\(\int \frac {e^{-x} (e^{10 e^{-x}} (-2 e^x-10 x)-e^{e^x+2 x} x^3+e^{5 e^{-x}} (4 e^x+10 x)+e^x (-6-x^3))}{x^3} \, dx\) [7560]

Optimal result

Integrand size = 73, antiderivative size = 33 \[ \int \frac {e^{-x} \left (e^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx=2-e^{e^x}+\frac {2+\left (1-e^{5 e^{-x}}\right )^2}{x^2}-x \]

[Out]

2+((1-exp(5/exp(x)))^2+2)/x^2-x-exp(exp(x))

Rubi [F]

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

[In]

Int[(E^(10/E^x)*(-2*E^x - 10*x) - E^(E^x + 2*x)*x^3 + E^(5/E^x)*(4*E^x + 10*x) + E^x*(-6 - x^3))/(E^x*x^3),x]

[Out]

-E^E^x + 3/x^2 - x + 4*Defer[Int][E^(5/E^x)/x^3, x] - 2*Defer[Int][E^(10/E^x)/x^3, x] + 10*Defer[Int][E^(5/E^x
- x)/x^2, x] - 10*Defer[Int][E^(10/E^x - x)/x^2, x]

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

Mathematica [A] (verified)

Time = 4.79 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x} \left (e^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx=-e^{e^x}+\frac {3}{x^2}-\frac {2 e^{5 e^{-x}}}{x^2}+\frac {e^{10 e^{-x}}}{x^2}-x \]

[In]

Integrate[(E^(10/E^x)*(-2*E^x - 10*x) - E^(E^x + 2*x)*x^3 + E^(5/E^x)*(4*E^x + 10*x) + E^x*(-6 - x^3))/(E^x*x^
3),x]

[Out]

-E^E^x + 3/x^2 - (2*E^(5/E^x))/x^2 + E^(10/E^x)/x^2 - x

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15

method	result	size
risch	\(-x +\frac {3}{x^{2}}-{\mathrm e}^{{\mathrm e}^{x}}+\frac {{\mathrm e}^{10 \,{\mathrm e}^{-x}}}{x^{2}}-\frac {2 \,{\mathrm e}^{5 \,{\mathrm e}^{-x}}}{x^{2}}\)	\(38\)
parallelrisch	\(-\frac {-3+\ln \left ({\mathrm e}^{x}\right ) x^{2}+{\mathrm e}^{{\mathrm e}^{x}} x^{2}-{\mathrm e}^{10 \,{\mathrm e}^{-x}}+2 \,{\mathrm e}^{5 \,{\mathrm e}^{-x}}}{x^{2}}\)	\(42\)

[In]

int((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*exp(x))
/exp(x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-x+3/x^2-exp(exp(x))+1/x^2*exp(10*exp(-x))-2/x^2*exp(5*exp(-x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {e^{-x} \left (e^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx=-\frac {{\left (x^{2} e^{\left (2 \, x + e^{x}\right )} + {\left (x^{3} - 3\right )} e^{\left (2 \, x\right )} - e^{\left (2 \, x + 10 \, e^{\left (-x\right )}\right )} + 2 \, e^{\left (2 \, x + 5 \, e^{\left (-x\right )}\right )}\right )} e^{\left (-2 \, x\right )}}{x^{2}} \]

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="fricas")

[Out]

-(x^2*e^(2*x + e^x) + (x^3 - 3)*e^(2*x) - e^(2*x + 10*e^(-x)) + 2*e^(2*x + 5*e^(-x)))*e^(-2*x)/x^2

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-x} \left (e^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx=- x - e^{e^{x}} + \frac {3}{x^{2}} + \frac {x^{2} e^{10 e^{- x}} - 2 x^{2} e^{5 e^{- x}}}{x^{4}} \]

[In]

integrate((-x**3*exp(x)**2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))**2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x**3-
6)*exp(x))/exp(x)/x**3,x)

[Out]

-x - exp(exp(x)) + 3/x**2 + (x**2*exp(10*exp(-x)) - 2*x**2*exp(5*exp(-x)))/x**4

Maxima [F]

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

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="maxima")

[Out]

-x + 3/x^2 - e^(e^x) - integrate(2*(5*x + e^x)*e^(-x + 10*e^(-x))/x^3, x) + integrate(2*(5*x + 2*e^x)*e^(-x +
5*e^(-x))/x^3, x)

Giac [F]

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

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="giac")

[Out]

integrate(-(x^3*e^(2*x + e^x) + (x^3 + 6)*e^x + 2*(5*x + e^x)*e^(10*e^(-x)) - 2*(5*x + 2*e^x)*e^(5*e^(-x)))*e^
(-x)/x^3, x)

Mupad [B] (verification not implemented)

Time = 13.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-x} \left (e^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx=\frac {{\mathrm {e}}^{10\,{\mathrm {e}}^{-x}}}{x^2}-{\mathrm {e}}^{{\mathrm {e}}^x}-\frac {2\,{\mathrm {e}}^{5\,{\mathrm {e}}^{-x}}}{x^2}-x+\frac {3}{x^2} \]

[In]

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

[Out]

exp(10*exp(-x))/x^2 - exp(exp(x)) - (2*exp(5*exp(-x)))/x^2 - x + 3/x^2