3.51.19 \(\int e^{2 e^{2 x}-2 x^2} (4 e^{2 x}-4 x) \, dx\)

Optimal. Leaf size=22 \[ \frac {105}{16}+e^4+e^{2 e^{2 x}-2 x^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6706} \begin {gather*} e^{2 e^{2 x}-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 {gather*} \begin {aligned} \text {integral} &=e^{2 e^{2 x}-2 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 15, normalized size = 0.68 \begin {gather*} e^{2 e^{2 x}-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.92, size = 13, normalized size = 0.59 \begin {gather*} e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.20, size = 13, normalized size = 0.59 \begin {gather*} e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 14, normalized size = 0.64

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.35, size = 13, normalized size = 0.59 \begin {gather*} e^{\left (-2 \, x^{2} + 2 \, e^{\left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.05, size = 13, normalized size = 0.59 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.14, size = 12, normalized size = 0.55 \begin {gather*} e^{- 2 x^{2} + 2 e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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