3.45.22 \(\int \frac {2+e^{2+e^{e^{e^x}-x} (-4+x)-x} (-16+8 x-x^2+e^{e^{e^x}-x} (80-56 x+13 x^2-x^3+e^{e^x+x} (-64+48 x-12 x^2+x^3)))}{16-8 x+x^2} \, dx\) [4422]

Optimal. Leaf size=34 \[ e^{2+e^{e^{e^x}-x} (-4+x)-x}+\frac {2-x}{-4+x} \]

[Out]

(2-x)/(x-4)+exp((x-4)*exp(exp(exp(x))-x)+2-x)

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Rubi [F]
time = 2.77, 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 {2+e^{2+e^{e^{e^x}-x} (-4+x)-x} \left (-16+8 x-x^2+e^{e^{e^x}-x} \left (80-56 x+13 x^2-x^3+e^{e^x+x} \left (-64+48 x-12 x^2+x^3\right )\right )\right )}{16-8 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + E^(2 + E^(E^E^x - x)*(-4 + x) - x)*(-16 + 8*x - x^2 + E^(E^E^x - x)*(80 - 56*x + 13*x^2 - x^3 + E^(E^
x + x)*(-64 + 48*x - 12*x^2 + x^3))))/(16 - 8*x + x^2),x]

[Out]

2/(4 - x) + 5*Defer[Int][E^(2 + E^E^x + E^(E^E^x - x)*(-4 + x) - 2*x), x] - Defer[Int][E^(2 + E^(E^E^x - x)*(-
4 + x) - x), x] - 4*Defer[Int][E^(2 + E^E^x + E^x + E^(E^E^x - x)*(-4 + x) - x), x] - Defer[Int][E^(2 + E^E^x
+ E^(E^E^x - x)*(-4 + x) - 2*x)*x, x] + Defer[Int][E^(2 + E^E^x + E^x + E^(E^E^x - x)*(-4 + x) - x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+e^{2+e^{e^{e^x}-x} (-4+x)-x} \left (-16+8 x-x^2+e^{e^{e^x}-x} \left (80-56 x+13 x^2-x^3+e^{e^x+x} \left (-64+48 x-12 x^2+x^3\right )\right )\right )}{(-4+x)^2} \, dx\\ &=\int \left (e^{2+e^{e^{e^x}-x} (-4+x)-2 x} \left (-e^x-e^{e^{e^x}} (-5+x)+e^{e^{e^x}+e^x+x} (-4+x)\right )+\frac {2}{(-4+x)^2}\right ) \, dx\\ &=\frac {2}{4-x}+\int e^{2+e^{e^{e^x}-x} (-4+x)-2 x} \left (-e^x-e^{e^{e^x}} (-5+x)+e^{e^{e^x}+e^x+x} (-4+x)\right ) \, dx\\ &=\frac {2}{4-x}+\int \left (-e^{2+e^{e^{e^x}-x} (-4+x)-x}-e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x} (-5+x)+e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x} (-4+x)\right ) \, dx\\ &=\frac {2}{4-x}-\int e^{2+e^{e^{e^x}-x} (-4+x)-x} \, dx-\int e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x} (-5+x) \, dx+\int e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x} (-4+x) \, dx\\ &=\frac {2}{4-x}-\int e^{2+e^{e^{e^x}-x} (-4+x)-x} \, dx-\int \left (-5 e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x}+e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x} x\right ) \, dx+\int \left (-4 e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x}+e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x} x\right ) \, dx\\ &=\frac {2}{4-x}-4 \int e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x} \, dx+5 \int e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x} \, dx-\int e^{2+e^{e^{e^x}-x} (-4+x)-x} \, dx-\int e^{2+e^{e^x}+e^{e^{e^x}-x} (-4+x)-2 x} x \, dx+\int e^{2+e^{e^x}+e^x+e^{e^{e^x}-x} (-4+x)-x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.36, size = 30, normalized size = 0.88 \begin {gather*} e^{2+e^{e^{e^x}-x} (-4+x)-x}-\frac {2}{-4+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^(2 + E^(E^E^x - x)*(-4 + x) - x)*(-16 + 8*x - x^2 + E^(E^E^x - x)*(80 - 56*x + 13*x^2 - x^3 +
E^(E^x + x)*(-64 + 48*x - 12*x^2 + x^3))))/(16 - 8*x + x^2),x]

[Out]

E^(2 + E^(E^E^x - x)*(-4 + x) - x) - 2/(-4 + x)

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Maple [A]
time = 1.15, size = 35, normalized size = 1.03

method	result	size
risch	\(-\frac {2}{x -4}+{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}-x} x -4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}-x}-x +2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x^3-12*x^2+48*x-64)*exp(x)*exp(exp(x))-x^3+13*x^2-56*x+80)*exp(exp(exp(x))-x)-x^2+8*x-16)*exp((x-4)*ex
p(exp(exp(x))-x)+2-x)+2)/(x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

-2/(x-4)+exp(exp(exp(exp(x))-x)*x-4*exp(exp(exp(x))-x)-x+2)

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Maxima [A]
time = 0.49, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2}{x - 4} + e^{\left (x e^{\left (-x + e^{\left (e^{x}\right )}\right )} - x - 4 \, e^{\left (-x + e^{\left (e^{x}\right )}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^3-12*x^2+48*x-64)*exp(x)*exp(exp(x))-x^3+13*x^2-56*x+80)*exp(exp(exp(x))-x)-x^2+8*x-16)*exp((x
-4)*exp(exp(exp(x))-x)+2-x)+2)/(x^2-8*x+16),x, algorithm="maxima")

[Out]

-2/(x - 4) + e^(x*e^(-x + e^(e^x)) - x - 4*e^(-x + e^(e^x)) + 2)

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Fricas [A]
time = 0.38, size = 41, normalized size = 1.21 \begin {gather*} \frac {{\left (x - 4\right )} e^{\left ({\left (x - 4\right )} e^{\left (-{\left (x e^{x} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - x + 2\right )} - 2}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^3-12*x^2+48*x-64)*exp(x)*exp(exp(x))-x^3+13*x^2-56*x+80)*exp(exp(exp(x))-x)-x^2+8*x-16)*exp((x
-4)*exp(exp(exp(x))-x)+2-x)+2)/(x^2-8*x+16),x, algorithm="fricas")

[Out]

((x - 4)*e^((x - 4)*e^(-(x*e^x - e^(x + e^x))*e^(-x)) - x + 2) - 2)/(x - 4)

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Sympy [A]
time = 1.07, size = 20, normalized size = 0.59 \begin {gather*} e^{- x + \left (x - 4\right ) e^{- x + e^{e^{x}}} + 2} - \frac {2}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x**3-12*x**2+48*x-64)*exp(x)*exp(exp(x))-x**3+13*x**2-56*x+80)*exp(exp(exp(x))-x)-x**2+8*x-16)*e
xp((x-4)*exp(exp(exp(x))-x)+2-x)+2)/(x**2-8*x+16),x)

[Out]

exp(-x + (x - 4)*exp(-x + exp(exp(x))) + 2) - 2/(x - 4)

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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^3-12*x^2+48*x-64)*exp(x)*exp(exp(x))-x^3+13*x^2-56*x+80)*exp(exp(exp(x))-x)-x^2+8*x-16)*exp((x
-4)*exp(exp(exp(x))-x)+2-x)+2)/(x^2-8*x+16),x, algorithm="giac")

[Out]

integrate(-((x^2 + (x^3 - 13*x^2 - (x^3 - 12*x^2 + 48*x - 64)*e^(x + e^x) + 56*x - 80)*e^(-x + e^(e^x)) - 8*x
+ 16)*e^((x - 4)*e^(-x + e^(e^x)) - x + 2) - 2)/(x^2 - 8*x + 16), x)

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Mupad [B]
time = 3.20, size = 37, normalized size = 1.09 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}-\frac {2}{x-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(exp(exp(x)) - x)*(x - 4) - x + 2)*(8*x + exp(exp(exp(x)) - x)*(13*x^2 - 56*x - x^3 + exp(exp(x))*
exp(x)*(48*x - 12*x^2 + x^3 - 64) + 80) - x^2 - 16) + 2)/(x^2 - 8*x + 16),x)

[Out]

exp(x*exp(-x)*exp(exp(exp(x))))*exp(-x)*exp(2)*exp(-4*exp(-x)*exp(exp(exp(x)))) - 2/(x - 4)

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