Integrand size = 105, antiderivative size = 28 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x}{-e^{3+e^x-x+x^2}+x^2}+\log (2 x) \]
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=-\frac {e^x x}{e^{3+e^x+x^2}-e^x x^2}+\log (x) \]
Integrate[(E^(6 + 2*E^x - 2*x + 2*x^2) - x^3 + x^4 + E^(3 + E^x - x + x^2) *(-x - 3*x^2 + E^x*x^2 + 2*x^3))/(E^(6 + 2*E^x - 2*x + 2*x^2)*x - 2*E^(3 + E^x - x + x^2)*x^3 + x^5),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.
\(\displaystyle \int \frac {x^4-x^3+e^{2 x^2-2 x+2 e^x+6}+e^{x^2-x+e^x+3} \left (2 x^3+e^x x^2-3 x^2-x\right )}{x^5+e^{2 x^2-2 x+2 e^x+6} x-2 e^{x^2-x+e^x+3} x^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x} \left (x^4-x^3+e^{2 x^2-2 x+2 e^x+6}+e^{x^2-x+e^x+3} \left (2 x^3+e^x x^2-3 x^2-x\right )\right )}{x \left (e^{x^2+e^x+3}-e^x x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{2 x} \left (2 x^2+e^x x-x-2\right ) x^2}{\left (e^x x^2-e^{x^2+e^x+3}\right )^2}-\frac {e^x \left (2 x^2+e^x x-x-1\right )}{e^x x^2-e^{x^2+e^x+3}}+\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {e^x}{e^{x^2+e^x+3}-e^x x^2}dx-2 \int \frac {e^{2 x} x^2}{\left (e^x x^2-e^{x^2+e^x+3}\right )^2}dx+\int \frac {e^x x}{e^x x^2-e^{x^2+e^x+3}}dx-\int \frac {e^{2 x} x}{e^x x^2-e^{x^2+e^x+3}}dx-2 \int \frac {e^x x^2}{e^x x^2-e^{x^2+e^x+3}}dx+2 \int \frac {e^{2 x} x^4}{\left (e^x x^2-e^{x^2+e^x+3}\right )^2}dx-\int \frac {e^{2 x} x^3}{\left (e^x x^2-e^{x^2+e^x+3}\right )^2}dx+\int \frac {e^{3 x} x^3}{\left (e^x x^2-e^{x^2+e^x+3}\right )^2}dx+\log (x)\) |
Int[(E^(6 + 2*E^x - 2*x + 2*x^2) - x^3 + x^4 + E^(3 + E^x - x + x^2)*(-x - 3*x^2 + E^x*x^2 + 2*x^3))/(E^(6 + 2*E^x - 2*x + 2*x^2)*x - 2*E^(3 + E^x - x + x^2)*x^3 + x^5),x]
3.20.100.3.1 Defintions of rubi rules used
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\ln \left (x \right )+\frac {x}{x^{2}-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3}}\) | \(25\) |
parallelrisch | \(\frac {x^{2} \ln \left (x \right )-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3} \ln \left (x \right )+x}{x^{2}-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3}}\) | \(44\) |
int((exp(exp(x)+x^2-x+3)^2+(exp(x)*x^2+2*x^3-3*x^2-x)*exp(exp(x)+x^2-x+3)+ x^4-x^3)/(x*exp(exp(x)+x^2-x+3)^2-2*x^3*exp(exp(x)+x^2-x+3)+x^5),x,method= _RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x^{2} \log \left (x\right ) - e^{\left (x^{2} - x + e^{x} + 3\right )} \log \left (x\right ) + x}{x^{2} - e^{\left (x^{2} - x + e^{x} + 3\right )}} \]
integrate((exp(exp(x)+x^2-x+3)^2+(exp(x)*x^2+2*x^3-3*x^2-x)*exp(exp(x)+x^2 -x+3)+x^4-x^3)/(x*exp(exp(x)+x^2-x+3)^2-2*x^3*exp(exp(x)+x^2-x+3)+x^5),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=- \frac {x}{- x^{2} + e^{x^{2} - x + e^{x} + 3}} + \log {\left (x \right )} \]
integrate((exp(exp(x)+x**2-x+3)**2+(exp(x)*x**2+2*x**3-3*x**2-x)*exp(exp(x )+x**2-x+3)+x**4-x**3)/(x*exp(exp(x)+x**2-x+3)**2-2*x**3*exp(exp(x)+x**2-x +3)+x**5),x)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x e^{x}}{x^{2} e^{x} - e^{\left (x^{2} + e^{x} + 3\right )}} + \log \left (x\right ) \]
integrate((exp(exp(x)+x^2-x+3)^2+(exp(x)*x^2+2*x^3-3*x^2-x)*exp(exp(x)+x^2 -x+3)+x^4-x^3)/(x*exp(exp(x)+x^2-x+3)^2-2*x^3*exp(exp(x)+x^2-x+3)+x^5),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 410, normalized size of antiderivative = 14.64 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {2 \, x^{6} e^{x} \log \left (x\right ) + x^{5} e^{\left (2 \, x\right )} \log \left (x\right ) - x^{5} e^{x} \log \left (x\right ) + 2 \, x^{5} e^{x} - 4 \, x^{4} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - 2 \, x^{4} e^{x} \log \left (x\right ) + x^{4} e^{\left (2 \, x\right )} - x^{4} e^{x} - 2 \, x^{3} e^{\left (x^{2} + x + e^{x} + 3\right )} \log \left (x\right ) + 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right ) + 4 \, x^{2} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - x^{2} e^{\left (x^{2} + x + e^{x} + 3\right )} + x^{2} e^{\left (x^{2} + e^{x} + 3\right )} - x e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right ) + x e^{\left (2 \, x^{2} + 2 \, e^{x} + 6\right )} \log \left (x\right ) + 2 \, x e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right )}{2 \, x^{6} e^{x} + x^{5} e^{\left (2 \, x\right )} - x^{5} e^{x} - 4 \, x^{4} e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, x^{4} e^{x} - 2 \, x^{3} e^{\left (x^{2} + x + e^{x} + 3\right )} + 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} + 2 \, x^{2} e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} + 4 \, x^{2} e^{\left (x^{2} + e^{x} + 3\right )} - x e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} + x e^{\left (2 \, x^{2} + 2 \, e^{x} + 6\right )} - 2 \, e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )}} \]
integrate((exp(exp(x)+x^2-x+3)^2+(exp(x)*x^2+2*x^3-3*x^2-x)*exp(exp(x)+x^2 -x+3)+x^4-x^3)/(x*exp(exp(x)+x^2-x+3)^2-2*x^3*exp(exp(x)+x^2-x+3)+x^5),x, algorithm=\
(2*x^6*e^x*log(x) + x^5*e^(2*x)*log(x) - x^5*e^x*log(x) + 2*x^5*e^x - 4*x^ 4*e^(x^2 + e^x + 3)*log(x) - 2*x^4*e^x*log(x) + x^4*e^(2*x) - x^4*e^x - 2* x^3*e^(x^2 + x + e^x + 3)*log(x) + 2*x^3*e^(x^2 + e^x + 3)*log(x) - 2*x^3* e^(x^2 + e^x + 3) - 2*x^3*e^x + 2*x^2*e^(2*x^2 - x + 2*e^x + 6)*log(x) + 4 *x^2*e^(x^2 + e^x + 3)*log(x) - x^2*e^(x^2 + x + e^x + 3) + x^2*e^(x^2 + e ^x + 3) - x*e^(2*x^2 - x + 2*e^x + 6)*log(x) + x*e^(2*x^2 + 2*e^x + 6)*log (x) + 2*x*e^(x^2 + e^x + 3) - 2*e^(2*x^2 - x + 2*e^x + 6)*log(x))/(2*x^6*e ^x + x^5*e^(2*x) - x^5*e^x - 4*x^4*e^(x^2 + e^x + 3) - 2*x^4*e^x - 2*x^3*e ^(x^2 + x + e^x + 3) + 2*x^3*e^(x^2 + e^x + 3) + 2*x^2*e^(2*x^2 - x + 2*e^ x + 6) + 4*x^2*e^(x^2 + e^x + 3) - x*e^(2*x^2 - x + 2*e^x + 6) + x*e^(2*x^ 2 + 2*e^x + 6) - 2*e^(2*x^2 - x + 2*e^x + 6))
Time = 9.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\ln \left (x\right )+\frac {x}{x^2-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3} \]