\(\int (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} (-4 x+64 e^{2 x} x^2)) \, dx\) [1363]

Optimal result

Integrand size = 59, antiderivative size = 26 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x+\left (-e^{4 \left (-1+e^2-4 e^{2 x}\right )}+x^2\right )^2 \]

[Out]

(x^2-exp(-8*exp(x)^2+2*exp(2)-2)^2)^2+x

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2320, 2225, 2326} \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x^4-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x+e^{-32 e^{2 x}-8 \left (1-e^2\right )} \]

[In]

Int[1 - 64*E^(-8 + 8*E^2 - 32*E^(2*x) + 2*x) + 4*x^3 + E^(-4 + 4*E^2 - 16*E^(2*x))*(-4*x + 64*E^(2*x)*x^2),x]

[Out]

E^(-32*E^(2*x) - 8*(1 - E^2)) + x - 2*E^(-16*E^(2*x) - 4*(1 - E^2))*x^2 + x^4

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = x+x^4-64 \int e^{-8+8 e^2-32 e^{2 x}+2 x} \, dx+\int e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right ) \, dx \\ & = x-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x^4-32 \text {Subst}\left (\int e^{-8 \left (1-e^2\right )-32 x} \, dx,x,e^{2 x}\right ) \\ & = e^{-32 e^{2 x}-8 \left (1-e^2\right )}+x-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=e^{-32 e^{2 x}+8 \left (-1+e^2\right )}+x-2 e^{4 \left (-1+e^2-4 e^{2 x}\right )} x^2+x^4 \]

[In]

Integrate[1 - 64*E^(-8 + 8*E^2 - 32*E^(2*x) + 2*x) + 4*x^3 + E^(-4 + 4*E^2 - 16*E^(2*x))*(-4*x + 64*E^(2*x)*x^
2),x]

[Out]

E^(-32*E^(2*x) + 8*(-1 + E^2)) + x - 2*E^(4*(-1 + E^2 - 4*E^(2*x)))*x^2 + x^4

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42

method	result	size
risch	\(x -2 x^{2} {\mathrm e}^{-16 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2}-4}+x^{4}+{\mathrm e}^{-32 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2}-8}\)	\(37\)
default	\(x -2 x^{2} {\mathrm e}^{-16 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2}-4}+x^{4}+{\mathrm e}^{-32 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2}-8}\)	\(41\)
parallelrisch	\(x -2 x^{2} {\mathrm e}^{-16 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2}-4}+x^{4}+{\mathrm e}^{-32 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2}-8}\)	\(41\)

[In]

int(-64*exp(x)^2*exp(-8*exp(x)^2+2*exp(2)-2)^4+(64*exp(x)^2*x^2-4*x)*exp(-8*exp(x)^2+2*exp(2)-2)^2+4*x^3+1,x,m
ethod=_RETURNVERBOSE)

[Out]

x-2*x^2*exp(-16*exp(2*x)+4*exp(2)-4)+x^4+exp(-32*exp(2*x)+8*exp(2)-8)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \]

[In]

integrate(-64*exp(x)^2*exp(-8*exp(x)^2+2*exp(2)-2)^4+(64*exp(x)^2*x^2-4*x)*exp(-8*exp(x)^2+2*exp(2)-2)^2+4*x^3
+1,x, algorithm="fricas")

[Out]

x^4 - 2*x^2*e^(4*e^2 - 16*e^(2*x) - 4) + x + e^(8*e^2 - 32*e^(2*x) - 8)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x^{4} - 2 x^{2} e^{- 16 e^{2 x} - 4 + 4 e^{2}} + x + e^{- 32 e^{2 x} - 8 + 8 e^{2}} \]

[In]

integrate(-64*exp(x)**2*exp(-8*exp(x)**2+2*exp(2)-2)**4+(64*exp(x)**2*x**2-4*x)*exp(-8*exp(x)**2+2*exp(2)-2)**
2+4*x**3+1,x)

[Out]

x**4 - 2*x**2*exp(-16*exp(2*x) - 4 + 4*exp(2)) + x + exp(-32*exp(2*x) - 8 + 8*exp(2))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \]

[In]

integrate(-64*exp(x)^2*exp(-8*exp(x)^2+2*exp(2)-2)^4+(64*exp(x)^2*x^2-4*x)*exp(-8*exp(x)^2+2*exp(2)-2)^2+4*x^3
+1,x, algorithm="maxima")

[Out]

x^4 - 2*x^2*e^(4*e^2 - 16*e^(2*x) - 4) + x + e^(8*e^2 - 32*e^(2*x) - 8)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \]

[In]

integrate(-64*exp(x)^2*exp(-8*exp(x)^2+2*exp(2)-2)^4+(64*exp(x)^2*x^2-4*x)*exp(-8*exp(x)^2+2*exp(2)-2)^2+4*x^3
+1,x, algorithm="giac")

[Out]

x^4 - 2*x^2*e^(4*e^2 - 16*e^(2*x) - 4) + x + e^(8*e^2 - 32*e^(2*x) - 8)

Mupad [B] (verification not implemented)

Time = 9.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \left (1-64 e^{-8+8 e^2-32 e^{2 x}+2 x}+4 x^3+e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right )\right ) \, dx=x+{\mathrm {e}}^{8\,{\mathrm {e}}^2-32\,{\mathrm {e}}^{2\,x}-8}-2\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^2-16\,{\mathrm {e}}^{2\,x}-4}+x^4 \]

[In]

int(4*x^3 - 64*exp(8*exp(2) - 32*exp(2*x) - 8)*exp(2*x) - exp(4*exp(2) - 16*exp(2*x) - 4)*(4*x - 64*x^2*exp(2*
x)) + 1,x)

[Out]

x + exp(8*exp(2) - 32*exp(2*x) - 8) - 2*x^2*exp(4*exp(2) - 16*exp(2*x) - 4) + x^4