Integrand size = 89, antiderivative size = 25 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 \left (1+\frac {e^x}{x}\right )^2 x}{-e^{2 x}+x} \] Output:
2*x*(exp(x)/x+1)^2/(x-exp(2*x))
Time = 3.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 \left (e^x+x\right )^2}{x \left (-e^{2 x}+x\right )} \] Input:
Integrate[(E^(2*x)*(-4*x + 4*x^2) + E^x*(-4*x^2 + 4*x^3) + E^(2*x)*(2*E^(2 *x) - 2*x^2 + 4*E^x*x^2 + 4*x^3))/(E^(4*x)*x^2 - 2*E^(2*x)*x^3 + x^4),x]
Output:
(2*(E^x + x)^2)/(x*(-E^(2*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.
\(\displaystyle \int \frac {e^{2 x} \left (4 x^2-4 x\right )+e^x \left (4 x^3-4 x^2\right )+e^{2 x} \left (4 x^3+4 e^x x^2-2 x^2+2 e^{2 x}\right )}{x^4-2 e^{2 x} x^3+e^{4 x} x^2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 e^x \left (2 e^x x^3+2 x^3+e^x x^2+2 e^{2 x} x^2-2 x^2-2 e^x x+e^{3 x}\right )}{\left (e^{2 x}-x\right )^2 x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {e^x \left (2 e^x x^3+2 x^3+e^x x^2+2 e^{2 x} x^2-2 x^2-2 e^x x+e^{3 x}\right )}{\left (e^{2 x}-x\right )^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {e^x (2 x-1) \left (e^x x+2 x+e^x\right )}{\left (e^{2 x}-x\right )^2 x}+\frac {e^x \left (2 x^2+e^x\right )}{\left (e^{2 x}-x\right ) x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{2} \int \frac {1}{\left (e^{2 x}-x\right ) x^2}dx+\int \frac {e^{2 x}}{\left (e^{2 x}-x\right ) x^2}dx+\frac {1}{2} \int \frac {1}{\left (e^{2 x}-x\right )^2}dx-2 \int \frac {e^x}{\left (e^{2 x}-x\right )^2}dx+\int \frac {1}{e^{2 x}-x}dx+2 \int \frac {e^x}{e^{2 x}-x}dx-\frac {1}{2} \int \frac {1}{\left (e^{2 x}-x\right )^2 x}dx+\int \frac {x}{\left (e^{2 x}-x\right )^2}dx+4 \int \frac {e^x x}{\left (e^{2 x}-x\right )^2}dx-\frac {x}{e^{2 x}-x}-\frac {1}{2 \left (e^{2 x}-x\right )}+\frac {1}{2 \left (e^{2 x}-x\right ) x}\right )\) |
Input:
Int[(E^(2*x)*(-4*x + 4*x^2) + E^x*(-4*x^2 + 4*x^3) + E^(2*x)*(2*E^(2*x) - 2*x^2 + 4*E^x*x^2 + 4*x^3))/(E^(4*x)*x^2 - 2*E^(2*x)*x^3 + x^4),x]
Output:
$Aborted
Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {2}{x}+\frac {2 x +4 \,{\mathrm e}^{x}+2}{x -{\mathrm e}^{2 x}}\) | \(26\) |
parallelrisch | \(-\frac {-2 x^{2}-4 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{2 x}}{x \left (x -{\mathrm e}^{2 x}\right )}\) | \(33\) |
norman | \(\frac {2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x}{x \left (x -{\mathrm e}^{2 x}\right )}\) | \(34\) |
parts | \(\frac {2 \,{\mathrm e}^{2 x}}{x \left (x -{\mathrm e}^{2 x}\right )}+\frac {4 \,{\mathrm e}^{x}}{x -{\mathrm e}^{2 x}}+\frac {2 \,{\mathrm e}^{2 x}}{x -{\mathrm e}^{2 x}}\) | \(51\) |
Input:
int(((2*exp(x)^2+4*exp(x)*x^2+4*x^3-2*x^2)*exp(2*x)+(4*x^2-4*x)*exp(x)^2+( 4*x^3-4*x^2)*exp(x))/(x^2*exp(2*x)^2-2*exp(2*x)*x^3+x^4),x,method=_RETURNV ERBOSE)
Output:
-2/x+2*(x+2*exp(x)+1)/(x-exp(2*x))
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}}{x^{2} - x e^{\left (2 \, x\right )}} \] Input:
integrate(((2*exp(x)^2+4*exp(x)*x^2+4*x^3-2*x^2)*exp(2*x)+(4*x^2-4*x)*exp( x)^2+(4*x^3-4*x^2)*exp(x))/(x^2*exp(2*x)^2-2*exp(2*x)*x^3+x^4),x, algorith m="fricas")
Output:
2*(x^2 + 2*x*e^x + e^(2*x))/(x^2 - x*e^(2*x))
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {- 2 x - 4 e^{x} - 2}{- x + e^{2 x}} - \frac {2}{x} \] Input:
integrate(((2*exp(x)**2+4*exp(x)*x**2+4*x**3-2*x**2)*exp(2*x)+(4*x**2-4*x) *exp(x)**2+(4*x**3-4*x**2)*exp(x))/(x**2*exp(2*x)**2-2*exp(2*x)*x**3+x**4) ,x)
Output:
(-2*x - 4*exp(x) - 2)/(-x + exp(2*x)) - 2/x
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}}{x^{2} - x e^{\left (2 \, x\right )}} \] Input:
integrate(((2*exp(x)^2+4*exp(x)*x^2+4*x^3-2*x^2)*exp(2*x)+(4*x^2-4*x)*exp( x)^2+(4*x^3-4*x^2)*exp(x))/(x^2*exp(2*x)^2-2*exp(2*x)*x^3+x^4),x, algorith m="maxima")
Output:
2*(x^2 + 2*x*e^x + e^(2*x))/(x^2 - x*e^(2*x))
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 \, {\left (x^{2} + 4 \, x e^{x} + e^{\left (2 \, x\right )}\right )}}{x^{2} - x e^{\left (2 \, x\right )}} \] Input:
integrate(((2*exp(x)^2+4*exp(x)*x^2+4*x^3-2*x^2)*exp(2*x)+(4*x^2-4*x)*exp( x)^2+(4*x^3-4*x^2)*exp(x))/(x^2*exp(2*x)^2-2*exp(2*x)*x^3+x^4),x, algorith m="giac")
Output:
2*(x^2 + 4*x*e^x + e^(2*x))/(x^2 - x*e^(2*x))
Time = 7.60 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2\,{\left (x+{\mathrm {e}}^x\right )}^2}{x\,\left (x-{\mathrm {e}}^{2\,x}\right )} \] Input:
int(-(exp(2*x)*(4*x - 4*x^2) + exp(x)*(4*x^2 - 4*x^3) - exp(2*x)*(2*exp(2* x) + 4*x^2*exp(x) - 2*x^2 + 4*x^3))/(x^2*exp(4*x) - 2*x^3*exp(2*x) + x^4), x)
Output:
(2*(x + exp(x))^2)/(x*(x - exp(2*x)))
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{2 x} \left (-4 x+4 x^2\right )+e^x \left (-4 x^2+4 x^3\right )+e^{2 x} \left (2 e^{2 x}-2 x^2+4 e^x x^2+4 x^3\right )}{e^{4 x} x^2-2 e^{2 x} x^3+x^4} \, dx=\frac {2 e^{x} \left (-e^{x} x -e^{x}-2 x \right )}{x \left (e^{2 x}-x \right )} \] Input:
int(((2*exp(x)^2+4*exp(x)*x^2+4*x^3-2*x^2)*exp(2*x)+(4*x^2-4*x)*exp(x)^2+( 4*x^3-4*x^2)*exp(x))/(x^2*exp(2*x)^2-2*exp(2*x)*x^3+x^4),x)
Output:
(2*e**x*( - e**x*x - e**x - 2*x))/(x*(e**(2*x) - x))