\(\int \frac {e^{-x} (4 e^x+e^{e^{e^{-1+x}}} (4 x+e^{e^{-1+x}} (-4 e^{-1+x} x+4 e^{-1+2 x} x)))}{x} \, dx\) [1496]

Optimal result

Integrand size = 56, antiderivative size = 24 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=e^{e^{e^{-1+x}}} \left (4-4 e^{-x}\right )+\log \left (x^4\right ) \]

[Out]

exp(exp(exp(-1+x)))*(4-4/exp(x))+ln(x^4)

Rubi [F]

\[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=\int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx \]

[In]

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

[Out]

4*E^E^E^(-1 + x) + 4*Log[x] + 4*Defer[Subst][Defer[Int][E^E^(x/E)/x^2, x], x, E^x] - (4*Defer[Subst][Defer[Int
][E^(E^x + x)/x, x], x, E^(-1 + x)])/E

Rubi steps \begin{align*} \text {integral}& = \int \left (4 e^{e^{e^{-1+x}}-x}+4 e^{-1+e^{e^{-1+x}}+e^{-1+x}+x}+\frac {4 \left (e-e^{e^{e^{-1+x}}+e^{-1+x}} x\right )}{e x}\right ) \, dx \\ & = 4 \int e^{e^{e^{-1+x}}-x} \, dx+4 \int e^{-1+e^{e^{-1+x}}+e^{-1+x}+x} \, dx+\frac {4 \int \frac {e-e^{e^{e^{-1+x}}+e^{-1+x}} x}{x} \, dx}{e} \\ & = 4 \text {Subst}\left (\int e^{-1+e^{\frac {x}{e}}+\frac {x}{e}} \, dx,x,e^x\right )+4 \text {Subst}\left (\int \frac {e^{e^{\frac {x}{e}}}}{x^2} \, dx,x,e^x\right )+\frac {4 \int \left (-e^{e^{e^{-1+x}}+e^{-1+x}}+\frac {e}{x}\right ) \, dx}{e} \\ & = 4 \log (x)+4 \text {Subst}\left (\int \frac {e^{e^{\frac {x}{e}}}}{x^2} \, dx,x,e^x\right )-\frac {4 \int e^{e^{e^{-1+x}}+e^{-1+x}} \, dx}{e}+(4 e) \text {Subst}\left (\int e^{-1+x} \, dx,x,e^{e^{-1+x}}\right ) \\ & = 4 e^{e^{e^{-1+x}}}+4 \log (x)+4 \text {Subst}\left (\int \frac {e^{e^{\frac {x}{e}}}}{x^2} \, dx,x,e^x\right )-\frac {4 \text {Subst}\left (\int \frac {e^{e^x+x}}{x} \, dx,x,e^{-1+x}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=4 \left (e^{e^{e^{-1+x}}-x} \left (-1+e^x\right )+\log (x)\right ) \]

[In]

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

[Out]

4*(E^(E^E^(-1 + x) - x)*(-1 + E^x) + Log[x])

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

method	result	size
risch	\(4 \ln \left (x \right )+4 \left ({\mathrm e}^{x}-1\right ) {\mathrm e}^{-x +{\mathrm e}^{{\mathrm e}^{-1+x}}}\)	\(22\)
parts	\(-4 \,{\mathrm e}^{-x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1} {\mathrm e}^{x}}}+4 \ln \left (x \right )+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1+x}}}\)	\(27\)
parallelrisch	\(\left (4 \,{\mathrm e}^{x} \ln \left (x \right )+4 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1+x}}}-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1+x}}}\right ) {\mathrm e}^{-x}\)	\(31\)
norman	\(\left (4 \,{\mathrm e}^{-1} {\mathrm e} \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1} {\mathrm e}^{x}}}-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-1} {\mathrm e}^{x}}}\right ) {\mathrm e}^{-x}+4 \ln \left (x \right )\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=4 \, {\left ({\left (e^{x} - 1\right )} e^{\left (e^{\left (e^{\left (x - 1\right )}\right )}\right )} + e^{x} \log \left (x\right )\right )} e^{\left (-x\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=\left (4 e^{x} - 4\right ) e^{- x} e^{e^{\frac {e^{x}}{e}}} + 4 \log {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=4 \, {\left (e^{x} - 1\right )} e^{\left (-x + e^{\left (e^{\left (x - 1\right )}\right )}\right )} + 4 \, \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=4 \, {\left (e^{\left (x + e^{\left (x - 1\right )} + e^{\left (e^{\left (x - 1\right )}\right )}\right )} - e^{\left (e^{\left (x - 1\right )} + e^{\left (e^{\left (x - 1\right )}\right )}\right )}\right )} e^{\left (-x - e^{\left (x - 1\right )}\right )} + 4 \, \log \left (x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (4 e^x+e^{e^{e^{-1+x}}} \left (4 x+e^{e^{-1+x}} \left (-4 e^{-1+x} x+4 e^{-1+2 x} x\right )\right )\right )}{x} \, dx=4\,\ln \left (x\right )+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}-x}\,\left (4\,{\mathrm {e}}^x-4\right ) \]

[In]

int((exp(-x)*(4*exp(x) + exp(exp(exp(x - 1)))*(4*x - exp(exp(x - 1))*(4*x*exp(x - 1) - 4*x*exp(x - 1)*exp(x)))
))/x,x)

[Out]

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