3.37.79 \(\int \frac {e^{-e^x} (18 x-6 x^2+e^x (-18 x^2+3 x^3)+e^{e^x} (-6+x+e^4 (18 x-6 x^2)+x \log (x)))}{3 x} \, dx\)

Optimal. Leaf size=33 \[ -e+\left (-2+\frac {x}{3}\right ) \left (-\left (\left (3 e^4+3 e^{-e^x}\right ) x\right )+\log (x)\right ) \]

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Rubi [F]  time = 0.77, 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^{-e^x} \left (18 x-6 x^2+e^x \left (-18 x^2+3 x^3\right )+e^{e^x} \left (-6+x+e^4 \left (18 x-6 x^2\right )+x \log (x)\right )\right )}{3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18*x - 6*x^2 + E^x*(-18*x^2 + 3*x^3) + E^E^x*(-6 + x + E^4*(18*x - 6*x^2) + x*Log[x]))/(3*E^E^x*x),x]

[Out]

-(E^4*(3 - x)^2) + 6*ExpIntegralEi[-E^x] - 2*Log[x] + (x*Log[x])/3 - 2*Defer[Int][x/E^E^x, x] - 6*Defer[Int][E
^(-E^x + x)*x, x] + Defer[Int][E^(-E^x + x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-e^x} \left (18 x-6 x^2+e^x \left (-18 x^2+3 x^3\right )+e^{e^x} \left (-6+x+e^4 \left (18 x-6 x^2\right )+x \log (x)\right )\right )}{x} \, dx\\ &=\frac {1}{3} \int \left (-6 e^4 (-3+x)-6 e^{-e^x} (-3+x)+\frac {-6+x}{x}+3 e^{-e^x+x} (-6+x) x+\log (x)\right ) \, dx\\ &=-e^4 (3-x)^2+\frac {1}{3} \int \frac {-6+x}{x} \, dx+\frac {1}{3} \int \log (x) \, dx-2 \int e^{-e^x} (-3+x) \, dx+\int e^{-e^x+x} (-6+x) x \, dx\\ &=-e^4 (3-x)^2-\frac {x}{3}+\frac {1}{3} x \log (x)+\frac {1}{3} \int \left (1-\frac {6}{x}\right ) \, dx-2 \int \left (-3 e^{-e^x}+e^{-e^x} x\right ) \, dx+\int \left (-6 e^{-e^x+x} x+e^{-e^x+x} x^2\right ) \, dx\\ &=-e^4 (3-x)^2-2 \log (x)+\frac {1}{3} x \log (x)-2 \int e^{-e^x} x \, dx+6 \int e^{-e^x} \, dx-6 \int e^{-e^x+x} x \, dx+\int e^{-e^x+x} x^2 \, dx\\ &=-e^4 (3-x)^2-2 \log (x)+\frac {1}{3} x \log (x)-2 \int e^{-e^x} x \, dx-6 \int e^{-e^x+x} x \, dx+6 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,e^x\right )+\int e^{-e^x+x} x^2 \, dx\\ &=-e^4 (3-x)^2+6 \text {Ei}\left (-e^x\right )-2 \log (x)+\frac {1}{3} x \log (x)-2 \int e^{-e^x} x \, dx-6 \int e^{-e^x+x} x \, dx+\int e^{-e^x+x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 36, normalized size = 1.09 \begin {gather*} -\frac {1}{3} e^{-e^x} (-6+x) \left (3 \left (1+e^{4+e^x}\right ) x-e^{e^x} \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*x - 6*x^2 + E^x*(-18*x^2 + 3*x^3) + E^E^x*(-6 + x + E^4*(18*x - 6*x^2) + x*Log[x]))/(3*E^E^x*x),
x]

[Out]

-1/3*((-6 + x)*(3*(1 + E^(4 + E^x))*x - E^E^x*Log[x]))/E^E^x

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fricas [A]  time = 0.64, size = 39, normalized size = 1.18 \begin {gather*} -\frac {1}{3} \, {\left (3 \, x^{2} + {\left (3 \, {\left (x^{2} - 6 \, x\right )} e^{4} - {\left (x - 6\right )} \log \relax (x)\right )} e^{\left (e^{x}\right )} - 18 \, x\right )} e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((x*log(x)+(-6*x^2+18*x)*exp(2)^2+x-6)*exp(exp(x))+(3*x^3-18*x^2)*exp(x)-6*x^2+18*x)/x/exp(exp(x
)),x, algorithm="fricas")

[Out]

-1/3*(3*x^2 + (3*(x^2 - 6*x)*e^4 - (x - 6)*log(x))*e^(e^x) - 18*x)*e^(-e^x)

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giac [B]  time = 0.15, size = 58, normalized size = 1.76 \begin {gather*} -\frac {1}{3} \, {\left (3 \, x^{2} e^{\left (x - e^{x}\right )} + 3 \, x^{2} e^{\left (x + 4\right )} - x e^{x} \log \relax (x) - 18 \, x e^{\left (x - e^{x}\right )} - 18 \, x e^{\left (x + 4\right )} + 6 \, e^{x} \log \relax (x)\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((x*log(x)+(-6*x^2+18*x)*exp(2)^2+x-6)*exp(exp(x))+(3*x^3-18*x^2)*exp(x)-6*x^2+18*x)/x/exp(exp(x
)),x, algorithm="giac")

[Out]

-1/3*(3*x^2*e^(x - e^x) + 3*x^2*e^(x + 4) - x*e^x*log(x) - 18*x*e^(x - e^x) - 18*x*e^(x + 4) + 6*e^x*log(x))*e
^(-x)

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maple [A]  time = 0.08, size = 39, normalized size = 1.18

method	result	size
risch	\(\frac {x \ln \relax (x )}{3}-x^{2} {\mathrm e}^{4}+6 x \,{\mathrm e}^{4}-2 \ln \relax (x )+\frac {\left (-3 x^{2}+18 x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}{3}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((x*ln(x)+(-6*x^2+18*x)*exp(2)^2+x-6)*exp(exp(x))+(3*x^3-18*x^2)*exp(x)-6*x^2+18*x)/x/exp(exp(x)),x,me
thod=_RETURNVERBOSE)

[Out]

1/3*x*ln(x)-x^2*exp(4)+6*x*exp(4)-2*ln(x)+1/3*(-3*x^2+18*x)*exp(-exp(x))

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maxima [A]  time = 0.44, size = 36, normalized size = 1.09 \begin {gather*} -x^{2} e^{4} + 6 \, x e^{4} - {\left (x^{2} - 6 \, x\right )} e^{\left (-e^{x}\right )} + \frac {1}{3} \, x \log \relax (x) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((x*log(x)+(-6*x^2+18*x)*exp(2)^2+x-6)*exp(exp(x))+(3*x^3-18*x^2)*exp(x)-6*x^2+18*x)/x/exp(exp(x
)),x, algorithm="maxima")

[Out]

-x^2*e^4 + 6*x*e^4 - (x^2 - 6*x)*e^(-e^x) + 1/3*x*log(x) - 2*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-{\mathrm {e}}^x}\,\left (6\,x-\frac {{\mathrm {e}}^x\,\left (18\,x^2-3\,x^3\right )}{3}+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (x+{\mathrm {e}}^4\,\left (18\,x-6\,x^2\right )+x\,\ln \relax (x)-6\right )}{3}-2\,x^2\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(x))*(6*x - (exp(x)*(18*x^2 - 3*x^3))/3 + (exp(exp(x))*(x + exp(4)*(18*x - 6*x^2) + x*log(x) - 6)
)/3 - 2*x^2))/x,x)

[Out]

int((exp(-exp(x))*(6*x - (exp(x)*(18*x^2 - 3*x^3))/3 + (exp(exp(x))*(x + exp(4)*(18*x - 6*x^2) + x*log(x) - 6)
)/3 - 2*x^2))/x, x)

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sympy [A]  time = 0.43, size = 36, normalized size = 1.09 \begin {gather*} - x^{2} e^{4} + \frac {x \log {\relax (x )}}{3} + 6 x e^{4} + \left (- x^{2} + 6 x\right ) e^{- e^{x}} - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((x*ln(x)+(-6*x**2+18*x)*exp(2)**2+x-6)*exp(exp(x))+(3*x**3-18*x**2)*exp(x)-6*x**2+18*x)/x/exp(e
xp(x)),x)

[Out]

-x**2*exp(4) + x*log(x)/3 + 6*x*exp(4) + (-x**2 + 6*x)*exp(-exp(x)) - 2*log(x)

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