Integrand size = 271, antiderivative size = 28 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\log \left (\log \left (-5+x+\left ((-4+x)^2+\frac {x^3 (-x+\log (2))}{3+x}\right )^2\right )\right ) \end {dmath*}
\begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx \end {dmath*}
Integrate[(-6885 - 1701*x + 1593*x^2 - 867*x^3 - 84*x^4 + 216*x^5 + 2*x^6 + 14*x^7 + 6*x^8 + (864*x^2 - 96*x^3 - 182*x^4 + 6*x^5 - 34*x^6 - 10*x^7)* Log[2] + (18*x^5 + 4*x^6)*Log[2]^2)/((6777 - 108*x - 2034*x^2 + 116*x^3 - 84*x^4 - 69*x^5 + 39*x^6 + 5*x^7 + x^8 + x^9 + (288*x^3 + 48*x^4 - 46*x^5 - 4*x^6 - 4*x^7 - 2*x^8)*Log[2] + (3*x^6 + x^7)*Log[2]^2)*Log[(2259 - 789* x - 415*x^2 + 177*x^3 - 87*x^4 + 6*x^5 + 11*x^6 - 2*x^7 + x^8 + (96*x^3 - 16*x^4 - 10*x^5 + 2*x^6 - 2*x^7)*Log[2] + x^6*Log[2]^2)/(9 + 6*x + x^2)]), x]
Integrate[(-6885 - 1701*x + 1593*x^2 - 867*x^3 - 84*x^4 + 216*x^5 + 2*x^6 + 14*x^7 + 6*x^8 + (864*x^2 - 96*x^3 - 182*x^4 + 6*x^5 - 34*x^6 - 10*x^7)* Log[2] + (18*x^5 + 4*x^6)*Log[2]^2)/((6777 - 108*x - 2034*x^2 + 116*x^3 - 84*x^4 - 69*x^5 + 39*x^6 + 5*x^7 + x^8 + x^9 + (288*x^3 + 48*x^4 - 46*x^5 - 4*x^6 - 4*x^7 - 2*x^8)*Log[2] + (3*x^6 + x^7)*Log[2]^2)*Log[(2259 - 789* x - 415*x^2 + 177*x^3 - 87*x^4 + 6*x^5 + 11*x^6 - 2*x^7 + x^8 + (96*x^3 - 16*x^4 - 10*x^5 + 2*x^6 - 2*x^7)*Log[2] + x^6*Log[2]^2)/(9 + 6*x + x^2)]), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {6 x^8+14 x^7+2 x^6+216 x^5-84 x^4-867 x^3+1593 x^2+\left (4 x^6+18 x^5\right ) \log ^2(2)+\left (-10 x^7-34 x^6+6 x^5-182 x^4-96 x^3+864 x^2\right ) \log (2)-1701 x-6885}{\left (x^9+x^8+5 x^7+39 x^6-69 x^5-84 x^4+116 x^3-2034 x^2+\left (x^7+3 x^6\right ) \log ^2(2)+\left (-2 x^8-4 x^7-4 x^6-46 x^5+48 x^4+288 x^3\right ) \log (2)-108 x+6777\right ) \log \left (\frac {x^8-2 x^7+11 x^6+x^6 \log ^2(2)+6 x^5-87 x^4+177 x^3-415 x^2+\left (-2 x^7+2 x^6-10 x^5-16 x^4+96 x^3\right ) \log (2)-789 x+2259}{x^2+6 x+9}\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-x^7+5 x^6 \left (1+\frac {2 \log (2)}{5}\right )-26 x^5 \left (1+\frac {1}{26} \left (\log ^2(2)+\log (256)\right )\right )+72 x^4 \left (1+\frac {1}{72} \log (2) (34+\log (8))\right )-129 x^3 \left (1+\frac {1}{129} \log (2) (86+\log (512))\right )+210 x^2 \left (1+\frac {9}{70} \log (2) (6+\log (2))\right )-215 x \left (1+\frac {81}{215} \log (2) (6+\log (2))\right )+1434 \left (1+\frac {81}{478} \log (2) (6+\log (2))\right )\right ) \left (6 x^8+14 x^7+2 x^6+216 x^5-84 x^4-867 x^3+1593 x^2+\left (4 x^6+18 x^5\right ) \log ^2(2)+\left (-10 x^7-34 x^6+6 x^5-182 x^4-96 x^3+864 x^2\right ) \log (2)-1701 x-6885\right )}{729 (3+\log (2))^2 \left (x^8-2 x^7 (1+\log (2))+11 x^6 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right )+6 x^5 \left (1-\frac {5 \log (2)}{3}\right )-87 x^4 \left (1+\frac {16 \log (2)}{87}\right )+177 x^3 \left (1+\frac {32 \log (2)}{59}\right )-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-2 x^7+11 x^6+x^6 \log ^2(2)+6 x^5-87 x^4+177 x^3-415 x^2+\left (-2 x^7+2 x^6-10 x^5-16 x^4+96 x^3\right ) \log (2)-789 x+2259}{x^2+6 x+9}\right )}+\frac {6 x^8+14 x^7+2 x^6+216 x^5-84 x^4-867 x^3+1593 x^2+\left (4 x^6+18 x^5\right ) \log ^2(2)+\left (-10 x^7-34 x^6+6 x^5-182 x^4-96 x^3+864 x^2\right ) \log (2)-1701 x-6885}{729 (x+3) (3+\log (2))^2 \log \left (\frac {x^8-2 x^7+11 x^6+x^6 \log ^2(2)+6 x^5-87 x^4+177 x^3-415 x^2+\left (-2 x^7+2 x^6-10 x^5-16 x^4+96 x^3\right ) \log (2)-789 x+2259}{x^2+6 x+9}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^4 \left (\log ^2(8)-\log (2) \log (512)\right )+x^3 \left (27 \log ^2(2)-\log (8) \log (512)\right )+729 (3+\log (2))^2\right ) \left (6 x^8-2 x^7 (\log (32)-7)+x^6 \left (2+4 \log ^2(2)-34 \log (2)\right )+6 x^5 \left (36+3 \log ^2(2)+\log (2)\right )-14 x^4 (6+\log (8192))-3 x^3 (289+32 \log (2))+27 x^2 (59+32 \log (2))-1701 x-6885\right )}{729 (x+3) (3+\log (2))^2 \left (x^8-2 x^7 (1+\log (2))+x^6 \left (11+\log ^2(2)+\log (4)\right )+x^5 (6-10 \log (2))-x^4 (87+16 \log (2))+3 x^3 (59+32 \log (2))-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-x^7 (2+\log (4))+x^6 \left (11+\log ^2(2)+\log (4)\right )+x^5 (6-10 \log (2))-x^4 (87+16 \log (2))+3 x^3 (59+32 \log (2))-415 x^2-789 x+2259}{(x+3)^2}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {\left (\left (\log ^2(8)-\log (2) \log (512)\right ) x^4+\left (27 \log ^2(2)-\log (8) \log (512)\right ) x^3+729 (3+\log (2))^2\right ) \left (-6 x^8-2 (7-\log (32)) x^7-2 \left (1-17 \log (2)+2 \log ^2(2)\right ) x^6-6 \left (36+\log (2)+3 \log ^2(2)\right ) x^5+14 (6+\log (8192)) x^4+3 (289+32 \log (2)) x^3-27 (59+32 \log (2)) x^2+1701 x+6885\right )}{(x+3) \left (x^8-2 (1+\log (2)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{729 (3+\log (2))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (\left (\log ^2(8)-\log (2) \log (512)\right ) x^4+\left (27 \log ^2(2)-\log (8) \log (512)\right ) x^3+729 (3+\log (2))^2\right ) \left (-6 x^8-2 (7-\log (32)) x^7-2 \left (1-17 \log (2)+2 \log ^2(2)\right ) x^6-6 \left (36+\log (2)+3 \log ^2(2)\right ) x^5+14 (6+\log (8192)) x^4+3 (289+32 \log (2)) x^3-27 (59+32 \log (2)) x^2+1701 x+6885\right )}{(x+3) \left (x^8-2 (1+\log (2)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{729 (3+\log (2))^2}\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle -\frac {\int \frac {\left (\sqrt [4]{\log ^2(8)-\log (2) \log (512)} x+3 \sqrt {3 (3+\log (2))}\right )^4 \left (-6 x^8-2 (7-\log (32)) x^7-2 \left (1-17 \log (2)+2 \log ^2(2)\right ) x^6-6 \left (36+\log (2)+3 \log ^2(2)\right ) x^5+14 (6+\log (8192)) x^4+3 (289+32 \log (2)) x^3-27 (59+32 \log (2)) x^2+1701 x+6885\right )}{(x+3) \left (x^8-2 (1+\log (2)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{729 (3+\log (2))^2}\) |
| \(\Big \downarrow \) 2 |
| \(\displaystyle -\frac {\int \frac {729 (3+\log (2))^2 \left (-6 x^8-2 (7-\log (32)) x^7-2 \left (1-17 \log (2)+2 \log ^2(2)\right ) x^6-6 \left (36+\log (2)+3 \log ^2(2)\right ) x^5+14 (6+\log (8192)) x^4+3 (289+32 \log (2)) x^3-27 (59+32 \log (2)) x^2+1701 x+6885\right )}{(x+3) \left (x^8-2 (1+\log (2)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{729 (3+\log (2))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {-6 x^8-2 (7-\log (32)) x^7-2 \left (1-17 \log (2)+2 \log ^2(2)\right ) x^6-6 \left (36+\log (2)+3 \log ^2(2)\right ) x^5+14 (6+\log (8192)) x^4+3 (289+32 \log (2)) x^3-27 (59+32 \log (2)) x^2+1701 x+6885}{(x+3) \left (x^8-2 (1+\log (2)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+2 (3-5 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {-648 \left (1+\frac {1}{9} \log (2) (6+\log (2))\right ) x^7+1134 \left (1+\frac {1}{567} \left (259 \log ^2(2)+\frac {18 \log ^3(2) \log (128)}{\log (4)}+18 \log (4096)+\log (2) (729+14 \log (8192))\right )\right ) x^6-5346 \left (1+\frac {1909 \log (2) \left (1+\frac {516 \log (2) \left (1+\frac {1}{516} \log (2) \left (71+27 \log (2)+\frac {\log (4) \left (144-55 \log (2)+27 \log ^2(2)-27 \log (32)\right ) \log (64)-459 \log (32) \log (64)+14 \left (3 \log ^3(2)+13 \log (64)+\log (8) \log (256)\right ) \log (8192)}{\log ^3(2) \log (64)}\right )\right )}{1909}\right )}{2673}\right ) x^5-2430 \left (1+\frac {-665 \log ^2(2)+48 \log ^3(2)+9 (60 \log (4)+135 \log (32)-56 \log (8192))+\log (2) (-1818+192 \log (4)+432 \log (32)-238 \log (8192)-7 \log (8) \log (8192))}{1215}\right ) x^4+28188 \left (1+\frac {367 \log ^2(2)+144 \log ^3(2)+12 \log (2) (245+24 \log (4))+24 \log (17592186044416)+\log (4) (999+8 \log (17592186044416))}{7047}\right ) x^3-43011 \left (1+\frac {5049 \log ^2(2)+864 \log ^3(2)+1395 \log (4)+8 \log (2) (1939+108 \log (4))-22 \log (17592186044416)}{14337}\right ) x^2+67230 \left (1+\frac {46808 \log (2)+3735 \log ^2(2)-891 \log (4)-514 \log (17592186044416)}{33615}\right ) x+63909 \left (1+\frac {-21656 \log (2)+2367 \log ^2(2)+6885 \log (4)+502 \log (17592186044416)}{21303}\right )}{9 \left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \left (9+\log ^2(2)+\log (64)\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}+\frac {2 \left (81+9 \log ^2(2)+\log (18014398509481984)\right )}{9 (x+3) \left (9+\log ^2(2)+\log (64)\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (81+9 \log ^2(2)+\log (18014398509481984)\right ) \int \frac {1}{(x+3) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \left (9+\log ^2(2)+\log (64)\right )}-8 \int \frac {x^7}{\left (-x^8+2 (1+\log (2)) x^7-11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6-6 \left (1-\frac {5 \log (2)}{3}\right ) x^5+87 \left (1+\frac {16 \log (2)}{87}\right ) x^4-177 \left (1+\frac {32 \log (2)}{59}\right ) x^3+415 x^2+789 x-2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx-\frac {\left (21303-21656 \log (2)+2367 \log ^2(2)+6885 \log (4)+502 \log (17592186044416)\right ) \int \frac {1}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{3 \left (9+\log ^2(2)+\log (64)\right )}-\frac {2 \left (33615+46808 \log (2)+3735 \log ^2(2)-891 \log (4)-514 \log (17592186044416)\right ) \int \frac {x}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \left (9+\log ^2(2)+\log (64)\right )}+\frac {\left (14337+5049 \log ^2(2)+864 \log ^3(2)+1395 \log (4)+8 \log (2) (1939+108 \log (4))-22 \log (17592186044416)\right ) \int \frac {x^2}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{3 \left (9+\log ^2(2)+\log (64)\right )}-\frac {4 \left (7047+367 \log ^2(2)+144 \log ^3(2)+12 \log (2) (245+24 \log (4))+24 \log (17592186044416)+\log (4) (999+8 \log (17592186044416))\right ) \int \frac {x^3}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \left (9+\log ^2(2)+\log (64)\right )}-\frac {2 \left (665 \log ^2(2)-48 \log ^3(2)-9 (135+60 \log (4)+135 \log (32)-56 \log (8192))+\log (2) (1818-192 \log (4)-432 \log (32)+238 \log (8192)+7 \log (8) \log (8192))\right ) \int \frac {x^4}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \left (9+\log ^2(2)+\log (64)\right )}+\frac {2 \left (2673+71 \log ^3(2)+27 \log ^4(2)+\log (2) (1909-55 \log (4))+3 \log ^2(2) (172+9 \log (4))+9 \log (4) (16-3 \log (32))-459 \log (32)+182 \log (8192)+\frac {14 \left (3 \log ^3(2)+\log (8) \log (256)\right ) \log (8192)}{\log (64)}\right ) \int \frac {x^5}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \left (9+\log ^2(2)+\log (64)\right )}-\frac {4 \log (2) \left (567+729 \log (2)+259 \log ^2(2)+63 \log ^3(2)+18 \log (4096)+14 \log (2) \log (8192)\right ) \int \frac {x^6}{\left (x^8-2 (1+\log (2)) x^7+11 \left (1+\frac {1}{11} \left (\log ^2(2)+\log (4)\right )\right ) x^6+6 \left (1-\frac {5 \log (2)}{3}\right ) x^5-87 \left (1+\frac {16 \log (2)}{87}\right ) x^4+177 \left (1+\frac {32 \log (2)}{59}\right ) x^3-415 x^2-789 x+2259\right ) \log \left (\frac {x^8-(2+\log (4)) x^7+\left (11+\log ^2(2)+\log (4)\right ) x^6+(6-10 \log (2)) x^5-(87+16 \log (2)) x^4+3 (59+32 \log (2)) x^3-415 x^2-789 x+2259}{(x+3)^2}\right )}dx}{9 \log (4) \left (9+\log ^2(2)+\log (64)\right )}\) |
Int[(-6885 - 1701*x + 1593*x^2 - 867*x^3 - 84*x^4 + 216*x^5 + 2*x^6 + 14*x ^7 + 6*x^8 + (864*x^2 - 96*x^3 - 182*x^4 + 6*x^5 - 34*x^6 - 10*x^7)*Log[2] + (18*x^5 + 4*x^6)*Log[2]^2)/((6777 - 108*x - 2034*x^2 + 116*x^3 - 84*x^4 - 69*x^5 + 39*x^6 + 5*x^7 + x^8 + x^9 + (288*x^3 + 48*x^4 - 46*x^5 - 4*x^ 6 - 4*x^7 - 2*x^8)*Log[2] + (3*x^6 + x^7)*Log[2]^2)*Log[(2259 - 789*x - 41 5*x^2 + 177*x^3 - 87*x^4 + 6*x^5 + 11*x^6 - 2*x^7 + x^8 + (96*x^3 - 16*x^4 - 10*x^5 + 2*x^6 - 2*x^7)*Log[2] + x^6*Log[2]^2)/(9 + 6*x + x^2)]),x]
3.2.98.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*a^p, x] /; F reeQ[{a, b, n, p}, x] && EqQ[b, 0]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
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. \(88\) vs. \(2(28)=56\).
Time = 3.97 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18
| method | result | size |
| norman | \(\ln \left (\ln \left (\frac {x^{6} \ln \left (2\right )^{2}+\left (-2 x^{7}+2 x^{6}-10 x^{5}-16 x^{4}+96 x^{3}\right ) \ln \left (2\right )+x^{8}-2 x^{7}+11 x^{6}+6 x^{5}-87 x^{4}+177 x^{3}-415 x^{2}-789 x +2259}{x^{2}+6 x +9}\right )\right )\) | \(89\) |
| risch | \(\ln \left (\ln \left (\frac {x^{6} \ln \left (2\right )^{2}+\left (-2 x^{7}+2 x^{6}-10 x^{5}-16 x^{4}+96 x^{3}\right ) \ln \left (2\right )+x^{8}-2 x^{7}+11 x^{6}+6 x^{5}-87 x^{4}+177 x^{3}-415 x^{2}-789 x +2259}{x^{2}+6 x +9}\right )\right )\) | \(89\) |
| parallelrisch | \(\ln \left (\ln \left (\frac {x^{6} \ln \left (2\right )^{2}+\left (-2 x^{7}+2 x^{6}-10 x^{5}-16 x^{4}+96 x^{3}\right ) \ln \left (2\right )+x^{8}-2 x^{7}+11 x^{6}+6 x^{5}-87 x^{4}+177 x^{3}-415 x^{2}-789 x +2259}{x^{2}+6 x +9}\right )\right )\) | \(89\) |
| default | \(\ln \left (\ln \left (\frac {x^{6} \ln \left (2\right )^{2}-2 x^{7} \ln \left (2\right )+x^{8}+2 x^{6} \ln \left (2\right )-2 x^{7}-10 x^{5} \ln \left (2\right )+11 x^{6}-16 x^{4} \ln \left (2\right )+6 x^{5}+96 x^{3} \ln \left (2\right )-87 x^{4}+177 x^{3}-415 x^{2}-789 x +2259}{x^{2}+6 x +9}\right )\right )\) | \(95\) |
int(((4*x^6+18*x^5)*ln(2)^2+(-10*x^7-34*x^6+6*x^5-182*x^4-96*x^3+864*x^2)* ln(2)+6*x^8+14*x^7+2*x^6+216*x^5-84*x^4-867*x^3+1593*x^2-1701*x-6885)/((x^ 7+3*x^6)*ln(2)^2+(-2*x^8-4*x^7-4*x^6-46*x^5+48*x^4+288*x^3)*ln(2)+x^9+x^8+ 5*x^7+39*x^6-69*x^5-84*x^4+116*x^3-2034*x^2-108*x+6777)/ln((x^6*ln(2)^2+(- 2*x^7+2*x^6-10*x^5-16*x^4+96*x^3)*ln(2)+x^8-2*x^7+11*x^6+6*x^5-87*x^4+177* x^3-415*x^2-789*x+2259)/(x^2+6*x+9)),x,method=_RETURNVERBOSE)
ln(ln((x^6*ln(2)^2+(-2*x^7+2*x^6-10*x^5-16*x^4+96*x^3)*ln(2)+x^8-2*x^7+11* x^6+6*x^5-87*x^4+177*x^3-415*x^2-789*x+2259)/(x^2+6*x+9)))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\log \left (\log \left (\frac {x^{8} + x^{6} \log \left (2\right )^{2} - 2 \, x^{7} + 11 \, x^{6} + 6 \, x^{5} - 87 \, x^{4} + 177 \, x^{3} - 415 \, x^{2} - 2 \, {\left (x^{7} - x^{6} + 5 \, x^{5} + 8 \, x^{4} - 48 \, x^{3}\right )} \log \left (2\right ) - 789 \, x + 2259}{x^{2} + 6 \, x + 9}\right )\right ) \end {dmath*}
integrate(((4*x^6+18*x^5)*log(2)^2+(-10*x^7-34*x^6+6*x^5-182*x^4-96*x^3+86 4*x^2)*log(2)+6*x^8+14*x^7+2*x^6+216*x^5-84*x^4-867*x^3+1593*x^2-1701*x-68 85)/((x^7+3*x^6)*log(2)^2+(-2*x^8-4*x^7-4*x^6-46*x^5+48*x^4+288*x^3)*log(2 )+x^9+x^8+5*x^7+39*x^6-69*x^5-84*x^4+116*x^3-2034*x^2-108*x+6777)/log((x^6 *log(2)^2+(-2*x^7+2*x^6-10*x^5-16*x^4+96*x^3)*log(2)+x^8-2*x^7+11*x^6+6*x^ 5-87*x^4+177*x^3-415*x^2-789*x+2259)/(x^2+6*x+9)),x, algorithm=\
log(log((x^8 + x^6*log(2)^2 - 2*x^7 + 11*x^6 + 6*x^5 - 87*x^4 + 177*x^3 - 415*x^2 - 2*(x^7 - x^6 + 5*x^5 + 8*x^4 - 48*x^3)*log(2) - 789*x + 2259)/(x ^2 + 6*x + 9)))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 1.57 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\log {\left (\log {\left (\frac {x^{8} - 2 x^{7} + x^{6} \log {\left (2 \right )}^{2} + 11 x^{6} + 6 x^{5} - 87 x^{4} + 177 x^{3} - 415 x^{2} - 789 x + \left (- 2 x^{7} + 2 x^{6} - 10 x^{5} - 16 x^{4} + 96 x^{3}\right ) \log {\left (2 \right )} + 2259}{x^{2} + 6 x + 9} \right )} \right )} \end {dmath*}
integrate(((4*x**6+18*x**5)*ln(2)**2+(-10*x**7-34*x**6+6*x**5-182*x**4-96* x**3+864*x**2)*ln(2)+6*x**8+14*x**7+2*x**6+216*x**5-84*x**4-867*x**3+1593* x**2-1701*x-6885)/((x**7+3*x**6)*ln(2)**2+(-2*x**8-4*x**7-4*x**6-46*x**5+4 8*x**4+288*x**3)*ln(2)+x**9+x**8+5*x**7+39*x**6-69*x**5-84*x**4+116*x**3-2 034*x**2-108*x+6777)/ln((x**6*ln(2)**2+(-2*x**7+2*x**6-10*x**5-16*x**4+96* x**3)*ln(2)+x**8-2*x**7+11*x**6+6*x**5-87*x**4+177*x**3-415*x**2-789*x+225 9)/(x**2+6*x+9)),x)
log(log((x**8 - 2*x**7 + x**6*log(2)**2 + 11*x**6 + 6*x**5 - 87*x**4 + 177 *x**3 - 415*x**2 - 789*x + (-2*x**7 + 2*x**6 - 10*x**5 - 16*x**4 + 96*x**3 )*log(2) + 2259)/(x**2 + 6*x + 9)))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (30) = 60\).
Time = 0.42 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\log \left (\log \left (x^{8} - 2 \, x^{7} {\left (\log \left (2\right ) + 1\right )} + {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) + 11\right )} x^{6} - 2 \, x^{5} {\left (5 \, \log \left (2\right ) - 3\right )} - x^{4} {\left (16 \, \log \left (2\right ) + 87\right )} + 3 \, x^{3} {\left (32 \, \log \left (2\right ) + 59\right )} - 415 \, x^{2} - 789 \, x + 2259\right ) - 2 \, \log \left (x + 3\right )\right ) \end {dmath*}
integrate(((4*x^6+18*x^5)*log(2)^2+(-10*x^7-34*x^6+6*x^5-182*x^4-96*x^3+86 4*x^2)*log(2)+6*x^8+14*x^7+2*x^6+216*x^5-84*x^4-867*x^3+1593*x^2-1701*x-68 85)/((x^7+3*x^6)*log(2)^2+(-2*x^8-4*x^7-4*x^6-46*x^5+48*x^4+288*x^3)*log(2 )+x^9+x^8+5*x^7+39*x^6-69*x^5-84*x^4+116*x^3-2034*x^2-108*x+6777)/log((x^6 *log(2)^2+(-2*x^7+2*x^6-10*x^5-16*x^4+96*x^3)*log(2)+x^8-2*x^7+11*x^6+6*x^ 5-87*x^4+177*x^3-415*x^2-789*x+2259)/(x^2+6*x+9)),x, algorithm=\
log(log(x^8 - 2*x^7*(log(2) + 1) + (log(2)^2 + 2*log(2) + 11)*x^6 - 2*x^5* (5*log(2) - 3) - x^4*(16*log(2) + 87) + 3*x^3*(32*log(2) + 59) - 415*x^2 - 789*x + 2259) - 2*log(x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (30) = 60\).
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.39 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\log \left (-\log \left (x^{8} - 2 \, x^{7} \log \left (2\right ) + x^{6} \log \left (2\right )^{2} - 2 \, x^{7} + 2 \, x^{6} \log \left (2\right ) + 11 \, x^{6} - 10 \, x^{5} \log \left (2\right ) + 6 \, x^{5} - 16 \, x^{4} \log \left (2\right ) - 87 \, x^{4} + 96 \, x^{3} \log \left (2\right ) + 177 \, x^{3} - 415 \, x^{2} - 789 \, x + 2259\right ) + \log \left (x^{2} + 6 \, x + 9\right )\right ) \end {dmath*}
integrate(((4*x^6+18*x^5)*log(2)^2+(-10*x^7-34*x^6+6*x^5-182*x^4-96*x^3+86 4*x^2)*log(2)+6*x^8+14*x^7+2*x^6+216*x^5-84*x^4-867*x^3+1593*x^2-1701*x-68 85)/((x^7+3*x^6)*log(2)^2+(-2*x^8-4*x^7-4*x^6-46*x^5+48*x^4+288*x^3)*log(2 )+x^9+x^8+5*x^7+39*x^6-69*x^5-84*x^4+116*x^3-2034*x^2-108*x+6777)/log((x^6 *log(2)^2+(-2*x^7+2*x^6-10*x^5-16*x^4+96*x^3)*log(2)+x^8-2*x^7+11*x^6+6*x^ 5-87*x^4+177*x^3-415*x^2-789*x+2259)/(x^2+6*x+9)),x, algorithm=\
log(-log(x^8 - 2*x^7*log(2) + x^6*log(2)^2 - 2*x^7 + 2*x^6*log(2) + 11*x^6 - 10*x^5*log(2) + 6*x^5 - 16*x^4*log(2) - 87*x^4 + 96*x^3*log(2) + 177*x^ 3 - 415*x^2 - 789*x + 2259) + log(x^2 + 6*x + 9))
Time = 16.55 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \begin {dmath*} \int \frac {-6885-1701 x+1593 x^2-867 x^3-84 x^4+216 x^5+2 x^6+14 x^7+6 x^8+\left (864 x^2-96 x^3-182 x^4+6 x^5-34 x^6-10 x^7\right ) \log (2)+\left (18 x^5+4 x^6\right ) \log ^2(2)}{\left (6777-108 x-2034 x^2+116 x^3-84 x^4-69 x^5+39 x^6+5 x^7+x^8+x^9+\left (288 x^3+48 x^4-46 x^5-4 x^6-4 x^7-2 x^8\right ) \log (2)+\left (3 x^6+x^7\right ) \log ^2(2)\right ) \log \left (\frac {2259-789 x-415 x^2+177 x^3-87 x^4+6 x^5+11 x^6-2 x^7+x^8+\left (96 x^3-16 x^4-10 x^5+2 x^6-2 x^7\right ) \log (2)+x^6 \log ^2(2)}{9+6 x+x^2}\right )} \, dx=\ln \left (\ln \left (\frac {x^6\,{\ln \left (2\right )}^2-789\,x-\ln \left (2\right )\,\left (2\,x^7-2\,x^6+10\,x^5+16\,x^4-96\,x^3\right )-415\,x^2+177\,x^3-87\,x^4+6\,x^5+11\,x^6-2\,x^7+x^8+2259}{x^2+6\,x+9}\right )\right ) \end {dmath*}
int((1593*x^2 - log(2)*(96*x^3 - 864*x^2 + 182*x^4 - 6*x^5 + 34*x^6 + 10*x ^7) - 1701*x - 867*x^3 - 84*x^4 + 216*x^5 + 2*x^6 + 14*x^7 + 6*x^8 + log(2 )^2*(18*x^5 + 4*x^6) - 6885)/(log((x^6*log(2)^2 - 789*x - log(2)*(16*x^4 - 96*x^3 + 10*x^5 - 2*x^6 + 2*x^7) - 415*x^2 + 177*x^3 - 87*x^4 + 6*x^5 + 1 1*x^6 - 2*x^7 + x^8 + 2259)/(6*x + x^2 + 9))*(log(2)^2*(3*x^6 + x^7) - log (2)*(46*x^5 - 48*x^4 - 288*x^3 + 4*x^6 + 4*x^7 + 2*x^8) - 108*x - 2034*x^2 + 116*x^3 - 84*x^4 - 69*x^5 + 39*x^6 + 5*x^7 + x^8 + x^9 + 6777)),x)