Integrand size = 389, antiderivative size = 34 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{16 x^2 \left (2-x+\frac {3}{4} (3+x)^2 \left (-e^x+x\right )-\log (2)\right )^2} \] Output:
5/16/x^2/(3/4*(x-exp(x))*(3+x)^2+2-ln(2)-x)^2
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).
Time = 2.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.74 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=-\frac {5 \left (29-50 x^2-24 x^3-3 x^4+x (8+\log (16))+\log (1048576)\right )^3}{x^2 \left (-29+50 x^2+24 x^3+3 x^4-20 \log (2)-4 x (2+\log (2))\right )^3 \left (8+23 x+18 x^2+3 x^3-3 e^x (3+x)^2-\log (16)\right )^2} \] Input:
Integrate[(80 + 460*x + 540*x^2 + 120*x^3 + E^x*(-270 - 630*x - 270*x^2 - 30*x^3) - 40*Log[2])/(-512*x^3 - 4416*x^4 - 16152*x^5 - 32615*x^6 - 39654* x^7 - 29709*x^8 - 13500*x^9 - 3537*x^10 - 486*x^11 - 27*x^12 + E^(3*x)*(19 683*x^3 + 39366*x^4 + 32805*x^5 + 14580*x^6 + 3645*x^7 + 486*x^8 + 27*x^9) + (768*x^3 + 4416*x^4 + 9804*x^5 + 10512*x^6 + 5544*x^7 + 1296*x^8 + 108* x^9)*Log[2] + (-384*x^3 - 1104*x^4 - 864*x^5 - 144*x^6)*Log[2]^2 + 64*x^3* Log[2]^3 + E^(2*x)*(-17496*x^3 - 73629*x^4 - 118098*x^5 - 95175*x^6 - 4266 0*x^7 - 10827*x^8 - 1458*x^9 - 81*x^10 + (8748*x^3 + 11664*x^4 + 5832*x^5 + 1296*x^6 + 108*x^7)*Log[2]) + E^x*(5184*x^3 + 33264*x^4 + 86625*x^5 + 11 8386*x^6 + 92079*x^7 + 41580*x^8 + 10719*x^9 + 1458*x^10 + 81*x^11 + (-518 4*x^3 - 18360*x^4 - 22176*x^5 - 11376*x^6 - 2592*x^7 - 216*x^8)*Log[2] + ( 1296*x^3 + 864*x^4 + 144*x^5)*Log[2]^2)),x]
Output:
(-5*(29 - 50*x^2 - 24*x^3 - 3*x^4 + x*(8 + Log[16]) + Log[1048576])^3)/(x^ 2*(-29 + 50*x^2 + 24*x^3 + 3*x^4 - 20*Log[2] - 4*x*(2 + Log[2]))^3*(8 + 23 *x + 18*x^2 + 3*x^3 - 3*E^x*(3 + x)^2 - Log[16])^2)
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.
| \(\displaystyle \int \frac {120 x^3+540 x^2+e^x \left (-30 x^3-270 x^2-630 x-270\right )+460 x+80-40 \log (2)}{-27 x^{12}-486 x^{11}-3537 x^{10}-13500 x^9-29709 x^8-39654 x^7-32615 x^6-16152 x^5-4416 x^4-512 x^3+64 x^3 \log ^3(2)+\left (-144 x^6-864 x^5-1104 x^4-384 x^3\right ) \log ^2(2)+e^{3 x} \left (27 x^9+486 x^8+3645 x^7+14580 x^6+32805 x^5+39366 x^4+19683 x^3\right )+\left (108 x^9+1296 x^8+5544 x^7+10512 x^6+9804 x^5+4416 x^4+768 x^3\right ) \log (2)+e^{2 x} \left (-81 x^{10}-1458 x^9-10827 x^8-42660 x^7-95175 x^6-118098 x^5-73629 x^4-17496 x^3+\left (108 x^7+1296 x^6+5832 x^5+11664 x^4+8748 x^3\right ) \log (2)\right )+e^x \left (81 x^{11}+1458 x^{10}+10719 x^9+41580 x^8+92079 x^7+118386 x^6+86625 x^5+33264 x^4+5184 x^3+\left (144 x^5+864 x^4+1296 x^3\right ) \log ^2(2)+\left (-216 x^8-2592 x^7-11376 x^6-22176 x^5-18360 x^4-5184 x^3\right ) \log (2)\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {120 x^3+540 x^2+e^x \left (-30 x^3-270 x^2-630 x-270\right )+460 x+80-40 \log (2)}{-27 x^{12}-486 x^{11}-3537 x^{10}-13500 x^9-29709 x^8-39654 x^7-32615 x^6-16152 x^5-4416 x^4+x^3 \left (64 \log ^3(2)-512\right )+\left (-144 x^6-864 x^5-1104 x^4-384 x^3\right ) \log ^2(2)+e^{3 x} \left (27 x^9+486 x^8+3645 x^7+14580 x^6+32805 x^5+39366 x^4+19683 x^3\right )+\left (108 x^9+1296 x^8+5544 x^7+10512 x^6+9804 x^5+4416 x^4+768 x^3\right ) \log (2)+e^{2 x} \left (-81 x^{10}-1458 x^9-10827 x^8-42660 x^7-95175 x^6-118098 x^5-73629 x^4-17496 x^3+\left (108 x^7+1296 x^6+5832 x^5+11664 x^4+8748 x^3\right ) \log (2)\right )+e^x \left (81 x^{11}+1458 x^{10}+10719 x^9+41580 x^8+92079 x^7+118386 x^6+86625 x^5+33264 x^4+5184 x^3+\left (144 x^5+864 x^4+1296 x^3\right ) \log ^2(2)+\left (-216 x^8-2592 x^7-11376 x^6-22176 x^5-18360 x^4-5184 x^3\right ) \log (2)\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 \left (-12 x^3-54 x^2+3 e^x \left (x^3+9 x^2+21 x+9\right )-46 x-8 \left (1-\frac {\log (2)}{2}\right )\right )}{x^3 \left (3 x^3+18 x^2+23 x-3 e^x (x+3)^2+8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int -\frac {12 x^3+54 x^2+46 x-3 e^x \left (x^3+9 x^2+21 x+9\right )-\log (16)+8}{x^3 \left (3 x^3+18 x^2+23 x-3 e^x (x+3)^2+4 (2-\log (2))\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -10 \int \frac {12 x^3+54 x^2+46 x-3 e^x \left (x^3+9 x^2+21 x+9\right )-\log (16)+8}{x^3 \left (3 x^3+18 x^2+23 x-3 e^x (x+3)^2+4 (2-\log (2))\right )^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -10 \int \frac {12 x^3+54 x^2+46 x-3 e^x \left (x^3+9 x^2+21 x+9\right )+8 \left (1-\frac {\log (2)}{2}\right )}{x^3 \left (3 x^3+18 x^2+23 x-3 e^x (x+3)^2+4 (2-\log (2))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -10 \int \left (\frac {x^2+6 x+3}{x^3 (x+3) \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^2}+\frac {3 x^4+24 x^3+50 x^2-4 (2+\log (2)) x-20 \log (2)-29}{x^2 (x+3) \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -10 \left (15 \int \frac {1}{\left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx-\frac {1}{3} (29+20 \log (2)) \int \frac {1}{x^2 \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx+\frac {1}{9} (5+\log (256)) \int \frac {1}{x \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx+3 \int \frac {x}{\left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx+\frac {8}{9} (5-\log (2)) \int \frac {1}{(x+3) \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^3}dx+\int \frac {1}{x^3 \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^2}dx+\frac {5}{3} \int \frac {1}{x^2 \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^2}dx-\frac {2}{9} \int \frac {1}{x \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^2}dx+\frac {2}{9} \int \frac {1}{(x+3) \left (-3 x^3+3 e^x x^2-18 x^2+18 e^x x-23 x+27 e^x-8 \left (1-\frac {\log (2)}{2}\right )\right )^2}dx\right )\) |
Input:
Int[(80 + 460*x + 540*x^2 + 120*x^3 + E^x*(-270 - 630*x - 270*x^2 - 30*x^3 ) - 40*Log[2])/(-512*x^3 - 4416*x^4 - 16152*x^5 - 32615*x^6 - 39654*x^7 - 29709*x^8 - 13500*x^9 - 3537*x^10 - 486*x^11 - 27*x^12 + E^(3*x)*(19683*x^ 3 + 39366*x^4 + 32805*x^5 + 14580*x^6 + 3645*x^7 + 486*x^8 + 27*x^9) + (76 8*x^3 + 4416*x^4 + 9804*x^5 + 10512*x^6 + 5544*x^7 + 1296*x^8 + 108*x^9)*L og[2] + (-384*x^3 - 1104*x^4 - 864*x^5 - 144*x^6)*Log[2]^2 + 64*x^3*Log[2] ^3 + E^(2*x)*(-17496*x^3 - 73629*x^4 - 118098*x^5 - 95175*x^6 - 42660*x^7 - 10827*x^8 - 1458*x^9 - 81*x^10 + (8748*x^3 + 11664*x^4 + 5832*x^5 + 1296 *x^6 + 108*x^7)*Log[2]) + E^x*(5184*x^3 + 33264*x^4 + 86625*x^5 + 118386*x ^6 + 92079*x^7 + 41580*x^8 + 10719*x^9 + 1458*x^10 + 81*x^11 + (-5184*x^3 - 18360*x^4 - 22176*x^5 - 11376*x^6 - 2592*x^7 - 216*x^8)*Log[2] + (1296*x ^3 + 864*x^4 + 144*x^5)*Log[2]^2)),x]
Output:
$Aborted
Time = 1.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {5}{x^{2} \left (3 \,{\mathrm e}^{x} x^{2}-3 x^{3}+18 \,{\mathrm e}^{x} x -18 x^{2}+27 \,{\mathrm e}^{x}+4 \ln \left (2\right )-23 x -8\right )^{2}}\) | \(43\) |
| parallelrisch | \(\frac {5}{x^{2} \left (64+368 x -18 x^{5} {\mathrm e}^{x}+108 \,{\mathrm e}^{2 x} x^{3}-216 \,{\mathrm e}^{x} x^{4}+216 \,{\mathrm e}^{x} \ln \left (2\right )+486 \,{\mathrm e}^{2 x} x^{2}-24 x^{3} \ln \left (2\right )-184 x \ln \left (2\right )-948 \,{\mathrm e}^{x} x^{3}+24 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+972 x \,{\mathrm e}^{2 x}-1848 \,{\mathrm e}^{x} x^{2}+729 \,{\mathrm e}^{2 x}-1530 \,{\mathrm e}^{x} x -144 x^{2} \ln \left (2\right )+16 \ln \left (2\right )^{2}+9 x^{6}+817 x^{2}+876 x^{3}-64 \ln \left (2\right )+462 x^{4}+108 x^{5}-432 \,{\mathrm e}^{x}+144 x \ln \left (2\right ) {\mathrm e}^{x}+9 \,{\mathrm e}^{2 x} x^{4}\right )}\) | \(166\) |
Input:
int(((-30*x^3-270*x^2-630*x-270)*exp(x)-40*ln(2)+120*x^3+540*x^2+460*x+80) /((27*x^9+486*x^8+3645*x^7+14580*x^6+32805*x^5+39366*x^4+19683*x^3)*exp(x) ^3+((108*x^7+1296*x^6+5832*x^5+11664*x^4+8748*x^3)*ln(2)-81*x^10-1458*x^9- 10827*x^8-42660*x^7-95175*x^6-118098*x^5-73629*x^4-17496*x^3)*exp(x)^2+((1 44*x^5+864*x^4+1296*x^3)*ln(2)^2+(-216*x^8-2592*x^7-11376*x^6-22176*x^5-18 360*x^4-5184*x^3)*ln(2)+81*x^11+1458*x^10+10719*x^9+41580*x^8+92079*x^7+11 8386*x^6+86625*x^5+33264*x^4+5184*x^3)*exp(x)+64*x^3*ln(2)^3+(-144*x^6-864 *x^5-1104*x^4-384*x^3)*ln(2)^2+(108*x^9+1296*x^8+5544*x^7+10512*x^6+9804*x ^5+4416*x^4+768*x^3)*ln(2)-27*x^12-486*x^11-3537*x^10-13500*x^9-29709*x^8- 39654*x^7-32615*x^6-16152*x^5-4416*x^4-512*x^3),x,method=_RETURNVERBOSE)
Output:
5/x^2/(3*exp(x)*x^2-3*x^3+18*exp(x)*x-18*x^2+27*exp(x)+4*ln(2)-23*x-8)^2
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.62 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{9 \, x^{8} + 108 \, x^{7} + 462 \, x^{6} + 876 \, x^{5} + 817 \, x^{4} + 16 \, x^{2} \log \left (2\right )^{2} + 368 \, x^{3} + 64 \, x^{2} + 9 \, {\left (x^{6} + 12 \, x^{5} + 54 \, x^{4} + 108 \, x^{3} + 81 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (3 \, x^{7} + 36 \, x^{6} + 158 \, x^{5} + 308 \, x^{4} + 255 \, x^{3} + 72 \, x^{2} - 4 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2\right )\right )} e^{x} - 8 \, {\left (3 \, x^{5} + 18 \, x^{4} + 23 \, x^{3} + 8 \, x^{2}\right )} \log \left (2\right )} \] Input:
integrate(((-30*x^3-270*x^2-630*x-270)*exp(x)-40*log(2)+120*x^3+540*x^2+46 0*x+80)/((27*x^9+486*x^8+3645*x^7+14580*x^6+32805*x^5+39366*x^4+19683*x^3) *exp(x)^3+((108*x^7+1296*x^6+5832*x^5+11664*x^4+8748*x^3)*log(2)-81*x^10-1 458*x^9-10827*x^8-42660*x^7-95175*x^6-118098*x^5-73629*x^4-17496*x^3)*exp( x)^2+((144*x^5+864*x^4+1296*x^3)*log(2)^2+(-216*x^8-2592*x^7-11376*x^6-221 76*x^5-18360*x^4-5184*x^3)*log(2)+81*x^11+1458*x^10+10719*x^9+41580*x^8+92 079*x^7+118386*x^6+86625*x^5+33264*x^4+5184*x^3)*exp(x)+64*x^3*log(2)^3+(- 144*x^6-864*x^5-1104*x^4-384*x^3)*log(2)^2+(108*x^9+1296*x^8+5544*x^7+1051 2*x^6+9804*x^5+4416*x^4+768*x^3)*log(2)-27*x^12-486*x^11-3537*x^10-13500*x ^9-29709*x^8-39654*x^7-32615*x^6-16152*x^5-4416*x^4-512*x^3),x, algorithm= "fricas")
Output:
5/(9*x^8 + 108*x^7 + 462*x^6 + 876*x^5 + 817*x^4 + 16*x^2*log(2)^2 + 368*x ^3 + 64*x^2 + 9*(x^6 + 12*x^5 + 54*x^4 + 108*x^3 + 81*x^2)*e^(2*x) - 6*(3* x^7 + 36*x^6 + 158*x^5 + 308*x^4 + 255*x^3 + 72*x^2 - 4*(x^4 + 6*x^3 + 9*x ^2)*log(2))*e^x - 8*(3*x^5 + 18*x^4 + 23*x^3 + 8*x^2)*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (31) = 62\).
Time = 0.67 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.00 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{9 x^{8} + 108 x^{7} + 462 x^{6} - 24 x^{5} \log {\left (2 \right )} + 876 x^{5} - 144 x^{4} \log {\left (2 \right )} + 817 x^{4} - 184 x^{3} \log {\left (2 \right )} + 368 x^{3} - 64 x^{2} \log {\left (2 \right )} + 16 x^{2} \log {\left (2 \right )}^{2} + 64 x^{2} + \left (9 x^{6} + 108 x^{5} + 486 x^{4} + 972 x^{3} + 729 x^{2}\right ) e^{2 x} + \left (- 18 x^{7} - 216 x^{6} - 948 x^{5} - 1848 x^{4} + 24 x^{4} \log {\left (2 \right )} - 1530 x^{3} + 144 x^{3} \log {\left (2 \right )} - 432 x^{2} + 216 x^{2} \log {\left (2 \right )}\right ) e^{x}} \] Input:
integrate(((-30*x**3-270*x**2-630*x-270)*exp(x)-40*ln(2)+120*x**3+540*x**2 +460*x+80)/((27*x**9+486*x**8+3645*x**7+14580*x**6+32805*x**5+39366*x**4+1 9683*x**3)*exp(x)**3+((108*x**7+1296*x**6+5832*x**5+11664*x**4+8748*x**3)* ln(2)-81*x**10-1458*x**9-10827*x**8-42660*x**7-95175*x**6-118098*x**5-7362 9*x**4-17496*x**3)*exp(x)**2+((144*x**5+864*x**4+1296*x**3)*ln(2)**2+(-216 *x**8-2592*x**7-11376*x**6-22176*x**5-18360*x**4-5184*x**3)*ln(2)+81*x**11 +1458*x**10+10719*x**9+41580*x**8+92079*x**7+118386*x**6+86625*x**5+33264* x**4+5184*x**3)*exp(x)+64*x**3*ln(2)**3+(-144*x**6-864*x**5-1104*x**4-384* x**3)*ln(2)**2+(108*x**9+1296*x**8+5544*x**7+10512*x**6+9804*x**5+4416*x** 4+768*x**3)*ln(2)-27*x**12-486*x**11-3537*x**10-13500*x**9-29709*x**8-3965 4*x**7-32615*x**6-16152*x**5-4416*x**4-512*x**3),x)
Output:
5/(9*x**8 + 108*x**7 + 462*x**6 - 24*x**5*log(2) + 876*x**5 - 144*x**4*log (2) + 817*x**4 - 184*x**3*log(2) + 368*x**3 - 64*x**2*log(2) + 16*x**2*log (2)**2 + 64*x**2 + (9*x**6 + 108*x**5 + 486*x**4 + 972*x**3 + 729*x**2)*ex p(2*x) + (-18*x**7 - 216*x**6 - 948*x**5 - 1848*x**4 + 24*x**4*log(2) - 15 30*x**3 + 144*x**3*log(2) - 432*x**2 + 216*x**2*log(2))*exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (29) = 58\).
Time = 0.94 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.26 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{9 \, x^{8} + 108 \, x^{7} + 462 \, x^{6} - 12 \, x^{5} {\left (2 \, \log \left (2\right ) - 73\right )} - x^{4} {\left (144 \, \log \left (2\right ) - 817\right )} - 184 \, x^{3} {\left (\log \left (2\right ) - 2\right )} + 16 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 4\right )} x^{2} + 9 \, {\left (x^{6} + 12 \, x^{5} + 54 \, x^{4} + 108 \, x^{3} + 81 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (3 \, x^{7} + 36 \, x^{6} + 158 \, x^{5} - 4 \, x^{4} {\left (\log \left (2\right ) - 77\right )} - 3 \, x^{3} {\left (8 \, \log \left (2\right ) - 85\right )} - 36 \, x^{2} {\left (\log \left (2\right ) - 2\right )}\right )} e^{x}} \] Input:
integrate(((-30*x^3-270*x^2-630*x-270)*exp(x)-40*log(2)+120*x^3+540*x^2+46 0*x+80)/((27*x^9+486*x^8+3645*x^7+14580*x^6+32805*x^5+39366*x^4+19683*x^3) *exp(x)^3+((108*x^7+1296*x^6+5832*x^5+11664*x^4+8748*x^3)*log(2)-81*x^10-1 458*x^9-10827*x^8-42660*x^7-95175*x^6-118098*x^5-73629*x^4-17496*x^3)*exp( x)^2+((144*x^5+864*x^4+1296*x^3)*log(2)^2+(-216*x^8-2592*x^7-11376*x^6-221 76*x^5-18360*x^4-5184*x^3)*log(2)+81*x^11+1458*x^10+10719*x^9+41580*x^8+92 079*x^7+118386*x^6+86625*x^5+33264*x^4+5184*x^3)*exp(x)+64*x^3*log(2)^3+(- 144*x^6-864*x^5-1104*x^4-384*x^3)*log(2)^2+(108*x^9+1296*x^8+5544*x^7+1051 2*x^6+9804*x^5+4416*x^4+768*x^3)*log(2)-27*x^12-486*x^11-3537*x^10-13500*x ^9-29709*x^8-39654*x^7-32615*x^6-16152*x^5-4416*x^4-512*x^3),x, algorithm= "maxima")
Output:
5/(9*x^8 + 108*x^7 + 462*x^6 - 12*x^5*(2*log(2) - 73) - x^4*(144*log(2) - 817) - 184*x^3*(log(2) - 2) + 16*(log(2)^2 - 4*log(2) + 4)*x^2 + 9*(x^6 + 12*x^5 + 54*x^4 + 108*x^3 + 81*x^2)*e^(2*x) - 6*(3*x^7 + 36*x^6 + 158*x^5 - 4*x^4*(log(2) - 77) - 3*x^3*(8*log(2) - 85) - 36*x^2*(log(2) - 2))*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (29) = 58\).
Time = 0.48 (sec) , antiderivative size = 191, normalized size of antiderivative = 5.62 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{9 \, x^{8} - 18 \, x^{7} e^{x} + 108 \, x^{7} + 9 \, x^{6} e^{\left (2 \, x\right )} - 216 \, x^{6} e^{x} + 462 \, x^{6} + 108 \, x^{5} e^{\left (2 \, x\right )} - 948 \, x^{5} e^{x} - 24 \, x^{5} \log \left (2\right ) + 24 \, x^{4} e^{x} \log \left (2\right ) + 876 \, x^{5} + 486 \, x^{4} e^{\left (2 \, x\right )} - 1848 \, x^{4} e^{x} - 144 \, x^{4} \log \left (2\right ) + 144 \, x^{3} e^{x} \log \left (2\right ) + 817 \, x^{4} + 972 \, x^{3} e^{\left (2 \, x\right )} - 1530 \, x^{3} e^{x} - 184 \, x^{3} \log \left (2\right ) + 216 \, x^{2} e^{x} \log \left (2\right ) + 16 \, x^{2} \log \left (2\right )^{2} + 368 \, x^{3} + 729 \, x^{2} e^{\left (2 \, x\right )} - 432 \, x^{2} e^{x} - 64 \, x^{2} \log \left (2\right ) + 64 \, x^{2}} \] Input:
integrate(((-30*x^3-270*x^2-630*x-270)*exp(x)-40*log(2)+120*x^3+540*x^2+46 0*x+80)/((27*x^9+486*x^8+3645*x^7+14580*x^6+32805*x^5+39366*x^4+19683*x^3) *exp(x)^3+((108*x^7+1296*x^6+5832*x^5+11664*x^4+8748*x^3)*log(2)-81*x^10-1 458*x^9-10827*x^8-42660*x^7-95175*x^6-118098*x^5-73629*x^4-17496*x^3)*exp( x)^2+((144*x^5+864*x^4+1296*x^3)*log(2)^2+(-216*x^8-2592*x^7-11376*x^6-221 76*x^5-18360*x^4-5184*x^3)*log(2)+81*x^11+1458*x^10+10719*x^9+41580*x^8+92 079*x^7+118386*x^6+86625*x^5+33264*x^4+5184*x^3)*exp(x)+64*x^3*log(2)^3+(- 144*x^6-864*x^5-1104*x^4-384*x^3)*log(2)^2+(108*x^9+1296*x^8+5544*x^7+1051 2*x^6+9804*x^5+4416*x^4+768*x^3)*log(2)-27*x^12-486*x^11-3537*x^10-13500*x ^9-29709*x^8-39654*x^7-32615*x^6-16152*x^5-4416*x^4-512*x^3),x, algorithm= "giac")
Output:
5/(9*x^8 - 18*x^7*e^x + 108*x^7 + 9*x^6*e^(2*x) - 216*x^6*e^x + 462*x^6 + 108*x^5*e^(2*x) - 948*x^5*e^x - 24*x^5*log(2) + 24*x^4*e^x*log(2) + 876*x^ 5 + 486*x^4*e^(2*x) - 1848*x^4*e^x - 144*x^4*log(2) + 144*x^3*e^x*log(2) + 817*x^4 + 972*x^3*e^(2*x) - 1530*x^3*e^x - 184*x^3*log(2) + 216*x^2*e^x*l og(2) + 16*x^2*log(2)^2 + 368*x^3 + 729*x^2*e^(2*x) - 432*x^2*e^x - 64*x^2 *log(2) + 64*x^2)
Timed out. \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\int -\frac {460\,x-40\,\ln \left (2\right )+540\,x^2+120\,x^3-{\mathrm {e}}^x\,\left (30\,x^3+270\,x^2+630\,x+270\right )+80}{{\ln \left (2\right )}^2\,\left (144\,x^6+864\,x^5+1104\,x^4+384\,x^3\right )-{\mathrm {e}}^x\,\left ({\ln \left (2\right )}^2\,\left (144\,x^5+864\,x^4+1296\,x^3\right )-\ln \left (2\right )\,\left (216\,x^8+2592\,x^7+11376\,x^6+22176\,x^5+18360\,x^4+5184\,x^3\right )+5184\,x^3+33264\,x^4+86625\,x^5+118386\,x^6+92079\,x^7+41580\,x^8+10719\,x^9+1458\,x^{10}+81\,x^{11}\right )-64\,x^3\,{\ln \left (2\right )}^3-{\mathrm {e}}^{3\,x}\,\left (27\,x^9+486\,x^8+3645\,x^7+14580\,x^6+32805\,x^5+39366\,x^4+19683\,x^3\right )-\ln \left (2\right )\,\left (108\,x^9+1296\,x^8+5544\,x^7+10512\,x^6+9804\,x^5+4416\,x^4+768\,x^3\right )+512\,x^3+4416\,x^4+16152\,x^5+32615\,x^6+39654\,x^7+29709\,x^8+13500\,x^9+3537\,x^{10}+486\,x^{11}+27\,x^{12}+{\mathrm {e}}^{2\,x}\,\left (17496\,x^3-\ln \left (2\right )\,\left (108\,x^7+1296\,x^6+5832\,x^5+11664\,x^4+8748\,x^3\right )+73629\,x^4+118098\,x^5+95175\,x^6+42660\,x^7+10827\,x^8+1458\,x^9+81\,x^{10}\right )} \,d x \] Input:
int(-(460*x - 40*log(2) + 540*x^2 + 120*x^3 - exp(x)*(630*x + 270*x^2 + 30 *x^3 + 270) + 80)/(log(2)^2*(384*x^3 + 1104*x^4 + 864*x^5 + 144*x^6) - exp (x)*(log(2)^2*(1296*x^3 + 864*x^4 + 144*x^5) - log(2)*(5184*x^3 + 18360*x^ 4 + 22176*x^5 + 11376*x^6 + 2592*x^7 + 216*x^8) + 5184*x^3 + 33264*x^4 + 8 6625*x^5 + 118386*x^6 + 92079*x^7 + 41580*x^8 + 10719*x^9 + 1458*x^10 + 81 *x^11) - 64*x^3*log(2)^3 - exp(3*x)*(19683*x^3 + 39366*x^4 + 32805*x^5 + 1 4580*x^6 + 3645*x^7 + 486*x^8 + 27*x^9) - log(2)*(768*x^3 + 4416*x^4 + 980 4*x^5 + 10512*x^6 + 5544*x^7 + 1296*x^8 + 108*x^9) + 512*x^3 + 4416*x^4 + 16152*x^5 + 32615*x^6 + 39654*x^7 + 29709*x^8 + 13500*x^9 + 3537*x^10 + 48 6*x^11 + 27*x^12 + exp(2*x)*(17496*x^3 - log(2)*(8748*x^3 + 11664*x^4 + 58 32*x^5 + 1296*x^6 + 108*x^7) + 73629*x^4 + 118098*x^5 + 95175*x^6 + 42660* x^7 + 10827*x^8 + 1458*x^9 + 81*x^10)),x)
Output:
int(-(460*x - 40*log(2) + 540*x^2 + 120*x^3 - exp(x)*(630*x + 270*x^2 + 30 *x^3 + 270) + 80)/(log(2)^2*(384*x^3 + 1104*x^4 + 864*x^5 + 144*x^6) - exp (x)*(log(2)^2*(1296*x^3 + 864*x^4 + 144*x^5) - log(2)*(5184*x^3 + 18360*x^ 4 + 22176*x^5 + 11376*x^6 + 2592*x^7 + 216*x^8) + 5184*x^3 + 33264*x^4 + 8 6625*x^5 + 118386*x^6 + 92079*x^7 + 41580*x^8 + 10719*x^9 + 1458*x^10 + 81 *x^11) - 64*x^3*log(2)^3 - exp(3*x)*(19683*x^3 + 39366*x^4 + 32805*x^5 + 1 4580*x^6 + 3645*x^7 + 486*x^8 + 27*x^9) - log(2)*(768*x^3 + 4416*x^4 + 980 4*x^5 + 10512*x^6 + 5544*x^7 + 1296*x^8 + 108*x^9) + 512*x^3 + 4416*x^4 + 16152*x^5 + 32615*x^6 + 39654*x^7 + 29709*x^8 + 13500*x^9 + 3537*x^10 + 48 6*x^11 + 27*x^12 + exp(2*x)*(17496*x^3 - log(2)*(8748*x^3 + 11664*x^4 + 58 32*x^5 + 1296*x^6 + 108*x^7) + 73629*x^4 + 118098*x^5 + 95175*x^6 + 42660* x^7 + 10827*x^8 + 1458*x^9 + 81*x^10)), x)
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 5.26 \[ \int \frac {80+460 x+540 x^2+120 x^3+e^x \left (-270-630 x-270 x^2-30 x^3\right )-40 \log (2)}{-512 x^3-4416 x^4-16152 x^5-32615 x^6-39654 x^7-29709 x^8-13500 x^9-3537 x^{10}-486 x^{11}-27 x^{12}+e^{3 x} \left (19683 x^3+39366 x^4+32805 x^5+14580 x^6+3645 x^7+486 x^8+27 x^9\right )+\left (768 x^3+4416 x^4+9804 x^5+10512 x^6+5544 x^7+1296 x^8+108 x^9\right ) \log (2)+\left (-384 x^3-1104 x^4-864 x^5-144 x^6\right ) \log ^2(2)+64 x^3 \log ^3(2)+e^{2 x} \left (-17496 x^3-73629 x^4-118098 x^5-95175 x^6-42660 x^7-10827 x^8-1458 x^9-81 x^{10}+\left (8748 x^3+11664 x^4+5832 x^5+1296 x^6+108 x^7\right ) \log (2)\right )+e^x \left (5184 x^3+33264 x^4+86625 x^5+118386 x^6+92079 x^7+41580 x^8+10719 x^9+1458 x^{10}+81 x^{11}+\left (-5184 x^3-18360 x^4-22176 x^5-11376 x^6-2592 x^7-216 x^8\right ) \log (2)+\left (1296 x^3+864 x^4+144 x^5\right ) \log ^2(2)\right )} \, dx=\frac {5}{x^{2} \left (64+368 x +108 e^{2 x} x^{3}+486 e^{2 x} x^{2}-216 e^{x} x^{4}-948 e^{x} x^{3}-144 \,\mathrm {log}\left (2\right ) x^{2}-184 \,\mathrm {log}\left (2\right ) x -18 e^{x} x^{5}+817 x^{2}+144 e^{x} \mathrm {log}\left (2\right ) x +16 \mathrm {log}\left (2\right )^{2}+108 x^{5}+876 x^{3}+9 e^{2 x} x^{4}-1848 e^{x} x^{2}+9 x^{6}+972 e^{2 x} x -24 \,\mathrm {log}\left (2\right ) x^{3}+24 e^{x} \mathrm {log}\left (2\right ) x^{2}-432 e^{x}-64 \,\mathrm {log}\left (2\right )+729 e^{2 x}+462 x^{4}+216 e^{x} \mathrm {log}\left (2\right )-1530 e^{x} x \right )} \] Input:
int(((-30*x^3-270*x^2-630*x-270)*exp(x)-40*log(2)+120*x^3+540*x^2+460*x+80 )/((27*x^9+486*x^8+3645*x^7+14580*x^6+32805*x^5+39366*x^4+19683*x^3)*exp(x )^3+((108*x^7+1296*x^6+5832*x^5+11664*x^4+8748*x^3)*log(2)-81*x^10-1458*x^ 9-10827*x^8-42660*x^7-95175*x^6-118098*x^5-73629*x^4-17496*x^3)*exp(x)^2+( (144*x^5+864*x^4+1296*x^3)*log(2)^2+(-216*x^8-2592*x^7-11376*x^6-22176*x^5 -18360*x^4-5184*x^3)*log(2)+81*x^11+1458*x^10+10719*x^9+41580*x^8+92079*x^ 7+118386*x^6+86625*x^5+33264*x^4+5184*x^3)*exp(x)+64*x^3*log(2)^3+(-144*x^ 6-864*x^5-1104*x^4-384*x^3)*log(2)^2+(108*x^9+1296*x^8+5544*x^7+10512*x^6+ 9804*x^5+4416*x^4+768*x^3)*log(2)-27*x^12-486*x^11-3537*x^10-13500*x^9-297 09*x^8-39654*x^7-32615*x^6-16152*x^5-4416*x^4-512*x^3),x)
Output:
5/(x**2*(9*e**(2*x)*x**4 + 108*e**(2*x)*x**3 + 486*e**(2*x)*x**2 + 972*e** (2*x)*x + 729*e**(2*x) + 24*e**x*log(2)*x**2 + 144*e**x*log(2)*x + 216*e** x*log(2) - 18*e**x*x**5 - 216*e**x*x**4 - 948*e**x*x**3 - 1848*e**x*x**2 - 1530*e**x*x - 432*e**x + 16*log(2)**2 - 24*log(2)*x**3 - 144*log(2)*x**2 - 184*log(2)*x - 64*log(2) + 9*x**6 + 108*x**5 + 462*x**4 + 876*x**3 + 817 *x**2 + 368*x + 64))