\(\int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} (4 e^{2 x}+8 e^x x^2+4 x^4)}{\sqrt {e}}} (16 e^{4 x}+16 x^7+e^{3 x} (16 x+24 x^2)+e^{2 x} (48 x^3+24 x^4)+e^x (48 x^5+8 x^6)+e^{2 x} (16 e^{2 x}+16 x^3+8 x^4+e^x (16 x+24 x^2))) \, dx\) [8107]

Optimal result

Integrand size = 178, antiderivative size = 26 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\frac {2 \left (e^{2 x}+\left (e^x+x^2\right )^2\right )^2}{\sqrt {e}}} \]

[Out]

exp(2*((x^2+exp(x))^2+exp(2*x))^2/exp(1/4)^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(26)=52\).

Time = 3.71 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6820, 12, 6838} \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=\exp \left (\frac {1}{2}-\frac {-4 x^8-16 e^x x^6-32 e^{2 x} x^4-32 e^{3 x} x^2-16 e^{4 x}+\sqrt {e}}{2 \sqrt {e}}\right ) \]

[In]

Int[E^(-1/2 + (4*E^(4*x) + 8*E^(3*x)*x^2 + 12*E^(2*x)*x^4 + 8*E^x*x^6 + 2*x^8 + E^(2*x)*(4*E^(2*x) + 8*E^x*x^2
+ 4*x^4))/Sqrt[E])*(16*E^(4*x) + 16*x^7 + E^(3*x)*(16*x + 24*x^2) + E^(2*x)*(48*x^3 + 24*x^4) + E^x*(48*x^5 +
8*x^6) + E^(2*x)*(16*E^(2*x) + 16*x^3 + 8*x^4 + E^x*(16*x + 24*x^2))),x]

[Out]

E^(1/2 - (Sqrt[E] - 16*E^(4*x) - 32*E^(3*x)*x^2 - 32*E^(2*x)*x^4 - 16*E^x*x^6 - 4*x^8)/(2*Sqrt[E]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int 8 \exp \left (\frac {-\sqrt {e}+16 e^{4 x}+32 e^{3 x} x^2+32 e^{2 x} x^4+16 e^x x^6+4 x^8}{2 \sqrt {e}}\right ) \left (4 e^{4 x}+2 x^7+4 e^{2 x} x^3 (2+x)+e^x x^5 (6+x)+2 e^{3 x} x (2+3 x)\right ) \, dx \\ & = 8 \int \exp \left (\frac {-\sqrt {e}+16 e^{4 x}+32 e^{3 x} x^2+32 e^{2 x} x^4+16 e^x x^6+4 x^8}{2 \sqrt {e}}\right ) \left (4 e^{4 x}+2 x^7+4 e^{2 x} x^3 (2+x)+e^x x^5 (6+x)+2 e^{3 x} x (2+3 x)\right ) \, dx \\ & = \exp \left (\frac {1}{2}-\frac {\sqrt {e}-16 e^{4 x}-32 e^{3 x} x^2-32 e^{2 x} x^4-16 e^x x^6-4 x^8}{2 \sqrt {e}}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(26)=52\).

Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{8 e^{-\frac {1}{2}+4 x}+16 e^{-\frac {1}{2}+3 x} x^2+16 e^{-\frac {1}{2}+2 x} x^4+8 e^{-\frac {1}{2}+x} x^6+\frac {2 x^8}{\sqrt {e}}} \]

[In]

Integrate[E^(-1/2 + (4*E^(4*x) + 8*E^(3*x)*x^2 + 12*E^(2*x)*x^4 + 8*E^x*x^6 + 2*x^8 + E^(2*x)*(4*E^(2*x) + 8*E
^x*x^2 + 4*x^4))/Sqrt[E])*(16*E^(4*x) + 16*x^7 + E^(3*x)*(16*x + 24*x^2) + E^(2*x)*(48*x^3 + 24*x^4) + E^x*(48
*x^5 + 8*x^6) + E^(2*x)*(16*E^(2*x) + 16*x^3 + 8*x^4 + E^x*(16*x + 24*x^2))),x]

[Out]

E^(8*E^(-1/2 + 4*x) + 16*E^(-1/2 + 3*x)*x^2 + 16*E^(-1/2 + 2*x)*x^4 + 8*E^(-1/2 + x)*x^6 + (2*x^8)/Sqrt[E])

Maple [A] (verified)

Time = 14.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58

method	result	size
risch	\({\mathrm e}^{2 \left (x^{8}+4 x^{6} {\mathrm e}^{x}+8 \,{\mathrm e}^{2 x} x^{4}+8 x^{2} {\mathrm e}^{3 x}+4 \,{\mathrm e}^{4 x}\right ) {\mathrm e}^{-\frac {1}{2}}}\)	\(41\)
parallelrisch	\({\mathrm e}^{\left (4 \,{\mathrm e}^{4 x}+\left (4 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{2}+4 x^{4}\right ) {\mathrm e}^{2 x}+8 x^{2} {\mathrm e}^{3 x}+12 \,{\mathrm e}^{2 x} x^{4}+8 x^{6} {\mathrm e}^{x}+2 x^{8}\right ) {\mathrm e}^{-\frac {1}{2}}}\)	\(76\)

[In]

int((8*exp(2*x)^2+(16*exp(x)^2+(24*x^2+16*x)*exp(x)+8*x^4+16*x^3)*exp(2*x)+8*exp(x)^4+(24*x^2+16*x)*exp(x)^3+(
24*x^4+48*x^3)*exp(x)^2+(8*x^6+48*x^5)*exp(x)+16*x^7)*exp((2*exp(2*x)^2+(4*exp(x)^2+8*exp(x)*x^2+4*x^4)*exp(2*
x)+2*exp(x)^4+8*x^2*exp(x)^3+12*exp(x)^2*x^4+8*x^6*exp(x)+2*x^8)/exp(1/4)^2)/exp(1/4)^2,x,method=_RETURNVERBOS
E)

[Out]

exp(2*(x^8+4*x^6*exp(x)+8*exp(2*x)*x^4+8*x^2*exp(3*x)+4*exp(4*x))*exp(-1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (\frac {1}{2} \, {\left (4 \, x^{8} + 16 \, x^{6} e^{x} + 32 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{2} e^{\left (3 \, x\right )} - e^{\frac {1}{2}} + 16 \, e^{\left (4 \, x\right )}\right )} e^{\left (-\frac {1}{2}\right )} + \frac {1}{2}\right )} \]

[In]

integrate((8*exp(2*x)^2+(16*exp(x)^2+(24*x^2+16*x)*exp(x)+8*x^4+16*x^3)*exp(2*x)+8*exp(x)^4+(24*x^2+16*x)*exp(
x)^3+(24*x^4+48*x^3)*exp(x)^2+(8*x^6+48*x^5)*exp(x)+16*x^7)*exp((2*exp(2*x)^2+(4*exp(x)^2+8*exp(x)*x^2+4*x^4)*
exp(2*x)+2*exp(x)^4+8*x^2*exp(x)^3+12*exp(x)^2*x^4+8*x^6*exp(x)+2*x^8)/exp(1/4)^2)/exp(1/4)^2,x, algorithm="fr
icas")

[Out]

e^(1/2*(4*x^8 + 16*x^6*e^x + 32*x^4*e^(2*x) + 32*x^2*e^(3*x) - e^(1/2) + 16*e^(4*x))*e^(-1/2) + 1/2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\frac {2 x^{8} + 8 x^{6} e^{x} + 12 x^{4} e^{2 x} + 8 x^{2} e^{3 x} + \left (4 x^{4} + 8 x^{2} e^{x} + 4 e^{2 x}\right ) e^{2 x} + 4 e^{4 x}}{e^{\frac {1}{2}}}} \]

[In]

integrate((8*exp(2*x)**2+(16*exp(x)**2+(24*x**2+16*x)*exp(x)+8*x**4+16*x**3)*exp(2*x)+8*exp(x)**4+(24*x**2+16*
x)*exp(x)**3+(24*x**4+48*x**3)*exp(x)**2+(8*x**6+48*x**5)*exp(x)+16*x**7)*exp((2*exp(2*x)**2+(4*exp(x)**2+8*ex
p(x)*x**2+4*x**4)*exp(2*x)+2*exp(x)**4+8*x**2*exp(x)**3+12*exp(x)**2*x**4+8*x**6*exp(x)+2*x**8)/exp(1/4)**2)/e
xp(1/4)**2,x)

[Out]

exp((2*x**8 + 8*x**6*exp(x) + 12*x**4*exp(2*x) + 8*x**2*exp(3*x) + (4*x**4 + 8*x**2*exp(x) + 4*exp(2*x))*exp(2
*x) + 4*exp(4*x))*exp(-1/2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 0.41 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (2 \, x^{8} e^{\left (-\frac {1}{2}\right )} + 8 \, x^{6} e^{\left (x - \frac {1}{2}\right )} + 16 \, x^{4} e^{\left (2 \, x - \frac {1}{2}\right )} + 16 \, x^{2} e^{\left (3 \, x - \frac {1}{2}\right )} + 8 \, e^{\left (4 \, x - \frac {1}{2}\right )}\right )} \]

[In]

integrate((8*exp(2*x)^2+(16*exp(x)^2+(24*x^2+16*x)*exp(x)+8*x^4+16*x^3)*exp(2*x)+8*exp(x)^4+(24*x^2+16*x)*exp(
x)^3+(24*x^4+48*x^3)*exp(x)^2+(8*x^6+48*x^5)*exp(x)+16*x^7)*exp((2*exp(2*x)^2+(4*exp(x)^2+8*exp(x)*x^2+4*x^4)*
exp(2*x)+2*exp(x)^4+8*x^2*exp(x)^3+12*exp(x)^2*x^4+8*x^6*exp(x)+2*x^8)/exp(1/4)^2)/exp(1/4)^2,x, algorithm="ma
xima")

[Out]

e^(2*x^8*e^(-1/2) + 8*x^6*e^(x - 1/2) + 16*x^4*e^(2*x - 1/2) + 16*x^2*e^(3*x - 1/2) + 8*e^(4*x - 1/2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (2 \, {\left (x^{8} e^{\left (4 \, x - 2\right )} + 4 \, x^{6} e^{\left (5 \, x - 2\right )} + 8 \, x^{4} e^{\left (6 \, x - 2\right )} + 8 \, x^{2} e^{\left (7 \, x - 2\right )} + 4 \, e^{\left (8 \, x - 2\right )}\right )} e^{\left (-4 \, x + \frac {3}{2}\right )}\right )} \]

[In]

integrate((8*exp(2*x)^2+(16*exp(x)^2+(24*x^2+16*x)*exp(x)+8*x^4+16*x^3)*exp(2*x)+8*exp(x)^4+(24*x^2+16*x)*exp(
x)^3+(24*x^4+48*x^3)*exp(x)^2+(8*x^6+48*x^5)*exp(x)+16*x^7)*exp((2*exp(2*x)^2+(4*exp(x)^2+8*exp(x)*x^2+4*x^4)*
exp(2*x)+2*exp(x)^4+8*x^2*exp(x)^3+12*exp(x)^2*x^4+8*x^6*exp(x)+2*x^8)/exp(1/4)^2)/exp(1/4)^2,x, algorithm="gi
ac")

[Out]

e^(2*(x^8*e^(4*x - 2) + 4*x^6*e^(5*x - 2) + 8*x^4*e^(6*x - 2) + 8*x^2*e^(7*x - 2) + 4*e^(8*x - 2))*e^(-4*x + 3
/2))

Mupad [B] (verification not implemented)

Time = 13.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx={\mathrm {e}}^{2\,x^8\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{16\,x^2\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{16\,x^4\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{8\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{8\,x^6\,{\mathrm {e}}^{-\frac {1}{2}}\,{\mathrm {e}}^x} \]

[In]

int(exp(exp(-1/2)*(4*exp(4*x) + 8*x^6*exp(x) + 8*x^2*exp(3*x) + 12*x^4*exp(2*x) + 2*x^8 + exp(2*x)*(4*exp(2*x)
+ 8*x^2*exp(x) + 4*x^4)))*exp(-1/2)*(16*exp(4*x) + exp(3*x)*(16*x + 24*x^2) + exp(x)*(48*x^5 + 8*x^6) + exp(2
*x)*(48*x^3 + 24*x^4) + exp(2*x)*(16*exp(2*x) + exp(x)*(16*x + 24*x^2) + 16*x^3 + 8*x^4) + 16*x^7),x)

[Out]

exp(2*x^8*exp(-1/2))*exp(16*x^2*exp(3*x)*exp(-1/2))*exp(16*x^4*exp(2*x)*exp(-1/2))*exp(8*exp(4*x)*exp(-1/2))*e
xp(8*x^6*exp(-1/2)*exp(x))