)*log(3) + log(3)^2 - 2*(x + log(5) - log(3))*log(x) + log(x)^2 - 2*(x + log(5) - log(3) - lo g(x))*log(-x + e^(e^2) - 1) + log(-x + e^(e^2) - 1)^2)
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 \, x^{4}}{x^{2} + 2 \, x \log \left (-\frac {5}{3 \, {\left (x^{2} - x e^{\left (e^{2}\right )} + x\right )}}\right ) + \log \left (-\frac {5}{3 \, {\left (x^{2} - x e^{\left (e^{2}\right )} + x\right )}}\right )^{2}} \] Input:
integrate(((16*x^3*exp(exp(2))-16*x^4-16*x^3)*log(5*exp(x)/(3*x*exp(exp(2) )-3*x^2-3*x))+(-8*x^4+8*x^3)*exp(exp(2))+8*x^5-8*x^4-8*x^3)/(exp(exp(2))-x -1)/log(5*exp(x)/(3*x*exp(exp(2))-3*x^2-3*x))^3,x, algorithm="giac")
Output:
4*x^4/(x^2 + 2*x*log(-5/3/(x^2 - x*e^(e^2) + x)) + log(-5/3/(x^2 - x*e^(e^ 2) + x))^2)
Time = 8.34 (sec) , antiderivative size = 551, normalized size of antiderivative = 16.70 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx =\text {Too large to display} \] Input:
int((log(-(5*exp(x))/(3*x - 3*x*exp(exp(2)) + 3*x^2))*(16*x^3 - 16*x^3*exp (exp(2)) + 16*x^4) + 8*x^3 + 8*x^4 - 8*x^5 - exp(exp(2))*(8*x^3 - 8*x^4))/ (log(-(5*exp(x))/(3*x - 3*x*exp(exp(2)) + 3*x^2))^3*(x - exp(exp(2)) + 1)) ,x)
Output:
80*x - ((8*x^4*(x - exp(exp(2)) + 1))/(x - exp(exp(2)) + x*exp(exp(2)) - x ^2 + 1) + (8*x^4*log(-(5*exp(x))/(3*x - 3*x*exp(exp(2)) + 3*x^2))*(x - exp (exp(2)) + 1)*(8*x + 4*exp(2*exp(2)) - 8*exp(exp(2)) - 3*x*exp(2*exp(2)) + 6*x^2*exp(exp(2)) - 5*x*exp(exp(2)) + 2*x^2 - 3*x^3 + 4))/(x - exp(exp(2) ) + x*exp(exp(2)) - x^2 + 1)^3)/log(-(5*exp(x))/(3*x - 3*x*exp(exp(2)) + 3 *x^2)) + (384*exp(2*exp(2)) - 64*exp(3*exp(2)) - 32*exp(4*exp(2)) - 448*ex p(exp(2)) + x^5*(24*exp(2*exp(2)) + 96*exp(exp(2)) + 264) - x^2*(840*exp(2 *exp(2)) + 456*exp(3*exp(2)) + 96*exp(4*exp(2)) - 1128*exp(exp(2)) - 264) + x^3*(984*exp(2*exp(2)) + 280*exp(3*exp(2)) + 32*exp(4*exp(2)) + 968*exp( exp(2)) - 728) - x^4*(288*exp(2*exp(2)) + 56*exp(3*exp(2)) + 888*exp(exp(2 )) + 432) + x*(304*exp(3*exp(2)) - 336*exp(2*exp(2)) + 96*exp(4*exp(2)) - 624*exp(exp(2)) + 560) + 160)/(exp(3*exp(2)) - 3*exp(2*exp(2)) + 3*exp(exp (2)) + x^4*(3*exp(2*exp(2)) + 9*exp(exp(2))) + x*(3*exp(2*exp(2)) - 3*exp( 3*exp(2)) + 3*exp(exp(2)) - 3) - x^3*(9*exp(2*exp(2)) + exp(3*exp(2)) + 3* exp(exp(2)) - 5) - x^5*(3*exp(exp(2)) + 3) + x^2*(6*exp(2*exp(2)) + 3*exp( 3*exp(2)) - 9*exp(exp(2))) + x^6 - 1) + 24*x^2 + (4*x^4 + (8*x^4*log(-(5*e xp(x))/(3*x - 3*x*exp(exp(2)) + 3*x^2))*(x - exp(exp(2)) + 1))/(x - exp(ex p(2)) + x*exp(exp(2)) - x^2 + 1))/log(-(5*exp(x))/(3*x - 3*x*exp(exp(2)) + 3*x^2))^2
Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-8 x^3-8 x^4+8 x^5+e^{e^2} \left (8 x^3-8 x^4\right )+\left (-16 x^3+16 e^{e^2} x^3-16 x^4\right ) \log \left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )}{\left (-1+e^{e^2}-x\right ) \log ^3\left (\frac {5 e^x}{-3 x+3 e^{e^2} x-3 x^2}\right )} \, dx=\frac {4 x^{4}}{\mathrm {log}\left (\frac {5 e^{x}}{3 e^{e^{2}} x -3 x^{2}-3 x}\right )^{2}} \] Input:
int(((16*x^3*exp(exp(2))-16*x^4-16*x^3)*log(5*exp(x)/(3*x*exp(exp(2))-3*x^ 2-3*x))+(-8*x^4+8*x^3)*exp(exp(2))+8*x^5-8*x^4-8*x^3)/(exp(exp(2))-x-1)/lo g(5*exp(x)/(3*x*exp(exp(2))-3*x^2-3*x))^3,x)
Output:
(4*x**4)/log((5*e**x)/(3*e**(e**2)*x - 3*x**2 - 3*x))**2