3.29.92 \(\int \frac {e^{-e^x} (e^{e^x} (-124+56 x-7 x^2)+e^2 (-16+8 x-x^2)+e^{2+x} (16 x-8 x^2+x^3))}{16-8 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ -7 x-e^{2-e^x} x+\frac {3 x}{-4+x}+\log (2) \]

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Rubi [F]  time = 0.36, 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 (e^{e^x} \left (-124+56 x-7 x^2\right )+e^2 \left (-16+8 x-x^2\right )+e^{2+x} \left (16 x-8 x^2+x^3\right )\right )}{16-8 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^E^x*(-124 + 56*x - 7*x^2) + E^2*(-16 + 8*x - x^2) + E^(2 + x)*(16*x - 8*x^2 + x^3))/(E^E^x*(16 - 8*x +
x^2)),x]

[Out]

-12/(4 - x) - 7*x - E^2*ExpIntegralEi[-E^x] + Defer[Int][E^(2 - E^x + x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^x} \left (e^{e^x} \left (-124+56 x-7 x^2\right )+e^2 \left (-16+8 x-x^2\right )+e^{2+x} \left (16 x-8 x^2+x^3\right )\right )}{(-4+x)^2} \, dx\\ &=\int \left (-e^{2-e^x}+e^{2-e^x+x} x+\frac {-124+56 x-7 x^2}{(-4+x)^2}\right ) \, dx\\ &=-\int e^{2-e^x} \, dx+\int e^{2-e^x+x} x \, dx+\int \frac {-124+56 x-7 x^2}{(-4+x)^2} \, dx\\ &=\int \left (-7-\frac {12}{(-4+x)^2}\right ) \, dx+\int e^{2-e^x+x} x \, dx-\operatorname {Subst}\left (\int \frac {e^{2-x}}{x} \, dx,x,e^x\right )\\ &=-\frac {12}{4-x}-7 x-e^2 \text {Ei}\left (-e^x\right )+\int e^{2-e^x+x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 0.96 \begin {gather*} \frac {12}{-4+x}-7 (-4+x)-e^{2-e^x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^x*(-124 + 56*x - 7*x^2) + E^2*(-16 + 8*x - x^2) + E^(2 + x)*(16*x - 8*x^2 + x^3))/(E^E^x*(16 -
8*x + x^2)),x]

[Out]

12/(-4 + x) - 7*(-4 + x) - E^(2 - E^x)*x

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fricas [A]  time = 0.55, size = 37, normalized size = 1.42 \begin {gather*} -\frac {{\left ({\left (x^{2} - 4 \, x\right )} e^{2} + {\left (7 \, x^{2} - 28 \, x - 12\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )}}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+56*x-124)*exp(exp(x))+(x^3-8*x^2+16*x)*exp(2)*exp(x)+(-x^2+8*x-16)*exp(2))/(x^2-8*x+16)/exp
(exp(x)),x, algorithm="fricas")

[Out]

-((x^2 - 4*x)*e^2 + (7*x^2 - 28*x - 12)*e^(e^x))*e^(-e^x)/(x - 4)

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giac [B]  time = 0.26, size = 53, normalized size = 2.04 \begin {gather*} -\frac {x^{2} e^{\left (x - e^{x} + 2\right )} + 7 \, x^{2} e^{x} - 4 \, x e^{\left (x - e^{x} + 2\right )} - 28 \, x e^{x} - 12 \, e^{x}}{x e^{x} - 4 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+56*x-124)*exp(exp(x))+(x^3-8*x^2+16*x)*exp(2)*exp(x)+(-x^2+8*x-16)*exp(2))/(x^2-8*x+16)/exp
(exp(x)),x, algorithm="giac")

[Out]

-(x^2*e^(x - e^x + 2) + 7*x^2*e^x - 4*x*e^(x - e^x + 2) - 28*x*e^x - 12*e^x)/(x*e^x - 4*e^x)

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maple [A]  time = 0.54, size = 22, normalized size = 0.85

method	result	size
risch	\(-7 x +\frac {12}{x -4}-x \,{\mathrm e}^{-{\mathrm e}^{x}+2}\)	\(22\)
norman	\(\frac {\left (124 \,{\mathrm e}^{{\mathrm e}^{x}}-x^{2} {\mathrm e}^{2}+4 \,{\mathrm e}^{2} x -7 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}\right ) {\mathrm e}^{-{\mathrm e}^{x}}}{x -4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7*x^2+56*x-124)*exp(exp(x))+(x^3-8*x^2+16*x)*exp(2)*exp(x)+(-x^2+8*x-16)*exp(2))/(x^2-8*x+16)/exp(exp(x
)),x,method=_RETURNVERBOSE)

[Out]

-7*x+12/(x-4)-x*exp(-exp(x)+2)

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maxima [A]  time = 0.57, size = 35, normalized size = 1.35 \begin {gather*} -\frac {7 \, x^{2} + {\left (x^{2} e^{2} - 4 \, x e^{2}\right )} e^{\left (-e^{x}\right )} - 28 \, x - 12}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+56*x-124)*exp(exp(x))+(x^3-8*x^2+16*x)*exp(2)*exp(x)+(-x^2+8*x-16)*exp(2))/(x^2-8*x+16)/exp
(exp(x)),x, algorithm="maxima")

[Out]

-(7*x^2 + (x^2*e^2 - 4*x*e^2)*e^(-e^x) - 28*x - 12)/(x - 4)

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mupad [B]  time = 1.83, size = 45, normalized size = 1.73 \begin {gather*} \frac {31\,x-7\,x^2}{x-4}-\frac {4\,x\,{\mathrm {e}}^2-x^2\,{\mathrm {e}}^2}{4\,{\mathrm {e}}^{{\mathrm {e}}^x}-x\,{\mathrm {e}}^{{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(x))*(exp(exp(x))*(7*x^2 - 56*x + 124) + exp(2)*(x^2 - 8*x + 16) - exp(2)*exp(x)*(16*x - 8*x^2 +
x^3)))/(x^2 - 8*x + 16),x)

[Out]

(31*x - 7*x^2)/(x - 4) - (4*x*exp(2) - x^2*exp(2))/(4*exp(exp(x)) - x*exp(exp(x)))

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sympy [A]  time = 0.28, size = 17, normalized size = 0.65 \begin {gather*} - 7 x - x e^{2} e^{- e^{x}} + \frac {12}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x**2+56*x-124)*exp(exp(x))+(x**3-8*x**2+16*x)*exp(2)*exp(x)+(-x**2+8*x-16)*exp(2))/(x**2-8*x+16
)/exp(exp(x)),x)

[Out]

-7*x - x*exp(2)*exp(-exp(x)) + 12/(x - 4)

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