Integrand size = 92, antiderivative size = 32 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=x-e^{-e^{\frac {-4-x+x^2+(5-5 x) x^2}{x}}} x \]
\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx \]
Integrate[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 5*x^3)/x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x),x]
Integrate[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 5*x^3)/x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*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.
\(\displaystyle \int \frac {e^{-e^{\frac {-5 x^3+6 x^2-x-4}{x}}} \left (e^{e^{\frac {-5 x^3+6 x^2-x-4}{x}}} x+e^{\frac {-5 x^3+6 x^2-x-4}{x}} \left (-10 x^3+6 x^2+4\right )-x\right )}{x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-e^{-5 x^2+6 x-\frac {4}{x}-1}} \left (e^{e^{\frac {-5 x^3+6 x^2-x-4}{x}}} x+e^{\frac {-5 x^3+6 x^2-x-4}{x}} \left (-10 x^3+6 x^2+4\right )-x\right )}{x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 \left (5 x^3-3 x^2-2\right ) \exp \left (-5 x^2-e^{-5 x^2+6 x-\frac {4}{x}-1}+6 x-\frac {4}{x}-1\right )}{x}-e^{-e^{-5 x^2+6 x-\frac {4}{x}-1}}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {\exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right )}{x}dx+6 \int \exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right ) xdx-10 \int \exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right ) x^2dx-\int e^{-e^{-5 x^2+6 x-1-\frac {4}{x}}}dx+x\) |
Int[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 5*x^3) /x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x),x]
3.22.69.3.1 Defintions of rubi rules used
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
risch | \(x -x \,{\mathrm e}^{-{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}\) | \(28\) |
parallelrisch | \(-\left (-x \,{\mathrm e}^{{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}+x \right ) {\mathrm e}^{-{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}\) | \(50\) |
norman | \(\left (x \,{\mathrm e}^{{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}-x \right ) {\mathrm e}^{-{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}\) | \(52\) |
int((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+6*x^2- x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx={\left (x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} - x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} \]
integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm=\
Timed out. \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\text {Timed out} \]
integrate((x*exp(exp((-5*x**3+6*x**2-x-4)/x))+(-10*x**3+6*x**2+4)*exp((-5* x**3+6*x**2-x-4)/x)-x)/x/exp(exp((-5*x**3+6*x**2-x-4)/x)),x)
\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \]
integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm=\
x - integrate((x*e^(5*x^2 + 4/x + 1) + 2*(5*x^3 - 3*x^2 - 2)*e^(6*x))*e^(- 5*x^2 - 4/x - e^(-5*x^2 + 6*x - 4/x - 1) - 1)/x, x)
\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \]
integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm=\
integrate(-(2*(5*x^3 - 3*x^2 - 2)*e^(-(5*x^3 - 6*x^2 + x + 4)/x) - x*e^(e^ (-(5*x^3 - 6*x^2 + x + 4)/x)) + x)*e^(-e^(-(5*x^3 - 6*x^2 + x + 4)/x))/x, x)
Time = 13.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=-x\,\left ({\mathrm {e}}^{-{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {4}{x}}\,{\mathrm {e}}^{-5\,x^2}}-1\right ) \]