Integrand size = 91, antiderivative size = 24 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=-e^{e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}}+x \] Output:
x-exp(exp(exp(-ln(x^2+x-5)+exp(x^2))))
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=-e^{e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}}+x \] Input:
Integrate[(-5 + x + x^2 + (E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2 /(-5 + x + x^2))*(1 + 2*x + E^x^2*(10*x - 2*x^2 - 2*x^3)))/(-5 + x + x^2)) /(-5 + x + x^2),x]
Output:
-E^E^(E^E^x^2/(-5 + x + 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.
\(\displaystyle \int \frac {x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{x^2+x-5}}+\frac {e^{e^{x^2}}}{x^2+x-5}} \left (e^{x^2} \left (-2 x^3-2 x^2+10 x\right )+2 x+1\right )}{x^2+x-5}+x-5}{x^2+x-5} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (1-\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{x^2+x-5}}+\frac {e^{e^{x^2}}}{x^2+x-5}} \left (2 e^{x^2} x^2-10 e^{x^2} x+2 e^{x^2} x^3-2 x-1\right )}{\left (x^2+x-5\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (1-\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{x^2+x-5}}+\frac {e^{e^{x^2}}}{x^2+x-5}} \left (2 e^{x^2} x^2-10 e^{x^2} x+2 e^{x^2} x^3-2 x-1\right )}{\left (x^2+x-5\right )^2}\right )dx\) |
Input:
Int[(-5 + x + x^2 + (E^(E^x^2 + E^(E^E^x^2/(-5 + x + x^2)) + E^E^x^2/(-5 + x + x^2))*(1 + 2*x + E^x^2*(10*x - 2*x^2 - 2*x^3)))/(-5 + x + x^2))/(-5 + x + x^2),x]
Output:
$Aborted
Time = 130.70 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
risch | \(x -{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x^{2}}}}{x^{2}+x -5}}}\) | \(21\) |
parallelrisch | \(x -{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-\ln \left (x^{2}+x -5\right )+{\mathrm e}^{x^{2}}}}}+\frac {9}{2}\) | \(23\) |
Input:
int((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-ln(x^2+x-5)+exp(x^2))*exp(e xp(-ln(x^2+x-5)+exp(x^2)))*exp(exp(exp(-ln(x^2+x-5)+exp(x^2))))+x^2+x-5)/( x^2+x-5),x,method=_RETURNVERBOSE)
Output:
x-exp(exp(1/(x^2+x-5)*exp(exp(x^2))))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx={\left (x e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} - \log \left (x^{2} + x - 5\right )\right )} - e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + e^{\left (e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} e^{\left (-e^{\left (x^{2}\right )} - e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + \log \left (x^{2} + x - 5\right )\right )} \] Input:
integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2) )*exp(exp(-log(x^2+x-5)+exp(x^2)))*exp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x ^2+x-5)/(x^2+x-5),x, algorithm="fricas")
Output:
(x*e^(e^(x^2) + e^(e^(x^2) - log(x^2 + x - 5)) - log(x^2 + x - 5)) - e^(e^ (x^2) + e^(e^(x^2) - log(x^2 + x - 5)) + e^(e^(e^(x^2) - log(x^2 + x - 5)) ) - log(x^2 + x - 5)))*e^(-e^(x^2) - e^(e^(x^2) - log(x^2 + x - 5)) + log( x^2 + x - 5))
Time = 32.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=x - e^{e^{\frac {e^{e^{x^{2}}}}{x^{2} + x - 5}}} \] Input:
integrate((((-2*x**3-2*x**2+10*x)*exp(x**2)+2*x+1)*exp(-ln(x**2+x-5)+exp(x **2))*exp(exp(-ln(x**2+x-5)+exp(x**2)))*exp(exp(exp(-ln(x**2+x-5)+exp(x**2 ))))+x**2+x-5)/(x**2+x-5),x)
Output:
x - exp(exp(exp(exp(x**2))/(x**2 + x - 5)))
Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=x - e^{\left (e^{\left (\frac {e^{\left (e^{\left (x^{2}\right )}\right )}}{x^{2} + x - 5}\right )}\right )} \] Input:
integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2) )*exp(exp(-log(x^2+x-5)+exp(x^2)))*exp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x ^2+x-5)/(x^2+x-5),x, algorithm="maxima")
Output:
x - e^(e^(e^(e^(x^2))/(x^2 + x - 5)))
\[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=\int { \frac {x^{2} - {\left (2 \, {\left (x^{3} + x^{2} - 5 \, x\right )} e^{\left (x^{2}\right )} - 2 \, x - 1\right )} e^{\left (e^{\left (x^{2}\right )} + e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )} + e^{\left (e^{\left (e^{\left (x^{2}\right )} - \log \left (x^{2} + x - 5\right )\right )}\right )} - \log \left (x^{2} + x - 5\right )\right )} + x - 5}{x^{2} + x - 5} \,d x } \] Input:
integrate((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2) )*exp(exp(-log(x^2+x-5)+exp(x^2)))*exp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x ^2+x-5)/(x^2+x-5),x, algorithm="giac")
Output:
integrate((x^2 - (2*(x^3 + x^2 - 5*x)*e^(x^2) - 2*x - 1)*e^(e^(x^2) + e^(e ^(x^2) - log(x^2 + x - 5)) + e^(e^(e^(x^2) - log(x^2 + x - 5))) - log(x^2 + x - 5)) + x - 5)/(x^2 + x - 5), x)
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^2}}}{x^2+x-5}}} \] Input:
int((x + x^2 + exp(exp(exp(exp(x^2) - log(x + x^2 - 5))))*exp(exp(x^2) - l og(x + x^2 - 5))*exp(exp(exp(x^2) - log(x + x^2 - 5)))*(2*x - exp(x^2)*(2* x^2 - 10*x + 2*x^3) + 1) - 5)/(x + x^2 - 5),x)
Output:
x - exp(exp(exp(exp(x^2))/(x + x^2 - 5)))
Time = 0.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-5+x+x^2+\frac {e^{e^{x^2}+e^{\frac {e^{e^{x^2}}}{-5+x+x^2}}+\frac {e^{e^{x^2}}}{-5+x+x^2}} \left (1+2 x+e^{x^2} \left (10 x-2 x^2-2 x^3\right )\right )}{-5+x+x^2}}{-5+x+x^2} \, dx=-e^{e^{\frac {e^{e^{x^{2}}}}{x^{2}+x -5}}}+x \] Input:
int((((-2*x^3-2*x^2+10*x)*exp(x^2)+2*x+1)*exp(-log(x^2+x-5)+exp(x^2))*exp( exp(-log(x^2+x-5)+exp(x^2)))*exp(exp(exp(-log(x^2+x-5)+exp(x^2))))+x^2+x-5 )/(x^2+x-5),x)
Output:
- e**(e**(e**(e**(x**2))/(x**2 + x - 5))) + x