Integrand size = 98, antiderivative size = 32 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^2+\log \left (e^{-e^{-e^{e^x}+\frac {e^{5+x}}{x+x^2}}}\right ) \] Output:
ln(1/exp(exp(exp(5)*exp(x)/(x^2+x)-exp(exp(x)))))+x^2
Time = 5.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=-e^{-e^{e^x}+\frac {e^{5+x}}{x (1+x)}}+x^2 \] Input:
Integrate[(2*x^3 + 4*x^4 + 2*x^5 + E^((E^(5 + x) + E^E^x*(-x - x^2))/(x + x^2))*(E^(5 + x)*(1 + x - x^2) + E^(E^x + x)*(x^2 + 2*x^3 + x^4)))/(x^2 + 2*x^3 + x^4),x]
Output:
-E^(-E^E^x + E^(5 + x)/(x*(1 + x))) + 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.
\(\displaystyle \int \frac {2 x^5+4 x^4+2 x^3+e^{\frac {e^{e^x} \left (-x^2-x\right )+e^{x+5}}{x^2+x}} \left (e^{x+5} \left (-x^2+x+1\right )+e^{x+e^x} \left (x^4+2 x^3+x^2\right )\right )}{x^4+2 x^3+x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^5+4 x^4+2 x^3+e^{\frac {e^{e^x} \left (-x^2-x\right )+e^{x+5}}{x^2+x}} \left (e^{x+5} \left (-x^2+x+1\right )+e^{x+e^x} \left (x^4+2 x^3+x^2\right )\right )}{x^2 \left (x^2+2 x+1\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {2 x^5+4 x^4+2 x^3+e^{\frac {e^{e^x} \left (-x^2-x\right )+e^{x+5}}{x^2+x}} \left (e^{x+5} \left (-x^2+x+1\right )+e^{x+e^x} \left (x^4+2 x^3+x^2\right )\right )}{x^2 (x+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {e^{x+5}}{x^2+x}+x-e^{e^x}} \left (e^{e^x} x^4+2 e^{e^x} x^3+e^{e^x} x^2-e^5 x^2+e^5 x+e^5\right )}{(x+1)^2 x^2}+2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{x-e^{e^x}+e^x+\frac {e^{x+5}}{x^2+x}}dx-\int \frac {e^{x-e^{e^x}+\frac {e^{x+5}}{x^2+x}+5}}{-x-1}dx+\int \frac {e^{x-e^{e^x}+\frac {e^{x+5}}{x^2+x}+5}}{x^2}dx-\int \frac {e^{x-e^{e^x}+\frac {e^{x+5}}{x^2+x}+5}}{x}dx-\int \frac {e^{x-e^{e^x}+\frac {e^{x+5}}{x^2+x}+5}}{(x+1)^2}dx+x^2\) |
Input:
Int[(2*x^3 + 4*x^4 + 2*x^5 + E^((E^(5 + x) + E^E^x*(-x - x^2))/(x + x^2))* (E^(5 + x)*(1 + x - x^2) + E^(E^x + x)*(x^2 + 2*x^3 + x^4)))/(x^2 + 2*x^3 + x^4),x]
Output:
$Aborted
Time = 72.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16
method | result | size |
risch | \(x^{2}-{\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{x}} x^{2}+x \,{\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{5+x}}{\left (1+x \right ) x}}\) | \(37\) |
parallelrisch | \(x^{2}-{\mathrm e}^{\frac {\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{5} {\mathrm e}^{x}}{\left (1+x \right ) x}}-6\) | \(37\) |
Input:
int((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((- x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x ^2),x,method=_RETURNVERBOSE)
Output:
x^2-exp(-(exp(exp(x))*x^2+x*exp(exp(x))-exp(5+x))/(1+x)/x)
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\left (-\frac {{\left ({\left (x^{2} + x\right )} e^{\left ({\left (x e^{5} + e^{\left (x + 5\right )}\right )} e^{\left (-5\right )} + 5\right )} - e^{\left (2 \, x + 10\right )}\right )} e^{\left (-x - 5\right )}}{x^{2} + x}\right )} \] Input:
integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*e xp(((-x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2 *x^3+x^2),x, algorithm="fricas")
Output:
x^2 - e^(-((x^2 + x)*e^((x*e^5 + e^(x + 5))*e^(-5) + 5) - e^(2*x + 10))*e^ (-x - 5)/(x^2 + x))
Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\frac {\left (- x^{2} - x\right ) e^{e^{x}} + e^{5} e^{x}}{x^{2} + x}} \] Input:
integrate((((x**4+2*x**3+x**2)*exp(x)*exp(exp(x))+(-x**2+x+1)*exp(5)*exp(x ))*exp(((-x**2-x)*exp(exp(x))+exp(5)*exp(x))/(x**2+x))+2*x**5+4*x**4+2*x** 3)/(x**4+2*x**3+x**2),x)
Output:
x**2 - exp(((-x**2 - x)*exp(exp(x)) + exp(5)*exp(x))/(x**2 + x))
Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\left (-\frac {e^{\left (x + 5\right )}}{x + 1} + \frac {e^{\left (x + 5\right )}}{x} - e^{\left (e^{x}\right )}\right )} \] Input:
integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*e xp(((-x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2 *x^3+x^2),x, algorithm="maxima")
Output:
x^2 - e^(-e^(x + 5)/(x + 1) + e^(x + 5)/x - e^(e^x))
\[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=\int { \frac {2 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} + {\left ({\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (x + e^{x}\right )} - {\left (x^{2} - x - 1\right )} e^{\left (x + 5\right )}\right )} e^{\left (-\frac {{\left (x^{2} + x\right )} e^{\left (e^{x}\right )} - e^{\left (x + 5\right )}}{x^{2} + x}\right )}}{x^{4} + 2 \, x^{3} + x^{2}} \,d x } \] Input:
integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*e xp(((-x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2 *x^3+x^2),x, algorithm="giac")
Output:
integrate((2*x^5 + 4*x^4 + 2*x^3 + ((x^4 + 2*x^3 + x^2)*e^(x + e^x) - (x^2 - x - 1)*e^(x + 5))*e^(-((x^2 + x)*e^(e^x) - e^(x + 5))/(x^2 + x)))/(x^4 + 2*x^3 + x^2), x)
Time = 3.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^2-{\mathrm {e}}^{\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{x^2+x}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x^2+x}}\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x^2+x}} \] Input:
int((exp(-(exp(exp(x))*(x + x^2) - exp(5)*exp(x))/(x + x^2))*(exp(5)*exp(x )*(x - x^2 + 1) + exp(exp(x))*exp(x)*(x^2 + 2*x^3 + x^4)) + 2*x^3 + 4*x^4 + 2*x^5)/(x^2 + 2*x^3 + x^4),x)
Output:
x^2 - exp((exp(5)*exp(x))/(x + x^2))*exp(-(x*exp(exp(x)))/(x + x^2))*exp(- (x^2*exp(exp(x)))/(x + x^2))
Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=\frac {e^{e^{e^{x}}} x^{2}-e^{\frac {e^{x} e^{5}}{x^{2}+x}}}{e^{e^{e^{x}}}} \] Input:
int((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((- x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x ^2),x)
Output:
(e**(e**(e**x))*x**2 - e**((e**x*e**5)/(x**2 + x)))/e**(e**(e**x))