\(\int \frac {e^{3-x} (81 x^2+63 x^3-17 x^4+x^5)+e^{-\frac {e^{-2+x}}{-9 x+x^2}} (e (-9+11 x-x^2)+e^{3-x} (81 x^2-18 x^3+x^4))}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} (81 x^2-18 x^3+x^4)+e^{3-x} (81 x^3-18 x^4+x^5)} \, dx\) [7760]

Optimal result

Integrand size = 146, antiderivative size = 23 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=x+\log \left (e^{-\frac {e^{-2+x}}{-9 x+x^2}}+x\right ) \]

[Out]

x+ln(exp(-exp(1)/(x^2-9*x)/exp(-x+3))+x)

Rubi [F]

\[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=\int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx \]

[In]

Int[(E^(3 - x)*(81*x^2 + 63*x^3 - 17*x^4 + x^5) + (E*(-9 + 11*x - x^2) + E^(3 - x)*(81*x^2 - 18*x^3 + x^4))/E^
(E^(-2 + x)/(-9*x + x^2)))/(E^(3 - x - E^(-2 + x)/(-9*x + x^2))*(81*x^2 - 18*x^3 + x^4) + E^(3 - x)*(81*x^3 -
18*x^4 + x^5)),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^2 (-9+x)^2 x^2+e^{2+\frac {e^{-2+x}}{(-9+x) x}} (-9+x)^2 x^2 (1+x)-e^x \left (9-11 x+x^2\right )}{e^2 (9-x)^2 x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx \\ & = \frac {\int \frac {e^2 (-9+x)^2 x^2+e^{2+\frac {e^{-2+x}}{(-9+x) x}} (-9+x)^2 x^2 (1+x)-e^x \left (9-11 x+x^2\right )}{(9-x)^2 x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{e^2} \\ & = \frac {\int \left (\frac {e^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}}+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}{1+e^{\frac {e^{-2+x}}{(-9+x) x}} x}-\frac {e^x \left (9-11 x+x^2\right )}{(-9+x)^2 x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}\right ) \, dx}{e^2} \\ & = -\frac {\int \frac {e^x \left (9-11 x+x^2\right )}{(-9+x)^2 x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{e^2}+\int \frac {1+e^{\frac {e^{-2+x}}{(-9+x) x}}+e^{\frac {e^{-2+x}}{(-9+x) x}} x}{1+e^{\frac {e^{-2+x}}{(-9+x) x}} x} \, dx \\ & = -\frac {\int \left (-\frac {e^x}{9 (-9+x)^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}+\frac {e^x}{9 (-9+x) \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}+\frac {e^x}{9 x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}-\frac {e^x}{9 x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}\right ) \, dx}{e^2}+\int \frac {1+e^{-\frac {e^{-2+x}}{(-9+x) x}}+x}{e^{-\frac {e^{-2+x}}{(-9+x) x}}+x} \, dx \\ & = \frac {\int \frac {e^x}{(-9+x)^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{(-9+x) \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\frac {\int \frac {e^x}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\int \left (\frac {1+x}{x}-\frac {1}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}\right ) \, dx \\ & = \frac {\int \frac {e^x}{(-9+x)^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{(-9+x) \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\frac {\int \frac {e^x}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\int \frac {1+x}{x} \, dx-\int \frac {1}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx \\ & = \frac {\int \frac {e^x}{(-9+x)^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{(-9+x) \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\frac {\int \frac {e^x}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\int \left (1+\frac {1}{x}\right ) \, dx-\int \frac {1}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx \\ & = x+\log (x)+\frac {\int \frac {e^x}{(-9+x)^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{(-9+x) \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\frac {\int \frac {e^x}{x^2 \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}+\frac {\int \frac {e^x}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx}{9 e^2}-\int \frac {1}{x \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(23)=46\).

Time = 0.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=\frac {e^2 x+\frac {e^x}{9 x-x^2}+e^2 \log \left (1+e^{\frac {e^{-2+x}}{(-9+x) x}} x\right )}{e^2} \]

[In]

Integrate[(E^(3 - x)*(81*x^2 + 63*x^3 - 17*x^4 + x^5) + (E*(-9 + 11*x - x^2) + E^(3 - x)*(81*x^2 - 18*x^3 + x^
4))/E^(E^(-2 + x)/(-9*x + x^2)))/(E^(3 - x - E^(-2 + x)/(-9*x + x^2))*(81*x^2 - 18*x^3 + x^4) + E^(3 - x)*(81*
x^3 - 18*x^4 + x^5)),x]

[Out]

(E^2*x + E^x/(9*x - x^2) + E^2*Log[1 + E^(E^(-2 + x)/((-9 + x)*x))*x])/E^2

Maple [A] (verified)

Time = 7.39 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

method	result	size
parallelrisch	\(36+\ln \left ({\mathrm e}^{-\frac {{\mathrm e} \,{\mathrm e}^{-3+x}}{x \left (x -9\right )}}+x \right )+x\)	\(28\)
risch	\(x -\frac {{\mathrm e}^{-2+x}}{x \left (x -9\right )}+\frac {{\mathrm e}^{-2+x}}{x^{2}-9 x}+\ln \left ({\mathrm e}^{-\frac {{\mathrm e}^{-2+x}}{x \left (x -9\right )}}+x \right )\)	\(49\)
norman	\(\frac {\left (x^{3} {\mathrm e}^{-x +3}-81 x \,{\mathrm e}^{-x +3}\right ) {\mathrm e}^{-3+x}}{x \left (x -9\right )}+\ln \left ({\mathrm e}^{-\frac {{\mathrm e} \,{\mathrm e}^{-3+x}}{x^{2}-9 x}}+x \right )\)	\(64\)

[In]

int((((x^4-18*x^3+81*x^2)*exp(-x+3)+(-x^2+11*x-9)*exp(1))*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-17*x^4+63*x^3+
81*x^2)*exp(-x+3))/((x^4-18*x^3+81*x^2)*exp(-x+3)*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-18*x^4+81*x^3)*exp(-x+
3)),x,method=_RETURNVERBOSE)

[Out]

36+ln(exp(-exp(1)/x/(x-9)/exp(-x+3))+x)+x

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=x + \log \left (x + e^{\left (-\frac {e^{\left (x - 2\right )}}{x^{2} - 9 \, x}\right )}\right ) \]

[In]

integrate((((x^4-18*x^3+81*x^2)*exp(-x+3)+(-x^2+11*x-9)*exp(1))*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-17*x^4+6
3*x^3+81*x^2)*exp(-x+3))/((x^4-18*x^3+81*x^2)*exp(-x+3)*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-18*x^4+81*x^3)*e
xp(-x+3)),x, algorithm="fricas")

[Out]

x + log(x + e^(-e^(x - 2)/(x^2 - 9*x)))

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=x + \log {\left (x + e^{- \frac {e e^{x - 3}}{x^{2} - 9 x}} \right )} \]

[In]

integrate((((x**4-18*x**3+81*x**2)*exp(-x+3)+(-x**2+11*x-9)*exp(1))*exp(-exp(1)/(x**2-9*x)/exp(-x+3))+(x**5-17
*x**4+63*x**3+81*x**2)*exp(-x+3))/((x**4-18*x**3+81*x**2)*exp(-x+3)*exp(-exp(1)/(x**2-9*x)/exp(-x+3))+(x**5-18
*x**4+81*x**3)*exp(-x+3)),x)

[Out]

x + log(x + exp(-E*exp(x - 3)/(x**2 - 9*x)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).

Time = 0.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=\frac {9 \, x^{2} e^{2} - 81 \, x e^{2} - e^{x}}{9 \, {\left (x e^{2} - 9 \, e^{2}\right )}} + \log \left (x\right ) + \log \left (\frac {x e^{\left (\frac {e^{x}}{9 \, {\left (x e^{2} - 9 \, e^{2}\right )}}\right )} + e^{\left (\frac {e^{\left (x - 2\right )}}{9 \, x}\right )}}{x}\right ) \]

[In]

integrate((((x^4-18*x^3+81*x^2)*exp(-x+3)+(-x^2+11*x-9)*exp(1))*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-17*x^4+6
3*x^3+81*x^2)*exp(-x+3))/((x^4-18*x^3+81*x^2)*exp(-x+3)*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-18*x^4+81*x^3)*e
xp(-x+3)),x, algorithm="maxima")

[Out]

1/9*(9*x^2*e^2 - 81*x*e^2 - e^x)/(x*e^2 - 9*e^2) + log(x) + log((x*e^(1/9*e^x/(x*e^2 - 9*e^2)) + e^(1/9*e^(x -
2)/x))/x)

Giac [F]

\[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=\int { \frac {{\left (x^{5} - 17 \, x^{4} + 63 \, x^{3} + 81 \, x^{2}\right )} e^{\left (-x + 3\right )} - {\left ({\left (x^{2} - 11 \, x + 9\right )} e - {\left (x^{4} - 18 \, x^{3} + 81 \, x^{2}\right )} e^{\left (-x + 3\right )}\right )} e^{\left (-\frac {e^{\left (x - 2\right )}}{x^{2} - 9 \, x}\right )}}{{\left (x^{4} - 18 \, x^{3} + 81 \, x^{2}\right )} e^{\left (-x - \frac {e^{\left (x - 2\right )}}{x^{2} - 9 \, x} + 3\right )} + {\left (x^{5} - 18 \, x^{4} + 81 \, x^{3}\right )} e^{\left (-x + 3\right )}} \,d x } \]

[In]

integrate((((x^4-18*x^3+81*x^2)*exp(-x+3)+(-x^2+11*x-9)*exp(1))*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-17*x^4+6
3*x^3+81*x^2)*exp(-x+3))/((x^4-18*x^3+81*x^2)*exp(-x+3)*exp(-exp(1)/(x^2-9*x)/exp(-x+3))+(x^5-18*x^4+81*x^3)*e
xp(-x+3)),x, algorithm="giac")

[Out]

integrate(((x^5 - 17*x^4 + 63*x^3 + 81*x^2)*e^(-x + 3) - ((x^2 - 11*x + 9)*e - (x^4 - 18*x^3 + 81*x^2)*e^(-x +
3))*e^(-e^(x - 2)/(x^2 - 9*x)))/((x^4 - 18*x^3 + 81*x^2)*e^(-x - e^(x - 2)/(x^2 - 9*x) + 3) + (x^5 - 18*x^4 +
81*x^3)*e^(-x + 3)), x)

Mupad [B] (verification not implemented)

Time = 13.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{3-x} \left (81 x^2+63 x^3-17 x^4+x^5\right )+e^{-\frac {e^{-2+x}}{-9 x+x^2}} \left (e \left (-9+11 x-x^2\right )+e^{3-x} \left (81 x^2-18 x^3+x^4\right )\right )}{e^{3-x-\frac {e^{-2+x}}{-9 x+x^2}} \left (81 x^2-18 x^3+x^4\right )+e^{3-x} \left (81 x^3-18 x^4+x^5\right )} \, dx=x+\ln \left (x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x-2}}{x\,\left (x-9\right )}}\right ) \]

[In]

int((exp((exp(x - 3)*exp(1))/(9*x - x^2))*(exp(3 - x)*(81*x^2 - 18*x^3 + x^4) - exp(1)*(x^2 - 11*x + 9)) + exp
(3 - x)*(81*x^2 + 63*x^3 - 17*x^4 + x^5))/(exp(3 - x)*(81*x^3 - 18*x^4 + x^5) + exp((exp(x - 3)*exp(1))/(9*x -
x^2))*exp(3 - x)*(81*x^2 - 18*x^3 + x^4)),x)

[Out]

x + log(x + exp(-exp(x - 2)/(x*(x - 9))))