3.19.82 \(\int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} (e^x (2+2 x)+e^{x+e^{4 x} x} (e^x (-1-2 x)+e^{5 x} (-x-4 x^2))) \, dx\)

Optimal. Leaf size=31 \[ 4-e^{1-e^x \left (2 x-e^{x+e^{4 x} x} x\right )} \]

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Rubi [A]  time = 0.38, antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 1, number of rules used = 1, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} -e^{-2 e^x x+e^{e^{4 x} x+2 x} x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 - 2*E^x*x + E^(2*x + E^(4*x)*x)*x)*(E^x*(2 + 2*x) + E^(x + E^(4*x)*x)*(E^x*(-1 - 2*x) + E^(5*x)*(-x -
4*x^2))),x]

[Out]

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

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^{1-2 e^x x+e^{2 x+e^{4 x} x} x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.92, size = 27, normalized size = 0.87 \begin {gather*} -e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(1 - 2*E^x*x + E^(2*x + E^(4*x)*x)*x)*(E^x*(2 + 2*x) + E^(x + E^(4*x)*x)*(E^x*(-1 - 2*x) + E^(5*x)
*(-x - 4*x^2))),x]

[Out]

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

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fricas [A]  time = 0.80, size = 23, normalized size = 0.74 \begin {gather*} -e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-2*x-1)*exp(x))*exp(x*exp(4*x)+x)+(2*x+2)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="fricas")

[Out]

-e^(x*e^(x*e^(4*x) + 2*x) - 2*x*e^x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left ({\left ({\left (4 \, x^{2} + x\right )} e^{\left (5 \, x\right )} + {\left (2 \, x + 1\right )} e^{x}\right )} e^{\left (x e^{\left (4 \, x\right )} + x\right )} - 2 \, {\left (x + 1\right )} e^{x}\right )} e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-2*x-1)*exp(x))*exp(x*exp(4*x)+x)+(2*x+2)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="giac")

[Out]

integrate(-(((4*x^2 + x)*e^(5*x) + (2*x + 1)*e^x)*e^(x*e^(4*x) + x) - 2*(x + 1)*e^x)*e^(x*e^(x*e^(4*x) + 2*x)
- 2*x*e^x + 1), x)

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maple [A]  time = 0.07, size = 22, normalized size = 0.71

method	result	size
risch	\(-{\mathrm e}^{x \,{\mathrm e}^{x \left (2+{\mathrm e}^{4 x}\right )}-2 \,{\mathrm e}^{x} x +1}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^2-x)*exp(x)*exp(4*x)+(-2*x-1)*exp(x))*exp(x*exp(4*x)+x)+(2*x+2)*exp(x))*exp(x*exp(x)*exp(x*exp(4*x
)+x)-2*exp(x)*x+1),x,method=_RETURNVERBOSE)

[Out]

-exp(x*exp(x*(2+exp(4*x)))-2*exp(x)*x+1)

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maxima [A]  time = 0.53, size = 23, normalized size = 0.74 \begin {gather*} -e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-2*x-1)*exp(x))*exp(x*exp(4*x)+x)+(2*x+2)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="maxima")

[Out]

-e^(x*e^(x*e^(4*x) + 2*x) - 2*x*e^x + 1)

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mupad [B]  time = 1.30, size = 24, normalized size = 0.77 \begin {gather*} -{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^x}\,\mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{4\,x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x*exp(x + x*exp(4*x))*exp(x) - 2*x*exp(x) + 1)*(exp(x + x*exp(4*x))*(exp(x)*(2*x + 1) + exp(5*x)*(x +
4*x^2)) - exp(x)*(2*x + 2)),x)

[Out]

-exp(-2*x*exp(x))*exp(1)*exp(x*exp(2*x)*exp(x*exp(4*x)))

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sympy [A]  time = 0.82, size = 26, normalized size = 0.84 \begin {gather*} - e^{x e^{x} e^{x e^{4 x} + x} - 2 x e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x*exp(x)*exp(x*exp(4*x) + x) - 2*x*exp(x) + 1)

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