\(\int \frac {1}{20} (4+e^x (-4-4 x)-24 x+e^{e^x} (15+5 e^{2 x}+15 e^x x)) \, dx\) [6997]

Optimal result

Integrand size = 39, antiderivative size = 33 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=\left (\frac {e^{e^x}}{4}-\frac {x}{5}\right ) \left (2+e^x+\frac {3 \left (-x+x^2\right )}{x}\right ) \]

[Out]

(2+exp(x)+3*(x^2-x)/x)*(1/4*exp(exp(x))-1/5*x)

Rubi [F]

\[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=\int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx \]

[In]

Int[(4 + E^x*(-4 - 4*x) - 24*x + E^E^x*(15 + 5*E^(2*x) + 15*E^x*x))/20,x]

[Out]

-1/4*E^E^x + E^x/5 + E^(E^x + x)/4 + x/5 - (3*x^2)/5 - (E^x*(1 + x))/5 + (3*ExpIntegralEi[E^x])/4 + (3*Defer[I
nt][E^(E^x + x)*x, x])/4

Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} \int \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx \\ & = \frac {x}{5}-\frac {3 x^2}{5}+\frac {1}{20} \int e^x (-4-4 x) \, dx+\frac {1}{20} \int e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right ) \, dx \\ & = \frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {1}{20} \int 5 e^{e^x} \left (3+e^{2 x}+3 e^x x\right ) \, dx+\frac {\int e^x \, dx}{5} \\ & = \frac {e^x}{5}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {1}{4} \int e^{e^x} \left (3+e^{2 x}+3 e^x x\right ) \, dx \\ & = \frac {e^x}{5}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {1}{4} \int \left (3 e^{e^x}+e^{e^x+2 x}+3 e^{e^x+x} x\right ) \, dx \\ & = \frac {e^x}{5}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {1}{4} \int e^{e^x+2 x} \, dx+\frac {3}{4} \int e^{e^x} \, dx+\frac {3}{4} \int e^{e^x+x} x \, dx \\ & = \frac {e^x}{5}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {1}{4} \text {Subst}\left (\int e^x x \, dx,x,e^x\right )+\frac {3}{4} \int e^{e^x+x} x \, dx+\frac {3}{4} \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right ) \\ & = \frac {e^x}{5}+\frac {e^{e^x+x}}{4}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {3 \operatorname {ExpIntegralEi}\left (e^x\right )}{4}-\frac {1}{4} \text {Subst}\left (\int e^x \, dx,x,e^x\right )+\frac {3}{4} \int e^{e^x+x} x \, dx \\ & = -\frac {e^{e^x}}{4}+\frac {e^x}{5}+\frac {e^{e^x+x}}{4}+\frac {x}{5}-\frac {3 x^2}{5}-\frac {1}{5} e^x (1+x)+\frac {3 \operatorname {ExpIntegralEi}\left (e^x\right )}{4}+\frac {3}{4} \int e^{e^x+x} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=\frac {1}{20} \left (4 x-4 e^x x-12 x^2+5 e^{e^x} \left (-1+e^x+3 x\right )\right ) \]

[In]

Integrate[(4 + E^x*(-4 - 4*x) - 24*x + E^E^x*(15 + 5*E^(2*x) + 15*E^x*x))/20,x]

[Out]

(4*x - 4*E^x*x - 12*x^2 + 5*E^E^x*(-1 + E^x + 3*x))/20

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {\left (15 x -5+5 \,{\mathrm e}^{x}\right ) {\mathrm e}^{{\mathrm e}^{x}}}{20}-\frac {{\mathrm e}^{x} x}{5}-\frac {3 x^{2}}{5}+\frac {x}{5}\)	\(29\)
default	\(-\frac {{\mathrm e}^{x} x}{5}+\frac {x}{5}+\frac {3 x \,{\mathrm e}^{{\mathrm e}^{x}}}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {3 x^{2}}{5}\)	\(33\)
norman	\(-\frac {{\mathrm e}^{x} x}{5}+\frac {x}{5}+\frac {3 x \,{\mathrm e}^{{\mathrm e}^{x}}}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {3 x^{2}}{5}\)	\(33\)
parallelrisch	\(-\frac {{\mathrm e}^{x} x}{5}+\frac {x}{5}+\frac {3 x \,{\mathrm e}^{{\mathrm e}^{x}}}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {3 x^{2}}{5}\)	\(33\)

[In]

int(1/20*(5*exp(x)^2+15*exp(x)*x+15)*exp(exp(x))+1/20*(-4-4*x)*exp(x)-6/5*x+1/5,x,method=_RETURNVERBOSE)

[Out]

1/20*(15*x-5+5*exp(x))*exp(exp(x))-1/5*exp(x)*x-3/5*x^2+1/5*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=-\frac {3}{5} \, x^{2} - \frac {1}{5} \, x e^{x} + \frac {1}{4} \, {\left (3 \, x + e^{x} - 1\right )} e^{\left (e^{x}\right )} + \frac {1}{5} \, x \]

[In]

integrate(1/20*(5*exp(x)^2+15*exp(x)*x+15)*exp(exp(x))+1/20*(-4-4*x)*exp(x)-6/5*x+1/5,x, algorithm="fricas")

[Out]

-3/5*x^2 - 1/5*x*e^x + 1/4*(3*x + e^x - 1)*e^(e^x) + 1/5*x

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=- \frac {3 x^{2}}{5} - \frac {x e^{x}}{5} + \frac {x}{5} + \frac {\left (3 x + e^{x} - 1\right ) e^{e^{x}}}{4} \]

[In]

integrate(1/20*(5*exp(x)**2+15*exp(x)*x+15)*exp(exp(x))+1/20*(-4-4*x)*exp(x)-6/5*x+1/5,x)

[Out]

-3*x**2/5 - x*exp(x)/5 + x/5 + (3*x + exp(x) - 1)*exp(exp(x))/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=-\frac {3}{5} \, x^{2} - \frac {1}{5} \, x e^{x} + \frac {1}{4} \, {\left (3 \, x + e^{x} - 1\right )} e^{\left (e^{x}\right )} + \frac {1}{5} \, x \]

[In]

integrate(1/20*(5*exp(x)^2+15*exp(x)*x+15)*exp(exp(x))+1/20*(-4-4*x)*exp(x)-6/5*x+1/5,x, algorithm="maxima")

[Out]

-3/5*x^2 - 1/5*x*e^x + 1/4*(3*x + e^x - 1)*e^(e^x) + 1/5*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=-\frac {3}{5} \, x^{2} + \frac {1}{4} \, {\left (3 \, x e^{\left (2 \, x + e^{x}\right )} + e^{\left (3 \, x + e^{x}\right )} - e^{\left (2 \, x + e^{x}\right )}\right )} e^{\left (-2 \, x\right )} - \frac {1}{5} \, x e^{x} + \frac {1}{5} \, x \]

[In]

integrate(1/20*(5*exp(x)^2+15*exp(x)*x+15)*exp(exp(x))+1/20*(-4-4*x)*exp(x)-6/5*x+1/5,x, algorithm="giac")

[Out]

-3/5*x^2 + 1/4*(3*x*e^(2*x + e^x) + e^(3*x + e^x) - e^(2*x + e^x))*e^(-2*x) - 1/5*x*e^x + 1/5*x

Mupad [B] (verification not implemented)

Time = 16.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.55 \[ \int \frac {1}{20} \left (4+e^x (-4-4 x)-24 x+e^{e^x} \left (15+5 e^{2 x}+15 e^x x\right )\right ) \, dx=-\frac {\left (4\,x-5\,{\mathrm {e}}^{{\mathrm {e}}^x}\right )\,\left (3\,x+{\mathrm {e}}^x-1\right )}{20} \]

[In]

int((exp(exp(x))*(5*exp(2*x) + 15*x*exp(x) + 15))/20 - (6*x)/5 - (exp(x)*(4*x + 4))/20 + 1/5,x)

[Out]

-((4*x - 5*exp(exp(x)))*(3*x + exp(x) - 1))/20