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

Optimal. Leaf size=15 \[ -1+e^{\left (e^{2 x}-x\right )^2} \]

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Rubi [A]  time = 0.15, antiderivative size = 19, normalized size of antiderivative = 1.27, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6706} \begin {gather*} e^{x^2-2 e^{2 x} x+e^{4 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^(4*x) - 2*E^(2*x)*x + 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^{e^{4 x}-2 e^{2 x} x+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} e^{\left (e^{2 x}-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^(2*x) - x)^2

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fricas [A]  time = 0.47, size = 16, normalized size = 1.07 \begin {gather*} e^{\left (x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 - 2*x*e^(2*x) + e^(4*x))

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giac [A]  time = 0.25, size = 16, normalized size = 1.07 \begin {gather*} e^{\left (x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 - 2*x*e^(2*x) + e^(4*x))

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maple [A]  time = 0.04, size = 17, normalized size = 1.13

method	result	size
norman	\({\mathrm e}^{{\mathrm e}^{4 x}-2 x \,{\mathrm e}^{2 x}+x^{2}}\)	\(17\)
risch	\({\mathrm e}^{{\mathrm e}^{4 x}-2 x \,{\mathrm e}^{2 x}+x^{2}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(x)^4-2*x*exp(x)^2+x^2)

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maxima [A]  time = 0.60, size = 16, normalized size = 1.07 \begin {gather*} e^{\left (x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^2 - 2*x*e^(2*x) + e^(4*x))

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mupad [B]  time = 1.35, size = 18, normalized size = 1.20 \begin {gather*} {\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.19, size = 17, normalized size = 1.13 \begin {gather*} e^{x^{2} - 2 x e^{2 x} + e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**2 - 2*x*exp(2*x) + exp(4*x))

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