Integrand size = 152, antiderivative size = 34 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=5-\frac {5}{2+x}+\frac {x}{1-e^{-\frac {e^{6-x}}{3 x}} x} \]
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\frac {1}{3} \left (3 x+\frac {3 x^2}{e^{\frac {e^{6-x}}{3 x}}-x}-\frac {15}{2+x}\right ) \]
Integrate[(15*x^2 + E^((2*E^(6 - x))/(3*x))*(27 + 12*x + 3*x^2) + E^(E^(6 - x)/(3*x))*(-30*x + E^(6 - x)*(4 + 8*x + 5*x^2 + x^3)))/(12*x^2 + 12*x^3 + 3*x^4 + E^((2*E^(6 - x))/(3*x))*(12 + 12*x + 3*x^2) + E^(E^(6 - x)/(3*x) )*(-24*x - 24*x^2 - 6*x^3)),x]
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.
| \(\displaystyle \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+27\right )+e^{\frac {e^{6-x}}{3 x}} \left (e^{6-x} \left (x^3+5 x^2+8 x+4\right )-30 x\right )}{3 x^4+12 x^3+12 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+12\right )+e^{\frac {e^{6-x}}{3 x}} \left (-6 x^3-24 x^2-24 x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+27\right )+e^{\frac {e^{6-x}}{3 x}} \left (e^{6-x} \left (x^3+5 x^2+8 x+4\right )-30 x\right )}{3 \left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {15 x^2+3 e^{\frac {2 e^{6-x}}{3 x}} \left (x^2+4 x+9\right )-e^{\frac {e^{6-x}}{3 x}} \left (30 x-e^{6-x} \left (x^3+5 x^2+8 x+4\right )\right )}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {15 x^2}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}-\frac {30 e^{\frac {e^{6-x}}{3 x}} x}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}+\frac {e^{-x+6+\frac {e^{6-x}}{3 x}} (x+1)}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}+\frac {3 e^{\frac {2 e^{6-x}}{3 x}} \left (x^2+4 x+9\right )}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (15 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+\int \frac {e^{-x+6+\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+3 \int \frac {e^{\frac {2 e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+\int \frac {e^{-x+6+\frac {e^{6-x}}{3 x}} x}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+60 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx+60 \int \frac {e^{\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx+15 \int \frac {e^{\frac {2 e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx-60 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)}dx-30 \int \frac {e^{\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)}dx\right )\) |
Int[(15*x^2 + E^((2*E^(6 - x))/(3*x))*(27 + 12*x + 3*x^2) + E^(E^(6 - x)/( 3*x))*(-30*x + E^(6 - x)*(4 + 8*x + 5*x^2 + x^3)))/(12*x^2 + 12*x^3 + 3*x^ 4 + E^((2*E^(6 - x))/(3*x))*(12 + 12*x + 3*x^2) + E^(E^(6 - x)/(3*x))*(-24 *x - 24*x^2 - 6*x^3)),x]
3.16.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.97 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(x -\frac {5}{2+x}-\frac {x^{2}}{x -{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}\) | \(33\) |
| parallelrisch | \(-\frac {3 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x^{2}+6 x^{2}+27 x -27 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}{3 \left (x^{2}-{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x +2 x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}\right )}\) | \(81\) |
| norman | \(\frac {-5 x +5 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x -{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x^{2}}{x^{2}-{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x +2 x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}\) | \(90\) |
int(((3*x^2+12*x+27)*exp(1/3*exp(-x+6)/x)^2+((x^3+5*x^2+8*x+4)*exp(-x+6)-3 0*x)*exp(1/3*exp(-x+6)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(-x+6)/x)^2+ (-6*x^3-24*x^2-24*x)*exp(1/3*exp(-x+6)/x)+3*x^4+12*x^3+12*x^2),x,method=_R ETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {{\left (x^{2} + 2 \, x - 5\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 5 \, x}{x^{2} - {\left (x + 2\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 2 \, x} \]
integrate(((3*x^2+12*x+27)*exp(1/3*exp(-x+6)/x)^2+((x^3+5*x^2+8*x+4)*exp(- x+6)-30*x)*exp(1/3*exp(-x+6)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(-x+6) /x)^2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(-x+6)/x)+3*x^4+12*x^3+12*x^2),x, al gorithm=\
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\frac {x^{2}}{- x + e^{\frac {e^{6 - x}}{3 x}}} + x - \frac {5}{x + 2} \]
integrate(((3*x**2+12*x+27)*exp(1/3*exp(-x+6)/x)**2+((x**3+5*x**2+8*x+4)*e xp(-x+6)-30*x)*exp(1/3*exp(-x+6)/x)+15*x**2)/((3*x**2+12*x+12)*exp(1/3*exp (-x+6)/x)**2+(-6*x**3-24*x**2-24*x)*exp(1/3*exp(-x+6)/x)+3*x**4+12*x**3+12 *x**2),x)
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {{\left (x^{2} + 2 \, x - 5\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 5 \, x}{x^{2} - {\left (x + 2\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 2 \, x} \]
integrate(((3*x^2+12*x+27)*exp(1/3*exp(-x+6)/x)^2+((x^3+5*x^2+8*x+4)*exp(- x+6)-30*x)*exp(1/3*exp(-x+6)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(-x+6) /x)^2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(-x+6)/x)+3*x^4+12*x^3+12*x^2),x, al gorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 2098 vs. \(2 (30) = 60\).
Time = 0.37 (sec) , antiderivative size = 2098, normalized size of antiderivative = 61.71 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\text {Too large to display} \]
integrate(((3*x^2+12*x+27)*exp(1/3*exp(-x+6)/x)^2+((x^3+5*x^2+8*x+4)*exp(- x+6)-30*x)*exp(1/3*exp(-x+6)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(-x+6) /x)^2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(-x+6)/x)+3*x^4+12*x^3+12*x^2),x, al gorithm=\
-(9*x^5*e^(2*x + 1/3*e^(-x + 6)/x) + 3*x^5*e^(x + 1/3*(18*x + e^(-x + 6))/ x) + 3*x^5*e^(x + 1/3*e^(-x + 6)/x + 6) + x^5*e^(1/3*(18*x + e^(-x + 6))/x + 6) + 5*x^4*e^12 + 45*x^4*e^(2*x) + 18*x^4*e^(2*x + 1/3*e^(-x + 6)/x) - 9*x^4*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x) - 3*x^4* e^(x + 1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/x) + 9*x^4*e^(x + 1/3*(1 8*x + e^(-x + 6))/x) + 9*x^4*e^(x + 1/3*e^(-x + 6)/x + 6) + 30*x^4*e^(x + 6) - 3*x^4*e^(1/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) - x^4*e^(1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) + 4*x^4*e^(1/3 *(18*x + e^(-x + 6))/x + 6) + 10*x^3*e^12 - 45*x^3*e^(2*x + 1/3*e^(-x + 6) /x) - 18*x^3*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x) - 45*x^3*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x) - 9*x^3*e^(x + 1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/x) + 6*x^3*e^(x + 1/3*(18*x + e^(-x + 6)) /x) - 24*x^3*e^(x + 1/3*e^(-x + 6)/x + 6) + 30*x^3*e^(x + 6) - 9*x^3*e^(1/ 2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) - 30*x^3*e^(1/2 *x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 6) - 5*x^3*e^(-1/2*x + 1/6*(3*x^2 + 2* e^(-x + 6))/x + 12) - 4*x^3*e^(1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/ x + 6) + 5*x^3*e^(1/3*(18*x + e^(-x + 6))/x + 6) - 5*x^3*e^(1/3*e^(-x + 6) /x + 12) + 5*x^2*e^12 + 45*x^2*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1 /3*e^(-x + 6)/x) - 6*x^2*e^(x + 1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6) /x) - 30*x^2*e^(x + 1/3*e^(-x + 6)/x + 6) + 24*x^2*e^(1/2*x + 1/6*(3*x^...
Time = 12.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {5\,x-5\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}+2\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}+x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}}{\left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}\right )\,\left (x+2\right )} \]
int((exp((2*exp(6 - x))/(3*x))*(12*x + 3*x^2 + 27) - exp(exp(6 - x)/(3*x)) *(30*x - exp(6 - x)*(8*x + 5*x^2 + x^3 + 4)) + 15*x^2)/(exp((2*exp(6 - x)) /(3*x))*(12*x + 3*x^2 + 12) - exp(exp(6 - x)/(3*x))*(24*x + 24*x^2 + 6*x^3 ) + 12*x^2 + 12*x^3 + 3*x^4),x)