Integrand size = 190, antiderivative size = 30 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=x+\frac {1}{9} \left (20+\left (-2 e^x+\frac {1}{x^2}-x\right )^2-x\right )^2 x^2 \]
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(30)=60\).
Time = 6.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.70 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=\frac {1}{9} \left (\frac {1}{x^6}-\frac {4}{x^3}+\frac {40}{x^2}-\frac {2}{x}-71 x+404 x^2+16 e^{4 x} x^2-44 x^3+41 x^4-2 x^5+x^6+e^{3 x} \left (-32+32 x^3\right )+e^{2 x} \left (\frac {24}{x^2}-48 x+160 x^2-8 x^3+24 x^4\right )+e^x \left (-160-\frac {8}{x^4}+\frac {24}{x}+8 x-24 x^2+160 x^3-8 x^4+8 x^5\right )\right ) \]
Integrate[(-6 + 12*x^3 - 80*x^4 + 2*x^5 - 71*x^7 + 808*x^8 - 132*x^9 + 164 *x^10 - 10*x^11 + 6*x^12 + E^(4*x)*(32*x^8 + 64*x^9) + E^(3*x)*(-96*x^7 + 96*x^9 + 96*x^10) + E^(2*x)*(-48*x^4 + 48*x^5 - 48*x^7 + 224*x^8 + 296*x^9 + 80*x^10 + 48*x^11) + E^x*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*x^8 + 456*x^9 + 128*x^10 + 32*x^11 + 8*x^12))/(9*x^7),x]
(x^(-6) - 4/x^3 + 40/x^2 - 2/x - 71*x + 404*x^2 + 16*E^(4*x)*x^2 - 44*x^3 + 41*x^4 - 2*x^5 + x^6 + E^(3*x)*(-32 + 32*x^3) + E^(2*x)*(24/x^2 - 48*x + 160*x^2 - 8*x^3 + 24*x^4) + E^x*(-160 - 8/x^4 + 24/x + 8*x - 24*x^2 + 160 *x^3 - 8*x^4 + 8*x^5))/9
Leaf count is larger than twice the leaf count of optimal. \(183\) vs. \(2(30)=60\).
Time = 0.95 (sec) , antiderivative size = 183, normalized size of antiderivative = 6.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {27, 25, 2010, 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.
| \(\displaystyle \int \frac {6 x^{12}-10 x^{11}+164 x^{10}-132 x^9+808 x^8-71 x^7+2 x^5-80 x^4+12 x^3+e^{4 x} \left (64 x^9+32 x^8\right )+e^{3 x} \left (96 x^{10}+96 x^9-96 x^7\right )+e^{2 x} \left (48 x^{11}+80 x^{10}+296 x^9+224 x^8-48 x^7+48 x^5-48 x^4\right )+e^x \left (8 x^{12}+32 x^{11}+128 x^{10}+456 x^9-40 x^8-152 x^7+24 x^6-24 x^5-8 x^3+32 x^2\right )-6}{9 x^7} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int -\frac {-6 x^{12}+10 x^{11}-164 x^{10}+132 x^9-808 x^8+71 x^7-2 x^5+80 x^4-12 x^3-32 e^{4 x} \left (2 x^9+x^8\right )+96 e^{3 x} \left (-x^{10}-x^9+x^7\right )+8 e^{2 x} \left (-6 x^{11}-10 x^{10}-37 x^9-28 x^8+6 x^7-6 x^5+6 x^4\right )-8 e^x \left (x^{12}+4 x^{11}+16 x^{10}+57 x^9-5 x^8-19 x^7+3 x^6-3 x^5-x^3+4 x^2\right )+6}{x^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{9} \int \frac {-6 x^{12}+10 x^{11}-164 x^{10}+132 x^9-808 x^8+71 x^7-2 x^5+80 x^4-12 x^3-32 e^{4 x} \left (2 x^9+x^8\right )+96 e^{3 x} \left (-x^{10}-x^9+x^7\right )+8 e^{2 x} \left (-6 x^{11}-10 x^{10}-37 x^9-28 x^8+6 x^7-6 x^5+6 x^4\right )-8 e^x \left (x^{12}+4 x^{11}+16 x^{10}+57 x^9-5 x^8-19 x^7+3 x^6-3 x^5-x^3+4 x^2\right )+6}{x^7}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {1}{9} \int \left (-32 e^{4 x} x (2 x+1)-96 e^{3 x} \left (x^3+x^2-1\right )-\frac {8 e^{2 x} \left (6 x^7+10 x^6+37 x^5+28 x^4-6 x^3+6 x-6\right )}{x^3}-\frac {8 e^x \left (x^{10}+4 x^9+16 x^8+57 x^7-5 x^6-19 x^5+3 x^4-3 x^3-x+4\right )}{x^5}+\frac {-6 x^{12}+10 x^{11}-164 x^{10}+132 x^9-808 x^8+71 x^7-2 x^5+80 x^4-12 x^3+6}{x^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} \left (x^6+\frac {1}{x^6}+8 e^x x^5-2 x^5-8 e^x x^4+24 e^{2 x} x^4+41 x^4-\frac {8 e^x}{x^4}+160 e^x x^3-8 e^{2 x} x^3+32 e^{3 x} x^3-44 x^3-\frac {4}{x^3}-24 e^x x^2+160 e^{2 x} x^2+16 e^{4 x} x^2+404 x^2+\frac {24 e^{2 x}}{x^2}+\frac {40}{x^2}+8 e^x x-48 e^{2 x} x-71 x-160 e^x-32 e^{3 x}+\frac {24 e^x}{x}-\frac {2}{x}\right )\) |
Int[(-6 + 12*x^3 - 80*x^4 + 2*x^5 - 71*x^7 + 808*x^8 - 132*x^9 + 164*x^10 - 10*x^11 + 6*x^12 + E^(4*x)*(32*x^8 + 64*x^9) + E^(3*x)*(-96*x^7 + 96*x^9 + 96*x^10) + E^(2*x)*(-48*x^4 + 48*x^5 - 48*x^7 + 224*x^8 + 296*x^9 + 80* x^10 + 48*x^11) + E^x*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*x^8 + 456*x^9 + 128*x^10 + 32*x^11 + 8*x^12))/(9*x^7),x]
(-160*E^x - 32*E^(3*x) + x^(-6) - (8*E^x)/x^4 - 4/x^3 + 40/x^2 + (24*E^(2* x))/x^2 - 2/x + (24*E^x)/x - 71*x + 8*E^x*x - 48*E^(2*x)*x + 404*x^2 - 24* E^x*x^2 + 160*E^(2*x)*x^2 + 16*E^(4*x)*x^2 - 44*x^3 + 160*E^x*x^3 - 8*E^(2 *x)*x^3 + 32*E^(3*x)*x^3 + 41*x^4 - 8*E^x*x^4 + 24*E^(2*x)*x^4 - 2*x^5 + 8 *E^x*x^5 + x^6)/9
3.24.8.3.1 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_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(27)=54\).
Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.83
| method | result | size |
| risch | \(\frac {x^{6}}{9}-\frac {2 x^{5}}{9}+\frac {41 x^{4}}{9}-\frac {44 x^{3}}{9}+\frac {404 x^{2}}{9}-\frac {71 x}{9}+\frac {-2 x^{5}+40 x^{4}-4 x^{3}+1}{9 x^{6}}+\frac {16 x^{2} {\mathrm e}^{4 x}}{9}+\frac {\left (32 x^{3}-32\right ) {\mathrm e}^{3 x}}{9}+\frac {8 \left (3 x^{6}-x^{5}+20 x^{4}-6 x^{3}+3\right ) {\mathrm e}^{2 x}}{9 x^{2}}+\frac {8 \left (x^{9}-x^{8}+20 x^{7}-3 x^{6}+x^{5}-20 x^{4}+3 x^{3}-1\right ) {\mathrm e}^{x}}{9 x^{4}}\) | \(145\) |
| default | \(\frac {8 \,{\mathrm e}^{2 x} x^{4}}{3}-\frac {71 x}{9}-\frac {8 \,{\mathrm e}^{2 x} x^{3}}{9}+\frac {1}{9 x^{6}}+\frac {40}{9 x^{2}}+\frac {160 \,{\mathrm e}^{2 x} x^{2}}{9}-\frac {16 x \,{\mathrm e}^{2 x}}{3}+\frac {8 \,{\mathrm e}^{x}}{3 x}-\frac {8 \,{\mathrm e}^{x} x^{4}}{9}-\frac {8 \,{\mathrm e}^{x} x^{2}}{3}+\frac {160 \,{\mathrm e}^{x} x^{3}}{9}+\frac {8 \,{\mathrm e}^{x} x}{9}-\frac {2}{9 x}-\frac {8 \,{\mathrm e}^{x}}{9 x^{4}}-\frac {4}{9 x^{3}}+\frac {16 x^{2} {\mathrm e}^{4 x}}{9}+\frac {32 x^{3} {\mathrm e}^{3 x}}{9}+\frac {8 x^{5} {\mathrm e}^{x}}{9}-\frac {32 \,{\mathrm e}^{3 x}}{9}+\frac {x^{6}}{9}-\frac {2 x^{5}}{9}+\frac {41 x^{4}}{9}-\frac {44 x^{3}}{9}+\frac {404 x^{2}}{9}-\frac {160 \,{\mathrm e}^{x}}{9}+\frac {8 \,{\mathrm e}^{2 x}}{3 x^{2}}\) | \(168\) |
| parts | \(\frac {8 \,{\mathrm e}^{2 x} x^{4}}{3}-\frac {71 x}{9}-\frac {8 \,{\mathrm e}^{2 x} x^{3}}{9}+\frac {1}{9 x^{6}}+\frac {40}{9 x^{2}}+\frac {160 \,{\mathrm e}^{2 x} x^{2}}{9}-\frac {16 x \,{\mathrm e}^{2 x}}{3}+\frac {8 \,{\mathrm e}^{x}}{3 x}-\frac {8 \,{\mathrm e}^{x} x^{4}}{9}-\frac {8 \,{\mathrm e}^{x} x^{2}}{3}+\frac {160 \,{\mathrm e}^{x} x^{3}}{9}+\frac {8 \,{\mathrm e}^{x} x}{9}-\frac {2}{9 x}-\frac {8 \,{\mathrm e}^{x}}{9 x^{4}}-\frac {4}{9 x^{3}}+\frac {16 x^{2} {\mathrm e}^{4 x}}{9}+\frac {32 x^{3} {\mathrm e}^{3 x}}{9}+\frac {8 x^{5} {\mathrm e}^{x}}{9}-\frac {32 \,{\mathrm e}^{3 x}}{9}+\frac {x^{6}}{9}-\frac {2 x^{5}}{9}+\frac {41 x^{4}}{9}-\frac {44 x^{3}}{9}+\frac {404 x^{2}}{9}-\frac {160 \,{\mathrm e}^{x}}{9}+\frac {8 \,{\mathrm e}^{2 x}}{3 x^{2}}\) | \(168\) |
| parallelrisch | \(\frac {1+24 \,{\mathrm e}^{2 x} x^{4}-24 x^{8} {\mathrm e}^{x}-160 x^{6} {\mathrm e}^{x}-8 \,{\mathrm e}^{x} x^{2}+24 x^{5} {\mathrm e}^{x}-2 x^{11}+x^{12}-71 x^{7}+404 x^{8}+41 x^{10}-44 x^{9}-2 x^{5}+40 x^{4}-4 x^{3}+8 x^{7} {\mathrm e}^{x}+160 x^{9} {\mathrm e}^{x}+8 \,{\mathrm e}^{x} x^{11}-8 \,{\mathrm e}^{x} x^{10}+32 \,{\mathrm e}^{3 x} x^{9}+16 x^{8} {\mathrm e}^{4 x}+160 \,{\mathrm e}^{2 x} x^{8}-48 \,{\mathrm e}^{2 x} x^{7}-32 \,{\mathrm e}^{3 x} x^{6}+24 \,{\mathrm e}^{2 x} x^{10}-8 \,{\mathrm e}^{2 x} x^{9}}{9 x^{6}}\) | \(179\) |
int(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+(48*x^1 1+80*x^10+296*x^9+224*x^8-48*x^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32*x^11+1 28*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+6*x^12-1 0*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x,method =_RETURNVERBOSE)
1/9*x^6-2/9*x^5+41/9*x^4-44/9*x^3+404/9*x^2-71/9*x+1/9*(-2*x^5+40*x^4-4*x^ 3+1)/x^6+16/9*x^2*exp(x)^4+1/9*(32*x^3-32)*exp(x)^3+8/9*(3*x^6-x^5+20*x^4- 6*x^3+3)/x^2*exp(x)^2+8/9*(x^9-x^8+20*x^7-3*x^6+x^5-20*x^4+3*x^3-1)/x^4*ex p(x)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.90 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=\frac {x^{12} - 2 \, x^{11} + 41 \, x^{10} - 44 \, x^{9} + 16 \, x^{8} e^{\left (4 \, x\right )} + 404 \, x^{8} - 71 \, x^{7} - 2 \, x^{5} + 40 \, x^{4} - 4 \, x^{3} + 32 \, {\left (x^{9} - x^{6}\right )} e^{\left (3 \, x\right )} + 8 \, {\left (3 \, x^{10} - x^{9} + 20 \, x^{8} - 6 \, x^{7} + 3 \, x^{4}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{11} - x^{10} + 20 \, x^{9} - 3 \, x^{8} + x^{7} - 20 \, x^{6} + 3 \, x^{5} - x^{2}\right )} e^{x} + 1}{9 \, x^{6}} \]
integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+( 48*x^11+80*x^10+296*x^9+224*x^8-48*x^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32* x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+6* x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm=\
1/9*(x^12 - 2*x^11 + 41*x^10 - 44*x^9 + 16*x^8*e^(4*x) + 404*x^8 - 71*x^7 - 2*x^5 + 40*x^4 - 4*x^3 + 32*(x^9 - x^6)*e^(3*x) + 8*(3*x^10 - x^9 + 20*x ^8 - 6*x^7 + 3*x^4)*e^(2*x) + 8*(x^11 - x^10 + 20*x^9 - 3*x^8 + x^7 - 20*x ^6 + 3*x^5 - x^2)*e^x + 1)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.43 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=\frac {x^{6}}{9} - \frac {2 x^{5}}{9} + \frac {41 x^{4}}{9} - \frac {44 x^{3}}{9} + \frac {404 x^{2}}{9} - \frac {71 x}{9} + \frac {- 2 x^{5} + 40 x^{4} - 4 x^{3} + 1}{9 x^{6}} + \frac {11664 x^{8} e^{4 x} + \left (23328 x^{9} - 23328 x^{6}\right ) e^{3 x} + \left (17496 x^{10} - 5832 x^{9} + 116640 x^{8} - 34992 x^{7} + 17496 x^{4}\right ) e^{2 x} + \left (5832 x^{11} - 5832 x^{10} + 116640 x^{9} - 17496 x^{8} + 5832 x^{7} - 116640 x^{6} + 17496 x^{5} - 5832 x^{2}\right ) e^{x}}{6561 x^{6}} \]
integrate(1/9*((64*x**9+32*x**8)*exp(x)**4+(96*x**10+96*x**9-96*x**7)*exp( x)**3+(48*x**11+80*x**10+296*x**9+224*x**8-48*x**7+48*x**5-48*x**4)*exp(x) **2+(8*x**12+32*x**11+128*x**10+456*x**9-40*x**8-152*x**7+24*x**6-24*x**5- 8*x**3+32*x**2)*exp(x)+6*x**12-10*x**11+164*x**10-132*x**9+808*x**8-71*x** 7+2*x**5-80*x**4+12*x**3-6)/x**7,x)
x**6/9 - 2*x**5/9 + 41*x**4/9 - 44*x**3/9 + 404*x**2/9 - 71*x/9 + (-2*x**5 + 40*x**4 - 4*x**3 + 1)/(9*x**6) + (11664*x**8*exp(4*x) + (23328*x**9 - 2 3328*x**6)*exp(3*x) + (17496*x**10 - 5832*x**9 + 116640*x**8 - 34992*x**7 + 17496*x**4)*exp(2*x) + (5832*x**11 - 5832*x**10 + 116640*x**9 - 17496*x* *8 + 5832*x**7 - 116640*x**6 + 17496*x**5 - 5832*x**2)*exp(x))/(6561*x**6)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 327, normalized size of antiderivative = 10.90 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=\frac {1}{9} \, x^{6} - \frac {2}{9} \, x^{5} + \frac {41}{9} \, x^{4} - \frac {44}{9} \, x^{3} + \frac {404}{9} \, x^{2} + \frac {2}{9} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {2}{9} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + \frac {32}{81} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {32}{81} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + \frac {10}{9} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {74}{9} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {56}{9} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {8}{9} \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {32}{9} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {128}{9} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {152}{3} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - \frac {40}{9} \, {\left (x - 1\right )} e^{x} - \frac {71}{9} \, x - \frac {2}{9 \, x} + \frac {40}{9 \, x^{2}} - \frac {4}{9 \, x^{3}} + \frac {1}{9 \, x^{6}} + \frac {8}{3} \, {\rm Ei}\left (x\right ) - \frac {32}{9} \, e^{\left (3 \, x\right )} - \frac {8}{3} \, e^{\left (2 \, x\right )} - \frac {152}{9} \, e^{x} - \frac {8}{3} \, \Gamma \left (-1, -x\right ) + \frac {32}{3} \, \Gamma \left (-1, -2 \, x\right ) + \frac {64}{3} \, \Gamma \left (-2, -2 \, x\right ) - \frac {8}{9} \, \Gamma \left (-3, -x\right ) - \frac {32}{9} \, \Gamma \left (-4, -x\right ) \]
integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+( 48*x^11+80*x^10+296*x^9+224*x^8-48*x^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32* x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+6* x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm=\
1/9*x^6 - 2/9*x^5 + 41/9*x^4 - 44/9*x^3 + 404/9*x^2 + 2/9*(8*x^2 - 4*x + 1 )*e^(4*x) + 2/9*(4*x - 1)*e^(4*x) + 32/81*(9*x^3 - 9*x^2 + 6*x - 2)*e^(3*x ) + 32/81*(9*x^2 - 6*x + 2)*e^(3*x) + 4/3*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3 )*e^(2*x) + 10/9*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 74/9*(2*x^2 - 2*x + 1 )*e^(2*x) + 56/9*(2*x - 1)*e^(2*x) + 8/9*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x + 32/9*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 128/9*(x^ 3 - 3*x^2 + 6*x - 6)*e^x + 152/3*(x^2 - 2*x + 2)*e^x - 40/9*(x - 1)*e^x - 71/9*x - 2/9/x + 40/9/x^2 - 4/9/x^3 + 1/9/x^6 + 8/3*Ei(x) - 32/9*e^(3*x) - 8/3*e^(2*x) - 152/9*e^x - 8/3*gamma(-1, -x) + 32/3*gamma(-1, -2*x) + 64/3 *gamma(-2, -2*x) - 8/9*gamma(-3, -x) - 32/9*gamma(-4, -x)
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.93 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=\frac {x^{12} + 8 \, x^{11} e^{x} - 2 \, x^{11} + 24 \, x^{10} e^{\left (2 \, x\right )} - 8 \, x^{10} e^{x} + 41 \, x^{10} + 32 \, x^{9} e^{\left (3 \, x\right )} - 8 \, x^{9} e^{\left (2 \, x\right )} + 160 \, x^{9} e^{x} - 44 \, x^{9} + 16 \, x^{8} e^{\left (4 \, x\right )} + 160 \, x^{8} e^{\left (2 \, x\right )} - 24 \, x^{8} e^{x} + 404 \, x^{8} - 48 \, x^{7} e^{\left (2 \, x\right )} + 8 \, x^{7} e^{x} - 71 \, x^{7} - 32 \, x^{6} e^{\left (3 \, x\right )} - 160 \, x^{6} e^{x} + 24 \, x^{5} e^{x} - 2 \, x^{5} + 24 \, x^{4} e^{\left (2 \, x\right )} + 40 \, x^{4} - 4 \, x^{3} - 8 \, x^{2} e^{x} + 1}{9 \, x^{6}} \]
integrate(1/9*((64*x^9+32*x^8)*exp(x)^4+(96*x^10+96*x^9-96*x^7)*exp(x)^3+( 48*x^11+80*x^10+296*x^9+224*x^8-48*x^7+48*x^5-48*x^4)*exp(x)^2+(8*x^12+32* x^11+128*x^10+456*x^9-40*x^8-152*x^7+24*x^6-24*x^5-8*x^3+32*x^2)*exp(x)+6* x^12-10*x^11+164*x^10-132*x^9+808*x^8-71*x^7+2*x^5-80*x^4+12*x^3-6)/x^7,x, algorithm=\
1/9*(x^12 + 8*x^11*e^x - 2*x^11 + 24*x^10*e^(2*x) - 8*x^10*e^x + 41*x^10 + 32*x^9*e^(3*x) - 8*x^9*e^(2*x) + 160*x^9*e^x - 44*x^9 + 16*x^8*e^(4*x) + 160*x^8*e^(2*x) - 24*x^8*e^x + 404*x^8 - 48*x^7*e^(2*x) + 8*x^7*e^x - 71*x ^7 - 32*x^6*e^(3*x) - 160*x^6*e^x + 24*x^5*e^x - 2*x^5 + 24*x^4*e^(2*x) + 40*x^4 - 4*x^3 - 8*x^2*e^x + 1)/x^6
Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 4.73 \[ \int \frac {-6+12 x^3-80 x^4+2 x^5-71 x^7+808 x^8-132 x^9+164 x^{10}-10 x^{11}+6 x^{12}+e^{4 x} \left (32 x^8+64 x^9\right )+e^{3 x} \left (-96 x^7+96 x^9+96 x^{10}\right )+e^{2 x} \left (-48 x^4+48 x^5-48 x^7+224 x^8+296 x^9+80 x^{10}+48 x^{11}\right )+e^x \left (32 x^2-8 x^3-24 x^5+24 x^6-152 x^7-40 x^8+456 x^9+128 x^{10}+32 x^{11}+8 x^{12}\right )}{9 x^7} \, dx=x^2\,\left (\frac {160\,{\mathrm {e}}^{2\,x}}{9}+\frac {16\,{\mathrm {e}}^{4\,x}}{9}-\frac {8\,{\mathrm {e}}^x}{3}+\frac {404}{9}\right )-\frac {160\,{\mathrm {e}}^x}{9}-x^3\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{9}-\frac {32\,{\mathrm {e}}^{3\,x}}{9}-\frac {160\,{\mathrm {e}}^x}{9}+\frac {44}{9}\right )-\frac {32\,{\mathrm {e}}^{3\,x}}{9}-x\,\left (\frac {16\,{\mathrm {e}}^{2\,x}}{3}-\frac {8\,{\mathrm {e}}^x}{9}+\frac {71}{9}\right )+x^4\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{3}-\frac {8\,{\mathrm {e}}^x}{9}+\frac {41}{9}\right )+x^5\,\left (\frac {8\,{\mathrm {e}}^x}{9}-\frac {2}{9}\right )+\frac {x^6}{9}+\frac {x^4\,\left (\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {40}{9}\right )-\frac {8\,x^2\,{\mathrm {e}}^x}{9}+x^5\,\left (\frac {8\,{\mathrm {e}}^x}{3}-\frac {2}{9}\right )-\frac {4\,x^3}{9}+\frac {1}{9}}{x^6} \]
int(((exp(4*x)*(32*x^8 + 64*x^9))/9 + (exp(2*x)*(48*x^5 - 48*x^4 - 48*x^7 + 224*x^8 + 296*x^9 + 80*x^10 + 48*x^11))/9 + (exp(3*x)*(96*x^9 - 96*x^7 + 96*x^10))/9 + (exp(x)*(32*x^2 - 8*x^3 - 24*x^5 + 24*x^6 - 152*x^7 - 40*x^ 8 + 456*x^9 + 128*x^10 + 32*x^11 + 8*x^12))/9 + (4*x^3)/3 - (80*x^4)/9 + ( 2*x^5)/9 - (71*x^7)/9 + (808*x^8)/9 - (44*x^9)/3 + (164*x^10)/9 - (10*x^11 )/9 + (2*x^12)/3 - 2/3)/x^7,x)
x^2*((160*exp(2*x))/9 + (16*exp(4*x))/9 - (8*exp(x))/3 + 404/9) - (160*exp (x))/9 - x^3*((8*exp(2*x))/9 - (32*exp(3*x))/9 - (160*exp(x))/9 + 44/9) - (32*exp(3*x))/9 - x*((16*exp(2*x))/3 - (8*exp(x))/9 + 71/9) + x^4*((8*exp( 2*x))/3 - (8*exp(x))/9 + 41/9) + x^5*((8*exp(x))/9 - 2/9) + x^6/9 + (x^4*( (8*exp(2*x))/3 + 40/9) - (8*x^2*exp(x))/9 + x^5*((8*exp(x))/3 - 2/9) - (4* x^3)/9 + 1/9)/x^6