Integrand size = 162, antiderivative size = 32 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=e^{e^x x} \left (x-\frac {e^5 \log (x)}{-x+e^{2 x} x^2}\right ) \end {dmath*}
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=e^{e^x x} \left (x-\frac {e^5 \log (x)}{x \left (-1+e^{2 x} x\right )}\right ) \end {dmath*}
Integrate[(E^(E^x*x)*(E^5 + x^2 + E^x*(x^3 + x^4) + E^(2*x)*(-(E^5*x) - 2* x^3 + E^x*(-2*x^4 - 2*x^5)) + E^(4*x)*(x^4 + E^x*(x^5 + x^6)) + (-E^5 + E^ (5 + x)*(x + x^2) + E^(2*x)*(E^5*(2*x + 2*x^2) + E^(5 + x)*(-x^2 - x^3)))* Log[x]))/(x^2 - 2*E^(2*x)*x^3 + E^(4*x)*x^4),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.
\(\displaystyle \int \frac {e^{e^x x} \left (x^2+e^x \left (x^4+x^3\right )+\left (e^{x+5} \left (x^2+x\right )+e^{2 x} \left (e^5 \left (2 x^2+2 x\right )+e^{x+5} \left (-x^3-x^2\right )\right )-e^5\right ) \log (x)+e^{4 x} \left (x^4+e^x \left (x^6+x^5\right )\right )+e^{2 x} \left (-2 x^3+e^x \left (-2 x^5-2 x^4\right )-e^5 x\right )+e^5\right )}{e^{4 x} x^4-2 e^{2 x} x^3+x^2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{e^x x} \left (x^2+e^x \left (x^4+x^3\right )+\left (e^{x+5} \left (x^2+x\right )+e^{2 x} \left (e^5 \left (2 x^2+2 x\right )+e^{x+5} \left (-x^3-x^2\right )\right )-e^5\right ) \log (x)+e^{4 x} \left (x^4+e^x \left (x^6+x^5\right )\right )+e^{2 x} \left (-2 x^3+e^x \left (-2 x^5-2 x^4\right )-e^5 x\right )+e^5\right )}{x^2 \left (1-e^{2 x} x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{e^x x+5} (2 x+1) \log (x)}{x^2 \left (e^{2 x} x-1\right )^2}-\frac {e^{e^x x+5} \left (e^x x^2 \log (x)+e^x x \log (x)-2 x \log (x)-2 \log (x)+1\right )}{x^2 \left (e^{2 x} x-1\right )}+e^{e^x x+x} x (x+1)+e^{e^x x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{e^x x+x} x^2dx-\int \frac {e^{e^x x+5}}{x^2 \left (e^{2 x} x-1\right )}dx-\int \frac {\int \frac {e^{e^x x+5}}{x^2 \left (e^{2 x} x-1\right )^2}dx}{x}dx-2 \int \frac {\int \frac {e^{e^x x+5}}{x^2 \left (e^{2 x} x-1\right )}dx}{x}dx+\log (x) \int \frac {e^{e^x x+5}}{x^2 \left (e^{2 x} x-1\right )^2}dx+2 \log (x) \int \frac {e^{e^x x+5}}{x^2 \left (e^{2 x} x-1\right )}dx+\int e^{e^x x}dx+\int e^{e^x x+x} xdx-2 \int \frac {\int \frac {e^{e^x x+5}}{x \left (e^{2 x} x-1\right )^2}dx}{x}dx+\int \frac {\int \frac {e^{e^x x+x+5}}{e^{2 x} x-1}dx}{x}dx-2 \int \frac {\int \frac {e^{e^x x+5}}{x \left (e^{2 x} x-1\right )}dx}{x}dx+\int \frac {\int \frac {e^{e^x x+x+5}}{x \left (e^{2 x} x-1\right )}dx}{x}dx+2 \log (x) \int \frac {e^{e^x x+5}}{x \left (e^{2 x} x-1\right )^2}dx-\log (x) \int \frac {e^{e^x x+x+5}}{e^{2 x} x-1}dx+2 \log (x) \int \frac {e^{e^x x+5}}{x \left (e^{2 x} x-1\right )}dx-\log (x) \int \frac {e^{e^x x+x+5}}{x \left (e^{2 x} x-1\right )}dx\) |
Int[(E^(E^x*x)*(E^5 + x^2 + E^x*(x^3 + x^4) + E^(2*x)*(-(E^5*x) - 2*x^3 + E^x*(-2*x^4 - 2*x^5)) + E^(4*x)*(x^4 + E^x*(x^5 + x^6)) + (-E^5 + E^(5 + x )*(x + x^2) + E^(2*x)*(E^5*(2*x + 2*x^2) + E^(5 + x)*(-x^2 - x^3)))*Log[x] ))/(x^2 - 2*E^(2*x)*x^3 + E^(4*x)*x^4),x]
3.8.27.3.1 Defintions of rubi rules used
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22
\[-\frac {\left (-{\mathrm e}^{2 x} x^{3}+{\mathrm e}^{5} \ln \left (x \right )+x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x} x}}{x \left (x \,{\mathrm e}^{2 x}-1\right )}\]
int(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x)*exp(5 )*exp(x)-exp(5))*ln(x)+((x^6+x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2*x^4)*e xp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(exp(x)*x)/ (x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x)
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {{\left (x^{3} e^{\left (2 \, x + 10\right )} - x^{2} e^{10} - e^{15} \log \left (x\right )\right )} e^{\left (x e^{x}\right )}}{x^{2} e^{\left (2 \, x + 10\right )} - x e^{10}} \end {dmath*}
integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x) *exp(5)*exp(x)-exp(5))*log(x)+((x^6+x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2 *x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(exp (x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm=\
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {\left (x^{3} e^{2 x} - x^{2} - e^{5} \log {\left (x \right )}\right ) e^{x e^{x}}}{x^{2} e^{2 x} - x} \end {dmath*}
integrate(((((-x**3-x**2)*exp(5)*exp(x)+(2*x**2+2*x)*exp(5))*exp(2*x)+(x** 2+x)*exp(5)*exp(x)-exp(5))*ln(x)+((x**6+x**5)*exp(x)+x**4)*exp(2*x)**2+((- 2*x**5-2*x**4)*exp(x)-x*exp(5)-2*x**3)*exp(2*x)+(x**4+x**3)*exp(x)+x**2+ex p(5))*exp(exp(x)*x)/(x**4*exp(2*x)**2-2*exp(2*x)*x**3+x**2),x)
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {{\left (x^{3} e^{\left (2 \, x\right )} - x^{2} - e^{5} \log \left (x\right )\right )} e^{\left (x e^{x}\right )}}{x^{2} e^{\left (2 \, x\right )} - x} \end {dmath*}
integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x) *exp(5)*exp(x)-exp(5))*log(x)+((x^6+x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2 *x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(exp (x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {x^{3} e^{\left (x e^{x} + 2 \, x\right )} - x^{2} e^{\left (x e^{x}\right )} - e^{\left (x e^{x} + 5\right )} \log \left (x\right )}{x^{2} e^{\left (2 \, x\right )} - x} \end {dmath*}
integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x) *exp(5)*exp(x)-exp(5))*log(x)+((x^6+x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2 *x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(exp (x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\int \frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^5+{\mathrm {e}}^{4\,x}\,\left ({\mathrm {e}}^x\,\left (x^6+x^5\right )+x^4\right )+{\mathrm {e}}^x\,\left (x^4+x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^x\,\left (2\,x^5+2\,x^4\right )+x\,{\mathrm {e}}^5+2\,x^3\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^{x+5}\,\left (x^2+x\right )-{\mathrm {e}}^5+{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^5\,\left (2\,x^2+2\,x\right )-{\mathrm {e}}^{x+5}\,\left (x^3+x^2\right )\right )\right )+x^2\right )}{x^4\,{\mathrm {e}}^{4\,x}-2\,x^3\,{\mathrm {e}}^{2\,x}+x^2} \,d x \end {dmath*}
int((exp(x*exp(x))*(exp(5) + exp(4*x)*(exp(x)*(x^5 + x^6) + x^4) + exp(x)* (x^3 + x^4) - exp(2*x)*(exp(x)*(2*x^4 + 2*x^5) + x*exp(5) + 2*x^3) + log(x )*(exp(2*x)*(exp(5)*(2*x + 2*x^2) - exp(5)*exp(x)*(x^2 + x^3)) - exp(5) + exp(5)*exp(x)*(x + x^2)) + x^2))/(x^4*exp(4*x) - 2*x^3*exp(2*x) + x^2),x)