Integrand size = 107, antiderivative size = 30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=2+3 \left (e^{e^{e^{2-x}}}+\left (3+\frac {2}{-3+x+x^2}\right )^2\right ) \] Output:
2+3*(2/(x^2+x-3)+3)^2+3*exp(exp(exp(2)/exp(x)))
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=3 e^{e^{e^{2-x}}}+\frac {12}{\left (-3+x+x^2\right )^2}+\frac {36}{-3+x+x^2} \] Input:
Integrate[(E^x*(84 + 132*x - 108*x^2 - 72*x^3) + E^(2 + E^E^(2 - x) + E^(2 - x))*(81 - 81*x - 54*x^2 + 51*x^3 + 18*x^4 - 9*x^5 - 3*x^6))/(E^x*(-27 + 27*x + 18*x^2 - 17*x^3 - 6*x^4 + 3*x^5 + x^6)),x]
Output:
3*E^E^E^(2 - x) + 12/(-3 + x + x^2)^2 + 36/(-3 + x + x^2)
Time = 10.84 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2463, 7239, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{x^6+3 x^5-6 x^4-17 x^3+18 x^2+27 x-27} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {12 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{169 \sqrt {13} \left (-2 x+\sqrt {13}-1\right )}-\frac {12 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{169 \sqrt {13} \left (2 x+\sqrt {13}+1\right )}-\frac {12 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{169 \left (-2 x+\sqrt {13}-1\right )^2}-\frac {12 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{169 \left (2 x+\sqrt {13}+1\right )^2}-\frac {8 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{13 \sqrt {13} \left (-2 x+\sqrt {13}-1\right )^3}-\frac {8 e^{-x} \left (e^x \left (-72 x^3-108 x^2+132 x+84\right )+e^{e^{e^{2-x}}+e^{2-x}+2} \left (-3 x^6-9 x^5+18 x^4+51 x^3-54 x^2-81 x+81\right )\right )}{13 \sqrt {13} \left (2 x+\sqrt {13}+1\right )^3}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {12 \left (6 x^3+9 x^2-11 x-7\right )}{\left (x^2+x-3\right )^3}-3 e^{-x+e^{e^{2-x}}+e^{2-x}+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {36}{-x^2-x+3}+\frac {12}{\left (-x^2-x+3\right )^2}+3 e^{e^{e^{2-x}}}\) |
Input:
Int[(E^x*(84 + 132*x - 108*x^2 - 72*x^3) + E^(2 + E^E^(2 - x) + E^(2 - x)) *(81 - 81*x - 54*x^2 + 51*x^3 + 18*x^4 - 9*x^5 - 3*x^6))/(E^x*(-27 + 27*x + 18*x^2 - 17*x^3 - 6*x^4 + 3*x^5 + x^6)),x]
Output:
3*E^E^E^(2 - x) + 12/(3 - x - x^2)^2 - 36/(3 - x - x^2)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 35.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
parts | \(-\frac {12 \left (-3 x^{2}-3 x +8\right )}{\left (x^{2}+x -3\right )^{2}}+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}\) | \(33\) |
risch | \(\frac {36 x^{2}+36 x -96}{x^{4}+2 x^{3}-5 x^{2}-6 x +9}+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2-x}}}\) | \(43\) |
parallelrisch | \(\frac {-96+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x^{4}+6 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x^{3}-15 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x^{2}+36 x^{2}-18 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x +36 x +27 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}}{x^{4}+2 x^{3}-5 x^{2}-6 x +9}\) | \(97\) |
Input:
int(((-3*x^6-9*x^5+18*x^4+51*x^3-54*x^2-81*x+81)*exp(2)*exp(exp(2)/exp(x)) *exp(exp(exp(2)/exp(x)))+(-72*x^3-108*x^2+132*x+84)*exp(x))/(x^6+3*x^5-6*x ^4-17*x^3+18*x^2+27*x-27)/exp(x),x,method=_RETURNVERBOSE)
Output:
-12*(-3*x^2-3*x+8)/(x^2+x-3)^2+3*exp(exp(exp(2)/exp(x)))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (28) = 56\).
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=\frac {3 \, {\left ({\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )} e^{\left ({\left (e^{2} + e^{\left (x + e^{\left (-x + 2\right )}\right )} + 2 \, e^{x}\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (3 \, x^{2} + 3 \, x - 8\right )} e^{\left (e^{\left (-x + 2\right )} + 2\right )}\right )} e^{\left (-e^{\left (-x + 2\right )} - 2\right )}}{x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9} \] Input:
integrate(((-3*x^6-9*x^5+18*x^4+51*x^3-54*x^2-81*x+81)*exp(2)*exp(exp(2)/e xp(x))*exp(exp(exp(2)/exp(x)))+(-72*x^3-108*x^2+132*x+84)*exp(x))/(x^6+3*x ^5-6*x^4-17*x^3+18*x^2+27*x-27)/exp(x),x, algorithm="fricas")
Output:
3*((x^4 + 2*x^3 - 5*x^2 - 6*x + 9)*e^((e^2 + e^(x + e^(-x + 2)) + 2*e^x)*e ^(-x)) + 4*(3*x^2 + 3*x - 8)*e^(e^(-x + 2) + 2))*e^(-e^(-x + 2) - 2)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=- \frac {- 36 x^{2} - 36 x + 96}{x^{4} + 2 x^{3} - 5 x^{2} - 6 x + 9} + 3 e^{e^{e^{2} e^{- x}}} \] Input:
integrate(((-3*x**6-9*x**5+18*x**4+51*x**3-54*x**2-81*x+81)*exp(2)*exp(exp (2)/exp(x))*exp(exp(exp(2)/exp(x)))+(-72*x**3-108*x**2+132*x+84)*exp(x))/( x**6+3*x**5-6*x**4-17*x**3+18*x**2+27*x-27)/exp(x),x)
Output:
-(-36*x**2 - 36*x + 96)/(x**4 + 2*x**3 - 5*x**2 - 6*x + 9) + 3*exp(exp(exp (2)*exp(-x)))
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (28) = 56\).
Time = 0.36 (sec) , antiderivative size = 159, normalized size of antiderivative = 5.30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=-\frac {36 \, {\left (18 \, x^{3} - 142 \, x^{2} - 84 \, x + 207\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + \frac {42 \, {\left (12 \, x^{3} + 18 \, x^{2} - 56 \, x - 31\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + \frac {54 \, {\left (10 \, x^{3} + 15 \, x^{2} + 66 \, x - 54\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} - \frac {66 \, {\left (6 \, x^{3} + 9 \, x^{2} - 28 \, x + 69\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + 3 \, e^{\left (e^{\left (e^{\left (-x + 2\right )}\right )}\right )} \] Input:
integrate(((-3*x^6-9*x^5+18*x^4+51*x^3-54*x^2-81*x+81)*exp(2)*exp(exp(2)/e xp(x))*exp(exp(exp(2)/exp(x)))+(-72*x^3-108*x^2+132*x+84)*exp(x))/(x^6+3*x ^5-6*x^4-17*x^3+18*x^2+27*x-27)/exp(x),x, algorithm="maxima")
Output:
-36/169*(18*x^3 - 142*x^2 - 84*x + 207)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9) + 42/169*(12*x^3 + 18*x^2 - 56*x - 31)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9) + 54/ 169*(10*x^3 + 15*x^2 + 66*x - 54)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9) - 66/169 *(6*x^3 + 9*x^2 - 28*x + 69)/(x^4 + 2*x^3 - 5*x^2 - 6*x + 9) + 3*e^(e^(e^( -x + 2)))
\[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=\int { -\frac {3 \, {\left (4 \, {\left (6 \, x^{3} + 9 \, x^{2} - 11 \, x - 7\right )} e^{x} + {\left (x^{6} + 3 \, x^{5} - 6 \, x^{4} - 17 \, x^{3} + 18 \, x^{2} + 27 \, x - 27\right )} e^{\left (e^{\left (-x + 2\right )} + e^{\left (e^{\left (-x + 2\right )}\right )} + 2\right )}\right )} e^{\left (-x\right )}}{x^{6} + 3 \, x^{5} - 6 \, x^{4} - 17 \, x^{3} + 18 \, x^{2} + 27 \, x - 27} \,d x } \] Input:
integrate(((-3*x^6-9*x^5+18*x^4+51*x^3-54*x^2-81*x+81)*exp(2)*exp(exp(2)/e xp(x))*exp(exp(exp(2)/exp(x)))+(-72*x^3-108*x^2+132*x+84)*exp(x))/(x^6+3*x ^5-6*x^4-17*x^3+18*x^2+27*x-27)/exp(x),x, algorithm="giac")
Output:
integrate(-3*(4*(6*x^3 + 9*x^2 - 11*x - 7)*e^x + (x^6 + 3*x^5 - 6*x^4 - 17 *x^3 + 18*x^2 + 27*x - 27)*e^(e^(-x + 2) + e^(e^(-x + 2)) + 2))*e^(-x)/(x^ 6 + 3*x^5 - 6*x^4 - 17*x^3 + 18*x^2 + 27*x - 27), x)
Time = 3.74 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=3\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^2}}+\frac {36\,x^2+36\,x-96}{{\left (x^2+x-3\right )}^2} \] Input:
int((exp(-x)*(exp(x)*(132*x - 108*x^2 - 72*x^3 + 84) - exp(2)*exp(exp(-x)* exp(2))*exp(exp(exp(-x)*exp(2)))*(81*x + 54*x^2 - 51*x^3 - 18*x^4 + 9*x^5 + 3*x^6 - 81)))/(27*x + 18*x^2 - 17*x^3 - 6*x^4 + 3*x^5 + x^6 - 27),x)
Output:
3*exp(exp(exp(-x)*exp(2))) + (36*x + 36*x^2 - 96)/(x + x^2 - 3)^2
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.87 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=\frac {3 e^{e^{\frac {e^{2}}{e^{x}}}} x^{4}+6 e^{e^{\frac {e^{2}}{e^{x}}}} x^{3}-15 e^{e^{\frac {e^{2}}{e^{x}}}} x^{2}-18 e^{e^{\frac {e^{2}}{e^{x}}}} x +27 e^{e^{\frac {e^{2}}{e^{x}}}}+36 x^{2}+36 x -96}{x^{4}+2 x^{3}-5 x^{2}-6 x +9} \] Input:
int(((-3*x^6-9*x^5+18*x^4+51*x^3-54*x^2-81*x+81)*exp(2)*exp(exp(2)/exp(x)) *exp(exp(exp(2)/exp(x)))+(-72*x^3-108*x^2+132*x+84)*exp(x))/(x^6+3*x^5-6*x ^4-17*x^3+18*x^2+27*x-27)/exp(x),x)
Output:
(3*(e**(e**(e**2/e**x))*x**4 + 2*e**(e**(e**2/e**x))*x**3 - 5*e**(e**(e**2 /e**x))*x**2 - 6*e**(e**(e**2/e**x))*x + 9*e**(e**(e**2/e**x)) + 12*x**2 + 12*x - 32))/(x**4 + 2*x**3 - 5*x**2 - 6*x + 9)