Integrand size = 144, antiderivative size = 25 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=x \left (-4+4 \left (1+x+\frac {x}{4+x-x^2}+\log (x)\right )^2\right ) \] Output:
(2*(ln(x)+x+1+x/(-x^2+x+4))*(2*ln(x)+2*x+2+2*x/(-x^2+x+4))-4)*x
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(25)=50\).
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=4 \left (-2 x+2 x^2+x^3+\frac {4+5 x}{\left (-4-x+x^2\right )^2}+\frac {-15-11 x}{-4-x+x^2}-2 \log (x)+\frac {2 \left (-4-5 x-5 x^2+x^4\right ) \log (x)}{-4-x+x^2}+x \log ^2(x)\right ) \] Input:
Integrate[(-512 - 2304*x - 2224*x^2 + 420*x^3 + 1044*x^4 + 52*x^5 - 180*x^ 6 - 12*x^7 + 12*x^8 + (-1024 - 2048*x - 288*x^2 + 1000*x^3 + 232*x^4 - 192 *x^5 - 32*x^6 + 16*x^7)*Log[x] + (-256 - 192*x + 144*x^2 + 92*x^3 - 36*x^4 - 12*x^5 + 4*x^6)*Log[x]^2)/(-64 - 48*x + 36*x^2 + 23*x^3 - 9*x^4 - 3*x^5 + x^6),x]
Output:
4*(-2*x + 2*x^2 + x^3 + (4 + 5*x)/(-4 - x + x^2)^2 + (-15 - 11*x)/(-4 - x + x^2) - 2*Log[x] + (2*(-4 - 5*x - 5*x^2 + x^4)*Log[x])/(-4 - x + x^2) + x *Log[x]^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 11.85 (sec) , antiderivative size = 6625, normalized size of antiderivative = 265.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2463, 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.
| \(\displaystyle \int \frac {12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-2304 x-512}{x^6-3 x^5-9 x^4+23 x^3+36 x^2-48 x-64} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {12 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{289 \sqrt {17} \left (-2 x+\sqrt {17}+1\right )}-\frac {12 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{289 \sqrt {17} \left (2 x+\sqrt {17}-1\right )}-\frac {12 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}-\frac {12 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{289 \left (2 x+\sqrt {17}-1\right )^2}-\frac {8 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{17 \sqrt {17} \left (-2 x+\sqrt {17}+1\right )^3}-\frac {8 \left (12 x^8-12 x^7-180 x^6+52 x^5+1044 x^4+420 x^3-2224 x^2-2304 x+\left (4 x^6-12 x^5-36 x^4+92 x^3+144 x^2-192 x-256\right ) \log ^2(x)+\left (16 x^7-32 x^6-192 x^5+232 x^4+1000 x^3-288 x^2-2048 x-1024\right ) \log (x)-512\right )}{17 \sqrt {17} \left (2 x+\sqrt {17}-1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {36 \left (17+\sqrt {17}\right ) x^7}{34391}+\frac {36 \left (17-\sqrt {17}\right ) x^7}{34391}-\frac {72 x^7}{2023}-\frac {16 \left (17+7 \sqrt {17}\right ) \log (x) x^6}{4913 \left (1-\sqrt {17}\right )}-\frac {16 \left (17-7 \sqrt {17}\right ) \log (x) x^6}{4913 \left (1+\sqrt {17}\right )}-\frac {16}{289} \log (x) x^6+\frac {6 \left (17+9 \sqrt {17}\right ) x^6}{4913}+\frac {8 \left (17+7 \sqrt {17}\right ) x^6}{14739 \left (1-\sqrt {17}\right )}-\frac {6 \left (17+\sqrt {17}\right ) x^6}{4913}-\frac {6}{289} \left (1+\sqrt {17}\right ) x^6+\frac {8 \left (17-7 \sqrt {17}\right ) x^6}{14739 \left (1+\sqrt {17}\right )}-\frac {6 \left (17-\sqrt {17}\right ) x^6}{4913}-\frac {6}{289} \left (1-\sqrt {17}\right ) x^6+\frac {6 \left (17-9 \sqrt {17}\right ) x^6}{4913}+\frac {44 x^6}{867}+\frac {12 \left (17+5 \sqrt {17}\right ) \log ^2(x) x^5}{24565}+\frac {12 \left (17-5 \sqrt {17}\right ) \log ^2(x) x^5}{24565}-\frac {24 \log ^2(x) x^5}{1445}-\frac {24 \left (17+5 \sqrt {17}\right ) \log (x) x^5}{122825}-\frac {96 \left (9+\sqrt {17}\right ) \log (x) x^5}{1445 \left (1+\sqrt {17}\right )}-\frac {768 \log (x) x^5}{1445 \left (1+\sqrt {17}\right )}-\frac {96 \left (9-\sqrt {17}\right ) \log (x) x^5}{1445 \left (1-\sqrt {17}\right )}-\frac {768 \log (x) x^5}{1445 \left (1-\sqrt {17}\right )}-\frac {24 \left (17-5 \sqrt {17}\right ) \log (x) x^5}{122825}+\frac {48 \log (x) x^5}{7225}-\frac {36 \left (17+9 \sqrt {17}\right ) x^5}{24565}+\frac {24 \left (17+5 \sqrt {17}\right ) x^5}{614125}-\frac {234 \left (17+\sqrt {17}\right ) x^5}{24565}+\frac {96 \left (9+\sqrt {17}\right ) x^5}{7225 \left (1+\sqrt {17}\right )}-\frac {54 \left (9+\sqrt {17}\right ) x^5}{1445}+\frac {9 \left (1+\sqrt {17}\right )^3 x^5}{1445 \sqrt {17}}+\frac {36 \left (1+\sqrt {17}\right ) x^5}{1445}+\frac {768 x^5}{7225 \left (1+\sqrt {17}\right )}-\frac {234 \left (17-\sqrt {17}\right ) x^5}{24565}+\frac {96 \left (9-\sqrt {17}\right ) x^5}{7225 \left (1-\sqrt {17}\right )}-\frac {54 \left (9-\sqrt {17}\right ) x^5}{1445}+\frac {36 \left (1-\sqrt {17}\right ) x^5}{1445}+\frac {768 x^5}{7225 \left (1-\sqrt {17}\right )}+\frac {24 \left (17-5 \sqrt {17}\right ) x^5}{614125}-\frac {36 \left (17-9 \sqrt {17}\right ) x^5}{24565}+\frac {36 \left (85-13 \sqrt {17}\right ) x^5}{24565}+\frac {26952 x^5}{36125}-\frac {6 \left (17+6 \sqrt {17}\right ) \log ^2(x) x^4}{4913}+\frac {3}{289} \left (2+\sqrt {17}\right ) \log ^2(x) x^4+\frac {3}{289} \left (2-\sqrt {17}\right ) \log ^2(x) x^4-\frac {6 \left (17-6 \sqrt {17}\right ) \log ^2(x) x^4}{4913}+\frac {36 \left (17+47 \sqrt {17}\right ) \log (x) x^4}{4913 \left (1-\sqrt {17}\right )}+\frac {4 \left (17+25 \sqrt {17}\right ) \log (x) x^4}{289 \left (1+\sqrt {17}\right )}+\frac {3 \left (17+6 \sqrt {17}\right ) \log (x) x^4}{4913}+\frac {12 \left (9+\sqrt {17}\right ) \log (x) x^4}{289 \left (1+\sqrt {17}\right )}-\frac {3}{578} \left (2+\sqrt {17}\right ) \log (x) x^4+\frac {36 \left (17-47 \sqrt {17}\right ) \log (x) x^4}{4913 \left (1+\sqrt {17}\right )}+\frac {12 \left (9-\sqrt {17}\right ) \log (x) x^4}{289 \left (1-\sqrt {17}\right )}-\frac {3}{578} \left (2-\sqrt {17}\right ) \log (x) x^4+\frac {4 \left (17-25 \sqrt {17}\right ) \log (x) x^4}{289 \left (1-\sqrt {17}\right )}+\frac {3 \left (17-6 \sqrt {17}\right ) \log (x) x^4}{4913}-\frac {9 \left (17+47 \sqrt {17}\right ) x^4}{4913 \left (1-\sqrt {17}\right )}-\frac {\left (17+25 \sqrt {17}\right ) x^4}{289 \left (1+\sqrt {17}\right )}+\frac {18 \left (17+9 \sqrt {17}\right ) x^4}{4913}-\frac {3 \left (17+6 \sqrt {17}\right ) x^4}{19652}-\frac {75 \left (17+\sqrt {17}\right ) x^4}{9826}-\frac {3 \left (9+\sqrt {17}\right ) x^4}{289 \left (1+\sqrt {17}\right )}+\frac {27}{578} \left (9+\sqrt {17}\right ) x^4+\frac {3 \left (2+\sqrt {17}\right ) x^4}{2312}+\frac {9 \left (1+\sqrt {17}\right )^4 x^4}{2312 \sqrt {17}}-\frac {9 \left (1+\sqrt {17}\right )^3 x^4}{1156 \sqrt {17}}-\frac {9}{578} \left (1+\sqrt {17}\right )^3 x^4+\frac {135}{289} \left (1+\sqrt {17}\right ) x^4-\frac {9 \left (17-47 \sqrt {17}\right ) x^4}{4913 \left (1+\sqrt {17}\right )}-\frac {75 \left (17-\sqrt {17}\right ) x^4}{9826}-\frac {3 \left (9-\sqrt {17}\right ) x^4}{289 \left (1-\sqrt {17}\right )}+\frac {27}{578} \left (9-\sqrt {17}\right ) x^4+\frac {3 \left (2-\sqrt {17}\right ) x^4}{2312}-\frac {9 \left (1-\sqrt {17}\right )^4 x^4}{2312 \sqrt {17}}-\frac {9}{578} \left (1-\sqrt {17}\right )^3 x^4+\frac {135}{289} \left (1-\sqrt {17}\right ) x^4-\frac {\left (17-25 \sqrt {17}\right ) x^4}{289 \left (1-\sqrt {17}\right )}-\frac {3 \left (17-6 \sqrt {17}\right ) x^4}{19652}+\frac {18 \left (17-9 \sqrt {17}\right ) x^4}{4913}-\frac {9 \left (85-13 \sqrt {17}\right ) x^4}{4913}-\frac {78 x^4}{289}-\frac {4 \left (119+23 \sqrt {17}\right ) \log ^2(x) x^3}{4913}-\frac {6}{289} \left (1+\sqrt {17}\right ) \log ^2(x) x^3-\frac {6}{289} \left (1-\sqrt {17}\right ) \log ^2(x) x^3-\frac {4 \left (119-23 \sqrt {17}\right ) \log ^2(x) x^3}{4913}+\frac {8 \left (119+23 \sqrt {17}\right ) \log (x) x^3}{14739}+\frac {128 \left (119+10 \sqrt {17}\right ) \log (x) x^3}{4913 \left (1+\sqrt {17}\right )}+\frac {8 \left (167+7 \sqrt {17}\right ) \log (x) x^3}{289 \left (1+\sqrt {17}\right )}+\frac {4}{289} \left (1+\sqrt {17}\right ) \log (x) x^3-\frac {64 \log (x) x^3}{51 \sqrt {17} \left (1+\sqrt {17}\right )}+\frac {4}{289} \left (1-\sqrt {17}\right ) \log (x) x^3+\frac {8 \left (167-7 \sqrt {17}\right ) \log (x) x^3}{289 \left (1-\sqrt {17}\right )}+\frac {128 \left (119-10 \sqrt {17}\right ) \log (x) x^3}{4913 \left (1-\sqrt {17}\right )}+\frac {64 \log (x) x^3}{51 \sqrt {17} \left (1-\sqrt {17}\right )}+\frac {8 \left (119-23 \sqrt {17}\right ) \log (x) x^3}{14739}-\frac {8 \left (119+23 \sqrt {17}\right ) x^3}{44217}-\frac {128 \left (119+10 \sqrt {17}\right ) x^3}{14739 \left (1+\sqrt {17}\right )}-\frac {152 \left (17+9 \sqrt {17}\right ) x^3}{4913}-\frac {8 \left (167+7 \sqrt {17}\right ) x^3}{867 \left (1+\sqrt {17}\right )}-\frac {486 \left (17+\sqrt {17}\right ) x^3}{4913}+\frac {270}{289} \left (9+\sqrt {17}\right ) x^3+\frac {3 \left (1+\sqrt {17}\right )^5 x^3}{1156 \sqrt {17}}-\frac {3 \left (1+\sqrt {17}\right )^4 x^3}{578 \sqrt {17}}-\frac {15 \left (1+\sqrt {17}\right )^4 x^3}{1156}+\frac {40 \left (1+\sqrt {17}\right )^3 x^3}{289 \sqrt {17}}+\frac {6}{289} \left (1+\sqrt {17}\right )^3 x^3-\frac {160}{867} \left (1+\sqrt {17}\right ) x^3+\frac {64 x^3}{153 \sqrt {17} \left (1+\sqrt {17}\right )}-\frac {486 \left (17-\sqrt {17}\right ) x^3}{4913}+\frac {270}{289} \left (9-\sqrt {17}\right ) x^3-\frac {3 \left (1-\sqrt {17}\right )^5 x^3}{1156 \sqrt {17}}+\frac {3 \left (1-\sqrt {17}\right )^4 x^3}{578 \sqrt {17}}-\frac {15 \left (1-\sqrt {17}\right )^4 x^3}{1156}+\frac {6}{289} \left (1-\sqrt {17}\right )^3 x^3-\frac {160}{867} \left (1-\sqrt {17}\right ) x^3-\frac {8 \left (167-7 \sqrt {17}\right ) x^3}{867 \left (1-\sqrt {17}\right )}-\frac {128 \left (119-10 \sqrt {17}\right ) x^3}{14739 \left (1-\sqrt {17}\right )}-\frac {64 x^3}{153 \sqrt {17} \left (1-\sqrt {17}\right )}-\frac {152 \left (17-9 \sqrt {17}\right ) x^3}{4913}+\frac {160 \left (85-13 \sqrt {17}\right ) x^3}{4913}-\frac {8 \left (119-23 \sqrt {17}\right ) x^3}{44217}-\frac {2088 x^3}{289}+\frac {3}{289} \left (1+7 \sqrt {17}\right ) \log ^2(x) x^2+\frac {48 \left (17+3 \sqrt {17}\right ) \log ^2(x) x^2}{4913}+\frac {48 \left (17-3 \sqrt {17}\right ) \log ^2(x) x^2}{4913}+\frac {3}{289} \left (1-7 \sqrt {17}\right ) \log ^2(x) x^2+\frac {4 \left (289+49 \sqrt {17}\right ) \log (x) x^2}{289 \left (1+\sqrt {17}\right )}+\frac {12 \left (63+31 \sqrt {17}\right ) \log (x) x^2}{289 \left (1+\sqrt {17}\right )}+\frac {768 \left (17+15 \sqrt {17}\right ) \log (x) x^2}{4913 \left (1+\sqrt {17}\right )}-\frac {3}{289} \left (1+7 \sqrt {17}\right ) \log (x) x^2-\frac {48 \left (17+3 \sqrt {17}\right ) \log (x) x^2}{4913}-\frac {384 \log (x) x^2}{17 \sqrt {17} \left (1+\sqrt {17}\right )^2}+\frac {768 \left (17-15 \sqrt {17}\right ) \log (x) x^2}{4913 \left (1-\sqrt {17}\right )}+\frac {12 \left (63-31 \sqrt {17}\right ) \log (x) x^2}{289 \left (1-\sqrt {17}\right )}+\frac {4 \left (289-49 \sqrt {17}\right ) \log (x) x^2}{289 \left (1-\sqrt {17}\right )}+\frac {384 \log (x) x^2}{17 \sqrt {17} \left (1-\sqrt {17}\right )^2}-\frac {48 \left (17-3 \sqrt {17}\right ) \log (x) x^2}{4913}-\frac {3}{289} \left (1-7 \sqrt {17}\right ) \log (x) x^2-\frac {9216 x^2}{17 \sqrt {17} \left (1-\sqrt {17}\right ) \left (-2 x-\sqrt {17}+1\right )^2}+\frac {9216 x^2}{17 \sqrt {17} \left (1+\sqrt {17}\right ) \left (-2 x+\sqrt {17}+1\right )^2}-\frac {2 \left (289+49 \sqrt {17}\right ) x^2}{289 \left (1+\sqrt {17}\right )}-\frac {6 \left (63+31 \sqrt {17}\right ) x^2}{289 \left (1+\sqrt {17}\right )}-\frac {384 \left (17+15 \sqrt {17}\right ) x^2}{4913 \left (1+\sqrt {17}\right )}-\frac {3024 \left (17+9 \sqrt {17}\right ) x^2}{4913}+\frac {3}{578} \left (1+7 \sqrt {17}\right ) x^2+\frac {24 \left (17+3 \sqrt {17}\right ) x^2}{4913}+\frac {1293 \left (17+\sqrt {17}\right ) x^2}{4913}-\frac {117}{289} \left (9+\sqrt {17}\right ) x^2+\frac {96 x^2}{17 \sqrt {17} \left (9+\sqrt {17}\right )}+\frac {9 \left (1+\sqrt {17}\right )^6 x^2}{4624 \sqrt {17}}-\frac {9 \left (1+\sqrt {17}\right )^5 x^2}{2312 \sqrt {17}}-\frac {27 \left (1+\sqrt {17}\right )^5 x^2}{2312}+\frac {495 \left (1+\sqrt {17}\right )^4 x^2}{2312 \sqrt {17}}+\frac {45 \left (1+\sqrt {17}\right )^4 x^2}{2312}-\frac {108 \left (1+\sqrt {17}\right )^3 x^2}{289 \sqrt {17}}+\frac {135}{289} \left (1+\sqrt {17}\right )^3 x^2-\frac {1566}{289} \left (1+\sqrt {17}\right ) x^2+\frac {1293 \left (17-\sqrt {17}\right ) x^2}{4913}-\frac {117}{289} \left (9-\sqrt {17}\right ) x^2-\frac {9 \left (1-\sqrt {17}\right )^6 x^2}{4624 \sqrt {17}}+\frac {9 \left (1-\sqrt {17}\right )^5 x^2}{2312 \sqrt {17}}-\frac {27 \left (1-\sqrt {17}\right )^5 x^2}{2312}-\frac {495 \left (1-\sqrt {17}\right )^4 x^2}{2312 \sqrt {17}}+\frac {45 \left (1-\sqrt {17}\right )^4 x^2}{2312}+\frac {135}{289} \left (1-\sqrt {17}\right )^3 x^2-\frac {1566}{289} \left (1-\sqrt {17}\right ) x^2-\frac {384 \left (17-15 \sqrt {17}\right ) x^2}{4913 \left (1-\sqrt {17}\right )}-\frac {6 \left (63-31 \sqrt {17}\right ) x^2}{289 \left (1-\sqrt {17}\right )}-\frac {2 \left (289-49 \sqrt {17}\right ) x^2}{289 \left (1-\sqrt {17}\right )}-\frac {192 x^2}{17 \sqrt {17} \left (1-\sqrt {17}\right )^2}+\frac {24 \left (17-3 \sqrt {17}\right ) x^2}{4913}+\frac {3}{578} \left (1-7 \sqrt {17}\right ) x^2-\frac {3024 \left (17-9 \sqrt {17}\right ) x^2}{4913}-\frac {432 \left (85-13 \sqrt {17}\right ) x^2}{4913}-\frac {1260 x^2}{289}+\frac {192 \left (17+\sqrt {17}\right ) \log ^2(x) x}{4913}+\frac {24}{289} \left (9+\sqrt {17}\right ) \log ^2(x) x+\frac {192 \left (17-\sqrt {17}\right ) \log ^2(x) x}{4913}+\frac {24}{289} \left (9-\sqrt {17}\right ) \log ^2(x) x+\frac {32 \left (85-13 \sqrt {17}\right ) \log (x) x}{17 \left (1-\sqrt {17}\right )^2 \left (-2 x-\sqrt {17}+1\right )}+\frac {32 \left (85+13 \sqrt {17}\right ) \log (x) x}{17 \left (1+\sqrt {17}\right )^2 \left (-2 x+\sqrt {17}+1\right )}+\frac {4}{289} \left (221+9 \sqrt {17}\right ) \log (x) x+\frac {60}{289} \left (11+5 \sqrt {17}\right ) \log (x) x+\frac {384 \left (17+\sqrt {17}\right ) \log (x) x}{4913}-\frac {48}{289} \left (9+\sqrt {17}\right ) \log (x) x-\frac {4096 \log (x) x}{17 \sqrt {17} \left (1+\sqrt {17}\right )^3}+\frac {384 \left (17-\sqrt {17}\right ) \log (x) x}{4913}-\frac {48}{289} \left (9-\sqrt {17}\right ) \log (x) x+\frac {4096 \log (x) x}{17 \sqrt {17} \left (1-\sqrt {17}\right )^3}+\frac {60}{289} \left (11-5 \sqrt {17}\right ) \log (x) x+\frac {4}{289} \left (221-9 \sqrt {17}\right ) \log (x) x+\frac {4}{289} \left (85+13 \sqrt {17}\right ) x-\frac {4}{289} \left (221+9 \sqrt {17}\right ) x+\frac {3912 \left (17+9 \sqrt {17}\right ) x}{4913}-\frac {60}{289} \left (11+5 \sqrt {17}\right ) x+\frac {19566 \left (17+\sqrt {17}\right ) x}{4913}-\frac {4650}{289} \left (9+\sqrt {17}\right ) x+\frac {9 \left (1+\sqrt {17}\right )^7 x}{4624 \sqrt {17}}-\frac {9 \left (1+\sqrt {17}\right )^6 x}{2312 \sqrt {17}}-\frac {63 \left (1+\sqrt {17}\right )^6 x}{4624}+\frac {801 \left (1+\sqrt {17}\right )^5 x}{2312 \sqrt {17}}+\frac {27 \left (1+\sqrt {17}\right )^5 x}{1156}-\frac {687 \left (1+\sqrt {17}\right )^4 x}{1156 \sqrt {17}}+\frac {675 \left (1+\sqrt {17}\right )^4 x}{1156}-\frac {3042 \left (1+\sqrt {17}\right )^3 x}{289 \sqrt {17}}-\frac {78}{289} \left (1+\sqrt {17}\right )^3 x-\frac {1260}{289} \left (1+\sqrt {17}\right ) x+\frac {4096 x}{17 \sqrt {17} \left (1+\sqrt {17}\right )^3}+\frac {19566 \left (17-\sqrt {17}\right ) x}{4913}-\frac {4650}{289} \left (9-\sqrt {17}\right ) x-\frac {9 \left (1-\sqrt {17}\right )^7 x}{4624 \sqrt {17}}+\frac {9 \left (1-\sqrt {17}\right )^6 x}{2312 \sqrt {17}}-\frac {63 \left (1-\sqrt {17}\right )^6 x}{4624}-\frac {801 \left (1-\sqrt {17}\right )^5 x}{2312 \sqrt {17}}+\frac {27 \left (1-\sqrt {17}\right )^5 x}{1156}+\frac {687 \left (1-\sqrt {17}\right )^4 x}{1156 \sqrt {17}}+\frac {675 \left (1-\sqrt {17}\right )^4 x}{1156}-\frac {78}{289} \left (1-\sqrt {17}\right )^3 x-\frac {1260}{289} \left (1-\sqrt {17}\right ) x-\frac {60}{289} \left (11-5 \sqrt {17}\right ) x-\frac {4}{289} \left (221-9 \sqrt {17}\right ) x+\frac {3912 \left (17-9 \sqrt {17}\right ) x}{4913}-\frac {12168 \left (85-13 \sqrt {17}\right ) x}{4913}+\frac {13344 x}{289}-\frac {\left (x+\frac {8}{1-\sqrt {17}}\right )^4 \log ^2(x)}{17 \sqrt {17}}+\frac {\left (x+\frac {8}{1+\sqrt {17}}\right )^4 \log ^2(x)}{17 \sqrt {17}}-\frac {4096 \log ^2(x)}{17 \sqrt {17} \left (1+\sqrt {17}\right )^4}+\frac {4096 \log ^2(x)}{17 \sqrt {17} \left (1-\sqrt {17}\right )^4}+\frac {3798 \left (17-\sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}-\frac {1890}{289} \left (9-\sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )+\frac {6672}{289} \left (1-\sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )+\frac {16 \left (85-13 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{17 \left (1-\sqrt {17}\right )^2}-\frac {6264}{289} \left (13-5 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )-\frac {390}{289} \left (49-9 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )+\frac {46572 \left (17-9 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}+\frac {5680 \left (85-13 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}+\frac {1620}{289} \left (101-29 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )-\frac {19818 \left (153-49 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}+\frac {126}{289} \left (297-65 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )-\frac {1986 \left (493-101 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}-\frac {144}{289} \left (701-181 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )+\frac {2316 \left (1105-297 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}-\frac {36 \left (3077-701 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}+\frac {36 \left (7497-1889 \sqrt {17}\right ) \log \left (-2 x-\sqrt {17}+1\right )}{4913}+\frac {40880 \log \left (-2 x-\sqrt {17}+1\right )}{289 \sqrt {17}}+\frac {6912}{289} \log \left (-2 x-\sqrt {17}+1\right )-\frac {24 \left (13+5 \sqrt {17}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {17}\right )\right ) \log \left (-2 x+\sqrt {17}+1\right )}{17 \left (1+\sqrt {17}\right )}+\frac {36 \left (7497+1889 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}-\frac {36 \left (3077+701 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}+\frac {2316 \left (1105+297 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}-\frac {144}{289} \left (701+181 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )-\frac {1986 \left (493+101 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}+\frac {126}{289} \left (297+65 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )-\frac {19818 \left (153+49 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}+\frac {1620}{289} \left (101+29 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )+\frac {16 \left (85+13 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{17 \left (1+\sqrt {17}\right )^2}+\frac {5680 \left (85+13 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}-\frac {390}{289} \left (49+9 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )+\frac {46572 \left (17+9 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}-\frac {6264}{289} \left (13+5 \sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )+\frac {3798 \left (17+\sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )}{4913}-\frac {1890}{289} \left (9+\sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )+\frac {6672}{289} \left (1+\sqrt {17}\right ) \log \left (-2 x+\sqrt {17}+1\right )-\frac {40880 \log \left (-2 x+\sqrt {17}+1\right )}{289 \sqrt {17}}+\frac {6912}{289} \log \left (-2 x+\sqrt {17}+1\right )+\frac {24 \left (13+5 \sqrt {17}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {17}\right )\right ) \log \left (-\left (\left (1+\sqrt {17}\right ) x\right )+\sqrt {17}+9\right )}{17 \left (1+\sqrt {17}\right )}+\frac {24 \left (13+5 \sqrt {17}\right ) \operatorname {PolyLog}\left (2,1-\frac {2 x}{1+\sqrt {17}}\right )}{17 \left (1+\sqrt {17}\right )}-\frac {24 \left (13+5 \sqrt {17}\right ) \operatorname {PolyLog}\left (2,1-\frac {\left (1+\sqrt {17}\right ) x}{9+\sqrt {17}}\right )}{17 \left (1+\sqrt {17}\right )}-\frac {4448 \left (17-\sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {6672 \left (9-\sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {6912 \left (1-\sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {1260 \left (13-5 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {3132 \left (49-9 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {1260 \left (17-9 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {4176 \left (85-13 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {156 \left (101-29 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {260 \left (153-49 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {540 \left (297-65 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {1080 \left (493-101 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {36 \left (701-181 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {84 \left (1105-297 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}-\frac {36 \left (1889-441 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {96 \left (3077-701 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {3072}{289 \left (-2 x-\sqrt {17}+1\right )}+\frac {96 \left (3077+701 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {36 \left (1889+441 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {84 \left (1105+297 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {36 \left (701+181 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {1080 \left (493+101 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {540 \left (297+65 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {260 \left (153+49 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {156 \left (101+29 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {4176 \left (85+13 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {3132 \left (49+9 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {1260 \left (17+9 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {1260 \left (13+5 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}-\frac {4448 \left (17+\sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {6672 \left (9+\sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {6912 \left (1+\sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {3072}{289 \left (-2 x+\sqrt {17}+1\right )}+\frac {2224 \left (17-9 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}-\frac {420 \left (85-13 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}-\frac {1044 \left (153-49 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}-\frac {52 \left (493-101 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}+\frac {180 \left (1105-297 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}+\frac {12 \left (3077-701 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}-\frac {12 \left (7497-1889 \sqrt {17}\right )}{289 \left (-2 x-\sqrt {17}+1\right )^2}-\frac {1024}{17 \sqrt {17} \left (-2 x-\sqrt {17}+1\right )^2}-\frac {12 \left (7497+1889 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}+\frac {12 \left (3077+701 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}+\frac {180 \left (1105+297 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}-\frac {52 \left (493+101 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}-\frac {1044 \left (153+49 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}-\frac {420 \left (85+13 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}+\frac {2224 \left (17+9 \sqrt {17}\right )}{289 \left (-2 x+\sqrt {17}+1\right )^2}+\frac {1024}{17 \sqrt {17} \left (-2 x+\sqrt {17}+1\right )^2}\) |
Input:
Int[(-512 - 2304*x - 2224*x^2 + 420*x^3 + 1044*x^4 + 52*x^5 - 180*x^6 - 12 *x^7 + 12*x^8 + (-1024 - 2048*x - 288*x^2 + 1000*x^3 + 232*x^4 - 192*x^5 - 32*x^6 + 16*x^7)*Log[x] + (-256 - 192*x + 144*x^2 + 92*x^3 - 36*x^4 - 12* x^5 + 4*x^6)*Log[x]^2)/(-64 - 48*x + 36*x^2 + 23*x^3 - 9*x^4 - 3*x^5 + x^6 ),x]
Output:
-1024/(17*Sqrt[17]*(1 - Sqrt[17] - 2*x)^2) - (12*(7497 - 1889*Sqrt[17]))/( 289*(1 - Sqrt[17] - 2*x)^2) + (12*(3077 - 701*Sqrt[17]))/(289*(1 - Sqrt[17 ] - 2*x)^2) + (180*(1105 - 297*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)^2) - ( 52*(493 - 101*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)^2) - (1044*(153 - 49*Sq rt[17]))/(289*(1 - Sqrt[17] - 2*x)^2) - (420*(85 - 13*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)^2) + (2224*(17 - 9*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)^2 ) + 3072/(289*(1 - Sqrt[17] - 2*x)) + (96*(3077 - 701*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (36*(1889 - 441*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (84*(1105 - 297*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (36*(701 - 181*S qrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (1080*(493 - 101*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (540*(297 - 65*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (260*(153 - 49*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (156*(101 - 29*S qrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (4176*(85 - 13*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (1260*(17 - 9*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (3132*(49 - 9*Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (1260*(13 - 5*Sqrt[1 7]))/(289*(1 - Sqrt[17] - 2*x)) + (6912*(1 - Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + (6672*(9 - Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) - (4448*(17 - Sqrt[17]))/(289*(1 - Sqrt[17] - 2*x)) + 1024/(17*Sqrt[17]*(1 + Sqrt[17] - 2*x)^2) + (2224*(17 + 9*Sqrt[17]))/(289*(1 + Sqrt[17] - 2*x)^2) - (420*(85 + 13*Sqrt[17]))/(289*(1 + Sqrt[17] - 2*x)^2) - (1044*(153 + 49*Sqrt[17...
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]
Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(45)=90\).
Time = 5.52 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32
| method | result | size |
| risch | \(4 x \ln \left (x \right )^{2}+\frac {8 \left (x^{4}-5 x^{2}-5 x -4\right ) \ln \left (x \right )}{x^{2}-x -4}-\frac {4 \left (-x^{7}+2 x^{4} \ln \left (x \right )+13 x^{5}-4 x^{3} \ln \left (x \right )+2 x^{4}-14 x^{2} \ln \left (x \right )-35 x^{3}+16 x \ln \left (x \right )-12 x^{2}+32 \ln \left (x \right )-32 x -64\right )}{\left (x^{2}-x -4\right )^{2}}\) | \(108\) |
| parallelrisch | \(\frac {384+192 x -8 x^{4} \ln \left (x \right )^{2}-8 x^{5} \ln \left (x \right )+8 x^{6} \ln \left (x \right )+4 x^{5} \ln \left (x \right )^{2}-80 x^{4} \ln \left (x \right )-28 x^{3} \ln \left (x \right )^{2}+32 x^{2} \ln \left (x \right )^{2}+64 x \ln \left (x \right )^{2}+4 x^{7}+16 x^{3} \ln \left (x \right )+128 x \ln \left (x \right )+224 x^{2} \ln \left (x \right )-8 x^{2}+124 x^{3}-52 x^{5}}{x^{4}-2 x^{3}-7 x^{2}+8 x +16}\) | \(130\) |
| default | \(4 x \ln \left (x \right )^{2}+8 x \ln \left (x \right )-8 x +4 x^{3}+8 x^{2}-\frac {4 \left (11 x^{3}+4 x^{2}-64 x -64\right )}{\left (x^{2}-x -4\right )^{2}}+8 x^{2} \ln \left (x \right )-\frac {8 \ln \left (x \right ) \left (\ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right ) \sqrt {17}\, x^{2}-\ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right ) \sqrt {17}\, x^{2}-\ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right ) \sqrt {17}\, x +\ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right ) \sqrt {17}\, x -4 \sqrt {17}\, \ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right )+4 \sqrt {17}\, \ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right )+17 x^{2}\right )}{17 \left (x^{2}-x -4\right )}+\frac {8 \sqrt {17}\, \ln \left (x \right ) \ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right )}{17}-\frac {8 \sqrt {17}\, \ln \left (x \right ) \ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right )}{17}\) | \(267\) |
| parts | \(4 x \ln \left (x \right )^{2}+8 x \ln \left (x \right )-8 x +4 x^{3}+8 x^{2}-\frac {4 \left (11 x^{3}+4 x^{2}-64 x -64\right )}{\left (x^{2}-x -4\right )^{2}}+8 x^{2} \ln \left (x \right )-\frac {8 \ln \left (x \right ) \left (\ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right ) \sqrt {17}\, x^{2}-\ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right ) \sqrt {17}\, x^{2}-\ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right ) \sqrt {17}\, x +\ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right ) \sqrt {17}\, x -4 \sqrt {17}\, \ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right )+4 \sqrt {17}\, \ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right )+17 x^{2}\right )}{17 \left (x^{2}-x -4\right )}+\frac {8 \sqrt {17}\, \ln \left (x \right ) \ln \left (\frac {1+\sqrt {17}-2 x}{1+\sqrt {17}}\right )}{17}-\frac {8 \sqrt {17}\, \ln \left (x \right ) \ln \left (\frac {-1+\sqrt {17}+2 x}{-1+\sqrt {17}}\right )}{17}\) | \(267\) |
| orering | \(\text {Expression too large to display}\) | \(1765\) |
Input:
int(((4*x^6-12*x^5-36*x^4+92*x^3+144*x^2-192*x-256)*ln(x)^2+(16*x^7-32*x^6 -192*x^5+232*x^4+1000*x^3-288*x^2-2048*x-1024)*ln(x)+12*x^8-12*x^7-180*x^6 +52*x^5+1044*x^4+420*x^3-2224*x^2-2304*x-512)/(x^6-3*x^5-9*x^4+23*x^3+36*x ^2-48*x-64),x,method=_RETURNVERBOSE)
Output:
4*x*ln(x)^2+8*(x^4-5*x^2-5*x-4)/(x^2-x-4)*ln(x)-4*(-x^7+2*x^4*ln(x)+13*x^5 -4*x^3*ln(x)+2*x^4-14*x^2*ln(x)-35*x^3+16*x*ln(x)-12*x^2+32*ln(x)-32*x-64) /(x^2-x-4)^2
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=\frac {4 \, {\left (x^{7} - 13 \, x^{5} - 2 \, x^{4} + 35 \, x^{3} + {\left (x^{5} - 2 \, x^{4} - 7 \, x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (x\right )^{2} + 12 \, x^{2} + 2 \, {\left (x^{6} - x^{5} - 10 \, x^{4} + 2 \, x^{3} + 28 \, x^{2} + 16 \, x\right )} \log \left (x\right ) + 32 \, x + 64\right )}}{x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16} \] Input:
integrate(((4*x^6-12*x^5-36*x^4+92*x^3+144*x^2-192*x-256)*log(x)^2+(16*x^7 -32*x^6-192*x^5+232*x^4+1000*x^3-288*x^2-2048*x-1024)*log(x)+12*x^8-12*x^7 -180*x^6+52*x^5+1044*x^4+420*x^3-2224*x^2-2304*x-512)/(x^6-3*x^5-9*x^4+23* x^3+36*x^2-48*x-64),x, algorithm="fricas")
Output:
4*(x^7 - 13*x^5 - 2*x^4 + 35*x^3 + (x^5 - 2*x^4 - 7*x^3 + 8*x^2 + 16*x)*lo g(x)^2 + 12*x^2 + 2*(x^6 - x^5 - 10*x^4 + 2*x^3 + 28*x^2 + 16*x)*log(x) + 32*x + 64)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (24) = 48\).
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=4 x^{3} + 8 x^{2} + 4 x \log {\left (x \right )}^{2} - 8 x + \frac {- 44 x^{3} - 16 x^{2} + 256 x + 256}{x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 16} - 8 \log {\left (x \right )} + \frac {\left (8 x^{4} - 40 x^{2} - 40 x - 32\right ) \log {\left (x \right )}}{x^{2} - x - 4} \] Input:
integrate(((4*x**6-12*x**5-36*x**4+92*x**3+144*x**2-192*x-256)*ln(x)**2+(1 6*x**7-32*x**6-192*x**5+232*x**4+1000*x**3-288*x**2-2048*x-1024)*ln(x)+12* x**8-12*x**7-180*x**6+52*x**5+1044*x**4+420*x**3-2224*x**2-2304*x-512)/(x* *6-3*x**5-9*x**4+23*x**3+36*x**2-48*x-64),x)
Output:
4*x**3 + 8*x**2 + 4*x*log(x)**2 - 8*x + (-44*x**3 - 16*x**2 + 256*x + 256) /(x**4 - 2*x**3 - 7*x**2 + 8*x + 16) - 8*log(x) + (8*x**4 - 40*x**2 - 40*x - 32)*log(x)/(x**2 - x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 16.20 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=4 \, x^{3} + 12 \, x^{2} - \frac {6 \, {\left (37898 \, x^{3} + 23495 \, x^{2} - 180872 \, x - 197904\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} + \frac {6 \, {\left (11264 \, x^{3} + 12293 \, x^{2} - 56696 \, x - 74032\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} + \frac {90 \, {\left (4134 \, x^{3} + 1891 \, x^{2} - 17512 \, x - 17232\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {26 \, {\left (924 \, x^{3} + 1793 \, x^{2} - 4696 \, x - 7536\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {522 \, {\left (386 \, x^{3} - x^{2} - 1096 \, x - 656\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {210 \, {\left (24 \, x^{3} + 253 \, x^{2} - 152 \, x - 496\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} + \frac {1112 \, {\left (14 \, x^{3} - 21 \, x^{2} + 104 \, x + 96\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {256 \, {\left (12 \, x^{3} - 18 \, x^{2} - 76 \, x + 41\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {1152 \, {\left (6 \, x^{3} - 9 \, x^{2} - 38 \, x - 124\right )}}{289 \, {\left (x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16\right )}} - \frac {4 \, {\left (x^{4} + x^{3} - {\left (x^{3} - x^{2} - 4 \, x\right )} \log \left (x\right )^{2} - 6 \, x^{2} - 2 \, {\left (x^{4} - 6 \, x^{2} - 4 \, x\right )} \log \left (x\right ) - 8 \, x\right )}}{x^{2} - x - 4} \] Input:
integrate(((4*x^6-12*x^5-36*x^4+92*x^3+144*x^2-192*x-256)*log(x)^2+(16*x^7 -32*x^6-192*x^5+232*x^4+1000*x^3-288*x^2-2048*x-1024)*log(x)+12*x^8-12*x^7 -180*x^6+52*x^5+1044*x^4+420*x^3-2224*x^2-2304*x-512)/(x^6-3*x^5-9*x^4+23* x^3+36*x^2-48*x-64),x, algorithm="maxima")
Output:
4*x^3 + 12*x^2 - 6/289*(37898*x^3 + 23495*x^2 - 180872*x - 197904)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) + 6/289*(11264*x^3 + 12293*x^2 - 56696*x - 74032 )/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) + 90/289*(4134*x^3 + 1891*x^2 - 17512*x - 17232)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 26/289*(924*x^3 + 1793*x^2 - 4696*x - 7536)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 522/289*(386*x^3 - x^2 - 1096*x - 656)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 210/289*(24*x^3 + 253*x^ 2 - 152*x - 496)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) + 1112/289*(14*x^3 - 21* x^2 + 104*x + 96)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 256/289*(12*x^3 - 18* x^2 - 76*x + 41)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 1152/289*(6*x^3 - 9*x^ 2 - 38*x - 124)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 4*(x^4 + x^3 - (x^3 - x ^2 - 4*x)*log(x)^2 - 6*x^2 - 2*(x^4 - 6*x^2 - 4*x)*log(x) - 8*x)/(x^2 - x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.44 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=4 \, x^{3} + 4 \, x \log \left (x\right )^{2} + 8 \, x^{2} + 8 \, {\left (x^{2} + x - \frac {x + 4}{x^{2} - x - 4}\right )} \log \left (x\right ) - 8 \, x - \frac {4 \, {\left (11 \, x^{3} + 4 \, x^{2} - 64 \, x - 64\right )}}{x^{4} - 2 \, x^{3} - 7 \, x^{2} + 8 \, x + 16} - 8 \, \log \left (x\right ) \] Input:
integrate(((4*x^6-12*x^5-36*x^4+92*x^3+144*x^2-192*x-256)*log(x)^2+(16*x^7 -32*x^6-192*x^5+232*x^4+1000*x^3-288*x^2-2048*x-1024)*log(x)+12*x^8-12*x^7 -180*x^6+52*x^5+1044*x^4+420*x^3-2224*x^2-2304*x-512)/(x^6-3*x^5-9*x^4+23* x^3+36*x^2-48*x-64),x, algorithm="giac")
Output:
4*x^3 + 4*x*log(x)^2 + 8*x^2 + 8*(x^2 + x - (x + 4)/(x^2 - x - 4))*log(x) - 8*x - 4*(11*x^3 + 4*x^2 - 64*x - 64)/(x^4 - 2*x^3 - 7*x^2 + 8*x + 16) - 8*log(x)
Time = 4.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.56 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=4\,x\,{\ln \left (x\right )}^2-8\,\ln \left (x\right )-8\,x+8\,x^2+4\,x^3+\frac {-44\,x^3-16\,x^2+256\,x+256}{x^4-2\,x^3-7\,x^2+8\,x+16}+\frac {\ln \left (x\right )\,\left (-8\,x^4+40\,x^2+40\,x+32\right )}{-x^2+x+4} \] Input:
int((2304*x + log(x)*(2048*x + 288*x^2 - 1000*x^3 - 232*x^4 + 192*x^5 + 32 *x^6 - 16*x^7 + 1024) + 2224*x^2 - 420*x^3 - 1044*x^4 - 52*x^5 + 180*x^6 + 12*x^7 - 12*x^8 + log(x)^2*(192*x - 144*x^2 - 92*x^3 + 36*x^4 + 12*x^5 - 4*x^6 + 256) + 512)/(48*x - 36*x^2 - 23*x^3 + 9*x^4 + 3*x^5 - x^6 + 64),x)
Output:
4*x*log(x)^2 - 8*log(x) - 8*x + 8*x^2 + 4*x^3 + (256*x - 16*x^2 - 44*x^3 + 256)/(8*x - 7*x^2 - 2*x^3 + x^4 + 16) + (log(x)*(40*x + 40*x^2 - 8*x^4 + 32))/(x - x^2 + 4)
Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.40 \[ \int \frac {-512-2304 x-2224 x^2+420 x^3+1044 x^4+52 x^5-180 x^6-12 x^7+12 x^8+\left (-1024-2048 x-288 x^2+1000 x^3+232 x^4-192 x^5-32 x^6+16 x^7\right ) \log (x)+\left (-256-192 x+144 x^2+92 x^3-36 x^4-12 x^5+4 x^6\right ) \log ^2(x)}{-64-48 x+36 x^2+23 x^3-9 x^4-3 x^5+x^6} \, dx=\frac {4 \mathrm {log}\left (x \right )^{2} x^{5}-8 \mathrm {log}\left (x \right )^{2} x^{4}-28 \mathrm {log}\left (x \right )^{2} x^{3}+32 \mathrm {log}\left (x \right )^{2} x^{2}+64 \mathrm {log}\left (x \right )^{2} x +8 \,\mathrm {log}\left (x \right ) x^{6}-8 \,\mathrm {log}\left (x \right ) x^{5}-80 \,\mathrm {log}\left (x \right ) x^{4}+16 \,\mathrm {log}\left (x \right ) x^{3}+224 \,\mathrm {log}\left (x \right ) x^{2}+128 \,\mathrm {log}\left (x \right ) x +4 x^{7}-52 x^{5}+82 x^{4}-40 x^{3}-582 x^{2}+848 x +1696}{x^{4}-2 x^{3}-7 x^{2}+8 x +16} \] Input:
int(((4*x^6-12*x^5-36*x^4+92*x^3+144*x^2-192*x-256)*log(x)^2+(16*x^7-32*x^ 6-192*x^5+232*x^4+1000*x^3-288*x^2-2048*x-1024)*log(x)+12*x^8-12*x^7-180*x ^6+52*x^5+1044*x^4+420*x^3-2224*x^2-2304*x-512)/(x^6-3*x^5-9*x^4+23*x^3+36 *x^2-48*x-64),x)
Output:
(2*(2*log(x)**2*x**5 - 4*log(x)**2*x**4 - 14*log(x)**2*x**3 + 16*log(x)**2 *x**2 + 32*log(x)**2*x + 4*log(x)*x**6 - 4*log(x)*x**5 - 40*log(x)*x**4 + 8*log(x)*x**3 + 112*log(x)*x**2 + 64*log(x)*x + 2*x**7 - 26*x**5 + 41*x**4 - 20*x**3 - 291*x**2 + 424*x + 848))/(x**4 - 2*x**3 - 7*x**2 + 8*x + 16)