\(\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} (4+6 x^2-10 x^3))}{x} \, dx\) [8169]

Optimal result

Integrand size = 92, antiderivative size = 32 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=x-e^{-e^{\frac {-4-x+x^2+(5-5 x) x^2}{x}}} x \]

[Out]

x-x/exp(exp(((-5*x+5)*x^2-4-x+x^2)/x))

Rubi [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx \]

[In]

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]

[Out]

x - Defer[Int][E^(-E^(-1 - 4/x + 6*x - 5*x^2)), x] + 4*Defer[Int][E^(-1 - E^(-1 - 4/x + 6*x - 5*x^2) - 4/x + 6
*x - 5*x^2)/x, x] + 6*Defer[Int][E^(-1 - E^(-1 - 4/x + 6*x - 5*x^2) - 4/x + 6*x - 5*x^2)*x, x] - 10*Defer[Int]
[E^(-1 - E^(-1 - 4/x + 6*x - 5*x^2) - 4/x + 6*x - 5*x^2)*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-e^{-1-\frac {4}{x}+6 x-5 x^2}} \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 \left (1-e^{-e^{-1-\frac {4}{x}+6 x-5 x^2}}-\frac {2 \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) \left (-2-3 x^2+5 x^3\right )}{x}\right ) \, dx \\ & = x-2 \int \frac {\exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) \left (-2-3 x^2+5 x^3\right )}{x} \, dx-\int e^{-e^{-1-\frac {4}{x}+6 x-5 x^2}} \, dx \\ & = x-2 \int \left (-\frac {2 \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right )}{x}-3 \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) x+5 \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) x^2\right ) \, dx-\int e^{-e^{-1-\frac {4}{x}+6 x-5 x^2}} \, dx \\ & = x+4 \int \frac {\exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right )}{x} \, dx+6 \int \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) x \, dx-10 \int \exp \left (-1-e^{-1-\frac {4}{x}+6 x-5 x^2}-\frac {4}{x}+6 x-5 x^2\right ) x^2 \, dx-\int e^{-e^{-1-\frac {4}{x}+6 x-5 x^2}} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx \]

[In]

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]

[Out]

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]

Maple [A] (verified)

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	\(\left (x \,{\mathrm e}^{{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}-x \right ) {\mathrm e}^{-{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}\)	\(52\)

[In]

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)

[Out]

x-x*exp(-exp(-(5*x^3-6*x^2+x+4)/x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx={\left (x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} - x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} \]

[In]

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="fricas")

[Out]

(x*e^(e^(-(5*x^3 - 6*x^2 + x + 4)/x)) - x)*e^(-e^(-(5*x^3 - 6*x^2 + x + 4)/x))

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \]

[In]

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="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \]

[In]

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="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 13.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=-x\,\left ({\mathrm {e}}^{-{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {4}{x}}\,{\mathrm {e}}^{-5\,x^2}}-1\right ) \]

[In]

int((exp(-exp(-(x - 6*x^2 + 5*x^3 + 4)/x))*(exp(-(x - 6*x^2 + 5*x^3 + 4)/x)*(6*x^2 - 10*x^3 + 4) - x + x*exp(e
xp(-(x - 6*x^2 + 5*x^3 + 4)/x))))/x,x)

[Out]

-x*(exp(-exp(6*x)*exp(-1)*exp(-4/x)*exp(-5*x^2)) - 1)