\(\int e^{-x+\frac {e^{-x} (-40 x-8 e x+e^x (e^5+20 x^2+4 e x^2))}{5+e}} (-8+8 x+8 e^x x) \, dx\) [6928]

Optimal result

Integrand size = 56, antiderivative size = 28 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=e^{-\frac {e^5}{-5-e}-8 e^{-x} x+4 x^2} \]

[Out]

exp(4*x^2-exp(5)/(-exp(1)-5)-8*x/exp(x))

Rubi [F]

\[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=\int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \left (-8+8 x+8 e^x x\right ) \, dx \]

[In]

Int[E^(-x + (-40*x - 8*E*x + E^x*(E^5 + 20*x^2 + 4*E*x^2))/(E^x*(5 + E)))*(-8 + 8*x + 8*E^x*x),x]

[Out]

-8*Defer[Int][E^(-x + (-40*x - 8*E*x + E^x*(E^5 + 20*x^2 + 4*E*x^2))/(E^x*(5 + E))), x] + 8*Defer[Int][E^((-8*
x)/E^x + (E^5 + 20*x^2 + 4*E*x^2)/(5 + E))*x, x] + 8*Defer[Int][E^(-x + (-40*x - 8*E*x + E^x*(E^5 + 20*x^2 + 4
*E*x^2))/(E^x*(5 + E)))*x, x]

Rubi steps \begin{align*} \text {integral}& = \int 8 \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \left (-1+x+e^x x\right ) \, dx \\ & = 8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \left (-1+x+e^x x\right ) \, dx \\ & = 8 \int \left (-\exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right )+\exp \left (\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x+\exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x\right ) \, dx \\ & = -\left (8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \, dx\right )+8 \int \exp \left (\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x \, dx+8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x \, dx \\ & = -\left (8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \, dx\right )+8 \int \exp \left (\frac {e^{-x} \left (-40 \left (1+\frac {e}{5}\right ) x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x \, dx+8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x \, dx \\ & = -\left (8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) \, dx\right )+8 \int e^{-8 e^{-x} x+\frac {e^5+20 x^2+4 e x^2}{5+e}} x \, dx+8 \int \exp \left (-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}\right ) x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=e^{\frac {e^5}{5+e}-8 e^{-x} x+4 x^2} \]

[In]

Integrate[E^(-x + (-40*x - 8*E*x + E^x*(E^5 + 20*x^2 + 4*E*x^2))/(E^x*(5 + E)))*(-8 + 8*x + 8*E^x*x),x]

[Out]

E^(E^5/(5 + E) - (8*x)/E^x + 4*x^2)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43

method	result	size
default	\({\mathrm e}^{\frac {\left (\left ({\mathrm e}^{5}+4 x^{2} {\mathrm e}+20 x^{2}\right ) {\mathrm e}^{x}-8 x \,{\mathrm e}-40 x \right ) {\mathrm e}^{-x}}{{\mathrm e}+5}}\)	\(40\)
norman	\({\mathrm e}^{\frac {\left (\left ({\mathrm e}^{5}+4 x^{2} {\mathrm e}+20 x^{2}\right ) {\mathrm e}^{x}-8 x \,{\mathrm e}-40 x \right ) {\mathrm e}^{-x}}{{\mathrm e}+5}}\)	\(40\)
parallelrisch	\({\mathrm e}^{\frac {\left (\left ({\mathrm e}^{5}+4 x^{2} {\mathrm e}+20 x^{2}\right ) {\mathrm e}^{x}-8 x \,{\mathrm e}-40 x \right ) {\mathrm e}^{-x}}{{\mathrm e}+5}}\)	\(40\)
risch	\({\mathrm e}^{-\frac {\left (-20 \,{\mathrm e}^{x} x^{2}-4 x^{2} {\mathrm e}^{1+x}+8 x \,{\mathrm e}-{\mathrm e}^{5+x}+40 x \right ) {\mathrm e}^{-x}}{{\mathrm e}+5}}\)	\(45\)

[In]

int((8*exp(x)*x+8*x-8)*exp(((exp(5)+4*x^2*exp(1)+20*x^2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))/exp(x),x,m
ethod=_RETURNVERBOSE)

[Out]

exp(((exp(5)+4*x^2*exp(1)+20*x^2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=e^{\left (x - \frac {{\left (8 \, x e - {\left (20 \, x^{2} + {\left (4 \, x^{2} - x\right )} e - 5 \, x + e^{5}\right )} e^{x} + 40 \, x\right )} e^{\left (-x\right )}}{e + 5}\right )} \]

[In]

integrate((8*exp(x)*x+8*x-8)*exp(((exp(5)+4*x^2*exp(1)+20*x^2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))/exp(
x),x, algorithm="fricas")

[Out]

e^(x - (8*x*e - (20*x^2 + (4*x^2 - x)*e - 5*x + e^5)*e^x + 40*x)*e^(-x)/(e + 5))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=e^{\frac {\left (- 40 x - 8 e x + \left (4 e x^{2} + 20 x^{2} + e^{5}\right ) e^{x}\right ) e^{- x}}{e + 5}} \]

[In]

integrate((8*exp(x)*x+8*x-8)*exp(((exp(5)+4*x**2*exp(1)+20*x**2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))/ex
p(x),x)

[Out]

exp((-40*x - 8*E*x + (4*E*x**2 + 20*x**2 + exp(5))*exp(x))*exp(-x)/(E + 5))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).

Time = 0.42 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=e^{\left (\frac {4 \, x^{2} e}{e + 5} + \frac {20 \, x^{2}}{e + 5} - \frac {40 \, x e^{\left (-x\right )}}{e + 5} - \frac {8 \, x e^{\left (-x + 1\right )}}{e + 5} + \frac {e^{5}}{e + 5}\right )} \]

[In]

integrate((8*exp(x)*x+8*x-8)*exp(((exp(5)+4*x^2*exp(1)+20*x^2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))/exp(
x),x, algorithm="maxima")

[Out]

e^(4*x^2*e/(e + 5) + 20*x^2/(e + 5) - 40*x*e^(-x)/(e + 5) - 8*x*e^(-x + 1)/(e + 5) + e^5/(e + 5))

Giac [F]

\[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx=\int { 8 \, {\left (x e^{x} + x - 1\right )} e^{\left (-x - \frac {{\left (8 \, x e - {\left (4 \, x^{2} e + 20 \, x^{2} + e^{5}\right )} e^{x} + 40 \, x\right )} e^{\left (-x\right )}}{e + 5}\right )} \,d x } \]

[In]

integrate((8*exp(x)*x+8*x-8)*exp(((exp(5)+4*x^2*exp(1)+20*x^2)*exp(x)-8*x*exp(1)-40*x)/(exp(1)+5)/exp(x))/exp(
x),x, algorithm="giac")

[Out]

integrate(8*(x*e^x + x - 1)*e^(-x - (8*x*e - (4*x^2*e + 20*x^2 + e^5)*e^x + 40*x)*e^(-x)/(e + 5)), x)

Mupad [B] (verification not implemented)

Time = 12.76 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int e^{-x+\frac {e^{-x} \left (-40 x-8 e x+e^x \left (e^5+20 x^2+4 e x^2\right )\right )}{5+e}} \left (-8+8 x+8 e^x x\right ) \, dx={\mathrm {e}}^{-\frac {8\,x\,{\mathrm {e}}^{-x}\,\mathrm {e}}{\mathrm {e}+5}}\,{\mathrm {e}}^{\frac {4\,x^2\,\mathrm {e}}{\mathrm {e}+5}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{\mathrm {e}+5}}\,{\mathrm {e}}^{-\frac {40\,x\,{\mathrm {e}}^{-x}}{\mathrm {e}+5}}\,{\mathrm {e}}^{\frac {20\,x^2}{\mathrm {e}+5}} \]

[In]

int(exp(-x)*exp(-(exp(-x)*(40*x + 8*x*exp(1) - exp(x)*(exp(5) + 4*x^2*exp(1) + 20*x^2)))/(exp(1) + 5))*(8*x +
8*x*exp(x) - 8),x)

[Out]

exp(-(8*x*exp(-x)*exp(1))/(exp(1) + 5))*exp((4*x^2*exp(1))/(exp(1) + 5))*exp(exp(5)/(exp(1) + 5))*exp(-(40*x*e
xp(-x))/(exp(1) + 5))*exp((20*x^2)/(exp(1) + 5))