3.26.84 \(\int (1+e^{\frac {2 (e^x+x)}{x}} (e^x (512-512 x)-512 x)) \, dx\)

Optimal. Leaf size=26 \[ 2 (-4+x)-x-256 e^{\frac {2 \left (e^x+x\right )}{x}} x^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.73, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2288} \begin {gather*} \frac {256 e^{x+\frac {2 \left (x+e^x\right )}{x}} (1-x)}{\frac {e^x+1}{x}-\frac {x+e^x}{x^2}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^((2*(E^x + x))/x)*(E^x*(512 - 512*x) - 512*x),x]

[Out]

x + (256*E^(x + (2*(E^x + x))/x)*(1 - x))/((1 + E^x)/x - (E^x + x)/x^2)

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} &=x+\int e^{\frac {2 \left (e^x+x\right )}{x}} \left (e^x (512-512 x)-512 x\right ) \, dx\\ &=x+\frac {256 e^{x+\frac {2 \left (e^x+x\right )}{x}} (1-x)}{\frac {1+e^x}{x}-\frac {e^x+x}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 0.73 \begin {gather*} x-256 e^{2+\frac {2 e^x}{x}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^((2*(E^x + x))/x)*(E^x*(512 - 512*x) - 512*x),x]

[Out]

x - 256*E^(2 + (2*E^x)/x)*x^2

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fricas [A]  time = 0.61, size = 17, normalized size = 0.65 \begin {gather*} -256 \, x^{2} e^{\left (\frac {2 \, {\left (x + e^{x}\right )}}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-512*x+512)*exp(x)-512*x)*exp(1/x*(exp(x)+x))^2+1,x, algorithm="fricas")

[Out]

-256*x^2*e^(2*(x + e^x)/x) + x

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giac [A]  time = 0.48, size = 17, normalized size = 0.65 \begin {gather*} -256 \, x^{2} e^{\left (\frac {2 \, {\left (x + e^{x}\right )}}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-512*x+512)*exp(x)-512*x)*exp(1/x*(exp(x)+x))^2+1,x, algorithm="giac")

[Out]

-256*x^2*e^(2*(x + e^x)/x) + x

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maple [A]  time = 0.07, size = 18, normalized size = 0.69

method	result	size
risch	\(x -256 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}+2 x}{x}} x^{2}\)	\(18\)
default	\(x -256 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}+2 x}{x}} x^{2}\)	\(19\)
norman	\(x -256 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}+2 x}{x}} x^{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-512*x+512)*exp(x)-512*x)*exp(1/x*(exp(x)+x))^2+1,x,method=_RETURNVERBOSE)

[Out]

x-256*exp(2/x*(exp(x)+x))*x^2

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maxima [A]  time = 0.72, size = 17, normalized size = 0.65 \begin {gather*} -256 \, x^{2} e^{\left (\frac {2 \, e^{x}}{x} + 2\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-512*x+512)*exp(x)-512*x)*exp(1/x*(exp(x)+x))^2+1,x, algorithm="maxima")

[Out]

-256*x^2*e^(2*e^x/x + 2) + x

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mupad [B]  time = 1.43, size = 17, normalized size = 0.65 \begin {gather*} x-256\,x^2\,{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp((2*(x + exp(x)))/x)*(512*x + exp(x)*(512*x - 512)),x)

[Out]

x - 256*x^2*exp(2)*exp((2*exp(x))/x)

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sympy [A]  time = 0.17, size = 17, normalized size = 0.65 \begin {gather*} - 256 x^{2} e^{\frac {2 \left (x + e^{x}\right )}{x}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-512*x+512)*exp(x)-512*x)*exp(1/x*(exp(x)+x))**2+1,x)

[Out]

-256*x**2*exp(2*(x + exp(x))/x) + x

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