Integrand size = 84, antiderivative size = 27 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=-2+e^{2 e^5+\frac {x^2}{e^{2 x^2}-x}} \] Output:
exp(2*exp(5)+x^2/(exp(x^2)^2-x))-2
Time = 0.79 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=e^{2 e^5+\frac {x^2}{e^{2 x^2}-x}} \] Input:
Integrate[(E^((2*E^(5 + 2*x^2) - 2*E^5*x + x^2)/(E^(2*x^2) - x))*(-x^2 + E ^(2*x^2)*(2*x - 4*x^3)))/(E^(4*x^2) - 2*E^(2*x^2)*x + x^2),x]
Output:
E^(2*E^5 + x^2/(E^(2*x^2) - 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.
\(\displaystyle \int \frac {e^{\frac {x^2+2 e^{2 x^2+5}-2 e^5 x}{e^{2 x^2}-x}} \left (e^{2 x^2} \left (2 x-4 x^3\right )-x^2\right )}{x^2-2 e^{2 x^2} x+e^{4 x^2}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {x^2+2 e^{2 x^2+5}-2 e^5 x}{e^{2 x^2}-x}} \left (e^{2 x^2} \left (2 x-4 x^3\right )-x^2\right )}{\left (e^{2 x^2}-x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^{\frac {x^2+2 e^{2 x^2+5}-2 e^5 x}{e^{2 x^2}-x}} \left (4 x^2-1\right ) x^2}{\left (e^{2 x^2}-x\right )^2}-\frac {2 e^{\frac {x^2+2 e^{2 x^2+5}-2 e^5 x}{e^{2 x^2}-x}} \left (2 x^2-1\right ) x}{e^{2 x^2}-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {e^{\frac {x^2-2 e^5 x+2 e^{2 x^2+5}}{e^{2 x^2}-x}} x}{e^{2 x^2}-x}dx+\int \frac {e^{\frac {x^2-2 e^5 x+2 e^{2 x^2+5}}{e^{2 x^2}-x}} x^2}{\left (e^{2 x^2}-x\right )^2}dx-4 \int \frac {e^{\frac {x^2-2 e^5 x+2 e^{2 x^2+5}}{e^{2 x^2}-x}} x^4}{\left (e^{2 x^2}-x\right )^2}dx-4 \int \frac {e^{\frac {x^2-2 e^5 x+2 e^{2 x^2+5}}{e^{2 x^2}-x}} x^3}{e^{2 x^2}-x}dx\) |
Input:
Int[(E^((2*E^(5 + 2*x^2) - 2*E^5*x + x^2)/(E^(2*x^2) - x))*(-x^2 + E^(2*x^ 2)*(2*x - 4*x^3)))/(E^(4*x^2) - 2*E^(2*x^2)*x + x^2),x]
Output:
$Aborted
Time = 0.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{2 x^{2}}-2 x \,{\mathrm e}^{5}+x^{2}}{{\mathrm e}^{2 x^{2}}-x}}\) | \(34\) |
risch | \({\mathrm e}^{\frac {-2 \,{\mathrm e}^{2 x^{2}+5}+2 x \,{\mathrm e}^{5}-x^{2}}{-{\mathrm e}^{2 x^{2}}+x}}\) | \(36\) |
Input:
int(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2) /(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp(x^2)^2+x^2),x,method=_RETURNVERBOSE)
Output:
exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2)/(exp(x^2)^2-x))
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=e^{\left (-\frac {x^{2} e^{5} - 2 \, x e^{10} + 2 \, e^{\left (2 \, x^{2} + 10\right )}}{x e^{5} - e^{\left (2 \, x^{2} + 5\right )}}\right )} \] Input:
integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5 )+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp(x^2)^2+x^2),x, algorithm="frica s")
Output:
e^(-(x^2*e^5 - 2*x*e^10 + 2*e^(2*x^2 + 10))/(x*e^5 - e^(2*x^2 + 5)))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=e^{\frac {x^{2} - 2 x e^{5} + 2 e^{5} e^{2 x^{2}}}{- x + e^{2 x^{2}}}} \] Input:
integrate(((-4*x**3+2*x)*exp(x**2)**2-x**2)*exp((2*exp(5)*exp(x**2)**2-2*x *exp(5)+x**2)/(exp(x**2)**2-x))/(exp(x**2)**4-2*x*exp(x**2)**2+x**2),x)
Output:
exp((x**2 - 2*x*exp(5) + 2*exp(5)*exp(2*x**2))/(-x + exp(2*x**2)))
\[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=\int { -\frac {{\left (x^{2} + 2 \, {\left (2 \, x^{3} - x\right )} e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-\frac {x^{2} - 2 \, x e^{5} + 2 \, e^{\left (2 \, x^{2} + 5\right )}}{x - e^{\left (2 \, x^{2}\right )}}\right )}}{x^{2} - 2 \, x e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )}} \,d x } \] Input:
integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5 )+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp(x^2)^2+x^2),x, algorithm="maxim a")
Output:
-integrate((x^2 + 2*(2*x^3 - x)*e^(2*x^2))*e^(-(x^2 - 2*x*e^5 + 2*e^(2*x^2 + 5))/(x - e^(2*x^2)))/(x^2 - 2*x*e^(2*x^2) + e^(4*x^2)), x)
\[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=\int { -\frac {{\left (x^{2} + 2 \, {\left (2 \, x^{3} - x\right )} e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-\frac {x^{2} - 2 \, x e^{5} + 2 \, e^{\left (2 \, x^{2} + 5\right )}}{x - e^{\left (2 \, x^{2}\right )}}\right )}}{x^{2} - 2 \, x e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )}} \,d x } \] Input:
integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5 )+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp(x^2)^2+x^2),x, algorithm="giac" )
Output:
integrate(-(x^2 + 2*(2*x^3 - x)*e^(2*x^2))*e^(-(x^2 - 2*x*e^5 + 2*e^(2*x^2 + 5))/(x - e^(2*x^2)))/(x^2 - 2*x*e^(2*x^2) + e^(4*x^2)), x)
Time = 4.47 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx={\mathrm {e}}^{-\frac {x^2}{x-{\mathrm {e}}^{2\,x^2}}}\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^5}{x-{\mathrm {e}}^{2\,x^2}}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^5\,{\mathrm {e}}^{2\,x^2}}{x-{\mathrm {e}}^{2\,x^2}}} \] Input:
int((exp(-(2*exp(5)*exp(2*x^2) - 2*x*exp(5) + x^2)/(x - exp(2*x^2)))*(exp( 2*x^2)*(2*x - 4*x^3) - x^2))/(exp(4*x^2) - 2*x*exp(2*x^2) + x^2),x)
Output:
exp(-x^2/(x - exp(2*x^2)))*exp((2*x*exp(5))/(x - exp(2*x^2)))*exp(-(2*exp( 5)*exp(2*x^2))/(x - exp(2*x^2)))
Time = 87.50 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx=e^{\frac {2 e^{2 x^{2}} e^{5}-2 e^{5} x +x^{2}}{e^{2 x^{2}}-x}} \] Input:
int(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2) /(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp(x^2)^2+x^2),x)
Output:
e**((2*e**(2*x**2)*e**5 - 2*e**5*x + x**2)/(e**(2*x**2) - x))