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

Optimal. Leaf size=23 \[ \frac {1}{2} e^{-2+2 e^{4 e^x}-2 x} x^4 \]

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Rubi [B]  time = 0.16, antiderivative size = 56, normalized size of antiderivative = 2.43, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2288} \begin {gather*} \frac {e^{-2 x+2 e^{4 e^x}-2} x^2 \left (x^2-4 e^{x+4 e^x} x^2\right )}{2 \left (1-4 e^{x+4 e^x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=\frac {e^{-2+2 e^{4 e^x}-2 x} x^2 \left (x^2-4 e^{4 e^x+x} x^2\right )}{2 \left (1-4 e^{4 e^x+x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} \frac {1}{2} e^{-2+2 e^{4 e^x}-2 x} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-2 + 2*E^(4*E^x) - 2*x)*x^4)/2

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fricas [A]  time = 0.67, size = 34, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (-2 \, {\left ({\left (x + 1\right )} e^{x} - e^{x} \log \relax (x) - e^{\left (x + 4 \, e^{x}\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 19, normalized size = 0.83

method	result	size
risch	\(\frac {x^{4} {\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{x}}-2-2 x}}{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{4} e^{\left (-2 \, x + 2 \, e^{\left (4 \, e^{x}\right )} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4*e^(-2*x + 2*e^(4*e^x) - 2)

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mupad [B]  time = 3.68, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^4\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 22.55, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^{4} e^{- 2 x} e^{2 e^{4 e^{x}} - 2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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