3.23.47 \(\int \frac {-9+6 x-x^2+e^{e^x} (-3+e^x (-3 x+x^2))}{9-6 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ 16-x+\frac {e^{e^x} x}{-3+x} \]

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Rubi [F]  time = 0.35, 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 {-9+6 x-x^2+e^{e^x} \left (-3+e^x \left (-3 x+x^2\right )\right )}{9-6 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9 + 6*x - x^2 + E^E^x*(-3 + E^x*(-3*x + x^2)))/(9 - 6*x + x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9+6 x-x^2+e^{e^x} \left (-3+e^x \left (-3 x+x^2\right )\right )}{(-3+x)^2} \, dx\\ &=\int \left (-\frac {9}{(-3+x)^2}-\frac {3 e^{e^x}}{(-3+x)^2}+\frac {6 x}{(-3+x)^2}+\frac {e^{e^x+x} x}{-3+x}-\frac {x^2}{(-3+x)^2}\right ) \, dx\\ &=-\frac {9}{3-x}-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+6 \int \frac {x}{(-3+x)^2} \, dx+\int \frac {e^{e^x+x} x}{-3+x} \, dx-\int \frac {x^2}{(-3+x)^2} \, dx\\ &=-\frac {9}{3-x}-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+6 \int \left (\frac {3}{(-3+x)^2}+\frac {1}{-3+x}\right ) \, dx-\int \left (1+\frac {9}{(-3+x)^2}+\frac {6}{-3+x}\right ) \, dx+\int \left (e^{e^x+x}+\frac {3 e^{e^x+x}}{-3+x}\right ) \, dx\\ &=-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx+\int e^{e^x+x} \, dx\\ &=-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x \left (-3-e^{e^x}+x\right )}{-3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + 6*x - x^2 + E^E^x*(-3 + E^x*(-3*x + x^2)))/(9 - 6*x + x^2),x]

[Out]

-((x*(-3 - E^E^x + x))/(-3 + x))

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fricas [A]  time = 0.84, size = 20, normalized size = 1.18 \begin {gather*} -\frac {x^{2} - x e^{\left (e^{x}\right )} - 3 \, x}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-3*x)*exp(x)-3)*exp(exp(x))-x^2+6*x-9)/(x^2-6*x+9),x, algorithm="fricas")

[Out]

-(x^2 - x*e^(e^x) - 3*x)/(x - 3)

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giac [B]  time = 0.55, size = 33, normalized size = 1.94 \begin {gather*} -\frac {x^{2} e^{x} - x e^{\left (x + e^{x}\right )} - 3 \, x e^{x}}{x e^{x} - 3 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-3*x)*exp(x)-3)*exp(exp(x))-x^2+6*x-9)/(x^2-6*x+9),x, algorithm="giac")

[Out]

-(x^2*e^x - x*e^(x + e^x) - 3*x*e^x)/(x*e^x - 3*e^x)

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maple [A]  time = 0.46, size = 15, normalized size = 0.88

method	result	size
risch	\(-x +\frac {{\mathrm e}^{{\mathrm e}^{x}} x}{x -3}\)	\(15\)
norman	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{x}}-x^{2}+9}{x -3}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-3*x)*exp(x)-3)*exp(exp(x))-x^2+6*x-9)/(x^2-6*x+9),x,method=_RETURNVERBOSE)

[Out]

-x+exp(exp(x))/(x-3)*x

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maxima [A]  time = 0.63, size = 14, normalized size = 0.82 \begin {gather*} -x + \frac {x e^{\left (e^{x}\right )}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-3*x)*exp(x)-3)*exp(exp(x))-x^2+6*x-9)/(x^2-6*x+9),x, algorithm="maxima")

[Out]

-x + x*e^(e^x)/(x - 3)

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mupad [B]  time = 0.13, size = 15, normalized size = 0.88 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^{{\mathrm {e}}^x}-x+3\right )}{x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(exp(x)*(3*x - x^2) + 3) - 6*x + x^2 + 9)/(x^2 - 6*x + 9),x)

[Out]

(x*(exp(exp(x)) - x + 3))/(x - 3)

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sympy [A]  time = 0.14, size = 10, normalized size = 0.59 \begin {gather*} - x + \frac {x e^{e^{x}}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-3*x)*exp(x)-3)*exp(exp(x))-x**2+6*x-9)/(x**2-6*x+9),x)

[Out]

-x + x*exp(exp(x))/(x - 3)

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