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

Optimal. Leaf size=24 \[ e^{e^{e^{-1+x}}} \left (4-4 e^{-x}\right )+\log \left (x^4\right ) \]

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Rubi [F]  time = 0.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 \operatorname {Subst}\left (\int e^{-1+e^{\frac {x}{e}}+\frac {x}{e}} \, dx,x,e^x\right )+4 \operatorname {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 \operatorname {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) \operatorname {Subst}\left (\int e^{-1+x} \, dx,x,e^{e^{-1+x}}\right )\\ &=4 e^{e^{e^{-1+x}}}+4 \log (x)+4 \operatorname {Subst}\left (\int \frac {e^{e^{\frac {x}{e}}}}{x^2} \, dx,x,e^x\right )-\frac {4 \operatorname {Subst}\left (\int \frac {e^{e^x+x}}{x} \, dx,x,e^{-1+x}\right )}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 24, normalized size = 1.00 \begin {gather*} 4 \left (e^{e^{e^{-1+x}}-x} \left (-1+e^x\right )+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 1.29, size = 23, normalized size = 0.96 \begin {gather*} 4 \, {\left ({\left (e^{x} - 1\right )} e^{\left (e^{\left (e^{\left (x - 1\right )}\right )}\right )} + e^{x} \log \relax (x)\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.37, size = 44, normalized size = 1.83 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x-1)*exp(x)-4*x*exp(x-1))*exp(exp(x-1))+4*x)*exp(exp(exp(x-1)))+4*exp(x))/exp(x)/x,x, alg
orithm="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)

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maple [A]  time = 0.11, size = 22, normalized size = 0.92

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 21, normalized size = 0.88 \begin {gather*} 4 \, {\left (e^{x} - 1\right )} e^{\left (-x + e^{\left (e^{\left (x - 1\right )}\right )}\right )} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.16, size = 23, normalized size = 0.96 \begin {gather*} 4\,\ln \relax (x)+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}-x}\,\left (4\,{\mathrm {e}}^x-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 81.61, size = 20, normalized size = 0.83 \begin {gather*} \left (4 - 4 e^{- x}\right ) e^{e^{\frac {e^{x}}{e}}} + 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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