Integrand size = 74, antiderivative size = 22 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (e^{x^2}+(-142-x)^2\right )} \] Output:
exp(x)/(4*(-142-x)^2+4*exp(x^2))
Time = 1.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (20164+e^{x^2}+284 x+x^2\right )} \] Input:
Integrate[(E^(x + x^2)*(1 - 2*x) + E^x*(19880 + 282*x + x^2))/(1626347584 + 4*E^(2*x^2) + 45812608*x + 483936*x^2 + 2272*x^3 + 4*x^4 + E^x^2*(161312 + 2272*x + 8*x^2)),x]
Output:
E^x/(4*(20164 + E^x^2 + 284*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.
\(\displaystyle \int \frac {e^{x^2+x} (1-2 x)+e^x \left (x^2+282 x+19880\right )}{4 x^4+2272 x^3+483936 x^2+4 e^{2 x^2}+e^{x^2} \left (8 x^2+2272 x+161312\right )+45812608 x+1626347584} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^x \left (x^2+e^{x^2} (1-2 x)+282 x+19880\right )}{4 \left (e^{x^2}+(x+142)^2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {e^x \left (x^2+282 x+e^{x^2} (1-2 x)+19880\right )}{\left ((x+142)^2+e^{x^2}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{4} \int \left (\frac {2 e^x \left (x^3+284 x^2+20163 x-142\right )}{\left (x^2+284 x+e^{x^2}+20164\right )^2}-\frac {e^x (2 x-1)}{x^2+284 x+e^{x^2}+20164}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-284 \int \frac {e^x}{\left (x^2+284 x+e^{x^2}+20164\right )^2}dx+40326 \int \frac {e^x x}{\left (x^2+284 x+e^{x^2}+20164\right )^2}dx+568 \int \frac {e^x x^2}{\left (x^2+284 x+e^{x^2}+20164\right )^2}dx+\int \frac {e^x}{x^2+284 x+e^{x^2}+20164}dx-2 \int \frac {e^x x}{x^2+284 x+e^{x^2}+20164}dx+2 \int \frac {e^x x^3}{\left (x^2+284 x+e^{x^2}+20164\right )^2}dx\right )\) |
Input:
Int[(E^(x + x^2)*(1 - 2*x) + E^x*(19880 + 282*x + x^2))/(1626347584 + 4*E^ (2*x^2) + 45812608*x + 483936*x^2 + 2272*x^3 + 4*x^4 + E^x^2*(161312 + 227 2*x + 8*x^2)),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
risch | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
parallelrisch | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
Input:
int(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^ 2+2272*x+161312)*exp(x^2)+4*x^4+2272*x^3+483936*x^2+45812608*x+1626347584) ,x,method=_RETURNVERBOSE)
Output:
1/4*exp(x)/(x^2+exp(x^2)+284*x+20164)
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{\left (x^{2} + x\right )}}{4 \, {\left ({\left (x^{2} + 284 \, x + 20164\right )} e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )}} \] Input:
integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2 +(8*x^2+2272*x+161312)*exp(x^2)+4*x^4+2272*x^3+483936*x^2+45812608*x+16263 47584),x, algorithm="fricas")
Output:
1/4*e^(x^2 + x)/((x^2 + 284*x + 20164)*e^(x^2) + e^(2*x^2))
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 x^{2} + 1136 x + 4 e^{x^{2}} + 80656} \] Input:
integrate(((1-2*x)*exp(x)*exp(x**2)+(x**2+282*x+19880)*exp(x))/(4*exp(x**2 )**2+(8*x**2+2272*x+161312)*exp(x**2)+4*x**4+2272*x**3+483936*x**2+4581260 8*x+1626347584),x)
Output:
exp(x)/(4*x**2 + 1136*x + 4*exp(x**2) + 80656)
Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \] Input:
integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2 +(8*x^2+2272*x+161312)*exp(x^2)+4*x^4+2272*x^3+483936*x^2+45812608*x+16263 47584),x, algorithm="maxima")
Output:
1/4*e^x/(x^2 + 284*x + e^(x^2) + 20164)
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \] Input:
integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2 +(8*x^2+2272*x+161312)*exp(x^2)+4*x^4+2272*x^3+483936*x^2+45812608*x+16263 47584),x, algorithm="giac")
Output:
1/4*e^x/(x^2 + 284*x + e^(x^2) + 20164)
Timed out. \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (x^2+282\,x+19880\right )-{\mathrm {e}}^{x^2+x}\,\left (2\,x-1\right )}{45812608\,x+4\,{\mathrm {e}}^{2\,x^2}+{\mathrm {e}}^{x^2}\,\left (8\,x^2+2272\,x+161312\right )+483936\,x^2+2272\,x^3+4\,x^4+1626347584} \,d x \] Input:
int((exp(x)*(282*x + x^2 + 19880) - exp(x^2)*exp(x)*(2*x - 1))/(45812608*x + 4*exp(2*x^2) + exp(x^2)*(2272*x + 8*x^2 + 161312) + 483936*x^2 + 2272*x ^3 + 4*x^4 + 1626347584),x)
Output:
int((exp(x)*(282*x + x^2 + 19880) - exp(x + x^2)*(2*x - 1))/(45812608*x + 4*exp(2*x^2) + exp(x^2)*(2272*x + 8*x^2 + 161312) + 483936*x^2 + 2272*x^3 + 4*x^4 + 1626347584), x)
Time = 54.65 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 e^{x^{2}}+4 x^{2}+1136 x +80656} \] Input:
int(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^ 2+2272*x+161312)*exp(x^2)+4*x^4+2272*x^3+483936*x^2+45812608*x+1626347584) ,x)
Output:
e**x/(4*(e**(x**2) + x**2 + 284*x + 20164))