Integrand size = 104, antiderivative size = 22 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \]
Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \]
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]
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 |
\(\displaystyle -8 e^3 \int \frac {1}{\left (-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )\right )^2}dx+4 e^3 \int \frac {x}{\left (-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )\right )^2}dx+4 e^3 \int \frac {1}{-x+\log \left (x^2\right )+3 \left (1+\frac {e^e}{3}\right )}dx+x\) |
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]
3.11.34.3.1 Defintions of rubi rules used
Time = 1.36 (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 | \(\frac {x \ln \left (x^{2}\right )+\left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right ) \ln \left (x^{2}\right )-x^{2}+\left ({\mathrm e}^{{\mathrm e}}+3\right ) \left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right )}{3-x +\ln \left (x^{2}\right )+{\mathrm e}^{{\mathrm e}}}\) | \(57\) |
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)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \]
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=\
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x + \frac {4 x e^{3}}{- x + \log {\left (x^{2} \right )} + 3 + e^{e}} \]
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)
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - x {\left (4 \, e^{3} + e^{e} + 3\right )} - 2 \, x \log \left (x\right )}{x - e^{e} - 2 \, \log \left (x\right ) - 3} \]
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=\
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=\frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \]
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=\
Time = 11.99 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.41 \[ \int \frac {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx=x+\frac {x\,\left ({\mathrm {e}}^{2\,\mathrm {e}}-3\,x+4\,{\mathrm {e}}^{\mathrm {e}+3}+12\,{\mathrm {e}}^3+6\,{\mathrm {e}}^{\mathrm {e}}-x\,{\mathrm {e}}^{\mathrm {e}}+9\right )-x\,\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left ({\mathrm {e}}^{\mathrm {e}}-x+3\right )}{\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left (\ln \left (x^2\right )-x+{\mathrm {e}}^{\mathrm {e}}+3\right )} \]
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)