3.46.22 \(\int e^{-17-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} (e^8 (1+3 x)+e^{3^{\frac {x}{e^8}}} (-e^8 x+3^{\frac {x}{e^8}} (3 x-x^2) \log (3))) \, dx\) [4522]

Optimal. Leaf size=24 \[ e^{-\left (\left (3-e^{3^{\frac {x}{e^8}}}\right ) (3-x)\right )} x \]

[Out]

x/exp((3-x)*(3-exp(exp(x*ln(3)/exp(4)^2))))

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Rubi [F]
time = 3.37, 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 e^{-17-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} \left (e^8 (1+3 x)+e^{3^{\frac {x}{e^8}}} \left (-e^8 x+3^{\frac {x}{e^8}} \left (3 x-x^2\right ) \log (3)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-17 - E^3^(x/E^8)*(-3 + x) + 3*x)*(E^8*(1 + 3*x) + E^3^(x/E^8)*(-(E^8*x) + 3^(x/E^8)*(3*x - x^2)*Log[3]
)),x]

[Out]

Defer[Int][E^(-((-3 + E^3^(x/E^8))*(-3 + x))), x] + 3*Defer[Int][x/E^((-3 + E^3^(x/E^8))*(-3 + x)), x] + Log[3
]*Defer[Int][3^(1 + x/E^8)*E^(-17 + 3^(x/E^8) - E^3^(x/E^8)*(-3 + x) + 3*x)*x, x] - Defer[Int][E^(-9 + 3^(x/E^
8) - E^3^(x/E^8)*(-3 + x) + 3*x)*x, x] - Log[3]*Defer[Int][3^(x/E^8)*E^(-17 + 3^(x/E^8) - E^3^(x/E^8)*(-3 + x)
+ 3*x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-9-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} (1+3 x)-e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \left (e^8-3^{1+\frac {x}{e^8}} \log (3)+3^{\frac {x}{e^8}} x \log (3)\right )\right ) \, dx\\ &=\int e^{-9-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} (1+3 x) \, dx-\int e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \left (e^8-3^{1+\frac {x}{e^8}} \log (3)+3^{\frac {x}{e^8}} x \log (3)\right ) \, dx\\ &=\int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} (1+3 x) \, dx-\int e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \left (e^8+3^{\frac {x}{e^8}} (-3+x) \log (3)\right ) \, dx\\ &=\int \left (e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )}+3 e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} x\right ) \, dx-\int \left (e^{-9+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x+3^{\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} (-3+x) x \log (3)\right ) \, dx\\ &=3 \int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} x \, dx-\log (3) \int 3^{\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} (-3+x) x \, dx+\int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} \, dx-\int e^{-9+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \, dx\\ &=3 \int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} x \, dx-\log (3) \int \left (-3^{1+\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x+3^{\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x^2\right ) \, dx+\int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} \, dx-\int e^{-9+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \, dx\\ &=3 \int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} x \, dx+\log (3) \int 3^{1+\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \, dx-\log (3) \int 3^{\frac {x}{e^8}} e^{-17+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x^2 \, dx+\int e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} \, dx-\int e^{-9+3^{\frac {x}{e^8}}-e^{3^{\frac {x}{e^8}}} (-3+x)+3 x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.18, size = 20, normalized size = 0.83 \begin {gather*} e^{-\left (\left (-3+e^{3^{\frac {x}{e^8}}}\right ) (-3+x)\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-17 - E^3^(x/E^8)*(-3 + x) + 3*x)*(E^8*(1 + 3*x) + E^3^(x/E^8)*(-(E^8*x) + 3^(x/E^8)*(3*x - x^2)*
Log[3])),x]

[Out]

x/E^((-3 + E^3^(x/E^8))*(-3 + x))

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Maple [A]
time = 0.77, size = 18, normalized size = 0.75

method	result	size
risch	\(x \,{\mathrm e}^{-\left (x -3\right ) \left ({\mathrm e}^{3^{{\mathrm e}^{-8} x}}-3\right )}\)	\(18\)
norman	\(x \,{\mathrm e}^{-\left (x -3\right ) {\mathrm e}^{{\mathrm e}^{x \ln \left (3\right ) {\mathrm e}^{-8}}}+3 x -9}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+3*x)*ln(3)*exp(x*ln(3)/exp(4)^2)-x*exp(4)^2)*exp(exp(x*ln(3)/exp(4)^2))+(3*x+1)*exp(4)^2)/exp(4)^2
/exp((x-3)*exp(exp(x*ln(3)/exp(4)^2))-3*x+9),x,method=_RETURNVERBOSE)

[Out]

x*exp(-(x-3)*(exp(3^(exp(-8)*x))-3))

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Maxima [A]
time = 0.63, size = 27, normalized size = 1.12 \begin {gather*} x e^{\left (-x e^{\left (3^{x e^{\left (-8\right )}}\right )} + 3 \, x + 3 \, e^{\left (3^{x e^{\left (-8\right )}}\right )} - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+3*x)*log(3)*exp(x*log(3)/exp(4)^2)-x*exp(4)^2)*exp(exp(x*log(3)/exp(4)^2))+(1+3*x)*exp(4)^2)
/exp(4)^2/exp((-3+x)*exp(exp(x*log(3)/exp(4)^2))-3*x+9),x, algorithm="maxima")

[Out]

x*e^(-x*e^(3^(x*e^(-8))) + 3*x + 3*e^(3^(x*e^(-8))) - 9)

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Fricas [A]
time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} x e^{\left (-{\left (x - 3\right )} e^{\left (3^{x e^{\left (-8\right )}}\right )} + 3 \, x - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+3*x)*log(3)*exp(x*log(3)/exp(4)^2)-x*exp(4)^2)*exp(exp(x*log(3)/exp(4)^2))+(1+3*x)*exp(4)^2)
/exp(4)^2/exp((-3+x)*exp(exp(x*log(3)/exp(4)^2))-3*x+9),x, algorithm="fricas")

[Out]

x*e^(-(x - 3)*e^(3^(x*e^(-8))) + 3*x - 9)

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Sympy [A]
time = 1.49, size = 22, normalized size = 0.92 \begin {gather*} x e^{3 x - \left (x - 3\right ) e^{e^{\frac {x \log {\left (3 \right )}}{e^{8}}}} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+3*x)*ln(3)*exp(x*ln(3)/exp(4)**2)-x*exp(4)**2)*exp(exp(x*ln(3)/exp(4)**2))+(1+3*x)*exp(4)**
2)/exp(4)**2/exp((-3+x)*exp(exp(x*ln(3)/exp(4)**2))-3*x+9),x)

[Out]

x*exp(3*x - (x - 3)*exp(exp(x*exp(-8)*log(3))) - 9)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+3*x)*log(3)*exp(x*log(3)/exp(4)^2)-x*exp(4)^2)*exp(exp(x*log(3)/exp(4)^2))+(1+3*x)*exp(4)^2)
/exp(4)^2/exp((-3+x)*exp(exp(x*log(3)/exp(4)^2))-3*x+9),x, algorithm="giac")

[Out]

integrate(((3*x + 1)*e^8 - ((x^2 - 3*x)*3^(x*e^(-8))*log(3) + x*e^8)*e^(3^(x*e^(-8))))*e^(-(x - 3)*e^(3^(x*e^(
-8))) + 3*x - 17), x)

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Mupad [B]
time = 3.60, size = 29, normalized size = 1.21 \begin {gather*} x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{3^{x\,{\mathrm {e}}^{-8}}}}\,{\mathrm {e}}^{-9}\,{\mathrm {e}}^{3\,{\mathrm {e}}^{3^{x\,{\mathrm {e}}^{-8}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(3*x - exp(exp(x*exp(-8)*log(3)))*(x - 3) - 9)*exp(-8)*(exp(exp(x*exp(-8)*log(3)))*(x*exp(8) - exp(x*e
xp(-8)*log(3))*log(3)*(3*x - x^2)) - exp(8)*(3*x + 1)),x)

[Out]

x*exp(3*x)*exp(-x*exp(3^(x*exp(-8))))*exp(-9)*exp(3*exp(3^(x*exp(-8))))

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