Integrand size = 171, antiderivative size = 32 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=3+e^5-x-9 \left (3+x-\frac {1}{x+\frac {x^4}{16}}\right )^2 \log ^2(x) \] Output:
3+exp(5)-9*(x-1/(1/16*x^4+x)+3)^2*ln(x)^2-x
Time = 1.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-\frac {x^3 \left (16+x^3\right )^2+9 \left (-16+48 x+16 x^2+3 x^4+x^5\right )^2 \log ^2(x)}{x^2 \left (16+x^3\right )^2} \] Input:
Integrate[(-4096*x^3 - 768*x^6 - 48*x^9 - x^12 + (-73728 + 442368*x - 5160 96*x^2 - 446976*x^3 - 18432*x^4 - 105984*x^5 - 82944*x^6 - 12096*x^7 - 720 0*x^8 - 5184*x^9 - 864*x^10 - 162*x^11 - 108*x^12 - 18*x^13)*Log[x] + (737 28 - 221184*x - 202752*x^3 - 142848*x^4 - 13824*x^5 - 41472*x^6 - 17280*x^ 7 - 864*x^8 - 2592*x^9 - 864*x^10 - 54*x^12 - 18*x^13)*Log[x]^2)/(4096*x^3 + 768*x^6 + 48*x^9 + x^12),x]
Output:
-((x^3*(16 + x^3)^2 + 9*(-16 + 48*x + 16*x^2 + 3*x^4 + x^5)^2*Log[x]^2)/(x ^2*(16 + x^3)^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)}{x^{12}+48 x^9+768 x^6+4096 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)}{x^3 \left (x^9+48 x^6+768 x^3+4096\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {5 \left (-2+\sqrt [3]{-1}\right ) \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{13824\ 2^{2/3} \sqrt {3} \left (-i+\sqrt {3}\right ) x^3 \left (x+2 \sqrt [3]{2}\right )}+\frac {15 \sqrt [3]{-1} \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{1024\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^8 x^3 \left (\sqrt [3]{-1} x-2 \sqrt [3]{2}\right )}-\frac {5 \left (3 i+\sqrt {3}\right ) \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{55296 x^3 \left (2^{2/3} \sqrt {3} x-2 \sqrt {3}-6 i\right )}-\frac {3 \left (-2+\sqrt [3]{-1}\right ) \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{512 \sqrt [3]{2} \left (-1+\sqrt [3]{-1}\right )^4 \left (1+\sqrt [3]{-1}\right )^7 x^3 \left (x+2 \sqrt [3]{2}\right )^2}-\frac {3 \left (2+(-1)^{2/3}\right ) \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{512 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^7 x^3 \left (\sqrt [3]{-1} x-2 \sqrt [3]{2}\right )^2}+\frac {(-1)^{2/3} \left (-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)\right )}{4608 \sqrt [3]{2} \left (-1+\sqrt [3]{-1}\right )^4 x^3 \left ((-1)^{2/3} x+2 \sqrt [3]{2}\right )^2}+\frac {-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)}{6912 x^3 \left (x+2 \sqrt [3]{2}\right )^3}+\frac {-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)}{6912 x^3 \left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )^3}+\frac {-x^{12}-48 x^9-768 x^6-4096 x^3+\left (-18 x^{13}-54 x^{12}-864 x^{10}-2592 x^9-864 x^8-17280 x^7-41472 x^6-13824 x^5-142848 x^4-202752 x^3-221184 x+73728\right ) \log ^2(x)+\left (-18 x^{13}-108 x^{12}-162 x^{11}-864 x^{10}-5184 x^9-7200 x^8-12096 x^7-82944 x^6-105984 x^5-18432 x^4-446976 x^3-516096 x^2+442368 x-73728\right ) \log (x)}{6912 x^3 \left ((-1)^{2/3} x+2 \sqrt [3]{2}\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {27 i \sqrt {3} \left (\sqrt {3}+3 i\right ) \left (-x^3 \left (x^3+16\right )^3-18 \left (x^5+3 x^4+16 x^2+48 x-16\right )^2 \left (x^3+16\right ) \log (x)-18 \left (x^{13}+3 x^{12}+48 x^{10}+144 x^9+48 x^8+960 x^7+2304 x^6+768 x^5+7936 x^4+11264 x^3+12288 x-4096\right ) \log ^2(x)\right )}{\left (1-\sqrt [3]{-1}\right )^4 \left (1+\sqrt [3]{-1}\right )^8 \left (-\sqrt {3}+i\right ) x^3 \left (x^3+16\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int -\frac {x^3 \left (x^3+16\right )^3+18 \left (-x^5-3 x^4-16 x^2-48 x+16\right )^2 \left (x^3+16\right ) \log (x)-18 \left (-x^{13}-3 x^{12}-48 x^{10}-144 x^9-48 x^8-960 x^7-2304 x^6-768 x^5-7936 x^4-11264 x^3-12288 x+4096\right ) \log ^2(x)}{x^3 \left (x^3+16\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {x^3 \left (x^3+16\right )^3+18 \left (-x^5-3 x^4-16 x^2-48 x+16\right )^2 \log (x) \left (x^3+16\right )-18 \left (-x^{13}-3 x^{12}-48 x^{10}-144 x^9-48 x^8-960 x^7-2304 x^6-768 x^5-7936 x^4-11264 x^3-12288 x+4096\right ) \log ^2(x)}{x^3 \left (x^3+16\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {18 \log (x) \left (x^5+3 x^4+16 x^2+48 x-16\right )^2}{x^3 \left (x^3+16\right )^2}+\frac {18 (x+2) \left (x^7-2 x^6+4 x^5+24 x^4-48 x^3+160 x^2-64 x+128\right ) \log ^2(x) \left (x^5+3 x^4+16 x^2+48 x-16\right )}{x^3 \left (x^3+16\right )^3}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -9 x^2 \log ^2(x)-54 x \log ^2(x)-\frac {3 \sqrt [3]{-1} \left (\sqrt [3]{-1}-6 \sqrt [3]{2}\right ) \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log ^2(x)}{2\ 2^{2/3}}+\frac {405 \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log ^2(x)}{2\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^8}-\frac {5 \left (2-\sqrt [3]{-1}\right ) \log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log ^2(x)}{2^{2/3} \sqrt {3} \left (i-\sqrt {3}\right )}-\frac {3 \left (1+6 \sqrt [3]{2}\right ) \log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log ^2(x)}{2\ 2^{2/3}}-\frac {\log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log ^2(x)}{2^{2/3}}+\frac {3 \sqrt [3]{-1} \left (1-6 \sqrt [3]{-2}\right ) \log \left (\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}+1\right ) \log ^2(x)}{2\ 2^{2/3}}+\frac {\sqrt [3]{2} \log \left (1-\frac {\left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log ^2(x)}{1-i \sqrt {3}}+\frac {\sqrt [3]{2} \log \left (1-\frac {\left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log ^2(x)}{1+i \sqrt {3}}-\frac {5 \left (1+i \sqrt {3}\right ) \log \left (1-\frac {\sqrt {3} x}{\sqrt [3]{2} \left (3 i+\sqrt {3}\right )}\right ) \log ^2(x)}{4\ 2^{2/3}}+\frac {54 \log ^2(x)}{x}+\frac {3 x \log ^2(x)}{2\ 2^{2/3} \left (x+2 \sqrt [3]{2}\right )}+\frac {x \log ^2(x)}{2^{2/3} \left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )}+\frac {x \log ^2(x)}{2^{2/3} \left ((-1)^{2/3} x+2 \sqrt [3]{2}\right )}-\frac {x \log ^2(x)}{2 \left (2^{2/3} x+4\right )}+\frac {288 \log ^2(x)}{x^3+16}-\frac {9 \log ^2(x)}{x^2}-\frac {\log ^2(x)}{\left (x+2 \sqrt [3]{2}\right )^2}-\frac {(-1)^{2/3} \log ^2(x)}{\left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )^2}+\frac {\sqrt [3]{-1} \log ^2(x)}{\left ((-1)^{2/3} x+2 \sqrt [3]{2}\right )^2}+54 \log \left (1+\frac {16}{x^3}\right ) \log (x)+\frac {\sqrt [3]{-1} \log \left (1-\frac {2 \sqrt [3]{-2}}{x}\right ) \log (x)}{2\ 2^{2/3}}-\frac {\log \left (1+\frac {2 \sqrt [3]{2}}{x}\right ) \log (x)}{2\ 2^{2/3}}-\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \log \left (1+\frac {2 (-1)^{2/3} \sqrt [3]{2}}{x}\right ) \log (x)+\frac {81 \left (-\frac {1}{2}\right )^{2/3} \left (2+(-1)^{2/3}\right ) \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)}{\left (1+\sqrt [3]{-1}\right )^7}-\frac {9 \sqrt [3]{-1} \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)}{2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-9 \sqrt [3]{-1} 2^{2/3} \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)+\frac {3}{2} \left (-\frac {1}{2}\right )^{2/3} \log \left (1-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)+9\ 2^{2/3} \log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log (x)-\sqrt [3]{2} \log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log (x)+\frac {5 \log \left (\frac {x}{2 \sqrt [3]{2}}+1\right ) \log (x)}{2\ 2^{2/3}}-\frac {3 \log \left (\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}+1\right ) \log (x)}{2^{2/3} \left (1-\sqrt [3]{-1}\right )^4}-\frac {5 \sqrt [3]{-1} \log \left (\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}+1\right ) \log (x)}{2\ 2^{2/3}}+9 (-2)^{2/3} \log \left (\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}+1\right ) \log (x)-\frac {\sqrt [3]{2} \log \left (1-\frac {\left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log (x)}{1-i \sqrt {3}}-\frac {\sqrt [3]{2} \log \left (1-\frac {\left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log (x)}{1+i \sqrt {3}}-54 \log \left (\frac {x^3}{16}+1\right ) \log (x)-\frac {3 \sqrt [3]{-1} \left (\sqrt [3]{-1}-6 \sqrt [3]{2}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)}{2^{2/3}}+\frac {405 \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right ) \log (x)}{2^{2/3} \left (1+\sqrt [3]{-1}\right )^8}-\frac {5 \sqrt [3]{2} \left (2-\sqrt [3]{-1}\right ) \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right ) \log (x)}{\sqrt {3} \left (i-\sqrt {3}\right )}-\frac {3 \left (1+6 \sqrt [3]{2}\right ) \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right ) \log (x)}{2^{2/3}}-\sqrt [3]{2} \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right ) \log (x)+\frac {3 \sqrt [3]{-1} \left (1-6 \sqrt [3]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}\right ) \log (x)}{2^{2/3}}+\frac {2 \sqrt [3]{2} \operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log (x)}{1-i \sqrt {3}}+\frac {2 \sqrt [3]{2} \operatorname {PolyLog}\left (2,\frac {\left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right ) \log (x)}{1+i \sqrt {3}}-\frac {5 \left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {3} x}{\sqrt [3]{2} \left (3 i+\sqrt {3}\right )}\right ) \log (x)}{2\ 2^{2/3}}-\frac {x \log (x)}{2\ 2^{2/3} \left (x+2 \sqrt [3]{2}\right )}+\frac {x \log (x)}{2 \left (2^{2/3} x+4\right )}-x+\frac {9 \sqrt [3]{-1} \log \left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )}{2\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \log \left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )-18 \operatorname {PolyLog}\left (2,-\frac {16}{x^3}\right )-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{-2}}{x}\right )}{2\ 2^{2/3}}+\frac {\operatorname {PolyLog}\left (2,-\frac {2 \sqrt [3]{2}}{x}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (-\frac {1}{2}\right )^{2/3} \operatorname {PolyLog}\left (2,-\frac {2 (-1)^{2/3} \sqrt [3]{2}}{x}\right )+\frac {81 \left (-\frac {1}{2}\right )^{2/3} \left (2+(-1)^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )}{\left (1+\sqrt [3]{-1}\right )^7}-\frac {9 \sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )}{2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-9 \sqrt [3]{-1} 2^{2/3} \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )+\frac {3}{2} \left (-\frac {1}{2}\right )^{2/3} \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )+9\ 2^{2/3} \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right )-\sqrt [3]{2} \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right )+\frac {5 \operatorname {PolyLog}\left (2,-\frac {x}{2 \sqrt [3]{2}}\right )}{2\ 2^{2/3}}-\frac {3 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}\right )}{2^{2/3} \left (1-\sqrt [3]{-1}\right )^4}-\frac {5 \sqrt [3]{-1} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}\right )}{2\ 2^{2/3}}+9 (-2)^{2/3} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}\right )-\frac {\sqrt [3]{2} \operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right )}{1-i \sqrt {3}}-\frac {\sqrt [3]{2} \operatorname {PolyLog}\left (2,\frac {\left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right )}{1+i \sqrt {3}}-18 \operatorname {PolyLog}\left (2,-\frac {x^3}{16}\right )+\frac {3 \sqrt [3]{-1} \left (\sqrt [3]{-1}-6 \sqrt [3]{2}\right ) \operatorname {PolyLog}\left (3,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )}{2^{2/3}}-\frac {405 \operatorname {PolyLog}\left (3,\frac {1}{2} \sqrt [3]{-\frac {1}{2}} x\right )}{2^{2/3} \left (1+\sqrt [3]{-1}\right )^8}+\frac {5 \sqrt [3]{2} \left (2-\sqrt [3]{-1}\right ) \operatorname {PolyLog}\left (3,-\frac {x}{2 \sqrt [3]{2}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right )}+\frac {3 \left (1+6 \sqrt [3]{2}\right ) \operatorname {PolyLog}\left (3,-\frac {x}{2 \sqrt [3]{2}}\right )}{2^{2/3}}+\sqrt [3]{2} \operatorname {PolyLog}\left (3,-\frac {x}{2 \sqrt [3]{2}}\right )-\frac {3 \sqrt [3]{-1} \left (1-6 \sqrt [3]{-2}\right ) \operatorname {PolyLog}\left (3,-\frac {(-1)^{2/3} x}{2 \sqrt [3]{2}}\right )}{2^{2/3}}-\frac {2 \sqrt [3]{2} \operatorname {PolyLog}\left (3,\frac {\left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right )}{1-i \sqrt {3}}-\frac {2 \sqrt [3]{2} \operatorname {PolyLog}\left (3,\frac {\left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{2}}\right )}{1+i \sqrt {3}}+\frac {5 \left (1+i \sqrt {3}\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {3} x}{\sqrt [3]{2} \left (3 i+\sqrt {3}\right )}\right )}{2\ 2^{2/3}}-2592 \int \frac {x \log ^2(x)}{\left (x^3+16\right )^2}dx\) |
Input:
Int[(-4096*x^3 - 768*x^6 - 48*x^9 - x^12 + (-73728 + 442368*x - 516096*x^2 - 446976*x^3 - 18432*x^4 - 105984*x^5 - 82944*x^6 - 12096*x^7 - 7200*x^8 - 5184*x^9 - 864*x^10 - 162*x^11 - 108*x^12 - 18*x^13)*Log[x] + (73728 - 2 21184*x - 202752*x^3 - 142848*x^4 - 13824*x^5 - 41472*x^6 - 17280*x^7 - 86 4*x^8 - 2592*x^9 - 864*x^10 - 54*x^12 - 18*x^13)*Log[x]^2)/(4096*x^3 + 768 *x^6 + 48*x^9 + x^12),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).
Time = 7.52 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31
| method | result | size |
| risch | \(-\frac {9 \left (x^{10}+6 x^{9}+9 x^{8}+32 x^{7}+192 x^{6}+256 x^{5}+160 x^{4}+1536 x^{3}+1792 x^{2}-1536 x +256\right ) \ln \left (x \right )^{2}}{x^{2} \left (x^{6}+32 x^{3}+256\right )}-x\) | \(74\) |
| parallelrisch | \(-\frac {288 x^{10} \ln \left (x \right )^{2}+1728 \ln \left (x \right )^{2} x^{9}+2592 x^{8} \ln \left (x \right )^{2}+32 x^{9}+9216 x^{7} \ln \left (x \right )^{2}+55296 x^{6} \ln \left (x \right )^{2}+73728 x^{5} \ln \left (x \right )^{2}+1024 x^{6}+46080 x^{4} \ln \left (x \right )^{2}+442368 x^{3} \ln \left (x \right )^{2}+516096 x^{2} \ln \left (x \right )^{2}+8192 x^{3}-442368 x \ln \left (x \right )^{2}+73728 \ln \left (x \right )^{2}}{32 x^{2} \left (x^{6}+32 x^{3}+256\right )}\) | \(128\) |
| orering | \(\text {Expression too large to display}\) | \(2888\) |
Input:
int(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824 *x^5-142848*x^4-202752*x^3-221184*x+73728)*ln(x)^2+(-18*x^13-108*x^12-162* x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18432*x^4-4 46976*x^3-516096*x^2+442368*x-73728)*ln(x)-x^12-48*x^9-768*x^6-4096*x^3)/( x^12+48*x^9+768*x^6+4096*x^3),x,method=_RETURNVERBOSE)
Output:
-9*(x^10+6*x^9+9*x^8+32*x^7+192*x^6+256*x^5+160*x^4+1536*x^3+1792*x^2-1536 *x+256)/x^2/(x^6+32*x^3+256)*ln(x)^2-x
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-\frac {x^{9} + 32 \, x^{6} + 256 \, x^{3} + 9 \, {\left (x^{10} + 6 \, x^{9} + 9 \, x^{8} + 32 \, x^{7} + 192 \, x^{6} + 256 \, x^{5} + 160 \, x^{4} + 1536 \, x^{3} + 1792 \, x^{2} - 1536 \, x + 256\right )} \log \left (x\right )^{2}}{x^{8} + 32 \, x^{5} + 256 \, x^{2}} \] Input:
integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6 -13824*x^5-142848*x^4-202752*x^3-221184*x+73728)*log(x)^2+(-18*x^13-108*x^ 12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-1843 2*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-409 6*x^3)/(x^12+48*x^9+768*x^6+4096*x^3),x, algorithm="fricas")
Output:
-(x^9 + 32*x^6 + 256*x^3 + 9*(x^10 + 6*x^9 + 9*x^8 + 32*x^7 + 192*x^6 + 25 6*x^5 + 160*x^4 + 1536*x^3 + 1792*x^2 - 1536*x + 256)*log(x)^2)/(x^8 + 32* x^5 + 256*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=- x + \frac {\left (- 9 x^{10} - 54 x^{9} - 81 x^{8} - 288 x^{7} - 1728 x^{6} - 2304 x^{5} - 1440 x^{4} - 13824 x^{3} - 16128 x^{2} + 13824 x - 2304\right ) \log {\left (x \right )}^{2}}{x^{8} + 32 x^{5} + 256 x^{2}} \] Input:
integrate(((-18*x**13-54*x**12-864*x**10-2592*x**9-864*x**8-17280*x**7-414 72*x**6-13824*x**5-142848*x**4-202752*x**3-221184*x+73728)*ln(x)**2+(-18*x **13-108*x**12-162*x**11-864*x**10-5184*x**9-7200*x**8-12096*x**7-82944*x* *6-105984*x**5-18432*x**4-446976*x**3-516096*x**2+442368*x-73728)*ln(x)-x* *12-48*x**9-768*x**6-4096*x**3)/(x**12+48*x**9+768*x**6+4096*x**3),x)
Output:
-x + (-9*x**10 - 54*x**9 - 81*x**8 - 288*x**7 - 1728*x**6 - 2304*x**5 - 14 40*x**4 - 13824*x**3 - 16128*x**2 + 13824*x - 2304)*log(x)**2/(x**8 + 32*x **5 + 256*x**2)
Exception generated. \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6 -13824*x^5-142848*x^4-202752*x^3-221184*x+73728)*log(x)^2+(-18*x^13-108*x^ 12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-1843 2*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-409 6*x^3)/(x^12+48*x^9+768*x^6+4096*x^3),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (29) = 58\).
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-9 \, {\left (x^{2} + 6 \, x + \frac {6 \, x^{5} - x^{4} - 32 \, x^{3} + 96 \, x^{2} - 32 \, x - 512}{x^{6} + 32 \, x^{3} + 256} - \frac {6 \, x - 1}{x^{2}} + 9\right )} \log \left (x\right )^{2} - x \] Input:
integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6 -13824*x^5-142848*x^4-202752*x^3-221184*x+73728)*log(x)^2+(-18*x^13-108*x^ 12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-1843 2*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-409 6*x^3)/(x^12+48*x^9+768*x^6+4096*x^3),x, algorithm="giac")
Output:
-9*(x^2 + 6*x + (6*x^5 - x^4 - 32*x^3 + 96*x^2 - 32*x - 512)/(x^6 + 32*x^3 + 256) - (6*x - 1)/x^2 + 9)*log(x)^2 - x
Time = 2.94 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\left (-\frac {9\,x^{10}+54\,x^9+288\,x^7+1728\,x^6-288\,x^5+1440\,x^4+13824\,x^3-4608\,x^2-13824\,x+2304}{x^8+32\,x^5+256\,x^2}-81\right )\,{\ln \left (x\right )}^2-x \] Input:
int(-(log(x)*(516096*x^2 - 442368*x + 446976*x^3 + 18432*x^4 + 105984*x^5 + 82944*x^6 + 12096*x^7 + 7200*x^8 + 5184*x^9 + 864*x^10 + 162*x^11 + 108* x^12 + 18*x^13 + 73728) + 4096*x^3 + 768*x^6 + 48*x^9 + x^12 + log(x)^2*(2 21184*x + 202752*x^3 + 142848*x^4 + 13824*x^5 + 41472*x^6 + 17280*x^7 + 86 4*x^8 + 2592*x^9 + 864*x^10 + 54*x^12 + 18*x^13 - 73728))/(4096*x^3 + 768* x^6 + 48*x^9 + x^12),x)
Output:
- x - log(x)^2*((13824*x^3 - 4608*x^2 - 13824*x + 1440*x^4 - 288*x^5 + 172 8*x^6 + 288*x^7 + 54*x^9 + 9*x^10 + 2304)/(256*x^2 + 32*x^5 + x^8) + 81)
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.94 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\frac {-9 \mathrm {log}\left (x \right )^{2} x^{10}-54 \mathrm {log}\left (x \right )^{2} x^{9}-81 \mathrm {log}\left (x \right )^{2} x^{8}-288 \mathrm {log}\left (x \right )^{2} x^{7}-1728 \mathrm {log}\left (x \right )^{2} x^{6}-2304 \mathrm {log}\left (x \right )^{2} x^{5}-1440 \mathrm {log}\left (x \right )^{2} x^{4}-13824 \mathrm {log}\left (x \right )^{2} x^{3}-16128 \mathrm {log}\left (x \right )^{2} x^{2}+13824 \mathrm {log}\left (x \right )^{2} x -2304 \mathrm {log}\left (x \right )^{2}-x^{9}-32 x^{6}-256 x^{3}}{x^{2} \left (x^{6}+32 x^{3}+256\right )} \] Input:
int(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824 *x^5-142848*x^4-202752*x^3-221184*x+73728)*log(x)^2+(-18*x^13-108*x^12-162 *x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18432*x^4- 446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-4096*x^3) /(x^12+48*x^9+768*x^6+4096*x^3),x)
Output:
( - 9*log(x)**2*x**10 - 54*log(x)**2*x**9 - 81*log(x)**2*x**8 - 288*log(x) **2*x**7 - 1728*log(x)**2*x**6 - 2304*log(x)**2*x**5 - 1440*log(x)**2*x**4 - 13824*log(x)**2*x**3 - 16128*log(x)**2*x**2 + 13824*log(x)**2*x - 2304* log(x)**2 - x**9 - 32*x**6 - 256*x**3)/(x**2*(x**6 + 32*x**3 + 256))