\(\int \frac {1}{6} e^{-x} (6 e^x+e^{5 e^{\frac {1}{2} (e^{x^2}+x)}} (2+e^{\frac {1}{2} (e^{x^2}+x)} (-5-15 e^x+e^{x^2} (-10 x-30 e^x x)))) \, dx\) [9919]

Optimal result

Integrand size = 72, antiderivative size = 36 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=x-\frac {e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (x+\frac {e^{-x} x}{3}\right )}{x} \]

[Out]

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

Rubi [F]

\[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=\int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx \]

[In]

Int[(6*E^x + E^(5*E^((E^x^2 + x)/2))*(2 + E^((E^x^2 + x)/2)*(-5 - 15*E^x + E^x^2*(-10*x - 30*E^x*x))))/(6*E^x)
,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \int e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx \\ & = \frac {1}{6} \int \left (6+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \left (2-5 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}-15 e^{\frac {e^{x^2}}{2}+\frac {3 x}{2}}-10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}+x^2} x-30 e^{\frac {e^{x^2}}{2}+\frac {3 x}{2}+x^2} x\right )\right ) \, dx \\ & = x+\frac {1}{6} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \left (2-5 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}-15 e^{\frac {e^{x^2}}{2}+\frac {3 x}{2}}-10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}+x^2} x-30 e^{\frac {e^{x^2}}{2}+\frac {3 x}{2}+x^2} x\right ) \, dx \\ & = x+\frac {1}{6} \int \left (2 e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x}-5 e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+x\right )}-15 \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+3 x\right )\right )-10 \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+x+2 x^2\right )\right ) x-30 \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+3 x+2 x^2\right )\right ) x\right ) \, dx \\ & = x+\frac {1}{3} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \, dx-\frac {5}{6} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+x\right )} \, dx-\frac {5}{3} \int \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+x+2 x^2\right )\right ) x \, dx-\frac {5}{2} \int \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+3 x\right )\right ) \, dx-5 \int \exp \left (5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x+\frac {1}{2} \left (e^{x^2}+3 x+2 x^2\right )\right ) x \, dx \\ & = x+\frac {1}{3} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \, dx-\frac {5}{6} \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}-x\right )} \, dx-\frac {5}{3} \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}-x+2 x^2\right )} x \, dx-\frac {5}{2} \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}+x\right )} \, dx-5 \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}+x+2 x^2\right )} x \, dx \\ & = x+\frac {1}{3} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \, dx-\frac {5}{3} \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}-x+2 x^2\right )} x \, dx-\frac {5}{3} \text {Subst}\left (\int e^{\frac {e^{4 x^2}}{2}+5 e^{\frac {e^{4 x^2}}{2}+x}-x} \, dx,x,\frac {x}{2}\right )-5 \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}+x+2 x^2\right )} x \, dx-5 \text {Subst}\left (\int e^{\frac {e^{4 x^2}}{2}+5 e^{\frac {e^{4 x^2}}{2}+x}+x} \, dx,x,\frac {x}{2}\right ) \\ & = x+\frac {1}{3} \int e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}-x} \, dx-\frac {5}{3} \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}-x+2 x^2\right )} x \, dx-\frac {5}{3} \text {Subst}\left (\int e^{\frac {1}{2} \left (e^{4 x^2}+10 e^{\frac {e^{4 x^2}}{2}+x}-2 x\right )} \, dx,x,\frac {x}{2}\right )-5 \int e^{\frac {1}{2} \left (10 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}+e^{x^2}+x+2 x^2\right )} x \, dx-5 \text {Subst}\left (\int e^{\frac {1}{2} \left (e^{4 x^2}+10 e^{\frac {e^{4 x^2}}{2}+x}+2 x\right )} \, dx,x,\frac {x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 3.92 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=\frac {1}{6} e^{5 e^{\frac {e^{x^2}}{2}+\frac {x}{2}}} \left (-6-2 e^{-x}\right )+x \]

[In]

Integrate[(6*E^x + E^(5*E^((E^x^2 + x)/2))*(2 + E^((E^x^2 + x)/2)*(-5 - 15*E^x + E^x^2*(-10*x - 30*E^x*x))))/(
6*E^x),x]

[Out]

(E^(5*E^(E^x^2/2 + x/2))*(-6 - 2/E^x))/6 + x

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
risch	\(x -\frac {\left (1+3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x +5 \,{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}+\frac {x}{2}}}}{3}\)	\(29\)
parallelrisch	\(-\frac {\left (-6 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{x}+6 \,{\mathrm e}^{5 \,{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}+\frac {x}{2}}} {\mathrm e}^{x}+2 \,{\mathrm e}^{5 \,{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}+\frac {x}{2}}}\right ) {\mathrm e}^{-x}}{6}\)	\(49\)

[In]

int(1/6*((((-30*exp(x)*x-10*x)*exp(x^2)-15*exp(x)-5)*exp(1/2*exp(x^2)+1/2*x)+2)*exp(5*exp(1/2*exp(x^2)+1/2*x))
+6*exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=\frac {1}{3} \, {\left (3 \, x e^{x} - {\left (3 \, e^{x} + 1\right )} e^{\left (5 \, e^{\left (\frac {1}{2} \, x + \frac {1}{2} \, e^{\left (x^{2}\right )}\right )}\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(1/6*((((-30*exp(x)*x-10*x)*exp(x^2)-15*exp(x)-5)*exp(1/2*exp(x^2)+1/2*x)+2)*exp(5*exp(1/2*exp(x^2)+1
/2*x))+6*exp(x))/exp(x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=x + \frac {\left (- 3 e^{x} - 1\right ) e^{- x} e^{5 e^{\frac {x}{2} + \frac {e^{x^{2}}}{2}}}}{3} \]

[In]

integrate(1/6*((((-30*exp(x)*x-10*x)*exp(x**2)-15*exp(x)-5)*exp(1/2*exp(x**2)+1/2*x)+2)*exp(5*exp(1/2*exp(x**2
)+1/2*x))+6*exp(x))/exp(x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=-\frac {1}{3} \, {\left (3 \, e^{x} + 1\right )} e^{\left (-x + 5 \, e^{\left (\frac {1}{2} \, x + \frac {1}{2} \, e^{\left (x^{2}\right )}\right )}\right )} + x \]

[In]

integrate(1/6*((((-30*exp(x)*x-10*x)*exp(x^2)-15*exp(x)-5)*exp(1/2*exp(x^2)+1/2*x)+2)*exp(5*exp(1/2*exp(x^2)+1
/2*x))+6*exp(x))/exp(x),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=\int { -\frac {1}{6} \, {\left ({\left (5 \, {\left (2 \, {\left (3 \, x e^{x} + x\right )} e^{\left (x^{2}\right )} + 3 \, e^{x} + 1\right )} e^{\left (\frac {1}{2} \, x + \frac {1}{2} \, e^{\left (x^{2}\right )}\right )} - 2\right )} e^{\left (5 \, e^{\left (\frac {1}{2} \, x + \frac {1}{2} \, e^{\left (x^{2}\right )}\right )}\right )} - 6 \, e^{x}\right )} e^{\left (-x\right )} \,d x } \]

[In]

integrate(1/6*((((-30*exp(x)*x-10*x)*exp(x^2)-15*exp(x)-5)*exp(1/2*exp(x^2)+1/2*x)+2)*exp(5*exp(1/2*exp(x^2)+1
/2*x))+6*exp(x))/exp(x),x, algorithm="giac")

[Out]

integrate(-1/6*((5*(2*(3*x*e^x + x)*e^(x^2) + 3*e^x + 1)*e^(1/2*x + 1/2*e^(x^2)) - 2)*e^(5*e^(1/2*x + 1/2*e^(x
^2))) - 6*e^x)*e^(-x), x)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{6} e^{-x} \left (6 e^x+e^{5 e^{\frac {1}{2} \left (e^{x^2}+x\right )}} \left (2+e^{\frac {1}{2} \left (e^{x^2}+x\right )} \left (-5-15 e^x+e^{x^2} \left (-10 x-30 e^x x\right )\right )\right )\right ) \, dx=x-{\mathrm {e}}^{5\,\sqrt {{\mathrm {e}}^{{\mathrm {e}}^{x^2}}}\,\sqrt {{\mathrm {e}}^x}-x}\,\left ({\mathrm {e}}^x+\frac {1}{3}\right ) \]

[In]

int(exp(-x)*(exp(x) - (exp(5*exp(x/2 + exp(x^2)/2))*(exp(x/2 + exp(x^2)/2)*(15*exp(x) + exp(x^2)*(10*x + 30*x*
exp(x)) + 5) - 2))/6),x)

[Out]

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