Integrand size = 201, antiderivative size = 37 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=4+x+\frac {(5-x) \left (-x+\frac {1}{x (3+x)^2 \left (-e^x+x\right )}\right )}{2 x^2} \]
Time = 5.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {1}{2} \left (-\frac {5}{x}+2 x+\frac {-5+x}{\left (e^x-x\right ) x^3 (3+x)^2}\right ) \]
Integrate[(-60*x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(135*x^2 + 135*x^3 + 99*x^4 + 59*x^5 + 18*x^6 + 2 *x^7) + E^x*(45 + 34*x - 2*x^2 - 271*x^3 - 270*x^4 - 198*x^5 - 118*x^6 - 3 6*x^7 - 4*x^8))/(54*x^6 + 54*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(54*x^4 + 54*x ^5 + 18*x^6 + 2*x^7) + E^x*(-108*x^5 - 108*x^6 - 36*x^7 - 4*x^8)),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.
\(\displaystyle \int \frac {2 x^9+18 x^8+59 x^7+99 x^6+135 x^5+135 x^4+5 x^3-21 x^2+e^{2 x} \left (2 x^7+18 x^6+59 x^5+99 x^4+135 x^3+135 x^2\right )+e^x \left (-4 x^8-36 x^7-118 x^6-198 x^5-270 x^4-271 x^3-2 x^2+34 x+45\right )-60 x}{2 x^9+18 x^8+54 x^7+54 x^6+e^x \left (-4 x^8-36 x^7-108 x^6-108 x^5\right )+e^{2 x} \left (2 x^7+18 x^6+54 x^5+54 x^4\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^9+18 x^8+59 x^7+99 x^6+135 x^5+135 x^4+5 x^3-21 x^2+e^{2 x} \left (2 x^7+18 x^6+59 x^5+99 x^4+135 x^3+135 x^2\right )+e^x \left (-4 x^8-36 x^7-118 x^6-198 x^5-270 x^4-271 x^3-2 x^2+34 x+45\right )-60 x}{2 \left (e^x-x\right )^2 x^4 (x+3)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {-2 x^9-18 x^8-59 x^7-99 x^6-135 x^5-135 x^4-5 x^3+21 x^2+60 x-e^{2 x} \left (2 x^7+18 x^6+59 x^5+99 x^4+135 x^3+135 x^2\right )-e^x \left (-4 x^8-36 x^7-118 x^6-198 x^5-270 x^4-271 x^3-2 x^2+34 x+45\right )}{\left (e^x-x\right )^2 x^4 (x+3)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {-2 x^9-18 x^8-59 x^7-99 x^6-135 x^5-135 x^4-5 x^3+21 x^2+60 x-e^{2 x} \left (2 x^7+18 x^6+59 x^5+99 x^4+135 x^3+135 x^2\right )-e^x \left (-4 x^8-36 x^7-118 x^6-198 x^5-270 x^4-271 x^3-2 x^2+34 x+45\right )}{\left (e^x-x\right )^2 x^4 (x+3)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {-2 x^2-5}{x^2}+\frac {x^2-6 x+5}{\left (e^x-x\right )^2 x^3 (x+3)^2}+\frac {x^3+2 x^2-34 x-45}{\left (e^x-x\right ) x^4 (x+3)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{3} \int \frac {1}{\left (e^x-x\right ) x^4}dx-\frac {5}{9} \int \frac {1}{\left (e^x-x\right )^2 x^3}dx-\frac {11}{27} \int \frac {1}{\left (e^x-x\right ) x^3}dx+\frac {28}{27} \int \frac {1}{\left (e^x-x\right )^2 x^2}dx-\frac {2}{9} \int \frac {1}{\left (e^x-x\right ) x^2}dx-\frac {20}{27} \int \frac {1}{\left (e^x-x\right )^2 x}dx+\frac {7}{27} \int \frac {1}{\left (e^x-x\right ) x}dx-\frac {16}{27} \int \frac {1}{\left (e^x-x\right ) (x+3)^3}dx+\frac {32}{27} \int \frac {1}{\left (e^x-x\right )^2 (x+3)^2}dx-\frac {5}{9} \int \frac {1}{\left (e^x-x\right ) (x+3)^2}dx+\frac {20}{27} \int \frac {1}{\left (e^x-x\right )^2 (x+3)}dx-\frac {7}{27} \int \frac {1}{\left (e^x-x\right ) (x+3)}dx+2 x-\frac {5}{x}\right )\) |
Int[(-60*x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(135*x^2 + 135*x^3 + 99*x^4 + 59*x^5 + 18*x^6 + 2*x^7) + E^x*(45 + 34*x - 2*x^2 - 271*x^3 - 270*x^4 - 198*x^5 - 118*x^6 - 36*x^7 - 4*x^8))/(54*x^6 + 54*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(54*x^4 + 54*x^5 + 1 8*x^6 + 2*x^7) + E^x*(-108*x^5 - 108*x^6 - 36*x^7 - 4*x^8)),x]
3.13.51.3.1 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 = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
risch | \(x -\frac {5}{2 x}-\frac {-5+x}{2 x^{3} \left (x^{2}+6 x +9\right ) \left (x -{\mathrm e}^{x}\right )}\) | \(34\) |
norman | \(\frac {\frac {5}{2}+x^{7}-69 x^{4}-\frac {59 x^{5}}{2}+69 \,{\mathrm e}^{x} x^{3}+\frac {59 \,{\mathrm e}^{x} x^{4}}{2}-\frac {x}{2}-\frac {45 x^{3}}{2}-x^{6} {\mathrm e}^{x}+\frac {45 \,{\mathrm e}^{x} x^{2}}{2}}{x^{3} \left (x -{\mathrm e}^{x}\right ) \left (3+x \right )^{2}}\) | \(69\) |
parallelrisch | \(\frac {2 x^{7}-2 x^{6} {\mathrm e}^{x}+5-59 x^{5}+59 \,{\mathrm e}^{x} x^{4}-138 x^{4}+138 \,{\mathrm e}^{x} x^{3}-45 x^{3}+45 \,{\mathrm e}^{x} x^{2}-x}{2 x^{3} \left (x^{3}-{\mathrm e}^{x} x^{2}+6 x^{2}-6 \,{\mathrm e}^{x} x +9 x -9 \,{\mathrm e}^{x}\right )}\) | \(89\) |
int(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-36*x^7- 118*x^6-198*x^5-270*x^4-271*x^3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+59*x^7+ 99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54*x^4)*ex p(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54*x^6), x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.59 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \]
integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-3 6*x^7-118*x^6-198*x^5-270*x^4-271*x^3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+5 9*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54*x ^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54 *x^6),x, algorithm=\
1/2*(2*x^7 + 12*x^6 + 13*x^5 - 30*x^4 - 45*x^3 - (2*x^6 + 12*x^5 + 13*x^4 - 30*x^3 - 45*x^2)*e^x - x + 5)/(x^6 + 6*x^5 + 9*x^4 - (x^5 + 6*x^4 + 9*x^ 3)*e^x)
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=x + \frac {x - 5}{- 2 x^{6} - 12 x^{5} - 18 x^{4} + \left (2 x^{5} + 12 x^{4} + 18 x^{3}\right ) e^{x}} - \frac {5}{2 x} \]
integrate(((2*x**7+18*x**6+59*x**5+99*x**4+135*x**3+135*x**2)*exp(x)**2+(- 4*x**8-36*x**7-118*x**6-198*x**5-270*x**4-271*x**3-2*x**2+34*x+45)*exp(x)+ 2*x**9+18*x**8+59*x**7+99*x**6+135*x**5+135*x**4+5*x**3-21*x**2-60*x)/((2* x**7+18*x**6+54*x**5+54*x**4)*exp(x)**2+(-4*x**8-36*x**7-108*x**6-108*x**5 )*exp(x)+2*x**9+18*x**8+54*x**7+54*x**6),x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.59 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \]
integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-3 6*x^7-118*x^6-198*x^5-270*x^4-271*x^3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+5 9*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54*x ^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54 *x^6),x, algorithm=\
1/2*(2*x^7 + 12*x^6 + 13*x^5 - 30*x^4 - 45*x^3 - (2*x^6 + 12*x^5 + 13*x^4 - 30*x^3 - 45*x^2)*e^x - x + 5)/(x^6 + 6*x^5 + 9*x^4 - (x^5 + 6*x^4 + 9*x^ 3)*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.81 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} - 2 \, x^{6} e^{x} + 12 \, x^{6} - 12 \, x^{5} e^{x} + 13 \, x^{5} - 13 \, x^{4} e^{x} - 30 \, x^{4} + 30 \, x^{3} e^{x} - 45 \, x^{3} + 45 \, x^{2} e^{x} - x + 5}{2 \, {\left (x^{6} - x^{5} e^{x} + 6 \, x^{5} - 6 \, x^{4} e^{x} + 9 \, x^{4} - 9 \, x^{3} e^{x}\right )}} \]
integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-3 6*x^7-118*x^6-198*x^5-270*x^4-271*x^3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+5 9*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54*x ^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54 *x^6),x, algorithm=\
1/2*(2*x^7 - 2*x^6*e^x + 12*x^6 - 12*x^5*e^x + 13*x^5 - 13*x^4*e^x - 30*x^ 4 + 30*x^3*e^x - 45*x^3 + 45*x^2*e^x - x + 5)/(x^6 - x^5*e^x + 6*x^5 - 6*x ^4*e^x + 9*x^4 - 9*x^3*e^x)
Time = 8.65 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=x-\frac {5}{2\,x}+\frac {-x^3+3\,x^2+13\,x-15}{2\,x^3\,\left (x-{\mathrm {e}}^x\right )\,\left (x-1\right )\,{\left (x+3\right )}^3} \]
int((exp(2*x)*(135*x^2 + 135*x^3 + 99*x^4 + 59*x^5 + 18*x^6 + 2*x^7) - 60* x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 - exp(x)*(2*x^2 - 34*x + 271*x^3 + 270*x^4 + 198*x^5 + 118*x^6 + 36*x^7 + 4*x^8 - 45))/(exp(2*x)*(54*x^4 + 54*x^5 + 18*x^6 + 2*x^7) - exp(x)*(108*x^ 5 + 108*x^6 + 36*x^7 + 4*x^8) + 54*x^6 + 54*x^7 + 18*x^8 + 2*x^9),x)