Integrand size = 104, antiderivative size = 22 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \] Output:
4*exp(3)*x/(3-x+ln(x^2)+exp(exp(1)))+x
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \] Input:
Integrate[(9 + 4*E^3 + E^(2*E) + E^E*(6 + 4*E^3 - 2*x) - 6*x + x^2 + (6 + 4*E^3 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2)/(9 + E^(2*E) + E^E*(6 - 2*x) - 6*x + x^2 + (6 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2),x]
Output:
x + (4*E^3*x)/(3 + E^E - 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.
\(\displaystyle \int \frac {x^2+\log ^2\left (x^2\right )+\left (-2 x+2 e^e+4 e^3+6\right ) \log \left (x^2\right )-6 x+e^e \left (-2 x+4 e^3+6\right )+e^{2 e}+4 e^3+9}{x^2+\log ^2\left (x^2\right )+\left (-2 x+2 e^e+6\right ) \log \left (x^2\right )-6 x+e^e (6-2 x)+e^{2 e}+9} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2+\log ^2\left (x^2\right )+\left (-2 x+2 e^e+4 e^3+6\right ) \log \left (x^2\right )-6 x+e^e \left (-2 x+4 e^3+6\right )+9 \left (1+\frac {1}{9} \left (4 e^3+e^{2 e}\right )\right )}{\left (\log \left (x^2\right )-x+3 \left (1+\frac {e^e}{3}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 e^3 (x-2)}{\left (\log \left (x^2\right )-x+3 \left (1+\frac {e^e}{3}\right )\right )^2}+\frac {4 e^3}{\log \left (x^2\right )-x+3 \left (1+\frac {e^e}{3}\right )}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 e^3 \int \frac {1}{\left (-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )\right )^2}dx+4 e^3 \int \frac {x}{\left (-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )\right )^2}dx+4 e^3 \int \frac {1}{-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )}dx+x\) |
Input:
Int[(9 + 4*E^3 + E^(2*E) + E^E*(6 + 4*E^3 - 2*x) - 6*x + x^2 + (6 + 4*E^3 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2)/(9 + E^(2*E) + E^E*(6 - 2*x) - 6*x + x^2 + (6 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2),x]
Output:
$Aborted
Time = 1.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{3} x}{3-x +\ln \left (x^{2}\right )+{\mathrm e}^{{\mathrm e}}}+x\) | \(22\) |
parallelrisch | \(\frac {8 x \,{\mathrm e}^{3}+2 x \,{\mathrm e}^{{\mathrm e}}-2 x^{2}+2 x \ln \left (x^{2}\right )+6 x}{6-2 x +2 \ln \left (x^{2}\right )+2 \,{\mathrm e}^{{\mathrm e}}}\) | \(44\) |
norman | \(\frac {x \ln \left (x^{2}\right )+\left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right ) \ln \left (x^{2}\right )-x^{2}+\left ({\mathrm e}^{{\mathrm e}}+3\right ) \left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right )}{3-x +\ln \left (x^{2}\right )+{\mathrm e}^{{\mathrm e}}}\) | \(57\) |
Input:
int((ln(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*ln(x^2)+exp(exp(1))^2+(4*exp (3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-6*x+9)/(ln(x^2)^2+(2*exp(exp(1))+6-2*x )*ln(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x,method=_RETURNVER BOSE)
Output:
4*exp(3)*x/(3-x+ln(x^2)+exp(exp(1)))+x
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \] Input:
integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^ 2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp( 1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algori thm="fricas")
Output:
(x^2 - 4*x*e^3 - x*e^e - x*log(x^2) - 3*x)/(x - e^e - log(x^2) - 3)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x + \frac {4 x e^{3}}{- x + \log {\left (x^{2} \right )} + 3 + e^{e}} \] Input:
integrate((ln(x**2)**2+(2*exp(exp(1))+4*exp(3)+6-2*x)*ln(x**2)+exp(exp(1)) **2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x**2-6*x+9)/(ln(x**2)**2+(2*exp( exp(1))+6-2*x)*ln(x**2)+exp(exp(1))**2+(6-2*x)*exp(exp(1))+x**2-6*x+9),x)
Output:
x + 4*x*exp(3)/(-x + log(x**2) + 3 + exp(E))
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - x {\left (4 \, e^{3} + e^{e} + 3\right )} - 2 \, x \log \left (x\right )}{x - e^{e} - 2 \, \log \left (x\right ) - 3} \] Input:
integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^ 2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp( 1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algori thm="maxima")
Output:
(x^2 - x*(4*e^3 + e^e + 3) - 2*x*log(x))/(x - e^e - 2*log(x) - 3)
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \] Input:
integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^ 2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp( 1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algori thm="giac")
Output:
(x^2 - 4*x*e^3 - x*e^e - x*log(x^2) - 3*x)/(x - e^e - log(x^2) - 3)
Time = 3.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.41 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {x\,\left ({\mathrm {e}}^{2\,\mathrm {e}}-3\,x+4\,{\mathrm {e}}^{\mathrm {e}+3}+12\,{\mathrm {e}}^3+6\,{\mathrm {e}}^{\mathrm {e}}-x\,{\mathrm {e}}^{\mathrm {e}}+9\right )-x\,\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left ({\mathrm {e}}^{\mathrm {e}}-x+3\right )}{\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left (\ln \left (x^2\right )-x+{\mathrm {e}}^{\mathrm {e}}+3\right )} \] Input:
int((exp(2*exp(1)) - 6*x + 4*exp(3) + exp(exp(1))*(4*exp(3) - 2*x + 6) + l og(x^2)^2 + log(x^2)*(4*exp(3) - 2*x + 2*exp(exp(1)) + 6) + x^2 + 9)/(exp( 2*exp(1)) - 6*x + log(x^2)*(2*exp(exp(1)) - 2*x + 6) - exp(exp(1))*(2*x - 6) + log(x^2)^2 + x^2 + 9),x)
Output:
x + (x*(exp(2*exp(1)) - 3*x + 4*exp(exp(1) + 3) + 12*exp(3) + 6*exp(exp(1) ) - x*exp(exp(1)) + 9) - x*(exp(exp(1)) + 3)*(exp(exp(1)) - x + 3))/((exp( exp(1)) + 3)*(log(x^2) - x + exp(exp(1)) + 3))
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.50 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {e^{2 e}+2 e^{e} \mathrm {log}\left (x^{2}\right )-2 e^{e} \mathrm {log}\left (x \right )+4 e^{e} e^{3}+6 e^{e}+\mathrm {log}\left (x^{2}\right )^{2}-2 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x \right )+4 \,\mathrm {log}\left (x^{2}\right ) e^{3}+6 \,\mathrm {log}\left (x^{2}\right )+2 \,\mathrm {log}\left (x \right ) x -6 \,\mathrm {log}\left (x \right )+12 e^{3}-x^{2}+9}{e^{e}+\mathrm {log}\left (x^{2}\right )-x +3} \] Input:
int((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^2+(4*e xp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp(1))+6- 2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x)
Output:
(e**(2*e) + 2*e**e*log(x**2) - 2*e**e*log(x) + 4*e**e*e**3 + 6*e**e + log( x**2)**2 - 2*log(x**2)*log(x) + 4*log(x**2)*e**3 + 6*log(x**2) + 2*log(x)* x - 6*log(x) + 12*e**3 - x**2 + 9)/(e**e + log(x**2) - x + 3)