Integrand size = 102, antiderivative size = 33 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {2}{\left (3-e^{4-x}-x\right ) x}+x+(4-x) x+x^2 \] Output:
2/x/(3-exp(4-x)-x)+x^2+x+(4-x)*x
Time = 4.71 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {5 e^4 x^2+e^x \left (-2-15 x^2+5 x^3\right )}{\left (e^4+e^x (-3+x)\right ) x} \] Input:
Integrate[(-6 + 4*x + 45*x^2 + 5*E^(8 - 2*x)*x^2 - 30*x^3 + 5*x^4 + E^(4 - x)*(2 - 2*x - 30*x^2 + 10*x^3))/(9*x^2 + E^(8 - 2*x)*x^2 - 6*x^3 + x^4 + E^(4 - x)*(-6*x^2 + 2*x^3)),x]
Output:
(5*E^4*x^2 + E^x*(-2 - 15*x^2 + 5*x^3))/((E^4 + 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.
\(\displaystyle \int \frac {5 x^4-30 x^3+5 e^{8-2 x} x^2+45 x^2+e^{4-x} \left (10 x^3-30 x^2-2 x+2\right )+4 x-6}{x^4-6 x^3+e^{8-2 x} x^2+9 x^2+e^{4-x} \left (2 x^3-6 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x} \left (5 x^4-30 x^3+5 e^{8-2 x} x^2+45 x^2+e^{4-x} \left (10 x^3-30 x^2-2 x+2\right )+4 x-6\right )}{x^2 \left (e^x x-3 e^x+e^4\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{2 x-4} (x-3) (x-1)}{x^2 \left (e^x x-3 e^x+e^4\right )}-\frac {2 e^{x-4} (x-1)}{x^2}+\frac {2 e^{2 x} (x-2)}{x \left (e^x x-3 e^x+e^4\right )^2}+5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \int \frac {e^{2 x-4}}{x^2 \left (e^x x-3 e^x+e^4\right )}dx+2 \int \frac {e^{2 x}}{\left (e^x x-3 e^x+e^4\right )^2}dx-4 \int \frac {e^{2 x}}{x \left (e^x x-3 e^x+e^4\right )^2}dx+2 \int \frac {e^{2 x-4}}{e^x x-3 e^x+e^4}dx-8 \int \frac {e^{2 x-4}}{x \left (e^x x-3 e^x+e^4\right )}dx+5 x-\frac {2 e^{x-4}}{x}\) |
Input:
Int[(-6 + 4*x + 45*x^2 + 5*E^(8 - 2*x)*x^2 - 30*x^3 + 5*x^4 + E^(4 - x)*(2 - 2*x - 30*x^2 + 10*x^3))/(9*x^2 + E^(8 - 2*x)*x^2 - 6*x^3 + x^4 + E^(4 - x)*(-6*x^2 + 2*x^3)),x]
Output:
$Aborted
Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64
method | result | size |
risch | \(5 x -\frac {2}{x \left (-3+{\mathrm e}^{-x +4}+x \right )}\) | \(21\) |
parallelrisch | \(\frac {5 \,{\mathrm e}^{-x +4} x^{2}+5 x^{3}-15 x^{2}-2}{x \left (-3+{\mathrm e}^{-x +4}+x \right )}\) | \(39\) |
norman | \(\frac {-2-45 x +15 \,{\mathrm e}^{-x +4} x +5 x^{3}+5 \,{\mathrm e}^{-x +4} x^{2}}{x \left (-3+{\mathrm e}^{-x +4}+x \right )}\) | \(46\) |
Input:
int((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2 +4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x,method =_RETURNVERBOSE)
Output:
5*x-2/x/(-3+exp(-x+4)+x)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {5 \, x^{3} + 5 \, x^{2} e^{\left (-x + 4\right )} - 15 \, x^{2} - 2}{x^{2} + x e^{\left (-x + 4\right )} - 3 \, x} \] Input:
integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+ 45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="fricas")
Output:
(5*x^3 + 5*x^2*e^(-x + 4) - 15*x^2 - 2)/(x^2 + x*e^(-x + 4) - 3*x)
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.52 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=5 x - \frac {2}{x^{2} + x e^{4 - x} - 3 x} \] Input:
integrate((5*x**2*exp(-x+4)**2+(10*x**3-30*x**2-2*x+2)*exp(-x+4)+5*x**4-30 *x**3+45*x**2+4*x-6)/(x**2*exp(-x+4)**2+(2*x**3-6*x**2)*exp(-x+4)+x**4-6*x **3+9*x**2),x)
Output:
5*x - 2/(x**2 + x*exp(4 - x) - 3*x)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {5 \, x^{2} e^{4} + {\left (5 \, x^{3} - 15 \, x^{2} - 2\right )} e^{x}}{x e^{4} + {\left (x^{2} - 3 \, x\right )} e^{x}} \] Input:
integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+ 45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="maxima")
Output:
(5*x^2*e^4 + (5*x^3 - 15*x^2 - 2)*e^x)/(x*e^4 + (x^2 - 3*x)*e^x)
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {5 \, x^{3} + 5 \, x^{2} e^{\left (-x + 4\right )} - 15 \, x^{2} - 2}{x^{2} + x e^{\left (-x + 4\right )} - 3 \, x} \] Input:
integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+ 45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="giac")
Output:
(5*x^3 + 5*x^2*e^(-x + 4) - 15*x^2 - 2)/(x^2 + x*e^(-x + 4) - 3*x)
Time = 4.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=5\,x-\frac {2}{x\,{\mathrm {e}}^{4-x}-3\,x+x^2} \] Input:
int((4*x - exp(4 - x)*(2*x + 30*x^2 - 10*x^3 - 2) + 5*x^2*exp(8 - 2*x) + 4 5*x^2 - 30*x^3 + 5*x^4 - 6)/(x^2*exp(8 - 2*x) - exp(4 - x)*(6*x^2 - 2*x^3) + 9*x^2 - 6*x^3 + x^4),x)
Output:
5*x - 2/(x*exp(4 - x) - 3*x + x^2)
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx=\frac {5 e^{x} x^{3}-15 e^{x} x^{2}-2 e^{x}+5 e^{4} x^{2}}{x \left (e^{x} x -3 e^{x}+e^{4}\right )} \] Input:
int((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2 +4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x)
Output:
(5*e**x*x**3 - 15*e**x*x**2 - 2*e**x + 5*e**4*x**2)/(x*(e**x*x - 3*e**x + e**4))