Integrand size = 126, antiderivative size = 36 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=\frac {2-\frac {(4-x) \left (-e^x+e^{-x^2}+2 x\right )}{e^5+x}}{x} \] Output:
(2-(2*x+1/exp(x^2)-exp(x))*(4-x)/(exp(5)+x))/x
Time = 10.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=\frac {e^{-x^2} \left (-4-e^{x+x^2} (-4+x)-2 e^{5+x^2} (-1+x)+x-6 e^{x^2} x\right )}{x \left (e^5+x\right )} \] Input:
Integrate[(8*x - x^2 + 8*x^3 - 2*x^4 + E^5*(4 + 8*x^2 - 2*x^3) + E^x^2*(-2 *E^10 + 6*x^2 + E^5*(-4*x + 2*x^2) + E^x*(-8*x + 5*x^2 - x^3 + E^5*(-4 + 4 *x - x^2))))/(E^x^2*(E^10*x^2 + 2*E^5*x^3 + x^4)),x]
Output:
(-4 - E^(x + x^2)*(-4 + x) - 2*E^(5 + x^2)*(-1 + x) + x - 6*E^x^2*x)/(E^x^ 2*x*(E^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.
| \(\displaystyle \int \frac {e^{-x^2} \left (-2 x^4+8 x^3-x^2+e^5 \left (-2 x^3+8 x^2+4\right )+e^{x^2} \left (6 x^2+e^5 \left (2 x^2-4 x\right )+e^x \left (-x^3+5 x^2+e^5 \left (-x^2+4 x-4\right )-8 x\right )-2 e^{10}\right )+8 x\right )}{x^4+2 e^5 x^3+e^{10} x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{-x^2} \left (-2 x^4+8 x^3-x^2+e^5 \left (-2 x^3+8 x^2+4\right )+e^{x^2} \left (6 x^2+e^5 \left (2 x^2-4 x\right )+e^x \left (-x^3+5 x^2+e^5 \left (-x^2+4 x-4\right )-8 x\right )-2 e^{10}\right )+8 x\right )}{x^2 \left (x^2+2 e^5 x+e^{10}\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^{-x^2} \left (-2 x^4+8 x^3-x^2+e^5 \left (-2 x^3+8 x^2+4\right )+e^{x^2} \left (6 x^2+e^5 \left (2 x^2-4 x\right )+e^x \left (-x^3+5 x^2+e^5 \left (-x^2+4 x-4\right )-8 x\right )-2 e^{10}\right )+8 x\right )}{x^2 \left (x+e^5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 e^{-x^2} x^2}{\left (x+e^5\right )^2}+\frac {8 e^{-x^2} x}{\left (x+e^5\right )^2}-\frac {e^{-x^2}}{\left (x+e^5\right )^2}+\frac {8 e^{-x^2}}{\left (x+e^5\right )^2 x}-\frac {2 e^{5-x^2} \left (x^3-4 x^2-2\right )}{\left (x+e^5\right )^2 x^2}+\frac {-e^x x^3+6 \left (1+\frac {e^5}{3}\right ) x^2+5 \left (1-\frac {e^5}{5}\right ) e^x x^2-8 \left (1-\frac {e^5}{2}\right ) e^x x-4 e^5 x-4 e^{x+5}-2 e^{10}}{\left (x+e^5\right )^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e^5 \int \frac {e^{-x^2}}{x+e^5}dx+8 \int \frac {e^{-x^2}}{x+e^5}dx+2 \left (4-e^{15}\right ) \int \frac {e^{-x^2-10}}{x+e^5}dx-8 \int \frac {e^{-x^2-10}}{x+e^5}dx+4 e^5 \left (2+4 e^{10}+e^{15}\right ) \int \frac {e^{-x^2-5}}{x+e^5}dx-16 e^5 \int \frac {e^{-x^2-5}}{x+e^5}dx-16 e^5 \int \frac {e^{5-x^2}}{x+e^5}dx+4 \int \frac {e^{5-x^2}}{x+e^5}dx-4 e^5 \int \frac {e^{10-x^2}}{x+e^5}dx-\frac {2 \left (2+4 e^{10}+e^{15}\right ) \sqrt {\pi } \text {erf}(x)}{e^5}+2 e^{10} \sqrt {\pi } \text {erf}(x)+8 e^5 \sqrt {\pi } \text {erf}(x)+\frac {4 \sqrt {\pi } \text {erf}(x)}{e^5}-\frac {4 e^{-x^2-5}}{x}+\frac {e^{-x^2}}{x+e^5}+\frac {8 e^{-x^2-5}}{x+e^5}+\frac {8 e^{5-x^2}}{x+e^5}+\frac {2 e^{10-x^2}}{x+e^5}-\frac {2 \left (2+4 e^{10}+e^{15}\right ) e^{-x^2-5}}{x+e^5}+\frac {4 e^{x-5}}{x}+\frac {2}{x}-\frac {\left (4+e^5\right ) e^{x-5}}{x+e^5}-\frac {2 \left (4+e^5\right )}{x+e^5}\) |
Input:
Int[(8*x - x^2 + 8*x^3 - 2*x^4 + E^5*(4 + 8*x^2 - 2*x^3) + E^x^2*(-2*E^10 + 6*x^2 + E^5*(-4*x + 2*x^2) + E^x*(-8*x + 5*x^2 - x^3 + E^5*(-4 + 4*x - x ^2))))/(E^x^2*(E^10*x^2 + 2*E^5*x^3 + x^4)),x]
Output:
$Aborted
Time = 7.50 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58
| method | result | size |
| norman | \(\frac {\left (-4+x +\left (-2 \,{\mathrm e}^{5}-6\right ) x \,{\mathrm e}^{x^{2}}+2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+4 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} {\mathrm e}^{x} x \right ) {\mathrm e}^{-x^{2}}}{x \left ({\mathrm e}^{5}+x \right )}\) | \(57\) |
| risch | \(\frac {\left (-2 \,{\mathrm e}^{5}-6\right ) x +2 \,{\mathrm e}^{5}}{\left ({\mathrm e}^{5}+x \right ) x}-\frac {\left (x -4\right ) {\mathrm e}^{x}}{\left ({\mathrm e}^{5}+x \right ) x}+\frac {\left (x -4\right ) {\mathrm e}^{-x^{2}}}{\left ({\mathrm e}^{5}+x \right ) x}\) | \(60\) |
| parallelrisch | \(\frac {\left (-4-2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}} x -{\mathrm e}^{x^{2}} {\mathrm e}^{x} x +2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}-6 \,{\mathrm e}^{x^{2}} x +4 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}+x \right ) {\mathrm e}^{-x^{2}}}{x \left ({\mathrm e}^{5}+x \right )}\) | \(61\) |
| parts | \(\frac {\left (-2 \,{\mathrm e}^{5}-6\right ) x +2 \,{\mathrm e}^{5}}{\left ({\mathrm e}^{5}+x \right ) x}-\frac {{\mathrm e}^{x} {\mathrm e}^{5}}{{\mathrm e}^{5}+x}+\left (-{\mathrm e}^{5}+1\right ) {\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )-{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x}}{{\mathrm e}^{5}+x}-{\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )\right )-\frac {5 \,{\mathrm e}^{x}}{{\mathrm e}^{5}+x}-5 \,{\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )-\frac {8 \,{\mathrm e}^{x} {\mathrm e}^{-5}}{{\mathrm e}^{5}+x}+8 \left ({\mathrm e}^{-5}\right )^{2} \operatorname {expIntegral}_{1}\left (-x \right )-8 \left ({\mathrm e}^{-5}\right )^{2} \left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )-4 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x} \left (2 x +{\mathrm e}^{5}\right ) {\mathrm e}^{-10}}{x \left ({\mathrm e}^{5}+x \right )}-\left ({\mathrm e}^{5}-2\right ) {\mathrm e}^{-10} {\mathrm e}^{-5} \operatorname {expIntegral}_{1}\left (-x \right )-{\mathrm e}^{-10} \left ({\mathrm e}^{5}+2\right ) {\mathrm e}^{-5} {\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )\right )+4 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{x} {\mathrm e}^{-5}}{{\mathrm e}^{5}+x}-\left ({\mathrm e}^{-5}\right )^{2} \operatorname {expIntegral}_{1}\left (-x \right )+\left ({\mathrm e}^{-5}\right )^{2} \left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{-{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (-{\mathrm e}^{5}-x \right )\right )+\frac {\left (x -4\right ) {\mathrm e}^{-x^{2}}}{\left ({\mathrm e}^{5}+x \right ) x}\) | \(320\) |
Input:
int(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*ex p(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp (5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x,method=_RETURNVERBOSE)
Output:
(-4+x+(-2*exp(5)-6)*x*exp(x^2)+2*exp(5)*exp(x^2)+4*exp(x)*exp(x^2)-exp(x^2 )*exp(x)*x)/x/(exp(5)+x)/exp(x^2)
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=-\frac {{\left ({\left (2 \, {\left (x - 1\right )} e^{5} + {\left (x - 4\right )} e^{x} + 6 \, x\right )} e^{\left (x^{2}\right )} - x + 4\right )} e^{\left (-x^{2}\right )}}{x^{2} + x e^{5}} \] Input:
integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4 *x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x ^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="fricas")
Output:
-((2*(x - 1)*e^5 + (x - 4)*e^x + 6*x)*e^(x^2) - x + 4)*e^(-x^2)/(x^2 + x*e ^5)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=\frac {\left (4 - x\right ) e^{x}}{x^{2} + x e^{5}} + \frac {\left (x - 4\right ) e^{- x^{2}}}{x^{2} + x e^{5}} + \frac {x \left (- 2 e^{5} - 6\right ) + 2 e^{5}}{x^{2} + x e^{5}} \] Input:
integrate(((((-x**2+4*x-4)*exp(5)-x**3+5*x**2-8*x)*exp(x)-2*exp(5)**2+(2*x **2-4*x)*exp(5)+6*x**2)*exp(x**2)+(-2*x**3+8*x**2+4)*exp(5)-2*x**4+8*x**3- x**2+8*x)/(x**2*exp(5)**2+2*x**3*exp(5)+x**4)/exp(x**2),x)
Output:
(4 - x)*exp(x)/(x**2 + x*exp(5)) + (x - 4)*exp(-x**2)/(x**2 + x*exp(5)) + (x*(-2*exp(5) - 6) + 2*exp(5))/(x**2 + x*exp(5))
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=-\frac {2 \, x {\left (e^{5} + 3\right )} - {\left (x - 4\right )} e^{\left (-x^{2}\right )} + {\left (x - 4\right )} e^{x} - 2 \, e^{5}}{x^{2} + x e^{5}} \] Input:
integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4 *x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x ^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="maxima")
Output:
-(2*x*(e^5 + 3) - (x - 4)*e^(-x^2) + (x - 4)*e^x - 2*e^5)/(x^2 + x*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.83 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=-\frac {x e^{\left (x^{2} + x\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + 6 \, x e^{\left (x^{2}\right )} - x - 4 \, e^{\left (x^{2} + x\right )} - 2 \, e^{\left (x^{2} + 5\right )} + 4}{x^{2} e^{\left (x^{2}\right )} + x e^{\left (x^{2} + 5\right )}} \] Input:
integrate(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4 *x)*exp(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x ^2*exp(5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x, algorithm="giac")
Output:
-(x*e^(x^2 + x) + 2*x*e^(x^2 + 5) + 6*x*e^(x^2) - x - 4*e^(x^2 + x) - 2*e^ (x^2 + 5) + 4)/(x^2*e^(x^2) + x*e^(x^2 + 5))
Time = 8.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=-\frac {6\,x-2\,{\mathrm {e}}^5+4\,{\mathrm {e}}^{-x^2}-4\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^{-x^2}+x\,{\mathrm {e}}^x}{x\,\left (x+{\mathrm {e}}^5\right )} \] Input:
int((exp(-x^2)*(8*x - exp(x^2)*(2*exp(10) + exp(5)*(4*x - 2*x^2) + exp(x)* (8*x + exp(5)*(x^2 - 4*x + 4) - 5*x^2 + x^3) - 6*x^2) + exp(5)*(8*x^2 - 2* x^3 + 4) - x^2 + 8*x^3 - 2*x^4))/(2*x^3*exp(5) + x^2*exp(10) + x^4),x)
Output:
-(6*x - 2*exp(5) + 4*exp(-x^2) - 4*exp(x) + 2*x*exp(5) - x*exp(-x^2) + x*e xp(x))/(x*(x + exp(5)))
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.50 \[ \int \frac {e^{-x^2} \left (8 x-x^2+8 x^3-2 x^4+e^5 \left (4+8 x^2-2 x^3\right )+e^{x^2} \left (-2 e^{10}+6 x^2+e^5 \left (-4 x+2 x^2\right )+e^x \left (-8 x+5 x^2-x^3+e^5 \left (-4+4 x-x^2\right )\right )\right )\right )}{e^{10} x^2+2 e^5 x^3+x^4} \, dx=\frac {-e^{x^{2}+x} e^{5} x +4 e^{x^{2}+x} e^{5}+2 e^{x^{2}} e^{10}+2 e^{x^{2}} e^{5} x^{2}+6 e^{x^{2}} x^{2}+e^{5} x -4 e^{5}}{e^{x^{2}} e^{5} x \left (e^{5}+x \right )} \] Input:
int(((((-x^2+4*x-4)*exp(5)-x^3+5*x^2-8*x)*exp(x)-2*exp(5)^2+(2*x^2-4*x)*ex p(5)+6*x^2)*exp(x^2)+(-2*x^3+8*x^2+4)*exp(5)-2*x^4+8*x^3-x^2+8*x)/(x^2*exp (5)^2+2*x^3*exp(5)+x^4)/exp(x^2),x)
Output:
( - e**(x**2 + x)*e**5*x + 4*e**(x**2 + x)*e**5 + 2*e**(x**2)*e**10 + 2*e* *(x**2)*e**5*x**2 + 6*e**(x**2)*x**2 + e**5*x - 4*e**5)/(e**(x**2)*e**5*x* (e**5 + x))