Integrand size = 91, antiderivative size = 34 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=\frac {\left (1+e^{e^x-\frac {2 (-2+x)}{x}}\right ) \left (-x+x^4\right )}{3 (2-x)} \] Output:
1/3*(x^4-x)*(1+exp(exp(x)-2*(-2+x)/x))/(2-x)
Time = 5.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=\frac {e^{e^x+\frac {4}{x}} \left (x-x^4\right )-e^2 \left (14-8 x+x^4\right )}{3 e^2 (-2+x)} \] Input:
Integrate[(-2*x + 8*x^4 - 3*x^5 + E^((4 - 2*x + E^x*x)/x)*(8 - 6*x - 8*x^3 + 12*x^4 - 3*x^5 + E^x*(-2*x^2 + x^3 + 2*x^5 - x^6)))/(12*x - 12*x^2 + 3* x^3),x]
Output:
(E^(E^x + 4/x)*(x - x^4) - E^2*(14 - 8*x + x^4))/(3*E^2*(-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.
\(\displaystyle \int \frac {-3 x^5+8 x^4+e^{\frac {e^x x-2 x+4}{x}} \left (-3 x^5+12 x^4-8 x^3+e^x \left (-x^6+2 x^5+x^3-2 x^2\right )-6 x+8\right )-2 x}{3 x^3-12 x^2+12 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-3 x^5+8 x^4+e^{\frac {e^x x-2 x+4}{x}} \left (-3 x^5+12 x^4-8 x^3+e^x \left (-x^6+2 x^5+x^3-2 x^2\right )-6 x+8\right )-2 x}{x \left (3 x^2-12 x+12\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 12 \int -\frac {3 x^5-8 x^4+2 x-e^{\frac {e^x x-2 x+4}{x}} \left (-3 x^5+12 x^4-8 x^3-6 x-e^x \left (x^6-2 x^5-x^3+2 x^2\right )+8\right )}{36 (2-x)^2 x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} \int \frac {3 x^5-8 x^4+2 x-e^{\frac {e^x x-2 x+4}{x}} \left (-3 x^5+12 x^4-8 x^3-6 x-e^x \left (x^6-2 x^5-x^3+2 x^2\right )+8\right )}{(2-x)^2 x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {3 e^{e^x-2+\frac {4}{x}} x^4}{(x-2)^2}+\frac {3 x^4}{(x-2)^2}-\frac {12 e^{e^x-2+\frac {4}{x}} x^3}{(x-2)^2}-\frac {8 x^3}{(x-2)^2}+\frac {8 e^{e^x-2+\frac {4}{x}} x^2}{(x-2)^2}+\frac {e^{x+e^x-2+\frac {4}{x}} \left (x^3-1\right ) x}{x-2}+\frac {6 e^{e^x-2+\frac {4}{x}}}{(x-2)^2}+\frac {2}{(x-2)^2}-\frac {8 e^{e^x-2+\frac {4}{x}}}{(x-2)^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\int e^{x+e^x-2+\frac {4}{x}} x^3dx-3 \int e^{e^x-2+\frac {4}{x}} x^2dx-2 \int e^{x+e^x-2+\frac {4}{x}} x^2dx+4 \int e^{e^x-2+\frac {4}{x}}dx-7 \int e^{x+e^x-2+\frac {4}{x}}dx+14 \int \frac {e^{e^x-2+\frac {4}{x}}}{(x-2)^2}dx+14 \int \frac {e^{e^x-2+\frac {4}{x}}}{x-2}dx-14 \int \frac {e^{x+e^x-2+\frac {4}{x}}}{x-2}dx+2 \int \frac {e^{e^x-2+\frac {4}{x}}}{x}dx-4 \int e^{x+e^x-2+\frac {4}{x}} xdx-x^3-2 x^2-4 x+\frac {14}{2-x}\right )\) |
Input:
Int[(-2*x + 8*x^4 - 3*x^5 + E^((4 - 2*x + E^x*x)/x)*(8 - 6*x - 8*x^3 + 12* x^4 - 3*x^5 + E^x*(-2*x^2 + x^3 + 2*x^5 - x^6)))/(12*x - 12*x^2 + 3*x^3),x ]
Output:
$Aborted
Time = 1.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41
method | result | size |
parallelrisch | \(-\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x^{4}+x^{4}-{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x -2}{3 \left (-2+x \right )}\) | \(48\) |
risch | \(-\frac {x^{3}}{3}-\frac {2 x^{2}}{3}-\frac {4 x}{3}-\frac {14}{3 \left (-2+x \right )}-\frac {\left (x^{3}-1\right ) x \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}}}{3 \left (-2+x \right )}\) | \(49\) |
norman | \(\frac {-\frac {x^{4}}{3}+\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x}{3}-\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x^{4}}{3}+\frac {2}{3}}{-2+x}\) | \(50\) |
Input:
int((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)* x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-12*x^2+12*x),x,method=_RETURNVERBOSE)
Output:
-1/3*(exp((exp(x)*x+4-2*x)/x)*x^4+x^4-exp((exp(x)*x+4-2*x)/x)*x-2)/(-2+x)
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=-\frac {x^{4} + {\left (x^{4} - x\right )} e^{\left (\frac {x e^{x} - 2 \, x + 4}{x}\right )} - 8 \, x + 14}{3 \, {\left (x - 2\right )}} \] Input:
integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((e xp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-12*x^2+12*x),x, algorithm="frica s")
Output:
-1/3*(x^4 + (x^4 - x)*e^((x*e^x - 2*x + 4)/x) - 8*x + 14)/(x - 2)
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=- \frac {x^{3}}{3} - \frac {2 x^{2}}{3} - \frac {4 x}{3} + \frac {\left (- x^{4} + x\right ) e^{\frac {x e^{x} - 2 x + 4}{x}}}{3 x - 6} - \frac {14}{3 x - 6} \] Input:
integrate((((-x**6+2*x**5+x**3-2*x**2)*exp(x)-3*x**5+12*x**4-8*x**3-6*x+8) *exp((exp(x)*x+4-2*x)/x)-3*x**5+8*x**4-2*x)/(3*x**3-12*x**2+12*x),x)
Output:
-x**3/3 - 2*x**2/3 - 4*x/3 + (-x**4 + x)*exp((x*exp(x) - 2*x + 4)/x)/(3*x - 6) - 14/(3*x - 6)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=-\frac {1}{3} \, x^{3} - \frac {2}{3} \, x^{2} - \frac {4}{3} \, x - \frac {{\left (x^{4} - x\right )} e^{\left (\frac {4}{x} + e^{x}\right )}}{3 \, {\left (x e^{2} - 2 \, e^{2}\right )}} - \frac {14}{3 \, {\left (x - 2\right )}} \] Input:
integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((e xp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-12*x^2+12*x),x, algorithm="maxim a")
Output:
-1/3*x^3 - 2/3*x^2 - 4/3*x - 1/3*(x^4 - x)*e^(4/x + e^x)/(x*e^2 - 2*e^2) - 14/3/(x - 2)
\[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=\int { -\frac {3 \, x^{5} - 8 \, x^{4} + {\left (3 \, x^{5} - 12 \, x^{4} + 8 \, x^{3} + {\left (x^{6} - 2 \, x^{5} - x^{3} + 2 \, x^{2}\right )} e^{x} + 6 \, x - 8\right )} e^{\left (\frac {x e^{x} - 2 \, x + 4}{x}\right )} + 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \,d x } \] Input:
integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((e xp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-12*x^2+12*x),x, algorithm="giac" )
Output:
integrate(-1/3*(3*x^5 - 8*x^4 + (3*x^5 - 12*x^4 + 8*x^3 + (x^6 - 2*x^5 - x ^3 + 2*x^2)*e^x + 6*x - 8)*e^((x*e^x - 2*x + 4)/x) + 2*x)/(x^3 - 4*x^2 + 4 *x), x)
Time = 3.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.88 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x+\frac {4}{x}-2}}{3\,\left (x-2\right )}-\frac {14}{3\,\left (x-2\right )}-\frac {2\,x^2}{3}-\frac {x^3}{3}-\frac {x^4\,{\mathrm {e}}^{{\mathrm {e}}^x+\frac {4}{x}-2}}{3\,\left (x-2\right )}-\frac {4\,x}{3} \] Input:
int(-(2*x - 8*x^4 + 3*x^5 + exp((x*exp(x) - 2*x + 4)/x)*(6*x + exp(x)*(2*x ^2 - x^3 - 2*x^5 + x^6) + 8*x^3 - 12*x^4 + 3*x^5 - 8))/(12*x - 12*x^2 + 3* x^3),x)
Output:
(x*exp(exp(x) + 4/x - 2))/(3*(x - 2)) - 14/(3*(x - 2)) - (2*x^2)/3 - x^3/3 - (x^4*exp(exp(x) + 4/x - 2))/(3*(x - 2)) - (4*x)/3
Time = 5.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59 \[ \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx=\frac {x \left (-e^{\frac {e^{x} x +4}{x}} x^{3}+e^{\frac {e^{x} x +4}{x}}-e^{2} x^{3}+e^{2}\right )}{3 e^{2} \left (x -2\right )} \] Input:
int((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)* x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-12*x^2+12*x),x)
Output:
(x*( - e**((e**x*x + 4)/x)*x**3 + e**((e**x*x + 4)/x) - e**2*x**3 + e**2)) /(3*e**2*(x - 2))