\(\int e^{2 e^{2 x}-2 x^2} (4 e^{2 x}-4 x) \, dx\) [5019]

Optimal result

Integrand size = 27, antiderivative size = 22 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=\frac {105}{16}+e^4+e^{2 e^{2 x}-2 x^2} \]

[Out]

exp(4)+105/16+exp(exp(x)^2-x^2)^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6838} \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{2 e^{2 x}-2 x^2} \]

[In]

Int[E^(2*E^(2*x) - 2*x^2)*(4*E^(2*x) - 4*x),x]

[Out]

E^(2*E^(2*x) - 2*x^2)

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}& = e^{2 e^{2 x}-2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{2 e^{2 x}-2 x^2} \]

[In]

Integrate[E^(2*E^(2*x) - 2*x^2)*(4*E^(2*x) - 4*x),x]

[Out]

E^(2*E^(2*x) - 2*x^2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64

method	result	size
norman	\({\mathrm e}^{2 \,{\mathrm e}^{2 x}-2 x^{2}}\)	\(14\)
risch	\({\mathrm e}^{2 \,{\mathrm e}^{2 x}-2 x^{2}}\)	\(14\)
parallelrisch	\({\mathrm e}^{2 \,{\mathrm e}^{2 x}-2 x^{2}}\)	\(14\)

[In]

int((4*exp(x)^2-4*x)*exp(exp(x)^2-x^2)^2,x,method=_RETURNVERBOSE)

[Out]

exp(exp(x)^2-x^2)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \]

[In]

integrate((4*exp(x)^2-4*x)*exp(exp(x)^2-x^2)^2,x, algorithm="fricas")

[Out]

e^(-2*x^2 + 2*e^(2*x))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{- 2 x^{2} + 2 e^{2 x}} \]

[In]

integrate((4*exp(x)**2-4*x)*exp(exp(x)**2-x**2)**2,x)

[Out]

exp(-2*x**2 + 2*exp(2*x))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \]

[In]

integrate((4*exp(x)^2-4*x)*exp(exp(x)^2-x^2)^2,x, algorithm="maxima")

[Out]

e^(-2*x^2 + 2*e^(2*x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx=e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \]

[In]

integrate((4*exp(x)^2-4*x)*exp(exp(x)^2-x^2)^2,x, algorithm="giac")

[Out]

e^(-2*x^2 + 2*e^(2*x))

Mupad [B] (verification not implemented)

Time = 11.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int e^{2 e^{2 x}-2 x^2} \left (4 e^{2 x}-4 x\right ) \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-2\,x^2} \]

[In]

int(-exp(2*exp(2*x) - 2*x^2)*(4*x - 4*exp(2*x)),x)

[Out]

exp(2*exp(2*x) - 2*x^2)