Integrand size = 289, antiderivative size = 27 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=x \left (5+\frac {x^{12}}{6561 \left (-e^{-x}-x+x^2\right )^4}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(27)=54\).
Time = 10.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.04 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\frac {32840 x+20 x^2+10 x^3+4 x^4+x^5+\frac {-56+189 x-216 x^2+84 x^3}{(-1+x)^4}+\frac {x^9 \left (-1+4 e^x (-1+x) x-6 e^{2 x} (-1+x)^2 x^2+4 e^{3 x} (-1+x)^3 x^3\right )}{(-1+x)^4 \left (-1+e^x (-1+x) x\right )^4}}{6561} \]
Integrate[(-32805 + E^x*(-164025*x + 164025*x^2) + E^(2*x)*(-328050*x^2 + 656100*x^3 - 328050*x^4) + E^(3*x)*(-328050*x^3 + 984150*x^4 - 984150*x^5 + 328050*x^6) + E^(4*x)*(-164025*x^4 + 656100*x^5 - 984150*x^6 + 656100*x^ 7 - 164025*x^8 - 13*x^12 - 4*x^13) + E^(5*x)*(-32805*x^5 + 164025*x^6 - 32 8050*x^7 + 328050*x^8 - 164025*x^9 + 32805*x^10 - 9*x^13 + 5*x^14))/(-6561 + E^x*(-32805*x + 32805*x^2) + E^(2*x)*(-65610*x^2 + 131220*x^3 - 65610*x ^4) + E^(3*x)*(-65610*x^3 + 196830*x^4 - 196830*x^5 + 65610*x^6) + E^(4*x) *(-32805*x^4 + 131220*x^5 - 196830*x^6 + 131220*x^7 - 32805*x^8) + E^(5*x) *(-6561*x^5 + 32805*x^6 - 65610*x^7 + 65610*x^8 - 32805*x^9 + 6561*x^10)), x]
(32840*x + 20*x^2 + 10*x^3 + 4*x^4 + x^5 + (-56 + 189*x - 216*x^2 + 84*x^3 )/(-1 + x)^4 + (x^9*(-1 + 4*E^x*(-1 + x)*x - 6*E^(2*x)*(-1 + x)^2*x^2 + 4* E^(3*x)*(-1 + x)^3*x^3))/((-1 + x)^4*(-1 + E^x*(-1 + x)*x)^4))/6561
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.
| \(\displaystyle \int \frac {e^x \left (164025 x^2-164025 x\right )+e^{2 x} \left (-328050 x^4+656100 x^3-328050 x^2\right )+e^{3 x} \left (328050 x^6-984150 x^5+984150 x^4-328050 x^3\right )+e^{4 x} \left (-4 x^{13}-13 x^{12}-164025 x^8+656100 x^7-984150 x^6+656100 x^5-164025 x^4\right )+e^{5 x} \left (5 x^{14}-9 x^{13}+32805 x^{10}-164025 x^9+328050 x^8-328050 x^7+164025 x^6-32805 x^5\right )-32805}{e^x \left (32805 x^2-32805 x\right )+e^{2 x} \left (-65610 x^4+131220 x^3-65610 x^2\right )+e^{3 x} \left (65610 x^6-196830 x^5+196830 x^4-65610 x^3\right )+e^{4 x} \left (-32805 x^8+131220 x^7-196830 x^6+131220 x^5-32805 x^4\right )+e^{5 x} \left (6561 x^{10}-32805 x^9+65610 x^8-65610 x^7+32805 x^6-6561 x^5\right )-6561} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-328050 e^{3 x} (x-1)^3 x^3+328050 e^{2 x} (x-1)^2 x^2+e^{4 x} \left (4 x^9+13 x^8+164025 x^4-656100 x^3+984150 x^2-656100 x+164025\right ) x^4-e^{5 x} \left (5 x^9-9 x^8+32805 x^5-164025 x^4+328050 x^3-328050 x^2+164025 x-32805\right ) x^5-164025 e^x (x-1) x+32805}{6561 \left (1-e^x (x-1) x\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e^{5 x} \left (-5 x^9+9 x^8-32805 x^5+164025 x^4-328050 x^3+328050 x^2-164025 x+32805\right ) x^5+e^{4 x} \left (4 x^9+13 x^8+164025 x^4-656100 x^3+984150 x^2-656100 x+164025\right ) x^4+328050 e^{3 x} (1-x)^3 x^3+328050 e^{2 x} (1-x)^2 x^2+164025 e^x (1-x) x+32805}{\left (e^x (1-x) x+1\right )^5}dx}{6561}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \left (x^2-4 x+8\right ) x^8}{(x-1)^5 \left (e^x x^2-e^x x-1\right )}-\frac {2 \left (8 x^2-7 x+19\right ) x^8}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^2}-\frac {4 \left (6 x^2+x+3\right ) x^8}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^3}-\frac {\left (16 x^2+11 x-7\right ) x^8}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^4}-\frac {4 \left (x^2+x-1\right ) x^8}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^5}+\frac {5 x^9-9 x^8+32805 x^5-164025 x^4+328050 x^3-328050 x^2+164025 x-32805}{(x-1)^5}\right )dx}{6561}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^5+4 x^4+10 x^3+20 x^2+32840 x-644 \int \frac {1}{\left (e^x x^2-e^x x-1\right )^5}dx-4 \int \frac {1}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^5}dx-44 \int \frac {1}{(x-1)^4 \left (e^x x^2-e^x x-1\right )^5}dx-212 \int \frac {1}{(x-1)^3 \left (e^x x^2-e^x x-1\right )^5}dx-592 \int \frac {1}{(x-1)^2 \left (e^x x^2-e^x x-1\right )^5}dx-1064 \int \frac {1}{(x-1) \left (e^x x^2-e^x x-1\right )^5}dx-360 \int \frac {x}{\left (e^x x^2-e^x x-1\right )^5}dx-180 \int \frac {x^2}{\left (e^x x^2-e^x x-1\right )^5}dx-76 \int \frac {x^3}{\left (e^x x^2-e^x x-1\right )^5}dx-24 \int \frac {x^4}{\left (e^x x^2-e^x x-1\right )^5}dx-4 \int \frac {x^5}{\left (e^x x^2-e^x x-1\right )^5}dx-2541 \int \frac {1}{\left (e^x x^2-e^x x-1\right )^4}dx-20 \int \frac {1}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^4}dx-203 \int \frac {1}{(x-1)^4 \left (e^x x^2-e^x x-1\right )^4}dx-920 \int \frac {1}{(x-1)^3 \left (e^x x^2-e^x x-1\right )^4}dx-2452 \int \frac {1}{(x-1)^2 \left (e^x x^2-e^x x-1\right )^4}dx-4256 \int \frac {1}{(x-1) \left (e^x x^2-e^x x-1\right )^4}dx-1400 \int \frac {x}{\left (e^x x^2-e^x x-1\right )^4}dx-690 \int \frac {x^2}{\left (e^x x^2-e^x x-1\right )^4}dx-288 \int \frac {x^3}{\left (e^x x^2-e^x x-1\right )^4}dx-91 \int \frac {x^4}{\left (e^x x^2-e^x x-1\right )^4}dx-16 \int \frac {x^5}{\left (e^x x^2-e^x x-1\right )^4}dx-3724 \int \frac {1}{\left (e^x x^2-e^x x-1\right )^3}dx-40 \int \frac {1}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^3}dx-372 \int \frac {1}{(x-1)^4 \left (e^x x^2-e^x x-1\right )^3}dx-1560 \int \frac {1}{(x-1)^3 \left (e^x x^2-e^x x-1\right )^3}dx-3888 \int \frac {1}{(x-1)^2 \left (e^x x^2-e^x x-1\right )^3}dx-6384 \int \frac {1}{(x-1) \left (e^x x^2-e^x x-1\right )^3}dx-2000 \int \frac {x}{\left (e^x x^2-e^x x-1\right )^3}dx-960 \int \frac {x^2}{\left (e^x x^2-e^x x-1\right )^3}dx-392 \int \frac {x^3}{\left (e^x x^2-e^x x-1\right )^3}dx-124 \int \frac {x^4}{\left (e^x x^2-e^x x-1\right )^3}dx-24 \int \frac {x^5}{\left (e^x x^2-e^x x-1\right )^3}dx-2366 \int \frac {1}{\left (e^x x^2-e^x x-1\right )^2}dx-40 \int \frac {1}{(x-1)^5 \left (e^x x^2-e^x x-1\right )^2}dx-338 \int \frac {1}{(x-1)^4 \left (e^x x^2-e^x x-1\right )^2}dx-1280 \int \frac {1}{(x-1)^3 \left (e^x x^2-e^x x-1\right )^2}dx-2872 \int \frac {1}{(x-1)^2 \left (e^x x^2-e^x x-1\right )^2}dx-4256 \int \frac {1}{(x-1) \left (e^x x^2-e^x x-1\right )^2}dx-1200 \int \frac {x}{\left (e^x x^2-e^x x-1\right )^2}dx-540 \int \frac {x^2}{\left (e^x x^2-e^x x-1\right )^2}dx-208 \int \frac {x^3}{\left (e^x x^2-e^x x-1\right )^2}dx-66 \int \frac {x^4}{\left (e^x x^2-e^x x-1\right )^2}dx-16 \int \frac {x^5}{\left (e^x x^2-e^x x-1\right )^2}dx-504 \int \frac {1}{e^x x^2-e^x x-1}dx-20 \int \frac {1}{(x-1)^5 \left (e^x x^2-e^x x-1\right )}dx-152 \int \frac {1}{(x-1)^4 \left (e^x x^2-e^x x-1\right )}dx-500 \int \frac {1}{(x-1)^3 \left (e^x x^2-e^x x-1\right )}dx-928 \int \frac {1}{(x-1)^2 \left (e^x x^2-e^x x-1\right )}dx-1064 \int \frac {1}{(x-1) \left (e^x x^2-e^x x-1\right )}dx-200 \int \frac {x}{e^x x^2-e^x x-1}dx-60 \int \frac {x^2}{e^x x^2-e^x x-1}dx-12 \int \frac {x^3}{e^x x^2-e^x x-1}dx-4 \int \frac {x^4}{e^x x^2-e^x x-1}dx-4 \int \frac {x^5}{e^x x^2-e^x x-1}dx-\frac {84}{1-x}+\frac {36}{(1-x)^2}-\frac {9}{(1-x)^3}+\frac {1}{(x-1)^4}}{6561}\) |
Int[(-32805 + E^x*(-164025*x + 164025*x^2) + E^(2*x)*(-328050*x^2 + 656100 *x^3 - 328050*x^4) + E^(3*x)*(-328050*x^3 + 984150*x^4 - 984150*x^5 + 3280 50*x^6) + E^(4*x)*(-164025*x^4 + 656100*x^5 - 984150*x^6 + 656100*x^7 - 16 4025*x^8 - 13*x^12 - 4*x^13) + E^(5*x)*(-32805*x^5 + 164025*x^6 - 328050*x ^7 + 328050*x^8 - 164025*x^9 + 32805*x^10 - 9*x^13 + 5*x^14))/(-6561 + E^x *(-32805*x + 32805*x^2) + E^(2*x)*(-65610*x^2 + 131220*x^3 - 65610*x^4) + E^(3*x)*(-65610*x^3 + 196830*x^4 - 196830*x^5 + 65610*x^6) + E^(4*x)*(-328 05*x^4 + 131220*x^5 - 196830*x^6 + 131220*x^7 - 32805*x^8) + E^(5*x)*(-656 1*x^5 + 32805*x^6 - 65610*x^7 + 65610*x^8 - 32805*x^9 + 6561*x^10)),x]
3.1.90.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(26)=52\).
Time = 0.76 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.22
| method | result | size |
| risch | \(\frac {x^{5}}{6561}+\frac {4 x^{4}}{6561}+\frac {10 x^{3}}{6561}+\frac {20 x^{2}}{6561}+\frac {32840 x}{6561}+\frac {\frac {28}{2187} x^{3}-\frac {8}{243} x^{2}+\frac {7}{243} x -\frac {56}{6561}}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}+\frac {x^{9} \left (4 \,{\mathrm e}^{3 x} x^{6}-12 \,{\mathrm e}^{3 x} x^{5}+12 \,{\mathrm e}^{3 x} x^{4}-6 \,{\mathrm e}^{2 x} x^{4}-4 x^{3} {\mathrm e}^{3 x}+12 \,{\mathrm e}^{2 x} x^{3}-6 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{x} x^{2}-4 \,{\mathrm e}^{x} x -1\right )}{6561 \left (x^{2}-2 x +1\right )^{2} \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x -1\right )^{4}}\) | \(168\) |
| parallelrisch | \(\frac {131220 x -524880 x^{7} {\mathrm e}^{3 x}+524880 \,{\mathrm e}^{3 x} x^{4}-1574640 \,{\mathrm e}^{3 x} x^{5}-524880 x^{6} {\mathrm e}^{4 x}+1574640 \,{\mathrm e}^{3 x} x^{6}-1574640 \,{\mathrm e}^{2 x} x^{4}+787320 x^{5} {\mathrm e}^{2 x}+524880 \,{\mathrm e}^{x} x^{2}-524880 \,{\mathrm e}^{x} x^{3}-524880 x^{8} {\mathrm e}^{4 x}+131220 x^{5} {\mathrm e}^{4 x}+787320 \,{\mathrm e}^{2 x} x^{3}+4 \,{\mathrm e}^{4 x} x^{13}+787320 \,{\mathrm e}^{4 x} x^{7}+131220 \,{\mathrm e}^{4 x} x^{9}}{26244 x^{8} {\mathrm e}^{4 x}-104976 \,{\mathrm e}^{4 x} x^{7}+157464 x^{6} {\mathrm e}^{4 x}-104976 \,{\mathrm e}^{3 x} x^{6}-104976 x^{5} {\mathrm e}^{4 x}+314928 \,{\mathrm e}^{3 x} x^{5}+26244 x^{4} {\mathrm e}^{4 x}-314928 \,{\mathrm e}^{3 x} x^{4}+157464 \,{\mathrm e}^{2 x} x^{4}+104976 x^{3} {\mathrm e}^{3 x}-314928 \,{\mathrm e}^{2 x} x^{3}+157464 \,{\mathrm e}^{2 x} x^{2}-104976 \,{\mathrm e}^{x} x^{2}+104976 \,{\mathrm e}^{x} x +26244}\) | \(260\) |
int(((5*x^14-9*x^13+32805*x^10-164025*x^9+328050*x^8-328050*x^7+164025*x^6 -32805*x^5)*exp(x)^5+(-4*x^13-13*x^12-164025*x^8+656100*x^7-984150*x^6+656 100*x^5-164025*x^4)*exp(x)^4+(328050*x^6-984150*x^5+984150*x^4-328050*x^3) *exp(x)^3+(-328050*x^4+656100*x^3-328050*x^2)*exp(x)^2+(164025*x^2-164025* x)*exp(x)-32805)/((6561*x^10-32805*x^9+65610*x^8-65610*x^7+32805*x^6-6561* x^5)*exp(x)^5+(-32805*x^8+131220*x^7-196830*x^6+131220*x^5-32805*x^4)*exp( x)^4+(65610*x^6-196830*x^5+196830*x^4-65610*x^3)*exp(x)^3+(-65610*x^4+1312 20*x^3-65610*x^2)*exp(x)^2+(32805*x^2-32805*x)*exp(x)-6561),x,method=_RETU RNVERBOSE)
1/6561*x^5+4/6561*x^4+10/6561*x^3+20/6561*x^2+32840/6561*x+(28/2187*x^3-8/ 243*x^2+7/243*x-56/6561)/(x^4-4*x^3+6*x^2-4*x+1)+1/6561*x^9*(4*x^6*exp(x)^ 3-12*x^5*exp(x)^3+12*x^4*exp(x)^3-6*exp(x)^2*x^4-4*x^3*exp(x)^3+12*exp(x)^ 2*x^3-6*exp(x)^2*x^2+4*exp(x)*x^2-4*exp(x)*x-1)/(x^2-2*x+1)^2/(exp(x)*x^2- exp(x)*x-1)^4
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 7.70 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\frac {{\left (x^{13} + 32805 \, x^{9} - 131276 \, x^{8} + 197054 \, x^{7} - 131556 \, x^{6} + 33029 \, x^{5} - 56 \, x^{4}\right )} e^{\left (4 \, x\right )} - 4 \, {\left (32805 \, x^{7} - 98471 \, x^{6} + 98583 \, x^{5} - 32973 \, x^{4} + 56 \, x^{3}\right )} e^{\left (3 \, x\right )} + 6 \, {\left (32805 \, x^{5} - 65666 \, x^{4} + 32917 \, x^{3} - 56 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (32805 \, x^{3} - 32861 \, x^{2} + 56 \, x\right )} e^{x} + 32805 \, x - 56}{6561 \, {\left ({\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )} e^{\left (4 \, x\right )} - 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} - x\right )} e^{x} + 1\right )}} \]
integrate(((5*x^14-9*x^13+32805*x^10-164025*x^9+328050*x^8-328050*x^7+1640 25*x^6-32805*x^5)*exp(x)^5+(-4*x^13-13*x^12-164025*x^8+656100*x^7-984150*x ^6+656100*x^5-164025*x^4)*exp(x)^4+(328050*x^6-984150*x^5+984150*x^4-32805 0*x^3)*exp(x)^3+(-328050*x^4+656100*x^3-328050*x^2)*exp(x)^2+(164025*x^2-1 64025*x)*exp(x)-32805)/((6561*x^10-32805*x^9+65610*x^8-65610*x^7+32805*x^6 -6561*x^5)*exp(x)^5+(-32805*x^8+131220*x^7-196830*x^6+131220*x^5-32805*x^4 )*exp(x)^4+(65610*x^6-196830*x^5+196830*x^4-65610*x^3)*exp(x)^3+(-65610*x^ 4+131220*x^3-65610*x^2)*exp(x)^2+(32805*x^2-32805*x)*exp(x)-6561),x, algor ithm=\
1/6561*((x^13 + 32805*x^9 - 131276*x^8 + 197054*x^7 - 131556*x^6 + 33029*x ^5 - 56*x^4)*e^(4*x) - 4*(32805*x^7 - 98471*x^6 + 98583*x^5 - 32973*x^4 + 56*x^3)*e^(3*x) + 6*(32805*x^5 - 65666*x^4 + 32917*x^3 - 56*x^2)*e^(2*x) - 4*(32805*x^3 - 32861*x^2 + 56*x)*e^x + 32805*x - 56)/((x^8 - 4*x^7 + 6*x^ 6 - 4*x^5 + x^4)*e^(4*x) - 4*(x^6 - 3*x^5 + 3*x^4 - x^3)*e^(3*x) + 6*(x^4 - 2*x^3 + x^2)*e^(2*x) - 4*(x^2 - x)*e^x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (22) = 44\).
Time = 0.53 (sec) , antiderivative size = 318, normalized size of antiderivative = 11.78 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\frac {x^{5}}{6561} + \frac {4 x^{4}}{6561} + \frac {10 x^{3}}{6561} + \frac {20 x^{2}}{6561} + \frac {32840 x}{6561} + \frac {84 x^{3} - 216 x^{2} + 189 x - 56}{6561 x^{4} - 26244 x^{3} + 39366 x^{2} - 26244 x + 6561} + \frac {- x^{9} + \left (4 x^{11} - 4 x^{10}\right ) e^{x} + \left (- 6 x^{13} + 12 x^{12} - 6 x^{11}\right ) e^{2 x} + \left (4 x^{15} - 12 x^{14} + 12 x^{13} - 4 x^{12}\right ) e^{3 x}}{6561 x^{4} - 26244 x^{3} + 39366 x^{2} - 26244 x + \left (- 26244 x^{6} + 131220 x^{5} - 262440 x^{4} + 262440 x^{3} - 131220 x^{2} + 26244 x\right ) e^{x} + \left (39366 x^{8} - 236196 x^{7} + 590490 x^{6} - 787320 x^{5} + 590490 x^{4} - 236196 x^{3} + 39366 x^{2}\right ) e^{2 x} + \left (- 26244 x^{10} + 183708 x^{9} - 551124 x^{8} + 918540 x^{7} - 918540 x^{6} + 551124 x^{5} - 183708 x^{4} + 26244 x^{3}\right ) e^{3 x} + \left (6561 x^{12} - 52488 x^{11} + 183708 x^{10} - 367416 x^{9} + 459270 x^{8} - 367416 x^{7} + 183708 x^{6} - 52488 x^{5} + 6561 x^{4}\right ) e^{4 x} + 6561} \]
integrate(((5*x**14-9*x**13+32805*x**10-164025*x**9+328050*x**8-328050*x** 7+164025*x**6-32805*x**5)*exp(x)**5+(-4*x**13-13*x**12-164025*x**8+656100* x**7-984150*x**6+656100*x**5-164025*x**4)*exp(x)**4+(328050*x**6-984150*x* *5+984150*x**4-328050*x**3)*exp(x)**3+(-328050*x**4+656100*x**3-328050*x** 2)*exp(x)**2+(164025*x**2-164025*x)*exp(x)-32805)/((6561*x**10-32805*x**9+ 65610*x**8-65610*x**7+32805*x**6-6561*x**5)*exp(x)**5+(-32805*x**8+131220* x**7-196830*x**6+131220*x**5-32805*x**4)*exp(x)**4+(65610*x**6-196830*x**5 +196830*x**4-65610*x**3)*exp(x)**3+(-65610*x**4+131220*x**3-65610*x**2)*ex p(x)**2+(32805*x**2-32805*x)*exp(x)-6561),x)
x**5/6561 + 4*x**4/6561 + 10*x**3/6561 + 20*x**2/6561 + 32840*x/6561 + (84 *x**3 - 216*x**2 + 189*x - 56)/(6561*x**4 - 26244*x**3 + 39366*x**2 - 2624 4*x + 6561) + (-x**9 + (4*x**11 - 4*x**10)*exp(x) + (-6*x**13 + 12*x**12 - 6*x**11)*exp(2*x) + (4*x**15 - 12*x**14 + 12*x**13 - 4*x**12)*exp(3*x))/( 6561*x**4 - 26244*x**3 + 39366*x**2 - 26244*x + (-26244*x**6 + 131220*x**5 - 262440*x**4 + 262440*x**3 - 131220*x**2 + 26244*x)*exp(x) + (39366*x**8 - 236196*x**7 + 590490*x**6 - 787320*x**5 + 590490*x**4 - 236196*x**3 + 3 9366*x**2)*exp(2*x) + (-26244*x**10 + 183708*x**9 - 551124*x**8 + 918540*x **7 - 918540*x**6 + 551124*x**5 - 183708*x**4 + 26244*x**3)*exp(3*x) + (65 61*x**12 - 52488*x**11 + 183708*x**10 - 367416*x**9 + 459270*x**8 - 367416 *x**7 + 183708*x**6 - 52488*x**5 + 6561*x**4)*exp(4*x) + 6561)
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 208, normalized size of antiderivative = 7.70 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\frac {{\left (x^{13} + 32805 \, x^{9} - 131276 \, x^{8} + 197054 \, x^{7} - 131556 \, x^{6} + 33029 \, x^{5} - 56 \, x^{4}\right )} e^{\left (4 \, x\right )} - 4 \, {\left (32805 \, x^{7} - 98471 \, x^{6} + 98583 \, x^{5} - 32973 \, x^{4} + 56 \, x^{3}\right )} e^{\left (3 \, x\right )} + 6 \, {\left (32805 \, x^{5} - 65666 \, x^{4} + 32917 \, x^{3} - 56 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (32805 \, x^{3} - 32861 \, x^{2} + 56 \, x\right )} e^{x} + 32805 \, x - 56}{6561 \, {\left ({\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )} e^{\left (4 \, x\right )} - 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} - x\right )} e^{x} + 1\right )}} \]
integrate(((5*x^14-9*x^13+32805*x^10-164025*x^9+328050*x^8-328050*x^7+1640 25*x^6-32805*x^5)*exp(x)^5+(-4*x^13-13*x^12-164025*x^8+656100*x^7-984150*x ^6+656100*x^5-164025*x^4)*exp(x)^4+(328050*x^6-984150*x^5+984150*x^4-32805 0*x^3)*exp(x)^3+(-328050*x^4+656100*x^3-328050*x^2)*exp(x)^2+(164025*x^2-1 64025*x)*exp(x)-32805)/((6561*x^10-32805*x^9+65610*x^8-65610*x^7+32805*x^6 -6561*x^5)*exp(x)^5+(-32805*x^8+131220*x^7-196830*x^6+131220*x^5-32805*x^4 )*exp(x)^4+(65610*x^6-196830*x^5+196830*x^4-65610*x^3)*exp(x)^3+(-65610*x^ 4+131220*x^3-65610*x^2)*exp(x)^2+(32805*x^2-32805*x)*exp(x)-6561),x, algor ithm=\
1/6561*((x^13 + 32805*x^9 - 131276*x^8 + 197054*x^7 - 131556*x^6 + 33029*x ^5 - 56*x^4)*e^(4*x) - 4*(32805*x^7 - 98471*x^6 + 98583*x^5 - 32973*x^4 + 56*x^3)*e^(3*x) + 6*(32805*x^5 - 65666*x^4 + 32917*x^3 - 56*x^2)*e^(2*x) - 4*(32805*x^3 - 32861*x^2 + 56*x)*e^x + 32805*x - 56)/((x^8 - 4*x^7 + 6*x^ 6 - 4*x^5 + x^4)*e^(4*x) - 4*(x^6 - 3*x^5 + 3*x^4 - x^3)*e^(3*x) + 6*(x^4 - 2*x^3 + x^2)*e^(2*x) - 4*(x^2 - x)*e^x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (24) = 48\).
Time = 0.37 (sec) , antiderivative size = 291, normalized size of antiderivative = 10.78 \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\frac {x^{13} e^{\left (4 \, x\right )} + 32805 \, x^{9} e^{\left (4 \, x\right )} - 131276 \, x^{8} e^{\left (4 \, x\right )} + 197054 \, x^{7} e^{\left (4 \, x\right )} - 131220 \, x^{7} e^{\left (3 \, x\right )} - 131556 \, x^{6} e^{\left (4 \, x\right )} + 393884 \, x^{6} e^{\left (3 \, x\right )} + 33029 \, x^{5} e^{\left (4 \, x\right )} - 394332 \, x^{5} e^{\left (3 \, x\right )} + 196830 \, x^{5} e^{\left (2 \, x\right )} - 56 \, x^{4} e^{\left (4 \, x\right )} + 131892 \, x^{4} e^{\left (3 \, x\right )} - 393996 \, x^{4} e^{\left (2 \, x\right )} - 224 \, x^{3} e^{\left (3 \, x\right )} + 197502 \, x^{3} e^{\left (2 \, x\right )} - 131220 \, x^{3} e^{x} - 336 \, x^{2} e^{\left (2 \, x\right )} + 131444 \, x^{2} e^{x} - 224 \, x e^{x} + 32805 \, x - 56}{6561 \, {\left (x^{8} e^{\left (4 \, x\right )} - 4 \, x^{7} e^{\left (4 \, x\right )} + 6 \, x^{6} e^{\left (4 \, x\right )} - 4 \, x^{6} e^{\left (3 \, x\right )} - 4 \, x^{5} e^{\left (4 \, x\right )} + 12 \, x^{5} e^{\left (3 \, x\right )} + x^{4} e^{\left (4 \, x\right )} - 12 \, x^{4} e^{\left (3 \, x\right )} + 6 \, x^{4} e^{\left (2 \, x\right )} + 4 \, x^{3} e^{\left (3 \, x\right )} - 12 \, x^{3} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + 4 \, x e^{x} + 1\right )}} \]
integrate(((5*x^14-9*x^13+32805*x^10-164025*x^9+328050*x^8-328050*x^7+1640 25*x^6-32805*x^5)*exp(x)^5+(-4*x^13-13*x^12-164025*x^8+656100*x^7-984150*x ^6+656100*x^5-164025*x^4)*exp(x)^4+(328050*x^6-984150*x^5+984150*x^4-32805 0*x^3)*exp(x)^3+(-328050*x^4+656100*x^3-328050*x^2)*exp(x)^2+(164025*x^2-1 64025*x)*exp(x)-32805)/((6561*x^10-32805*x^9+65610*x^8-65610*x^7+32805*x^6 -6561*x^5)*exp(x)^5+(-32805*x^8+131220*x^7-196830*x^6+131220*x^5-32805*x^4 )*exp(x)^4+(65610*x^6-196830*x^5+196830*x^4-65610*x^3)*exp(x)^3+(-65610*x^ 4+131220*x^3-65610*x^2)*exp(x)^2+(32805*x^2-32805*x)*exp(x)-6561),x, algor ithm=\
1/6561*(x^13*e^(4*x) + 32805*x^9*e^(4*x) - 131276*x^8*e^(4*x) + 197054*x^7 *e^(4*x) - 131220*x^7*e^(3*x) - 131556*x^6*e^(4*x) + 393884*x^6*e^(3*x) + 33029*x^5*e^(4*x) - 394332*x^5*e^(3*x) + 196830*x^5*e^(2*x) - 56*x^4*e^(4* x) + 131892*x^4*e^(3*x) - 393996*x^4*e^(2*x) - 224*x^3*e^(3*x) + 197502*x^ 3*e^(2*x) - 131220*x^3*e^x - 336*x^2*e^(2*x) + 131444*x^2*e^x - 224*x*e^x + 32805*x - 56)/(x^8*e^(4*x) - 4*x^7*e^(4*x) + 6*x^6*e^(4*x) - 4*x^6*e^(3* x) - 4*x^5*e^(4*x) + 12*x^5*e^(3*x) + x^4*e^(4*x) - 12*x^4*e^(3*x) + 6*x^4 *e^(2*x) + 4*x^3*e^(3*x) - 12*x^3*e^(2*x) + 6*x^2*e^(2*x) - 4*x^2*e^x + 4* x*e^x + 1)
Timed out. \[ \int \frac {-32805+e^x \left (-164025 x+164025 x^2\right )+e^{2 x} \left (-328050 x^2+656100 x^3-328050 x^4\right )+e^{3 x} \left (-328050 x^3+984150 x^4-984150 x^5+328050 x^6\right )+e^{4 x} \left (-164025 x^4+656100 x^5-984150 x^6+656100 x^7-164025 x^8-13 x^{12}-4 x^{13}\right )+e^{5 x} \left (-32805 x^5+164025 x^6-328050 x^7+328050 x^8-164025 x^9+32805 x^{10}-9 x^{13}+5 x^{14}\right )}{-6561+e^x \left (-32805 x+32805 x^2\right )+e^{2 x} \left (-65610 x^2+131220 x^3-65610 x^4\right )+e^{3 x} \left (-65610 x^3+196830 x^4-196830 x^5+65610 x^6\right )+e^{4 x} \left (-32805 x^4+131220 x^5-196830 x^6+131220 x^7-32805 x^8\right )+e^{5 x} \left (-6561 x^5+32805 x^6-65610 x^7+65610 x^8-32805 x^9+6561 x^{10}\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,x}\,\left (4\,x^{13}+13\,x^{12}+164025\,x^8-656100\,x^7+984150\,x^6-656100\,x^5+164025\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (328050\,x^4-656100\,x^3+328050\,x^2\right )+{\mathrm {e}}^x\,\left (164025\,x-164025\,x^2\right )+{\mathrm {e}}^{5\,x}\,\left (-5\,x^{14}+9\,x^{13}-32805\,x^{10}+164025\,x^9-328050\,x^8+328050\,x^7-164025\,x^6+32805\,x^5\right )+{\mathrm {e}}^{3\,x}\,\left (-328050\,x^6+984150\,x^5-984150\,x^4+328050\,x^3\right )+32805}{{\mathrm {e}}^{4\,x}\,\left (32805\,x^8-131220\,x^7+196830\,x^6-131220\,x^5+32805\,x^4\right )+{\mathrm {e}}^{5\,x}\,\left (-6561\,x^{10}+32805\,x^9-65610\,x^8+65610\,x^7-32805\,x^6+6561\,x^5\right )+{\mathrm {e}}^{2\,x}\,\left (65610\,x^4-131220\,x^3+65610\,x^2\right )+{\mathrm {e}}^x\,\left (32805\,x-32805\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (-65610\,x^6+196830\,x^5-196830\,x^4+65610\,x^3\right )+6561} \,d x \]
int((exp(4*x)*(164025*x^4 - 656100*x^5 + 984150*x^6 - 656100*x^7 + 164025* x^8 + 13*x^12 + 4*x^13) + exp(2*x)*(328050*x^2 - 656100*x^3 + 328050*x^4) + exp(x)*(164025*x - 164025*x^2) + exp(5*x)*(32805*x^5 - 164025*x^6 + 3280 50*x^7 - 328050*x^8 + 164025*x^9 - 32805*x^10 + 9*x^13 - 5*x^14) + exp(3*x )*(328050*x^3 - 984150*x^4 + 984150*x^5 - 328050*x^6) + 32805)/(exp(4*x)*( 32805*x^4 - 131220*x^5 + 196830*x^6 - 131220*x^7 + 32805*x^8) + exp(5*x)*( 6561*x^5 - 32805*x^6 + 65610*x^7 - 65610*x^8 + 32805*x^9 - 6561*x^10) + ex p(2*x)*(65610*x^2 - 131220*x^3 + 65610*x^4) + exp(x)*(32805*x - 32805*x^2) + exp(3*x)*(65610*x^3 - 196830*x^4 + 196830*x^5 - 65610*x^6) + 6561),x)
int((exp(4*x)*(164025*x^4 - 656100*x^5 + 984150*x^6 - 656100*x^7 + 164025* x^8 + 13*x^12 + 4*x^13) + exp(2*x)*(328050*x^2 - 656100*x^3 + 328050*x^4) + exp(x)*(164025*x - 164025*x^2) + exp(5*x)*(32805*x^5 - 164025*x^6 + 3280 50*x^7 - 328050*x^8 + 164025*x^9 - 32805*x^10 + 9*x^13 - 5*x^14) + exp(3*x )*(328050*x^3 - 984150*x^4 + 984150*x^5 - 328050*x^6) + 32805)/(exp(4*x)*( 32805*x^4 - 131220*x^5 + 196830*x^6 - 131220*x^7 + 32805*x^8) + exp(5*x)*( 6561*x^5 - 32805*x^6 + 65610*x^7 - 65610*x^8 + 32805*x^9 - 6561*x^10) + ex p(2*x)*(65610*x^2 - 131220*x^3 + 65610*x^4) + exp(x)*(32805*x - 32805*x^2) + exp(3*x)*(65610*x^3 - 196830*x^4 + 196830*x^5 - 65610*x^6) + 6561), x)