Integrand size = 177, antiderivative size = 28 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {1}{3} \left (3+e^5+\frac {\left (-7+\frac {5}{x}-x\right )^4}{\log ^8(4)}\right )^2 \] Output:
1/3*(3+exp(5)+1/256*(5/x-7-x)^4/ln(2)^8)^2
Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(28)=56\).
Time = 0.03 (sec) , antiderivative size = 216, normalized size of antiderivative = 7.71 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {390625-4375000 x+20812500 x^2-53900000 x^3+17248 x^{13}+1332 x^{14}+56 x^{15}+x^{16}-7000 x^5 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )+56 x^{11} \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )+1250 x^4 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )+2 x^{12} \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )-280 x^7 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )+56 x^9 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )+100 x^6 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )+4 x^{10} \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 x^8 \log ^{16}(4)} \] Input:
Integrate[(-3125000 + 30625000*x - 124875000*x^2 + 269500000*x^3 - 3190250 00*x^4 + 183897000*x^5 - 22013600*x^6 - 14862680*x^7 - 2972536*x^9 + 88054 4*x^10 + 1471176*x^11 + 510440*x^12 + 86240*x^13 + 7992*x^14 + 392*x^15 + 8*x^16 + (-15000*x^4 + 63000*x^5 - 82200*x^6 + 28560*x^7 + 5712*x^9 + 3288 *x^10 + 504*x^11 + 24*x^12 + E^5*(-5000*x^4 + 21000*x^5 - 27400*x^6 + 9520 *x^7 + 1904*x^9 + 1096*x^10 + 168*x^11 + 8*x^12))*Log[4]^8)/(3*x^9*Log[4]^ 16),x]
Output:
(390625 - 4375000*x + 20812500*x^2 - 53900000*x^3 + 17248*x^13 + 1332*x^14 + 56*x^15 + x^16 - 7000*x^5*(8757 + 3*Log[4]^8 + E^5*Log[4]^8) + 56*x^11* (8757 + 3*Log[4]^8 + E^5*Log[4]^8) + 1250*x^4*(63805 + 3*Log[4]^8 + E^5*Lo g[4]^8) + 2*x^12*(63805 + 3*Log[4]^8 + E^5*Log[4]^8) - 280*x^7*(-53081 + 1 02*Log[4]^8 + 34*E^5*Log[4]^8) + 56*x^9*(-53081 + 102*Log[4]^8 + 34*E^5*Lo g[4]^8) + 100*x^6*(110068 + 411*Log[4]^8 + 137*E^5*Log[4]^8) + 4*x^10*(110 068 + 411*Log[4]^8 + 137*E^5*Log[4]^8))/(3*x^8*Log[4]^16)
Leaf count is larger than twice the leaf count of optimal. \(235\) vs. \(2(28)=56\).
Time = 0.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 8.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {27, 27, 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.
| \(\displaystyle \int \frac {8 x^{16}+392 x^{15}+7992 x^{14}+86240 x^{13}+510440 x^{12}+1471176 x^{11}+880544 x^{10}-2972536 x^9-14862680 x^7-22013600 x^6+183897000 x^5-319025000 x^4+269500000 x^3-124875000 x^2+\left (24 x^{12}+504 x^{11}+3288 x^{10}+5712 x^9+28560 x^7-82200 x^6+63000 x^5-15000 x^4+e^5 \left (8 x^{12}+168 x^{11}+1096 x^{10}+1904 x^9+9520 x^7-27400 x^6+21000 x^5-5000 x^4\right )\right ) \log ^8(4)+30625000 x-3125000}{3 x^9 \log ^{16}(4)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {8 \left (-x^{16}-49 x^{15}-999 x^{14}-10780 x^{13}-63805 x^{12}-183897 x^{11}-110068 x^{10}+371567 x^9+1857835 x^7+2751700 x^6-22987125 x^5+39878125 x^4-33687500 x^3+15609375 x^2-3828125 x+\left (-3 x^{12}-63 x^{11}-411 x^{10}-714 x^9-3570 x^7+10275 x^6-7875 x^5+1875 x^4+e^5 \left (-x^{12}-21 x^{11}-137 x^{10}-238 x^9-1190 x^7+3425 x^6-2625 x^5+625 x^4\right )\right ) \log ^8(4)+390625\right )}{x^9}dx}{3 \log ^{16}(4)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {8 \int \frac {-x^{16}-49 x^{15}-999 x^{14}-10780 x^{13}-63805 x^{12}-183897 x^{11}-110068 x^{10}+371567 x^9+1857835 x^7+2751700 x^6-22987125 x^5+39878125 x^4-33687500 x^3+15609375 x^2-3828125 x+\left (-3 x^{12}-63 x^{11}-411 x^{10}-714 x^9-3570 x^7+10275 x^6-7875 x^5+1875 x^4+e^5 \left (-x^{12}-21 x^{11}-137 x^{10}-238 x^9-1190 x^7+3425 x^6-2625 x^5+625 x^4\right )\right ) \log ^8(4)+390625}{x^9}dx}{3 \log ^{16}(4)}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {8 \int \left (-x^7-49 x^6-999 x^5-10780 x^4-\left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right ) x^3-21 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right ) x^2-\left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right ) x-7 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )-\frac {35 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )}{x^2}+\frac {25 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{x^3}-\frac {2625 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{x^4}+\frac {625 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{x^5}-\frac {33687500}{x^6}+\frac {15609375}{x^7}-\frac {3828125}{x^8}+\frac {390625}{x^9}\right )dx}{3 \log ^{16}(4)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 \left (-\frac {x^8}{8}-\frac {390625}{8 x^8}-7 x^7+\frac {546875}{x^7}-\frac {333 x^6}{2}-\frac {5203125}{2 x^6}-2156 x^5+\frac {6737500}{x^5}-\frac {1}{4} x^4 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )-\frac {625 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{4 x^4}-7 x^3 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )+\frac {875 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{x^3}-\frac {1}{2} x^2 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )-\frac {25 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{2 x^2}+7 x \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )-\frac {35 \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )}{x}\right )}{3 \log ^{16}(4)}\) |
Input:
Int[(-3125000 + 30625000*x - 124875000*x^2 + 269500000*x^3 - 319025000*x^4 + 183897000*x^5 - 22013600*x^6 - 14862680*x^7 - 2972536*x^9 + 880544*x^10 + 1471176*x^11 + 510440*x^12 + 86240*x^13 + 7992*x^14 + 392*x^15 + 8*x^16 + (-15000*x^4 + 63000*x^5 - 82200*x^6 + 28560*x^7 + 5712*x^9 + 3288*x^10 + 504*x^11 + 24*x^12 + E^5*(-5000*x^4 + 21000*x^5 - 27400*x^6 + 9520*x^7 + 1904*x^9 + 1096*x^10 + 168*x^11 + 8*x^12))*Log[4]^8)/(3*x^9*Log[4]^16),x]
Output:
(-8*(-390625/(8*x^8) + 546875/x^7 - 5203125/(2*x^6) + 6737500/x^5 - 2156*x ^5 - (333*x^6)/2 - 7*x^7 - x^8/8 - (35*(53081 - 102*Log[4]^8 - 34*E^5*Log[ 4]^8))/x + 7*x*(53081 - 102*Log[4]^8 - 34*E^5*Log[4]^8) + (875*(8757 + 3*L og[4]^8 + E^5*Log[4]^8))/x^3 - 7*x^3*(8757 + 3*Log[4]^8 + E^5*Log[4]^8) - (625*(63805 + 3*Log[4]^8 + E^5*Log[4]^8))/(4*x^4) - (x^4*(63805 + 3*Log[4] ^8 + E^5*Log[4]^8))/4 - (25*(110068 + 411*Log[4]^8 + 137*E^5*Log[4]^8))/(2 *x^2) - (x^2*(110068 + 411*Log[4]^8 + 137*E^5*Log[4]^8))/2))/(3*Log[4]^16)
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. \(225\) vs. \(2(26)=52\).
Time = 0.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 8.07
| method | result | size |
| default | \(\frac {64 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{4}+1792 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{3}+192 \ln \left (2\right )^{8} x^{4}+17536 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{2}+5376 \ln \left (2\right )^{8} x^{3}+60928 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x +52608 \ln \left (2\right )^{8} x^{2}+182784 \ln \left (2\right )^{8} x +\frac {x^{8}}{8}+7 x^{7}+\frac {333 x^{6}}{2}+2156 x^{5}+\frac {63805 x^{4}}{4}+61299 x^{3}+55034 x^{2}-371567 x -\frac {6737500}{x^{5}}-\frac {546875}{x^{7}}-\frac {672000 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+2016000 \ln \left (2\right )^{8}+22987125}{3 x^{3}}-\frac {-876800 \,{\mathrm e}^{5} \ln \left (2\right )^{8}-2630400 \ln \left (2\right )^{8}-2751700}{2 x^{2}}-\frac {304640 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+913920 \ln \left (2\right )^{8}-1857835}{x}+\frac {5203125}{2 x^{6}}+\frac {390625}{8 x^{8}}-\frac {-160000 \,{\mathrm e}^{5} \ln \left (2\right )^{8}-480000 \ln \left (2\right )^{8}-39878125}{4 x^{4}}}{24576 \ln \left (2\right )^{16}}\) | \(226\) |
| gosper | \(\frac {390625-4375000 x -7311360 \ln \left (2\right )^{8} x^{7}+420864 \ln \left (2\right )^{8} x^{10}+1462272 \ln \left (2\right )^{8} x^{9}+x^{16}+1332 x^{14}+490392 x^{11}+14862680 x^{7}+11006800 x^{6}+440272 x^{10}-2972536 x^{9}+17248 x^{13}+127610 x^{12}+56 x^{15}-61299000 x^{5}+20812500 x^{2}-53900000 x^{3}+79756250 x^{4}+1536 \ln \left (2\right )^{8} x^{12}-1792000 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{5}+320000 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{4}+3507200 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{6}-2437120 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{7}+140288 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{10}+487424 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{9}+14336 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{11}+512 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{12}+960000 \ln \left (2\right )^{8} x^{4}-5376000 \ln \left (2\right )^{8} x^{5}+43008 \ln \left (2\right )^{8} x^{11}+10521600 \ln \left (2\right )^{8} x^{6}}{196608 \ln \left (2\right )^{16} x^{8}}\) | \(243\) |
| parallelrisch | \(\frac {390625-4375000 x -7311360 \ln \left (2\right )^{8} x^{7}+420864 \ln \left (2\right )^{8} x^{10}+1462272 \ln \left (2\right )^{8} x^{9}+x^{16}+1332 x^{14}+490392 x^{11}+14862680 x^{7}+11006800 x^{6}+440272 x^{10}-2972536 x^{9}+17248 x^{13}+127610 x^{12}+56 x^{15}-61299000 x^{5}+20812500 x^{2}-53900000 x^{3}+79756250 x^{4}+1536 \ln \left (2\right )^{8} x^{12}-1792000 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{5}+320000 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{4}+3507200 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{6}-2437120 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{7}+140288 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{10}+487424 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{9}+14336 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{11}+512 \,{\mathrm e}^{5} \ln \left (2\right )^{8} x^{12}+960000 \ln \left (2\right )^{8} x^{4}-5376000 \ln \left (2\right )^{8} x^{5}+43008 \ln \left (2\right )^{8} x^{11}+10521600 \ln \left (2\right )^{8} x^{6}}{196608 \ln \left (2\right )^{16} x^{8}}\) | \(243\) |
| risch | \(\frac {{\mathrm e}^{5} x^{4}}{384 \ln \left (2\right )^{8}}+\frac {7 \,{\mathrm e}^{5} x^{3}}{96 \ln \left (2\right )^{8}}+\frac {x^{4}}{128 \ln \left (2\right )^{8}}+\frac {137 \,{\mathrm e}^{5} x^{2}}{192 \ln \left (2\right )^{8}}+\frac {7 x^{3}}{32 \ln \left (2\right )^{8}}+\frac {119 \,{\mathrm e}^{5} x}{48 \ln \left (2\right )^{8}}+\frac {137 x^{2}}{64 \ln \left (2\right )^{8}}+\frac {119 x}{16 \ln \left (2\right )^{8}}+\frac {x^{8}}{196608 \ln \left (2\right )^{16}}+\frac {7 x^{7}}{24576 \ln \left (2\right )^{16}}+\frac {111 x^{6}}{16384 \ln \left (2\right )^{16}}+\frac {539 x^{5}}{6144 \ln \left (2\right )^{16}}+\frac {63805 x^{4}}{98304 \ln \left (2\right )^{16}}+\frac {20433 x^{3}}{8192 \ln \left (2\right )^{16}}+\frac {27517 x^{2}}{12288 \ln \left (2\right )^{16}}-\frac {371567 x}{24576 \ln \left (2\right )^{16}}+\frac {\left (-2437120 \,{\mathrm e}^{5} \ln \left (2\right )^{8}-7311360 \ln \left (2\right )^{8}+14862680\right ) x^{7}+\left (3507200 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+10521600 \ln \left (2\right )^{8}+11006800\right ) x^{6}+\left (-1792000 \,{\mathrm e}^{5} \ln \left (2\right )^{8}-5376000 \ln \left (2\right )^{8}-61299000\right ) x^{5}+\left (320000 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+960000 \ln \left (2\right )^{8}+79756250\right ) x^{4}-53900000 x^{3}+20812500 x^{2}-4375000 x +390625}{196608 \ln \left (2\right )^{16} x^{8}}\) | \(252\) |
| norman | \(\frac {\frac {390625}{196608 \ln \left (2\right )}-\frac {546875 x}{24576 \ln \left (2\right )}+\frac {1734375 x^{2}}{16384 \ln \left (2\right )}-\frac {1684375 x^{3}}{6144 \ln \left (2\right )}+\frac {539 x^{13}}{6144 \ln \left (2\right )}+\frac {111 x^{14}}{16384 \ln \left (2\right )}+\frac {7 x^{15}}{24576 \ln \left (2\right )}+\frac {x^{16}}{196608 \ln \left (2\right )}-\frac {35 \left (-53081+8704 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+26112 \ln \left (2\right )^{8}\right ) x^{7}}{24576 \ln \left (2\right )}+\frac {7 \left (-53081+8704 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+26112 \ln \left (2\right )^{8}\right ) x^{9}}{24576 \ln \left (2\right )}-\frac {875 \left (8757+256 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+768 \ln \left (2\right )^{8}\right ) x^{5}}{24576 \ln \left (2\right )}+\frac {7 \left (8757+256 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+768 \ln \left (2\right )^{8}\right ) x^{11}}{24576 \ln \left (2\right )}+\frac {625 \left (256 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+768 \ln \left (2\right )^{8}+63805\right ) x^{4}}{98304 \ln \left (2\right )}+\frac {\left (256 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+768 \ln \left (2\right )^{8}+63805\right ) x^{12}}{98304 \ln \left (2\right )}+\frac {25 \left (8768 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+26304 \ln \left (2\right )^{8}+27517\right ) x^{6}}{12288 \ln \left (2\right )}+\frac {\left (8768 \,{\mathrm e}^{5} \ln \left (2\right )^{8}+26304 \ln \left (2\right )^{8}+27517\right ) x^{10}}{12288 \ln \left (2\right )}}{x^{8} \ln \left (2\right )^{15}}\) | \(277\) |
Input:
int(1/196608*(256*((8*x^12+168*x^11+1096*x^10+1904*x^9+9520*x^7-27400*x^6+ 21000*x^5-5000*x^4)*exp(5)+24*x^12+504*x^11+3288*x^10+5712*x^9+28560*x^7-8 2200*x^6+63000*x^5-15000*x^4)*ln(2)^8+8*x^16+392*x^15+7992*x^14+86240*x^13 +510440*x^12+1471176*x^11+880544*x^10-2972536*x^9-14862680*x^7-22013600*x^ 6+183897000*x^5-319025000*x^4+269500000*x^3-124875000*x^2+30625000*x-31250 00)/x^9/ln(2)^16,x,method=_RETURNVERBOSE)
Output:
1/24576/ln(2)^16*(64*exp(5)*ln(2)^8*x^4+1792*exp(5)*ln(2)^8*x^3+192*ln(2)^ 8*x^4+17536*exp(5)*ln(2)^8*x^2+5376*ln(2)^8*x^3+60928*exp(5)*ln(2)^8*x+526 08*ln(2)^8*x^2+182784*ln(2)^8*x+1/8*x^8+7*x^7+333/2*x^6+2156*x^5+63805/4*x ^4+61299*x^3+55034*x^2-371567*x-6737500/x^5-546875/x^7-1/3*(672000*exp(5)* ln(2)^8+2016000*ln(2)^8+22987125)/x^3-1/2*(-876800*exp(5)*ln(2)^8-2630400* ln(2)^8-2751700)/x^2-(304640*exp(5)*ln(2)^8+913920*ln(2)^8-1857835)/x+5203 125/2/x^6+390625/8/x^8-1/4*(-160000*exp(5)*ln(2)^8-480000*ln(2)^8-39878125 )/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.11 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {x^{16} + 56 \, x^{15} + 1332 \, x^{14} + 17248 \, x^{13} + 127610 \, x^{12} + 490392 \, x^{11} + 440272 \, x^{10} - 2972536 \, x^{9} + 512 \, {\left (3 \, x^{12} + 84 \, x^{11} + 822 \, x^{10} + 2856 \, x^{9} - 14280 \, x^{7} + 20550 \, x^{6} - 10500 \, x^{5} + 1875 \, x^{4} + {\left (x^{12} + 28 \, x^{11} + 274 \, x^{10} + 952 \, x^{9} - 4760 \, x^{7} + 6850 \, x^{6} - 3500 \, x^{5} + 625 \, x^{4}\right )} e^{5}\right )} \log \left (2\right )^{8} + 14862680 \, x^{7} + 11006800 \, x^{6} - 61299000 \, x^{5} + 79756250 \, x^{4} - 53900000 \, x^{3} + 20812500 \, x^{2} - 4375000 \, x + 390625}{196608 \, x^{8} \log \left (2\right )^{16}} \] Input:
integrate(1/196608*(256*((8*x^12+168*x^11+1096*x^10+1904*x^9+9520*x^7-2740 0*x^6+21000*x^5-5000*x^4)*exp(5)+24*x^12+504*x^11+3288*x^10+5712*x^9+28560 *x^7-82200*x^6+63000*x^5-15000*x^4)*log(2)^8+8*x^16+392*x^15+7992*x^14+862 40*x^13+510440*x^12+1471176*x^11+880544*x^10-2972536*x^9-14862680*x^7-2201 3600*x^6+183897000*x^5-319025000*x^4+269500000*x^3-124875000*x^2+30625000* x-3125000)/x^9/log(2)^16,x, algorithm="fricas")
Output:
1/196608*(x^16 + 56*x^15 + 1332*x^14 + 17248*x^13 + 127610*x^12 + 490392*x ^11 + 440272*x^10 - 2972536*x^9 + 512*(3*x^12 + 84*x^11 + 822*x^10 + 2856* x^9 - 14280*x^7 + 20550*x^6 - 10500*x^5 + 1875*x^4 + (x^12 + 28*x^11 + 274 *x^10 + 952*x^9 - 4760*x^7 + 6850*x^6 - 3500*x^5 + 625*x^4)*e^5)*log(2)^8 + 14862680*x^7 + 11006800*x^6 - 61299000*x^5 + 79756250*x^4 - 53900000*x^3 + 20812500*x^2 - 4375000*x + 390625)/(x^8*log(2)^16)
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (22) = 44\).
Time = 8.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 8.07 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {\frac {x^{8}}{8} + 7 x^{7} + \frac {333 x^{6}}{2} + 2156 x^{5} + x^{4} \cdot \left (192 \log {\left (2 \right )}^{8} + 64 e^{5} \log {\left (2 \right )}^{8} + \frac {63805}{4}\right ) + x^{3} \cdot \left (5376 \log {\left (2 \right )}^{8} + 1792 e^{5} \log {\left (2 \right )}^{8} + 61299\right ) + x^{2} \cdot \left (52608 \log {\left (2 \right )}^{8} + 55034 + 17536 e^{5} \log {\left (2 \right )}^{8}\right ) + x \left (-371567 + 182784 \log {\left (2 \right )}^{8} + 60928 e^{5} \log {\left (2 \right )}^{8}\right ) + \frac {x^{7} \left (- 2437120 e^{5} \log {\left (2 \right )}^{8} - 7311360 \log {\left (2 \right )}^{8} + 14862680\right ) + x^{6} \cdot \left (10521600 \log {\left (2 \right )}^{8} + 11006800 + 3507200 e^{5} \log {\left (2 \right )}^{8}\right ) + x^{5} \left (-61299000 - 1792000 e^{5} \log {\left (2 \right )}^{8} - 5376000 \log {\left (2 \right )}^{8}\right ) + x^{4} \cdot \left (960000 \log {\left (2 \right )}^{8} + 320000 e^{5} \log {\left (2 \right )}^{8} + 79756250\right ) - 53900000 x^{3} + 20812500 x^{2} - 4375000 x + 390625}{8 x^{8}}}{24576 \log {\left (2 \right )}^{16}} \] Input:
integrate(1/196608*(256*((8*x**12+168*x**11+1096*x**10+1904*x**9+9520*x**7 -27400*x**6+21000*x**5-5000*x**4)*exp(5)+24*x**12+504*x**11+3288*x**10+571 2*x**9+28560*x**7-82200*x**6+63000*x**5-15000*x**4)*ln(2)**8+8*x**16+392*x **15+7992*x**14+86240*x**13+510440*x**12+1471176*x**11+880544*x**10-297253 6*x**9-14862680*x**7-22013600*x**6+183897000*x**5-319025000*x**4+269500000 *x**3-124875000*x**2+30625000*x-3125000)/x**9/ln(2)**16,x)
Output:
(x**8/8 + 7*x**7 + 333*x**6/2 + 2156*x**5 + x**4*(192*log(2)**8 + 64*exp(5 )*log(2)**8 + 63805/4) + x**3*(5376*log(2)**8 + 1792*exp(5)*log(2)**8 + 61 299) + x**2*(52608*log(2)**8 + 55034 + 17536*exp(5)*log(2)**8) + x*(-37156 7 + 182784*log(2)**8 + 60928*exp(5)*log(2)**8) + (x**7*(-2437120*exp(5)*lo g(2)**8 - 7311360*log(2)**8 + 14862680) + x**6*(10521600*log(2)**8 + 11006 800 + 3507200*exp(5)*log(2)**8) + x**5*(-61299000 - 1792000*exp(5)*log(2)* *8 - 5376000*log(2)**8) + x**4*(960000*log(2)**8 + 320000*exp(5)*log(2)**8 + 79756250) - 53900000*x**3 + 20812500*x**2 - 4375000*x + 390625)/(8*x**8 ))/(24576*log(2)**16)
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (25) = 50\).
Time = 0.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 6.39 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {x^{8} + 56 \, x^{7} + 1332 \, x^{6} + 2 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 63805\right )} x^{4} + 17248 \, x^{5} + 56 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 8757\right )} x^{3} + 16 \, {\left (8768 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 27517\right )} x^{2} + 56 \, {\left (8704 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} - 53081\right )} x - \frac {5 \, {\left (56 \, {\left (8704 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} - 53081\right )} x^{7} - 80 \, {\left (8768 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 27517\right )} x^{6} + 1400 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 8757\right )} x^{5} - 250 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \left (2\right )^{8} + 63805\right )} x^{4} + 10780000 \, x^{3} - 4162500 \, x^{2} + 875000 \, x - 78125\right )}}{x^{8}}}{196608 \, \log \left (2\right )^{16}} \] Input:
integrate(1/196608*(256*((8*x^12+168*x^11+1096*x^10+1904*x^9+9520*x^7-2740 0*x^6+21000*x^5-5000*x^4)*exp(5)+24*x^12+504*x^11+3288*x^10+5712*x^9+28560 *x^7-82200*x^6+63000*x^5-15000*x^4)*log(2)^8+8*x^16+392*x^15+7992*x^14+862 40*x^13+510440*x^12+1471176*x^11+880544*x^10-2972536*x^9-14862680*x^7-2201 3600*x^6+183897000*x^5-319025000*x^4+269500000*x^3-124875000*x^2+30625000* x-3125000)/x^9/log(2)^16,x, algorithm="maxima")
Output:
1/196608*(x^8 + 56*x^7 + 1332*x^6 + 2*(256*(e^5 + 3)*log(2)^8 + 63805)*x^4 + 17248*x^5 + 56*(256*(e^5 + 3)*log(2)^8 + 8757)*x^3 + 16*(8768*(e^5 + 3) *log(2)^8 + 27517)*x^2 + 56*(8704*(e^5 + 3)*log(2)^8 - 53081)*x - 5*(56*(8 704*(e^5 + 3)*log(2)^8 - 53081)*x^7 - 80*(8768*(e^5 + 3)*log(2)^8 + 27517) *x^6 + 1400*(256*(e^5 + 3)*log(2)^8 + 8757)*x^5 - 250*(256*(e^5 + 3)*log(2 )^8 + 63805)*x^4 + 10780000*x^3 - 4162500*x^2 + 875000*x - 78125)/x^8)/log (2)^16
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 8.54 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {512 \, x^{4} e^{5} \log \left (2\right )^{8} + 1536 \, x^{4} \log \left (2\right )^{8} + 14336 \, x^{3} e^{5} \log \left (2\right )^{8} + 43008 \, x^{3} \log \left (2\right )^{8} + 140288 \, x^{2} e^{5} \log \left (2\right )^{8} + 420864 \, x^{2} \log \left (2\right )^{8} + 487424 \, x e^{5} \log \left (2\right )^{8} + 1462272 \, x \log \left (2\right )^{8} + x^{8} + 56 \, x^{7} + 1332 \, x^{6} + 17248 \, x^{5} + 127610 \, x^{4} + 490392 \, x^{3} + 440272 \, x^{2} - 2972536 \, x - \frac {5 \, {\left (487424 \, x^{7} e^{5} \log \left (2\right )^{8} + 1462272 \, x^{7} \log \left (2\right )^{8} - 701440 \, x^{6} e^{5} \log \left (2\right )^{8} - 2104320 \, x^{6} \log \left (2\right )^{8} + 358400 \, x^{5} e^{5} \log \left (2\right )^{8} + 1075200 \, x^{5} \log \left (2\right )^{8} - 64000 \, x^{4} e^{5} \log \left (2\right )^{8} - 192000 \, x^{4} \log \left (2\right )^{8} - 2972536 \, x^{7} - 2201360 \, x^{6} + 12259800 \, x^{5} - 15951250 \, x^{4} + 10780000 \, x^{3} - 4162500 \, x^{2} + 875000 \, x - 78125\right )}}{x^{8}}}{196608 \, \log \left (2\right )^{16}} \] Input:
integrate(1/196608*(256*((8*x^12+168*x^11+1096*x^10+1904*x^9+9520*x^7-2740 0*x^6+21000*x^5-5000*x^4)*exp(5)+24*x^12+504*x^11+3288*x^10+5712*x^9+28560 *x^7-82200*x^6+63000*x^5-15000*x^4)*log(2)^8+8*x^16+392*x^15+7992*x^14+862 40*x^13+510440*x^12+1471176*x^11+880544*x^10-2972536*x^9-14862680*x^7-2201 3600*x^6+183897000*x^5-319025000*x^4+269500000*x^3-124875000*x^2+30625000* x-3125000)/x^9/log(2)^16,x, algorithm="giac")
Output:
1/196608*(512*x^4*e^5*log(2)^8 + 1536*x^4*log(2)^8 + 14336*x^3*e^5*log(2)^ 8 + 43008*x^3*log(2)^8 + 140288*x^2*e^5*log(2)^8 + 420864*x^2*log(2)^8 + 4 87424*x*e^5*log(2)^8 + 1462272*x*log(2)^8 + x^8 + 56*x^7 + 1332*x^6 + 1724 8*x^5 + 127610*x^4 + 490392*x^3 + 440272*x^2 - 2972536*x - 5*(487424*x^7*e ^5*log(2)^8 + 1462272*x^7*log(2)^8 - 701440*x^6*e^5*log(2)^8 - 2104320*x^6 *log(2)^8 + 358400*x^5*e^5*log(2)^8 + 1075200*x^5*log(2)^8 - 64000*x^4*e^5 *log(2)^8 - 192000*x^4*log(2)^8 - 2972536*x^7 - 2201360*x^6 + 12259800*x^5 - 15951250*x^4 + 10780000*x^3 - 4162500*x^2 + 875000*x - 78125)/x^8)/log( 2)^16
Time = 0.16 (sec) , antiderivative size = 241, normalized size of antiderivative = 8.61 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {539\,x^5}{6144\,{\ln \left (2\right )}^{16}}+\frac {111\,x^6}{16384\,{\ln \left (2\right )}^{16}}+\frac {7\,x^7}{24576\,{\ln \left (2\right )}^{16}}+\frac {x^8}{196608\,{\ln \left (2\right )}^{16}}+\frac {x^4\,\left (256\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+768\,{\ln \left (2\right )}^8+63805\right )}{98304\,{\ln \left (2\right )}^{16}}+\frac {x^3\,\left (5376\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+16128\,{\ln \left (2\right )}^8+183897\right )}{73728\,{\ln \left (2\right )}^{16}}+\frac {x^2\,\left (35072\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+105216\,{\ln \left (2\right )}^8+110068\right )}{49152\,{\ln \left (2\right )}^{16}}+\frac {x\,\left (60928\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+182784\,{\ln \left (2\right )}^8-371567\right )}{24576\,{\ln \left (2\right )}^{16}}-\frac {\left (304640\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+913920\,{\ln \left (2\right )}^8-1857835\right )\,x^7+\left (-438400\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8-1315200\,{\ln \left (2\right )}^8-1375850\right )\,x^6+\left (224000\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8+672000\,{\ln \left (2\right )}^8+7662375\right )\,x^5+\left (-40000\,{\mathrm {e}}^5\,{\ln \left (2\right )}^8-120000\,{\ln \left (2\right )}^8-\frac {39878125}{4}\right )\,x^4+6737500\,x^3-\frac {5203125\,x^2}{2}+546875\,x-\frac {390625}{8}}{24576\,x^8\,{\ln \left (2\right )}^{16}} \] Input:
int(((3828125*x)/24576 + (log(2)^8*(63000*x^5 - 15000*x^4 - 82200*x^6 + 28 560*x^7 + 5712*x^9 + 3288*x^10 + 504*x^11 + 24*x^12 + exp(5)*(21000*x^5 - 5000*x^4 - 27400*x^6 + 9520*x^7 + 1904*x^9 + 1096*x^10 + 168*x^11 + 8*x^12 )))/768 - (5203125*x^2)/8192 + (8421875*x^3)/6144 - (39878125*x^4)/24576 + (7662375*x^5)/8192 - (687925*x^6)/6144 - (1857835*x^7)/24576 - (371567*x^ 9)/24576 + (27517*x^10)/6144 + (61299*x^11)/8192 + (63805*x^12)/24576 + (2 695*x^13)/6144 + (333*x^14)/8192 + (49*x^15)/24576 + x^16/24576 - 390625/2 4576)/(x^9*log(2)^16),x)
Output:
(539*x^5)/(6144*log(2)^16) + (111*x^6)/(16384*log(2)^16) + (7*x^7)/(24576* log(2)^16) + x^8/(196608*log(2)^16) + (x^4*(256*exp(5)*log(2)^8 + 768*log( 2)^8 + 63805))/(98304*log(2)^16) + (x^3*(5376*exp(5)*log(2)^8 + 16128*log( 2)^8 + 183897))/(73728*log(2)^16) + (x^2*(35072*exp(5)*log(2)^8 + 105216*l og(2)^8 + 110068))/(49152*log(2)^16) + (x*(60928*exp(5)*log(2)^8 + 182784* log(2)^8 - 371567))/(24576*log(2)^16) - (546875*x + x^7*(304640*exp(5)*log (2)^8 + 913920*log(2)^8 - 1857835) - x^6*(438400*exp(5)*log(2)^8 + 1315200 *log(2)^8 + 1375850) + x^5*(224000*exp(5)*log(2)^8 + 672000*log(2)^8 + 766 2375) - x^4*(40000*exp(5)*log(2)^8 + 120000*log(2)^8 + 39878125/4) - (5203 125*x^2)/2 + 6737500*x^3 - 390625/8)/(24576*x^8*log(2)^16)
Time = 0.17 (sec) , antiderivative size = 250, normalized size of antiderivative = 8.93 \[ \int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{3 x^9 \log ^{16}(4)} \, dx=\frac {390625-4375000 x +440272 x^{10}+17248 x^{13}+512 \mathrm {log}\left (2\right )^{8} e^{5} x^{12}+14336 \mathrm {log}\left (2\right )^{8} e^{5} x^{11}+140288 \mathrm {log}\left (2\right )^{8} e^{5} x^{10}+487424 \mathrm {log}\left (2\right )^{8} e^{5} x^{9}-2437120 \mathrm {log}\left (2\right )^{8} e^{5} x^{7}+3507200 \mathrm {log}\left (2\right )^{8} e^{5} x^{6}-1792000 \mathrm {log}\left (2\right )^{8} e^{5} x^{5}+320000 \mathrm {log}\left (2\right )^{8} e^{5} x^{4}+20812500 x^{2}-61299000 x^{5}+14862680 x^{7}+490392 x^{11}-53900000 x^{3}+11006800 x^{6}-2972536 x^{9}+x^{16}+1332 x^{14}+56 x^{15}+79756250 x^{4}+1536 \mathrm {log}\left (2\right )^{8} x^{12}+43008 \mathrm {log}\left (2\right )^{8} x^{11}+420864 \mathrm {log}\left (2\right )^{8} x^{10}+1462272 \mathrm {log}\left (2\right )^{8} x^{9}-7311360 \mathrm {log}\left (2\right )^{8} x^{7}+10521600 \mathrm {log}\left (2\right )^{8} x^{6}-5376000 \mathrm {log}\left (2\right )^{8} x^{5}+960000 \mathrm {log}\left (2\right )^{8} x^{4}+127610 x^{12}}{196608 \mathrm {log}\left (2\right )^{16} x^{8}} \] Input:
int(1/196608*(256*((8*x^12+168*x^11+1096*x^10+1904*x^9+9520*x^7-27400*x^6+ 21000*x^5-5000*x^4)*exp(5)+24*x^12+504*x^11+3288*x^10+5712*x^9+28560*x^7-8 2200*x^6+63000*x^5-15000*x^4)*log(2)^8+8*x^16+392*x^15+7992*x^14+86240*x^1 3+510440*x^12+1471176*x^11+880544*x^10-2972536*x^9-14862680*x^7-22013600*x ^6+183897000*x^5-319025000*x^4+269500000*x^3-124875000*x^2+30625000*x-3125 000)/x^9/log(2)^16,x)
Output:
(512*log(2)**8*e**5*x**12 + 14336*log(2)**8*e**5*x**11 + 140288*log(2)**8* e**5*x**10 + 487424*log(2)**8*e**5*x**9 - 2437120*log(2)**8*e**5*x**7 + 35 07200*log(2)**8*e**5*x**6 - 1792000*log(2)**8*e**5*x**5 + 320000*log(2)**8 *e**5*x**4 + 1536*log(2)**8*x**12 + 43008*log(2)**8*x**11 + 420864*log(2)* *8*x**10 + 1462272*log(2)**8*x**9 - 7311360*log(2)**8*x**7 + 10521600*log( 2)**8*x**6 - 5376000*log(2)**8*x**5 + 960000*log(2)**8*x**4 + x**16 + 56*x **15 + 1332*x**14 + 17248*x**13 + 127610*x**12 + 490392*x**11 + 440272*x** 10 - 2972536*x**9 + 14862680*x**7 + 11006800*x**6 - 61299000*x**5 + 797562 50*x**4 - 53900000*x**3 + 20812500*x**2 - 4375000*x + 390625)/(196608*log( 2)**16*x**8)