Integrand size = 119, antiderivative size = 32 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2}{\frac {5 e^{-4-\frac {2+x+\left (2-e^x\right ) x}{x}}}{x}+x} \] Output:
2/(x+5/exp(4)/exp((2+x+x*(-exp(x)+2))/x)/x)
Time = 0.79 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2 e^{7+\frac {2}{x}} x}{5 e^{e^x}+e^{7+\frac {2}{x}} x^2} \] Input:
Integrate[(-2*E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^3 + E^((2 + 3*x - E^x*x)/x )*(-10*E^(4 + x)*x^2 + E^4*(-20 + 10*x)))/(25*x + 10*E^(4 + (2 + 3*x - E^x *x)/x)*x^3 + E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^5),x]
Output:
(2*E^(7 + 2/x)*x)/(5*E^E^x + E^(7 + 2/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.
\(\displaystyle \int \frac {e^{\frac {-e^x x+3 x+2}{x}} \left (e^4 (10 x-20)-10 e^{x+4} x^2\right )-2 e^{\frac {2 \left (-e^x x+3 x+2\right )}{x}+8} x^3}{e^{\frac {2 \left (-e^x x+3 x+2\right )}{x}+8} x^5+10 e^{\frac {-e^x x+3 x+2}{x}+4} x^3+25 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 e^{\frac {2}{x}+7} \left (-e^{\frac {2}{x}+7} x^3-5 e^{x+e^x} x^2+5 e^{e^x} x-10 e^{e^x}\right )}{x \left (e^{\frac {2}{x}+7} x^2+5 e^{e^x}\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {e^{7+\frac {2}{x}} \left (e^{7+\frac {2}{x}} x^3+5 e^{x+e^x} x^2-5 e^{e^x} x+10 e^{e^x}\right )}{x \left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{7+\frac {2}{x}} \left (e^{7+\frac {2}{x}} x^3+5 e^{x+e^x} x^2-5 e^{e^x} x+10 e^{e^x}\right )}{x \left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {5 e^{x+e^x+7+\frac {2}{x}} x}{\left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}+\frac {e^{7+\frac {2}{x}} \left (e^{7+\frac {2}{x}} x^3-5 e^{e^x} x+10 e^{e^x}\right )}{\left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (-10 \int \frac {e^{e^x+7+\frac {2}{x}}}{\left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}dx+10 \int \frac {e^{e^x+7+\frac {2}{x}}}{x \left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}dx+5 \int \frac {e^{x+e^x+7+\frac {2}{x}} x}{\left (e^{7+\frac {2}{x}} x^2+5 e^{e^x}\right )^2}dx+\int \frac {e^{7+\frac {2}{x}}}{e^{7+\frac {2}{x}} x^2+5 e^{e^x}}dx\right )\) |
Input:
Int[(-2*E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^3 + E^((2 + 3*x - E^x*x)/x)*(-10 *E^(4 + x)*x^2 + E^4*(-20 + 10*x)))/(25*x + 10*E^(4 + (2 + 3*x - E^x*x)/x) *x^3 + E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^5),x]
Output:
$Aborted
Time = 2.76 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {2}{x}-\frac {10}{x \left ({\mathrm e}^{-\frac {{\mathrm e}^{x} x -7 x -2}{x}} x^{2}+5\right )}\) | \(35\) |
parallelrisch | \(\frac {2 x \,{\mathrm e}^{4} {\mathrm e}^{-\frac {{\mathrm e}^{x} x -3 x -2}{x}}}{x^{2} {\mathrm e}^{4} {\mathrm e}^{-\frac {{\mathrm e}^{x} x -3 x -2}{x}}+5}\) | \(46\) |
Input:
int((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10 *x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+ 2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x,method=_RETURNVERBO SE)
Output:
2/x-10/x/(exp(-(exp(x)*x-7*x-2)/x)*x^2+5)
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2 \, x e^{\left (\frac {{\left ({\left (7 \, x + 2\right )} e^{4} - x e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{x}\right )}}{x^{2} e^{\left (\frac {{\left ({\left (7 \, x + 2\right )} e^{4} - x e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{x}\right )} + 5} \] Input:
integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp( x)+(10*x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)* x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm="f ricas")
Output:
2*x*e^(((7*x + 2)*e^4 - x*e^(x + 4))*e^(-4)/x)/(x^2*e^(((7*x + 2)*e^4 - x* e^(x + 4))*e^(-4)/x) + 5)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=- \frac {10}{x^{3} e^{4} e^{\frac {- x e^{x} + 3 x + 2}{x}} + 5 x} + \frac {2}{x} \] Input:
integrate((-2*x**3*exp(4)**2*exp((-exp(x)*x+3*x+2)/x)**2+(-10*x**2*exp(4)* exp(x)+(10*x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x**5*exp(4)**2*exp((-e xp(x)*x+3*x+2)/x)**2+10*x**3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x)
Output:
-10/(x**3*exp(4)*exp((-x*exp(x) + 3*x + 2)/x) + 5*x) + 2/x
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2 \, x e^{\left (\frac {2}{x} + 7\right )}}{x^{2} e^{\left (\frac {2}{x} + 7\right )} + 5 \, e^{\left (e^{x}\right )}} \] Input:
integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp( x)+(10*x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)* x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm="m axima")
Output:
2*x*e^(2/x + 7)/(x^2*e^(2/x + 7) + 5*e^(e^x))
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2 \, x e^{\left (\frac {7 \, x + 2}{x}\right )}}{x^{2} e^{\left (\frac {7 \, x + 2}{x}\right )} + 5 \, e^{\left (e^{x}\right )}} \] Input:
integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp( x)+(10*x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)* x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm="g iac")
Output:
2*x*e^((7*x + 2)/x)/(x^2*e^((7*x + 2)/x) + 5*e^(e^x))
Time = 0.61 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2}{x}-\frac {10\,\left (x^2\,{\mathrm {e}}^x-2\,x+2\right )}{x\,\left ({\mathrm {e}}^{\frac {2}{x}-{\mathrm {e}}^x+3}+\frac {5\,{\mathrm {e}}^{-4}}{x^2}\right )\,\left (x^4\,{\mathrm {e}}^{x+4}+2\,x^2\,{\mathrm {e}}^4-2\,x^3\,{\mathrm {e}}^4\right )} \] Input:
int((exp((3*x - x*exp(x) + 2)/x)*(exp(4)*(10*x - 20) - 10*x^2*exp(4)*exp(x )) - 2*x^3*exp(8)*exp((2*(3*x - x*exp(x) + 2))/x))/(25*x + 10*x^3*exp(4)*e xp((3*x - x*exp(x) + 2)/x) + x^5*exp(8)*exp((2*(3*x - x*exp(x) + 2))/x)),x )
Output:
2/x - (10*(x^2*exp(x) - 2*x + 2))/(x*(exp(2/x - exp(x) + 3) + (5*exp(-4))/ x^2)*(x^4*exp(x + 4) + 2*x^2*exp(4) - 2*x^3*exp(4)))
Time = 3.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx=\frac {2 e^{\frac {2}{x}} e^{7} x}{5 e^{e^{x}}+e^{\frac {2}{x}} e^{7} x^{2}} \] Input:
int((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10 *x-20)*exp(4))*exp((-exp(x)*x+3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+ 2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x)
Output:
(2*e**(2/x)*e**7*x)/(5*e**(e**x) + e**(2/x)*e**7*x**2)