3.58.97 \(\int (3+e^4+18 x+e^{2 e^x-2 x} (1-2 x+2 e^x x)) \, dx\)

Optimal. Leaf size=21 \[ x \left (3+e^4+e^{2 e^x-2 x}+9 x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 2.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2288} \begin {gather*} 9 x^2+\left (3+e^4\right ) x+\frac {e^{2 e^x-2 x} \left (x-e^x x\right )}{1-e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3 + E^4 + 18*x + E^(2*E^x - 2*x)*(1 - 2*x + 2*E^x*x),x]

[Out]

(3 + E^4)*x + 9*x^2 + (E^(2*E^x - 2*x)*(x - E^x*x))/(1 - E^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} &=\left (3+e^4\right ) x+9 x^2+\int e^{2 e^x-2 x} \left (1-2 x+2 e^x x\right ) \, dx\\ &=\left (3+e^4\right ) x+9 x^2+\frac {e^{2 e^x-2 x} \left (x-e^x x\right )}{1-e^x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 1.00 \begin {gather*} x \left (3+e^4+e^{2 e^x-2 x}+9 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3 + E^4 + 18*x + E^(2*E^x - 2*x)*(1 - 2*x + 2*E^x*x),x]

[Out]

x*(3 + E^4 + E^(2*E^x - 2*x) + 9*x)

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fricas [A]  time = 0.93, size = 24, normalized size = 1.14 \begin {gather*} 9 \, x^{2} + x e^{4} + x e^{\left (-2 \, x + 2 \, e^{x}\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x+1-2*x)*exp(exp(x)-x)^2+exp(4)+18*x+3,x, algorithm="fricas")

[Out]

9*x^2 + x*e^4 + x*e^(-2*x + 2*e^x) + 3*x

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giac [A]  time = 0.34, size = 24, normalized size = 1.14 \begin {gather*} 9 \, x^{2} + x e^{4} + x e^{\left (-2 \, x + 2 \, e^{x}\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x+1-2*x)*exp(exp(x)-x)^2+exp(4)+18*x+3,x, algorithm="giac")

[Out]

9*x^2 + x*e^4 + x*e^(-2*x + 2*e^x) + 3*x

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maple [A]  time = 0.08, size = 24, normalized size = 1.14

method	result	size
norman	\(\left ({\mathrm e}^{4}+3\right ) x +{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x} x +9 x^{2}\)	\(24\)
default	\(3 x +{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x} x +9 x^{2}+x \,{\mathrm e}^{4}\)	\(25\)
risch	\(3 x +{\mathrm e}^{2 \,{\mathrm e}^{x}-2 x} x +9 x^{2}+x \,{\mathrm e}^{4}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)*x+1-2*x)*exp(exp(x)-x)^2+exp(4)+18*x+3,x,method=_RETURNVERBOSE)

[Out]

(exp(4)+3)*x+exp(exp(x)-x)^2*x+9*x^2

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maxima [A]  time = 0.35, size = 24, normalized size = 1.14 \begin {gather*} 9 \, x^{2} + x e^{4} + x e^{\left (-2 \, x + 2 \, e^{x}\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x+1-2*x)*exp(exp(x)-x)^2+exp(4)+18*x+3,x, algorithm="maxima")

[Out]

9*x^2 + x*e^4 + x*e^(-2*x + 2*e^x) + 3*x

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mupad [B]  time = 4.07, size = 18, normalized size = 0.86 \begin {gather*} x\,\left (9\,x+{\mathrm {e}}^4+{\mathrm {e}}^{2\,{\mathrm {e}}^x-2\,x}+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(18*x + exp(4) + exp(2*exp(x) - 2*x)*(2*x*exp(x) - 2*x + 1) + 3,x)

[Out]

x*(9*x + exp(4) + exp(2*exp(x) - 2*x) + 3)

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sympy [A]  time = 0.14, size = 22, normalized size = 1.05 \begin {gather*} 9 x^{2} + x e^{- 2 x + 2 e^{x}} + x \left (3 + e^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)*x+1-2*x)*exp(exp(x)-x)**2+exp(4)+18*x+3,x)

[Out]

9*x**2 + x*exp(-2*x + 2*exp(x)) + x*(3 + exp(4))

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