Integrand size = 175, antiderivative size = 32 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\left (e^{e^{-\frac {x}{10}+x^2} x}-\frac {e^x}{x}-e^x x\right )^2 \] Output:
(exp(x/exp(-x^2+1/10*x))-exp(x)/x-exp(x)*x)^2
Time = 6.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\frac {\left (e^{e^{-\frac {x}{10}+x^2} x} x-e^x \left (1+x^2\right )\right )^2}{x^2} \] Input:
Integrate[(E^((-x + 10*x^2)/10)*(E^(2*x + (x - 10*x^2)/10)*(-10 + 10*x + 2 0*x^3 + 10*x^4 + 10*x^5) + E^(2*E^((-x + 10*x^2)/10)*x)*(10*x^3 - x^4 + 20 *x^5) + E^(x + E^((-x + 10*x^2)/10)*x)*(-10*x^2 + x^3 - 30*x^4 + x^5 - 20* x^6 + E^((x - 10*x^2)/10)*(10*x - 10*x^2 - 10*x^3 - 10*x^4))))/(5*x^3),x]
Output:
(E^(E^(-1/10*x + x^2)*x)*x - E^x*(1 + x^2))^2/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.
| \(\displaystyle \int \frac {e^{\frac {1}{10} \left (10 x^2-x\right )} \left (e^{\frac {1}{10} \left (x-10 x^2\right )+2 x} \left (10 x^5+10 x^4+20 x^3+10 x-10\right )+e^{2 e^{\frac {1}{10} \left (10 x^2-x\right )} x} \left (20 x^5-x^4+10 x^3\right )+e^{e^{\frac {1}{10} \left (10 x^2-x\right )} x+x} \left (-20 x^6+x^5-30 x^4+x^3-10 x^2+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (-10 x^4-10 x^3-10 x^2+10 x\right )\right )\right )}{5 x^3} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {e^{\frac {1}{10} \left (10 x^2-x\right )} \left (10 e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-x^5-x^4-2 x^3-x+1\right )-e^{2 e^{\frac {1}{10} \left (10 x^2-x\right )} x} \left (20 x^5-x^4+10 x^3\right )+e^{e^{\frac {1}{10} \left (10 x^2-x\right )} x+x} \left (20 x^6-x^5+30 x^4-x^3+10 x^2-10 e^{\frac {1}{10} \left (x-10 x^2\right )} \left (-x^4-x^3-x^2+x\right )\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{\frac {1}{10} \left (10 x^2-x\right )} \left (10 e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-x^5-x^4-2 x^3-x+1\right )-e^{2 e^{\frac {1}{10} \left (10 x^2-x\right )} x} \left (20 x^5-x^4+10 x^3\right )+e^{e^{\frac {1}{10} \left (10 x^2-x\right )} x+x} \left (20 x^6-x^5+30 x^4-x^3+10 x^2-10 e^{\frac {1}{10} \left (x-10 x^2\right )} \left (-x^4-x^3-x^2+x\right )\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{\frac {1}{10} x (10 x-1)-x^2} \left (e^x x^2-e^{e^{x^2-\frac {x}{10}} x} x+e^x\right ) \left (20 e^{x \left (x+e^{x^2-\frac {x}{10}}\right )} x^4-10 e^{11 x/10} x^3-e^{x \left (x+e^{x^2-\frac {x}{10}}\right )} x^3-10 e^{11 x/10} x^2+10 e^{x \left (x+e^{x^2-\frac {x}{10}}\right )} x^2-10 e^{11 x/10} x+10 e^{11 x/10}\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-x/10} \left (e^{e^{x^2-\frac {x}{10}} x} x-e^x \left (x^2+1\right )\right ) \left (10 e^{11 x/10} \left (x^3+x^2+x-1\right )-e^{x \left (x+e^{x^2-\frac {x}{10}}\right )} x^2 \left (20 x^2-x+10\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x^2+e^{x^2-\frac {x}{10}} x-\frac {x}{10}} \left (20 x^2-x+10\right ) \left (e^x x^2-e^{e^{x^2-\frac {x}{10}} x} x+e^x\right )}{x}-\frac {10 e^x \left (e^x x^2-e^{e^{x^2-\frac {x}{10}} x} x+e^x\right ) \left (x^3+x^2+x-1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (-10 \int e^{e^{x^2-\frac {x}{10}} x+x}dx+\int e^{x^2+e^{x^2-\frac {x}{10}} x+\frac {9 x}{10}}dx+10 \int e^{x^2+2 e^{x^2-\frac {x}{10}} x-\frac {x}{10}}dx+10 \int \frac {e^{e^{x^2-\frac {x}{10}} x+x}}{x^2}dx-10 \int \frac {e^{e^{x^2-\frac {x}{10}} x+x}}{x}dx-10 \int \frac {e^{x^2+e^{x^2-\frac {x}{10}} x+\frac {9 x}{10}}}{x}dx-10 \int e^{e^{x^2-\frac {x}{10}} x+x} xdx-30 \int e^{x^2+e^{x^2-\frac {x}{10}} x+\frac {9 x}{10}} xdx-\int e^{x^2+2 e^{x^2-\frac {x}{10}} x-\frac {x}{10}} xdx+\int e^{x^2+e^{x^2-\frac {x}{10}} x+\frac {9 x}{10}} x^2dx+20 \int e^{x^2+2 e^{x^2-\frac {x}{10}} x-\frac {x}{10}} x^2dx-20 \int e^{x^2+e^{x^2-\frac {x}{10}} x+\frac {9 x}{10}} x^3dx+5 e^{2 x} x^2+\frac {5 e^{2 x}}{x^2}+10 e^{2 x}\right )\) |
Input:
Int[(E^((-x + 10*x^2)/10)*(E^(2*x + (x - 10*x^2)/10)*(-10 + 10*x + 20*x^3 + 10*x^4 + 10*x^5) + E^(2*E^((-x + 10*x^2)/10)*x)*(10*x^3 - x^4 + 20*x^5) + E^(x + E^((-x + 10*x^2)/10)*x)*(-10*x^2 + x^3 - 30*x^4 + x^5 - 20*x^6 + E^((x - 10*x^2)/10)*(10*x - 10*x^2 - 10*x^3 - 10*x^4))))/(5*x^3),x]
Output:
$Aborted
Time = 0.75 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(\frac {\left (x^{4}+2 x^{2}+1\right ) {\mathrm e}^{2 x}}{x^{2}}+{\mathrm e}^{2 x \,{\mathrm e}^{\frac {x \left (10 x -1\right )}{10}}}-\frac {2 \left (x^{2}+1\right ) {\mathrm e}^{x \left ({\mathrm e}^{\frac {x \left (10 x -1\right )}{10}}+1\right )}}{x}\) | \(57\) |
| parallelrisch | \(\frac {50 \,{\mathrm e}^{2 x} x^{4}-100 \,{\mathrm e}^{x} {\mathrm e}^{x \,{\mathrm e}^{x^{2}-\frac {1}{10} x}} x^{3}+100 \,{\mathrm e}^{2 x} x^{2}+50 \,{\mathrm e}^{2 x \,{\mathrm e}^{x^{2}-\frac {1}{10} x}} x^{2}-100 \,{\mathrm e}^{x} x \,{\mathrm e}^{x \,{\mathrm e}^{x^{2}-\frac {1}{10} x}}+50 \,{\mathrm e}^{2 x}}{50 x^{2}}\) | \(95\) |
Input:
int(1/5*((20*x^5-x^4+10*x^3)*exp(x/exp(-x^2+1/10*x))^2+((-10*x^4-10*x^3-10 *x^2+10*x)*exp(-x^2+1/10*x)-20*x^6+x^5-30*x^4+x^3-10*x^2)*exp(x)*exp(x/exp (-x^2+1/10*x))+(10*x^5+10*x^4+20*x^3+10*x-10)*exp(-x^2+1/10*x)*exp(x)^2)/x ^3/exp(-x^2+1/10*x),x,method=_RETURNVERBOSE)
Output:
(x^4+2*x^2+1)/x^2*exp(2*x)+exp(2*x*exp(1/10*x*(10*x-1)))-2*(x^2+1)/x*exp(x *(exp(1/10*x*(10*x-1))+1))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\frac {{\left (x^{2} e^{\left (2 \, x e^{\left (x^{2} - \frac {1}{10} \, x\right )} + 2 \, x\right )} - 2 \, {\left (x^{3} + x\right )} e^{\left (x e^{\left (x^{2} - \frac {1}{10} \, x\right )} + 3 \, x\right )} + {\left (x^{4} + 2 \, x^{2} + 1\right )} e^{\left (4 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x^{2}} \] Input:
integrate(1/5*((20*x^5-x^4+10*x^3)*exp(x/exp(-x^2+1/10*x))^2+((-10*x^4-10* x^3-10*x^2+10*x)*exp(-x^2+1/10*x)-20*x^6+x^5-30*x^4+x^3-10*x^2)*exp(x)*exp (x/exp(-x^2+1/10*x))+(10*x^5+10*x^4+20*x^3+10*x-10)*exp(-x^2+1/10*x)*exp(x )^2)/x^3/exp(-x^2+1/10*x),x, algorithm="fricas")
Output:
(x^2*e^(2*x*e^(x^2 - 1/10*x) + 2*x) - 2*(x^3 + x)*e^(x*e^(x^2 - 1/10*x) + 3*x) + (x^4 + 2*x^2 + 1)*e^(4*x))*e^(-2*x)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\frac {x e^{2 x e^{x^{2} - \frac {x}{10}}} + \left (- 2 x^{2} e^{x} - 2 e^{x}\right ) e^{x e^{x^{2} - \frac {x}{10}}}}{x} + \frac {\left (x^{4} + 2 x^{2} + 1\right ) e^{2 x}}{x^{2}} \] Input:
integrate(1/5*((20*x**5-x**4+10*x**3)*exp(x/exp(-x**2+1/10*x))**2+((-10*x* *4-10*x**3-10*x**2+10*x)*exp(-x**2+1/10*x)-20*x**6+x**5-30*x**4+x**3-10*x* *2)*exp(x)*exp(x/exp(-x**2+1/10*x))+(10*x**5+10*x**4+20*x**3+10*x-10)*exp( -x**2+1/10*x)*exp(x)**2)/x**3/exp(-x**2+1/10*x),x)
Output:
(x*exp(2*x*exp(x**2 - x/10)) + (-2*x**2*exp(x) - 2*exp(x))*exp(x*exp(x**2 - x/10)))/x + (x**4 + 2*x**2 + 1)*exp(2*x)/x**2
\[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\int { \frac {{\left (10 \, {\left (x^{5} + x^{4} + 2 \, x^{3} + x - 1\right )} e^{\left (-x^{2} + \frac {21}{10} \, x\right )} + {\left (20 \, x^{5} - x^{4} + 10 \, x^{3}\right )} e^{\left (2 \, x e^{\left (x^{2} - \frac {1}{10} \, x\right )}\right )} - {\left (20 \, x^{6} - x^{5} + 30 \, x^{4} - x^{3} + 10 \, x^{2} + 10 \, {\left (x^{4} + x^{3} + x^{2} - x\right )} e^{\left (-x^{2} + \frac {1}{10} \, x\right )}\right )} e^{\left (x e^{\left (x^{2} - \frac {1}{10} \, x\right )} + x\right )}\right )} e^{\left (x^{2} - \frac {1}{10} \, x\right )}}{5 \, x^{3}} \,d x } \] Input:
integrate(1/5*((20*x^5-x^4+10*x^3)*exp(x/exp(-x^2+1/10*x))^2+((-10*x^4-10* x^3-10*x^2+10*x)*exp(-x^2+1/10*x)-20*x^6+x^5-30*x^4+x^3-10*x^2)*exp(x)*exp (x/exp(-x^2+1/10*x))+(10*x^5+10*x^4+20*x^3+10*x-10)*exp(-x^2+1/10*x)*exp(x )^2)/x^3/exp(-x^2+1/10*x),x, algorithm="maxima")
Output:
1/2*(2*x^2 - 2*x + 1)*e^(2*x) + 1/2*(2*x - 1)*e^(2*x) + e^(2*x*e^(x^2 - 1/ 10*x)) + 2*e^(2*x) + 4*gamma(-1, -2*x) + 8*gamma(-2, -2*x) - 1/5*integrate (((20*x^5 - x^4 + 30*x^3 - x^2 + 10*x)*e^(x^2 + 9/10*x) + 10*(x^3 + x^2 + x - 1)*e^x)*e^(x*e^(x^2 - 1/10*x))/x^2, x)
\[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\int { \frac {{\left (10 \, {\left (x^{5} + x^{4} + 2 \, x^{3} + x - 1\right )} e^{\left (-x^{2} + \frac {21}{10} \, x\right )} + {\left (20 \, x^{5} - x^{4} + 10 \, x^{3}\right )} e^{\left (2 \, x e^{\left (x^{2} - \frac {1}{10} \, x\right )}\right )} - {\left (20 \, x^{6} - x^{5} + 30 \, x^{4} - x^{3} + 10 \, x^{2} + 10 \, {\left (x^{4} + x^{3} + x^{2} - x\right )} e^{\left (-x^{2} + \frac {1}{10} \, x\right )}\right )} e^{\left (x e^{\left (x^{2} - \frac {1}{10} \, x\right )} + x\right )}\right )} e^{\left (x^{2} - \frac {1}{10} \, x\right )}}{5 \, x^{3}} \,d x } \] Input:
integrate(1/5*((20*x^5-x^4+10*x^3)*exp(x/exp(-x^2+1/10*x))^2+((-10*x^4-10* x^3-10*x^2+10*x)*exp(-x^2+1/10*x)-20*x^6+x^5-30*x^4+x^3-10*x^2)*exp(x)*exp (x/exp(-x^2+1/10*x))+(10*x^5+10*x^4+20*x^3+10*x-10)*exp(-x^2+1/10*x)*exp(x )^2)/x^3/exp(-x^2+1/10*x),x, algorithm="giac")
Output:
integrate(1/5*(10*(x^5 + x^4 + 2*x^3 + x - 1)*e^(-x^2 + 21/10*x) + (20*x^5 - x^4 + 10*x^3)*e^(2*x*e^(x^2 - 1/10*x)) - (20*x^6 - x^5 + 30*x^4 - x^3 + 10*x^2 + 10*(x^4 + x^3 + x^2 - x)*e^(-x^2 + 1/10*x))*e^(x*e^(x^2 - 1/10*x ) + x))*e^(x^2 - 1/10*x)/x^3, x)
Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\frac {{\left ({\mathrm {e}}^x+x^2\,{\mathrm {e}}^x-x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {x}{10}}\,{\mathrm {e}}^{x^2}}\right )}^2}{x^2} \] Input:
int((exp(x^2 - x/10)*((exp(2*x*exp(x^2 - x/10))*(10*x^3 - x^4 + 20*x^5))/5 + (exp(2*x)*exp(x/10 - x^2)*(10*x + 20*x^3 + 10*x^4 + 10*x^5 - 10))/5 - ( exp(x*exp(x^2 - x/10))*exp(x)*(exp(x/10 - x^2)*(10*x^2 - 10*x + 10*x^3 + 1 0*x^4) + 10*x^2 - x^3 + 30*x^4 - x^5 + 20*x^6))/5))/x^3,x)
Output:
(exp(x) + x^2*exp(x) - x*exp(x*exp(-x/10)*exp(x^2)))^2/x^2
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.38 \[ \int \frac {e^{\frac {1}{10} \left (-x+10 x^2\right )} \left (e^{2 x+\frac {1}{10} \left (x-10 x^2\right )} \left (-10+10 x+20 x^3+10 x^4+10 x^5\right )+e^{2 e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (10 x^3-x^4+20 x^5\right )+e^{x+e^{\frac {1}{10} \left (-x+10 x^2\right )} x} \left (-10 x^2+x^3-30 x^4+x^5-20 x^6+e^{\frac {1}{10} \left (x-10 x^2\right )} \left (10 x-10 x^2-10 x^3-10 x^4\right )\right )\right )}{5 x^3} \, dx=\frac {e^{\frac {2 e^{x^{2}} x}{e^{\frac {x}{10}}}} x^{2}-2 e^{\frac {e^{x^{2}} x +e^{\frac {x}{10}} x}{e^{\frac {x}{10}}}} x^{3}-2 e^{\frac {e^{x^{2}} x +e^{\frac {x}{10}} x}{e^{\frac {x}{10}}}} x +e^{2 x} x^{4}+2 e^{2 x} x^{2}+e^{2 x}}{x^{2}} \] Input:
int(1/5*((20*x^5-x^4+10*x^3)*exp(x/exp(-x^2+1/10*x))^2+((-10*x^4-10*x^3-10 *x^2+10*x)*exp(-x^2+1/10*x)-20*x^6+x^5-30*x^4+x^3-10*x^2)*exp(x)*exp(x/exp (-x^2+1/10*x))+(10*x^5+10*x^4+20*x^3+10*x-10)*exp(-x^2+1/10*x)*exp(x)^2)/x ^3/exp(-x^2+1/10*x),x)
Output:
(e**((2*e**(x**2)*x)/e**(x/10))*x**2 - 2*e**((e**(x**2)*x + e**(x/10)*x)/e **(x/10))*x**3 - 2*e**((e**(x**2)*x + e**(x/10)*x)/e**(x/10))*x + e**(2*x) *x**4 + 2*e**(2*x)*x**2 + e**(2*x))/x**2