Integrand size = 114, antiderivative size = 25 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40}{3 \left (-1+\frac {e^{x (2+x)}}{1+e^x}\right )+x} \] Output:
10/(3/4*exp(x*(2+x))/(1+exp(x))-3/4+1/4*x)
Time = 3.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40 \left (1+e^x\right )}{-3+3 e^{x (2+x)}+e^x (-3+x)+x} \] Input:
Integrate[(-40 - 80*E^x - 40*E^(2*x) + E^(2*x + x^2)*(-240 + E^x*(-120 - 2 40*x) - 240*x))/(9 + 9*E^(4*x + 2*x^2) - 6*x + x^2 + E^(2*x)*(9 - 6*x + x^ 2) + E^x*(18 - 12*x + 2*x^2) + E^(2*x + x^2)*(-18 + 6*x + E^x*(-18 + 6*x)) ),x]
Output:
(40*(1 + E^x))/(-3 + 3*E^(x*(2 + x)) + E^x*(-3 + x) + 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.
| \(\displaystyle \int \frac {e^{x^2+2 x} \left (e^x (-240 x-120)-240 x-240\right )-80 e^x-40 e^{2 x}-40}{x^2+9 e^{2 x^2+4 x}+e^{2 x} \left (x^2-6 x+9\right )+e^x \left (2 x^2-12 x+18\right )+e^{x^2+2 x} \left (6 x+e^x (6 x-18)-18\right )-6 x+9} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {40 \left (-6 e^{x (x+2)} (x+1)-2 e^x-e^{2 x}-e^{x (x+3)} (6 x+3)-1\right )}{\left (-e^x (x-3)-3 e^{x (x+2)}-x+3\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 40 \int -\frac {6 e^{x (x+2)} (x+1)+2 e^x+e^{2 x}+3 e^{x (x+3)} (2 x+1)+1}{\left (e^x (3-x)-3 e^{x (x+2)}-x+3\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -40 \int \frac {6 e^{x (x+2)} (x+1)+2 e^x+e^{2 x}+3 e^{x (x+3)} (2 x+1)+1}{\left (e^x (3-x)-3 e^{x (x+2)}-x+3\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -40 \int \left (\frac {6 e^{x^2+2 x} (x+1)}{\left (-e^x x-x+3 e^x-3 e^{x (x+2)}+3\right )^2}+\frac {3 e^{x^2+3 x} (2 x+1)}{\left (-e^x x-x+3 e^x-3 e^{x (x+2)}+3\right )^2}+\frac {2 e^x}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}+\frac {e^{2 x}}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}+\frac {1}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -40 \left (6 \int \frac {e^{x^2+2 x}}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+3 \int \frac {e^{x^2+3 x}}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+6 \int \frac {e^{x^2+2 x} x}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+6 \int \frac {e^{x^2+3 x} x}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+\int \frac {1}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+2 \int \frac {e^x}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx+\int \frac {e^{2 x}}{\left (e^x x+x-3 e^x+3 e^{x (x+2)}-3\right )^2}dx\right )\) |
Input:
Int[(-40 - 80*E^x - 40*E^(2*x) + E^(2*x + x^2)*(-240 + E^x*(-120 - 240*x) - 240*x))/(9 + 9*E^(4*x + 2*x^2) - 6*x + x^2 + E^(2*x)*(9 - 6*x + x^2) + E ^x*(18 - 12*x + 2*x^2) + E^(2*x + x^2)*(-18 + 6*x + E^x*(-18 + 6*x))),x]
Output:
$Aborted
Time = 0.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {40 \,{\mathrm e}^{x}+40}{{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{x \left (2+x \right )}+x -3}\) | \(28\) |
| norman | \(\frac {40 \,{\mathrm e}^{x}+40}{{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{x^{2}+2 x}+x -3}\) | \(31\) |
| parallelrisch | \(\frac {120+120 \,{\mathrm e}^{x}}{3 \,{\mathrm e}^{x} x -9 \,{\mathrm e}^{x}+9 \,{\mathrm e}^{x^{2}+2 x}+3 x -9}\) | \(32\) |
Input:
int((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp(x)-40 )/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9)*exp( x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x,method=_RETURNVERBOSE)
Output:
40*(exp(x)+1)/(exp(x)*x-3*exp(x)+3*exp(x*(2+x))+x-3)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40 \, {\left (e^{x} + 1\right )}}{{\left (x - 3\right )} e^{x} + x + 3 \, e^{\left (x^{2} + 2 \, x\right )} - 3} \] Input:
integrate((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp (x)-40)/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9 )*exp(x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x, algorithm="fricas")
Output:
40*(e^x + 1)/((x - 3)*e^x + x + 3*e^(x^2 + 2*x) - 3)
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40 e^{x} + 40}{x e^{x} + x - 3 e^{x} + 3 e^{x^{2} + 2 x} - 3} \] Input:
integrate((((-240*x-120)*exp(x)-240*x-240)*exp(x**2+2*x)-40*exp(x)**2-80*e xp(x)-40)/(9*exp(x**2+2*x)**2+((6*x-18)*exp(x)+6*x-18)*exp(x**2+2*x)+(x**2 -6*x+9)*exp(x)**2+(2*x**2-12*x+18)*exp(x)+x**2-6*x+9),x)
Output:
(40*exp(x) + 40)/(x*exp(x) + x - 3*exp(x) + 3*exp(x**2 + 2*x) - 3)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40 \, {\left (e^{x} + 1\right )}}{{\left (x - 3\right )} e^{x} + x + 3 \, e^{\left (x^{2} + 2 \, x\right )} - 3} \] Input:
integrate((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp (x)-40)/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9 )*exp(x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x, algorithm="maxima")
Output:
40*(e^x + 1)/((x - 3)*e^x + x + 3*e^(x^2 + 2*x) - 3)
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\frac {40 \, {\left (e^{x} + 1\right )}}{x e^{x} + x + 3 \, e^{\left (x^{2} + 2 \, x\right )} - 3 \, e^{x} - 3} \] Input:
integrate((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp (x)-40)/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9 )*exp(x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x, algorithm="giac")
Output:
40*(e^x + 1)/(x*e^x + x + 3*e^(x^2 + 2*x) - 3*e^x - 3)
Timed out. \[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\int -\frac {40\,{\mathrm {e}}^{2\,x}+80\,{\mathrm {e}}^x+{\mathrm {e}}^{x^2+2\,x}\,\left (240\,x+{\mathrm {e}}^x\,\left (240\,x+120\right )+240\right )+40}{9\,{\mathrm {e}}^{2\,x^2+4\,x}-6\,x+{\mathrm {e}}^{x^2+2\,x}\,\left (6\,x+{\mathrm {e}}^x\,\left (6\,x-18\right )-18\right )+{\mathrm {e}}^{2\,x}\,\left (x^2-6\,x+9\right )+{\mathrm {e}}^x\,\left (2\,x^2-12\,x+18\right )+x^2+9} \,d x \] Input:
int(-(40*exp(2*x) + 80*exp(x) + exp(2*x + x^2)*(240*x + exp(x)*(240*x + 12 0) + 240) + 40)/(9*exp(4*x + 2*x^2) - 6*x + exp(2*x + x^2)*(6*x + exp(x)*( 6*x - 18) - 18) + exp(2*x)*(x^2 - 6*x + 9) + exp(x)*(2*x^2 - 12*x + 18) + x^2 + 9),x)
Output:
int(-(40*exp(2*x) + 80*exp(x) + exp(2*x + x^2)*(240*x + exp(x)*(240*x + 12 0) + 240) + 40)/(9*exp(4*x + 2*x^2) - 6*x + exp(2*x + x^2)*(6*x + exp(x)*( 6*x - 18) - 18) + exp(2*x)*(x^2 - 6*x + 9) + exp(x)*(2*x^2 - 12*x + 18) + x^2 + 9), x)
\[ \int \frac {-40-80 e^x-40 e^{2 x}+e^{2 x+x^2} \left (-240+e^x (-120-240 x)-240 x\right )}{9+9 e^{4 x+2 x^2}-6 x+x^2+e^{2 x} \left (9-6 x+x^2\right )+e^x \left (18-12 x+2 x^2\right )+e^{2 x+x^2} \left (-18+6 x+e^x (-18+6 x)\right )} \, dx=\int \frac {\left (\left (-240 x -120\right ) {\mathrm e}^{x}-240 x -240\right ) {\mathrm e}^{x^{2}+2 x}-40 \left ({\mathrm e}^{x}\right )^{2}-80 \,{\mathrm e}^{x}-40}{9 \left ({\mathrm e}^{x^{2}+2 x}\right )^{2}+\left (\left (6 x -18\right ) {\mathrm e}^{x}+6 x -18\right ) {\mathrm e}^{x^{2}+2 x}+\left (x^{2}-6 x +9\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (2 x^{2}-12 x +18\right ) {\mathrm e}^{x}+x^{2}-6 x +9}d x \] Input:
int((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp(x)-40 )/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9)*exp( x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x)
Output:
int((((-240*x-120)*exp(x)-240*x-240)*exp(x^2+2*x)-40*exp(x)^2-80*exp(x)-40 )/(9*exp(x^2+2*x)^2+((6*x-18)*exp(x)+6*x-18)*exp(x^2+2*x)+(x^2-6*x+9)*exp( x)^2+(2*x^2-12*x+18)*exp(x)+x^2-6*x+9),x)