Integrand size = 149, antiderivative size = 31 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {16}{25} \left (5-\frac {2}{x}-x-x^2\right )^2 \left (x-\frac {2}{\log (x)}\right )^2 \] Output:
16/25*(5-x^2-x-2/x)^2*(x-2/ln(x))^2
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(31)=62\).
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.48 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {32}{25} \left (\frac {1}{2} x \left (-20+29 x-6 x^2-9 x^3+2 x^4+x^5\right )+\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x^2 \log ^2(x)}-\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x \log (x)}\right ) \] Input:
Integrate[(-512 + 2560*x - 3712*x^2 + 768*x^3 + 1152*x^4 - 256*x^5 - 128*x ^6 + (-512 + 1536*x - 1280*x^2 + 1472*x^3 - 1536*x^4 - 192*x^5 + 384*x^6 + 64*x^7)*Log[x] + (256*x - 1856*x^3 + 768*x^4 + 1728*x^5 - 512*x^6 - 320*x ^7)*Log[x]^2 + (-320*x^3 + 928*x^4 - 288*x^5 - 576*x^6 + 160*x^7 + 96*x^8) *Log[x]^3)/(25*x^3*Log[x]^3),x]
Output:
(32*((x*(-20 + 29*x - 6*x^2 - 9*x^3 + 2*x^4 + x^5))/2 + (2*(2 - 5*x + x^2 + x^3)^2)/(x^2*Log[x]^2) - (2*(2 - 5*x + x^2 + x^3)^2)/(x*Log[x])))/25
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.
| \(\displaystyle \int \frac {-128 x^6-256 x^5+1152 x^4+768 x^3-3712 x^2+\left (-320 x^7-512 x^6+1728 x^5+768 x^4-1856 x^3+256 x\right ) \log ^2(x)+\left (96 x^8+160 x^7-576 x^6-288 x^5+928 x^4-320 x^3\right ) \log ^3(x)+\left (64 x^7+384 x^6-192 x^5-1536 x^4+1472 x^3-1280 x^2+1536 x-512\right ) \log (x)+2560 x-512}{25 x^3 \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \int -\frac {32 \left (4 x^6+8 x^5-36 x^4-24 x^3+116 x^2-80 x+\left (-3 x^8-5 x^7+18 x^6+9 x^5-29 x^4+10 x^3\right ) \log ^3(x)-2 \left (-5 x^7-8 x^6+27 x^5+12 x^4-29 x^3+4 x\right ) \log ^2(x)+2 \left (-x^7-6 x^6+3 x^5+24 x^4-23 x^3+20 x^2-24 x+8\right ) \log (x)+16\right )}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {32}{25} \int \frac {4 x^6+8 x^5-36 x^4-24 x^3+116 x^2-80 x+\left (-3 x^8-5 x^7+18 x^6+9 x^5-29 x^4+10 x^3\right ) \log ^3(x)-2 \left (-5 x^7-8 x^6+27 x^5+12 x^4-29 x^3+4 x\right ) \log ^2(x)+2 \left (-x^7-6 x^6+3 x^5+24 x^4-23 x^3+20 x^2-24 x+8\right ) \log (x)+16}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {32}{25} \int \frac {\left (x^3+x^2-5 x+2\right ) \left (-3 \log ^3(x) x^5-2 \log ^3(x) x^4+10 \log ^2(x) x^4-2 \log (x) x^4+5 \log ^3(x) x^3+6 \log ^2(x) x^3-10 \log (x) x^3+4 x^3-10 \log ^2(x) x^2+6 \log (x) x^2+4 x^2-4 \log ^2(x) x-4 \log (x) x-20 x+8 \log (x)+8\right )}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {32}{25} \int \left (-3 x^5-5 x^4+18 x^3+9 x^2-29 x+10+\frac {2 \left (5 x^6+8 x^5-27 x^4-12 x^3+29 x^2-4\right )}{\log (x) x^2}-\frac {2 \left (x^7+6 x^6-3 x^5-24 x^4+23 x^3-20 x^2+24 x-8\right )}{\log ^2(x) x^3}+\frac {4 \left (x^3+x^2-5 x+2\right )^2}{\log ^3(x) x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {32}{25} \left (2 \int \frac {5 x^6+8 x^5-27 x^4-12 x^3+29 x^2-4}{x^2 \log (x)}dx-2 \int \frac {x^7+6 x^6-3 x^5-24 x^4+23 x^3-20 x^2+24 x-8}{x^3 \log ^2(x)}dx+32 \operatorname {ExpIntegralEi}(-2 \log (x))-40 \operatorname {ExpIntegralEi}(-\log (x))-72 \operatorname {ExpIntegralEi}(2 \log (x))+36 \operatorname {ExpIntegralEi}(3 \log (x))+32 \operatorname {ExpIntegralEi}(4 \log (x))-12 \operatorname {LogIntegral}(x)-\frac {x^6}{2}-x^5+\frac {9 x^4}{2}-\frac {2 x^4}{\log ^2(x)}-\frac {8 x^4}{\log (x)}+3 x^3-\frac {4 x^3}{\log ^2(x)}-\frac {12 x^3}{\log (x)}-\frac {29 x^2}{2}+\frac {18 x^2}{\log ^2(x)}-\frac {8}{x^2 \log ^2(x)}+\frac {36 x^2}{\log (x)}+\frac {16}{x^2 \log (x)}+10 x+\frac {12 x}{\log ^2(x)}-\frac {58}{\log ^2(x)}+\frac {40}{x \log ^2(x)}+\frac {12 x}{\log (x)}-\frac {40}{x \log (x)}\right )\) |
Input:
Int[(-512 + 2560*x - 3712*x^2 + 768*x^3 + 1152*x^4 - 256*x^5 - 128*x^6 + ( -512 + 1536*x - 1280*x^2 + 1472*x^3 - 1536*x^4 - 192*x^5 + 384*x^6 + 64*x^ 7)*Log[x] + (256*x - 1856*x^3 + 768*x^4 + 1728*x^5 - 512*x^6 - 320*x^7)*Lo g[x]^2 + (-320*x^3 + 928*x^4 - 288*x^5 - 576*x^6 + 160*x^7 + 96*x^8)*Log[x ]^3)/(25*x^3*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(29)=58\).
Time = 10.82 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.74
| method | result | size |
| risch | \(\frac {16 x^{6}}{25}+\frac {32 x^{5}}{25}-\frac {144 x^{4}}{25}-\frac {96 x^{3}}{25}+\frac {464 x^{2}}{25}-\frac {64 x}{5}+\frac {64}{25}-\frac {64 \left (x^{7} \ln \left (x \right )+2 x^{6} \ln \left (x \right )-x^{6}-9 x^{5} \ln \left (x \right )-2 x^{5}-6 x^{4} \ln \left (x \right )+9 x^{4}+29 x^{3} \ln \left (x \right )+6 x^{3}-20 x^{2} \ln \left (x \right )-29 x^{2}+4 x \ln \left (x \right )+20 x -4\right )}{25 x^{2} \ln \left (x \right )^{2}}\) | \(116\) |
| parallelrisch | \(-\frac {-16 x^{8} \ln \left (x \right )^{2}-32 x^{7} \ln \left (x \right )^{2}+64 x^{7} \ln \left (x \right )+144 x^{6} \ln \left (x \right )^{2}+128 x^{6} \ln \left (x \right )+96 x^{5} \ln \left (x \right )^{2}-64 x^{6}-576 x^{5} \ln \left (x \right )-464 x^{4} \ln \left (x \right )^{2}-128 x^{5}-384 x^{4} \ln \left (x \right )+320 x^{3} \ln \left (x \right )^{2}-256+576 x^{4}+1856 x^{3} \ln \left (x \right )+384 x^{3}-1280 x^{2} \ln \left (x \right )-1856 x^{2}+256 x \ln \left (x \right )+1280 x}{25 x^{2} \ln \left (x \right )^{2}}\) | \(141\) |
| default | \(-\frac {64 x}{5}+\frac {1856}{25 \ln \left (x \right )^{2}}-\frac {256}{5 x \ln \left (x \right )^{2}}-\frac {256}{25 x \ln \left (x \right )}+\frac {256}{5 \ln \left (x \right )}+\frac {256}{25 x^{2} \ln \left (x \right )^{2}}-\frac {64 x^{5}}{25 \ln \left (x \right )}-\frac {576 x^{2}}{25 \ln \left (x \right )^{2}}+\frac {576 x^{3}}{25 \ln \left (x \right )}-\frac {128 x^{4}}{25 \ln \left (x \right )}+\frac {128 x^{3}}{25 \ln \left (x \right )^{2}}+\frac {64 x^{4}}{25 \ln \left (x \right )^{2}}-\frac {384 x}{25 \ln \left (x \right )^{2}}+\frac {384 x^{2}}{25 \ln \left (x \right )}-\frac {1856 x}{25 \ln \left (x \right )}+\frac {16 x^{6}}{25}+\frac {464 x^{2}}{25}-\frac {96 x^{3}}{25}-\frac {144 x^{4}}{25}+\frac {32 x^{5}}{25}\) | \(146\) |
| parts | \(-\frac {64 x}{5}+\frac {1856}{25 \ln \left (x \right )^{2}}-\frac {256}{5 x \ln \left (x \right )^{2}}-\frac {256}{25 x \ln \left (x \right )}+\frac {256}{5 \ln \left (x \right )}+\frac {256}{25 x^{2} \ln \left (x \right )^{2}}-\frac {64 x^{5}}{25 \ln \left (x \right )}-\frac {576 x^{2}}{25 \ln \left (x \right )^{2}}+\frac {576 x^{3}}{25 \ln \left (x \right )}-\frac {128 x^{4}}{25 \ln \left (x \right )}+\frac {128 x^{3}}{25 \ln \left (x \right )^{2}}+\frac {64 x^{4}}{25 \ln \left (x \right )^{2}}-\frac {384 x}{25 \ln \left (x \right )^{2}}+\frac {384 x^{2}}{25 \ln \left (x \right )}-\frac {1856 x}{25 \ln \left (x \right )}+\frac {16 x^{6}}{25}+\frac {464 x^{2}}{25}-\frac {96 x^{3}}{25}-\frac {144 x^{4}}{25}+\frac {32 x^{5}}{25}\) | \(146\) |
Input:
int(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*ln(x)^3+(-320*x ^7-512*x^6+1728*x^5+768*x^4-1856*x^3+256*x)*ln(x)^2+(64*x^7+384*x^6-192*x^ 5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*ln(x)-128*x^6-256*x^5+1152*x^4+76 8*x^3-3712*x^2+2560*x-512)/x^3/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
16/25*x^6+32/25*x^5-144/25*x^4-96/25*x^3+464/25*x^2-64/5*x+64/25-64/25*(x^ 7*ln(x)+2*x^6*ln(x)-x^6-9*x^5*ln(x)-2*x^5-6*x^4*ln(x)+9*x^4+29*x^3*ln(x)+6 *x^3-20*x^2*ln(x)-29*x^2+4*x*ln(x)+20*x-4)/x^2/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {16 \, {\left (4 \, x^{6} + 8 \, x^{5} - 36 \, x^{4} - 24 \, x^{3} + {\left (x^{8} + 2 \, x^{7} - 9 \, x^{6} - 6 \, x^{5} + 29 \, x^{4} - 20 \, x^{3}\right )} \log \left (x\right )^{2} + 116 \, x^{2} - 4 \, {\left (x^{7} + 2 \, x^{6} - 9 \, x^{5} - 6 \, x^{4} + 29 \, x^{3} - 20 \, x^{2} + 4 \, x\right )} \log \left (x\right ) - 80 \, x + 16\right )}}{25 \, x^{2} \log \left (x\right )^{2}} \] Input:
integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+ (-320*x^7-512*x^6+1728*x^5+768*x^4-1856*x^3+256*x)*log(x)^2+(64*x^7+384*x^ 6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+11 52*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="fricas")
Output:
16/25*(4*x^6 + 8*x^5 - 36*x^4 - 24*x^3 + (x^8 + 2*x^7 - 9*x^6 - 6*x^5 + 29 *x^4 - 20*x^3)*log(x)^2 + 116*x^2 - 4*(x^7 + 2*x^6 - 9*x^5 - 6*x^4 + 29*x^ 3 - 20*x^2 + 4*x)*log(x) - 80*x + 16)/(x^2*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.74 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {16 x^{6}}{25} + \frac {32 x^{5}}{25} - \frac {144 x^{4}}{25} - \frac {96 x^{3}}{25} + \frac {464 x^{2}}{25} - \frac {64 x}{5} + \frac {64 x^{6} + 128 x^{5} - 576 x^{4} - 384 x^{3} + 1856 x^{2} - 1280 x + \left (- 64 x^{7} - 128 x^{6} + 576 x^{5} + 384 x^{4} - 1856 x^{3} + 1280 x^{2} - 256 x\right ) \log {\left (x \right )} + 256}{25 x^{2} \log {\left (x \right )}^{2}} \] Input:
integrate(1/25*((96*x**8+160*x**7-576*x**6-288*x**5+928*x**4-320*x**3)*ln( x)**3+(-320*x**7-512*x**6+1728*x**5+768*x**4-1856*x**3+256*x)*ln(x)**2+(64 *x**7+384*x**6-192*x**5-1536*x**4+1472*x**3-1280*x**2+1536*x-512)*ln(x)-12 8*x**6-256*x**5+1152*x**4+768*x**3-3712*x**2+2560*x-512)/x**3/ln(x)**3,x)
Output:
16*x**6/25 + 32*x**5/25 - 144*x**4/25 - 96*x**3/25 + 464*x**2/25 - 64*x/5 + (64*x**6 + 128*x**5 - 576*x**4 - 384*x**3 + 1856*x**2 - 1280*x + (-64*x* *7 - 128*x**6 + 576*x**5 + 384*x**4 - 1856*x**3 + 1280*x**2 - 256*x)*log(x ) + 256)/(25*x**2*log(x)**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 5.84 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x + \frac {256}{5 \, \log \left (x\right )} + \frac {1856}{25 \, \log \left (x\right )^{2}} - \frac {64}{5} \, {\rm Ei}\left (5 \, \log \left (x\right )\right ) - \frac {512}{25} \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) + \frac {1728}{25} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) + \frac {768}{25} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + \frac {256}{25} \, {\rm Ei}\left (-\log \left (x\right )\right ) - \frac {1856}{25} \, {\rm Ei}\left (\log \left (x\right )\right ) + \frac {1024}{25} \, \Gamma \left (-1, 2 \, \log \left (x\right )\right ) + \frac {1472}{25} \, \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {3072}{25} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - \frac {576}{25} \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + \frac {1536}{25} \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + \frac {64}{5} \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) - \frac {1536}{25} \, \Gamma \left (-1, \log \left (x\right )\right ) + \frac {2048}{25} \, \Gamma \left (-2, 2 \, \log \left (x\right )\right ) - \frac {768}{25} \, \Gamma \left (-2, -\log \left (x\right )\right ) - \frac {4608}{25} \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + \frac {2304}{25} \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + \frac {2048}{25} \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) - \frac {512}{5} \, \Gamma \left (-2, \log \left (x\right )\right ) \] Input:
integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+ (-320*x^7-512*x^6+1728*x^5+768*x^4-1856*x^3+256*x)*log(x)^2+(64*x^7+384*x^ 6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+11 52*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="maxima")
Output:
16/25*x^6 + 32/25*x^5 - 144/25*x^4 - 96/25*x^3 + 464/25*x^2 - 64/5*x + 256 /5/log(x) + 1856/25/log(x)^2 - 64/5*Ei(5*log(x)) - 512/25*Ei(4*log(x)) + 1 728/25*Ei(3*log(x)) + 768/25*Ei(2*log(x)) + 256/25*Ei(-log(x)) - 1856/25*E i(log(x)) + 1024/25*gamma(-1, 2*log(x)) + 1472/25*gamma(-1, -log(x)) - 307 2/25*gamma(-1, -2*log(x)) - 576/25*gamma(-1, -3*log(x)) + 1536/25*gamma(-1 , -4*log(x)) + 64/5*gamma(-1, -5*log(x)) - 1536/25*gamma(-1, log(x)) + 204 8/25*gamma(-2, 2*log(x)) - 768/25*gamma(-2, -log(x)) - 4608/25*gamma(-2, - 2*log(x)) + 2304/25*gamma(-2, -3*log(x)) + 2048/25*gamma(-2, -4*log(x)) - 512/5*gamma(-2, log(x))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.68 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x - \frac {64 \, {\left (x^{7} \log \left (x\right ) + 2 \, x^{6} \log \left (x\right ) - x^{6} - 9 \, x^{5} \log \left (x\right ) - 2 \, x^{5} - 6 \, x^{4} \log \left (x\right ) + 9 \, x^{4} + 29 \, x^{3} \log \left (x\right ) + 6 \, x^{3} - 20 \, x^{2} \log \left (x\right ) - 29 \, x^{2} + 4 \, x \log \left (x\right ) + 20 \, x - 4\right )}}{25 \, x^{2} \log \left (x\right )^{2}} \] Input:
integrate(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+ (-320*x^7-512*x^6+1728*x^5+768*x^4-1856*x^3+256*x)*log(x)^2+(64*x^7+384*x^ 6-192*x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+11 52*x^4+768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x, algorithm="giac")
Output:
16/25*x^6 + 32/25*x^5 - 144/25*x^4 - 96/25*x^3 + 464/25*x^2 - 64/5*x - 64/ 25*(x^7*log(x) + 2*x^6*log(x) - x^6 - 9*x^5*log(x) - 2*x^5 - 6*x^4*log(x) + 9*x^4 + 29*x^3*log(x) + 6*x^3 - 20*x^2*log(x) - 29*x^2 + 4*x*log(x) + 20 *x - 4)/(x^2*log(x)^2)
Time = 2.71 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.10 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {\frac {16\,{\left (2\,x^3+2\,x^2-10\,x+4\right )}^2}{25}-\ln \left (x\right )\,\left (\frac {16\,\left (x^4+x^3-5\,x^2+4\,x\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}\right )}{x^2\,{\ln \left (x\right )}^2}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (x^4+x^3-5\,x^2+4\,x\right )}{25\,x^2} \] Input:
int(((512*x)/5 - (log(x)^3*(320*x^3 - 928*x^4 + 288*x^5 + 576*x^6 - 160*x^ 7 - 96*x^8))/25 + (log(x)*(1536*x - 1280*x^2 + 1472*x^3 - 1536*x^4 - 192*x ^5 + 384*x^6 + 64*x^7 - 512))/25 - (3712*x^2)/25 + (768*x^3)/25 + (1152*x^ 4)/25 - (256*x^5)/25 - (128*x^6)/25 + (log(x)^2*(256*x - 1856*x^3 + 768*x^ 4 + 1728*x^5 - 512*x^6 - 320*x^7))/25 - 512/25)/(x^3*log(x)^3),x)
Output:
((16*(2*x^2 - 10*x + 2*x^3 + 4)^2)/25 - log(x)*((16*(4*x - 5*x^2 + x^3 + x ^4)*(2*x^2 - 10*x + 2*x^3 + 4))/25 + (16*(x^3 - 5*x^2 + x^4)*(2*x^2 - 10*x + 2*x^3 + 4))/25))/(x^2*log(x)^2) + (16*(x^3 - 5*x^2 + x^4)*(4*x - 5*x^2 + x^3 + x^4))/(25*x^2)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.48 \[ \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {\frac {16 \mathrm {log}\left (x \right )^{2} x^{8}}{25}+\frac {32 \mathrm {log}\left (x \right )^{2} x^{7}}{25}-\frac {144 \mathrm {log}\left (x \right )^{2} x^{6}}{25}-\frac {96 \mathrm {log}\left (x \right )^{2} x^{5}}{25}+\frac {464 \mathrm {log}\left (x \right )^{2} x^{4}}{25}-\frac {64 \mathrm {log}\left (x \right )^{2} x^{3}}{5}-\frac {64 \,\mathrm {log}\left (x \right ) x^{7}}{25}-\frac {128 \,\mathrm {log}\left (x \right ) x^{6}}{25}+\frac {576 \,\mathrm {log}\left (x \right ) x^{5}}{25}+\frac {384 \,\mathrm {log}\left (x \right ) x^{4}}{25}-\frac {1856 \,\mathrm {log}\left (x \right ) x^{3}}{25}+\frac {256 \,\mathrm {log}\left (x \right ) x^{2}}{5}-\frac {256 \,\mathrm {log}\left (x \right ) x}{25}+\frac {64 x^{6}}{25}+\frac {128 x^{5}}{25}-\frac {576 x^{4}}{25}-\frac {384 x^{3}}{25}+\frac {1856 x^{2}}{25}-\frac {256 x}{5}+\frac {256}{25}}{\mathrm {log}\left (x \right )^{2} x^{2}} \] Input:
int(1/25*((96*x^8+160*x^7-576*x^6-288*x^5+928*x^4-320*x^3)*log(x)^3+(-320* x^7-512*x^6+1728*x^5+768*x^4-1856*x^3+256*x)*log(x)^2+(64*x^7+384*x^6-192* x^5-1536*x^4+1472*x^3-1280*x^2+1536*x-512)*log(x)-128*x^6-256*x^5+1152*x^4 +768*x^3-3712*x^2+2560*x-512)/x^3/log(x)^3,x)
Output:
(16*(log(x)**2*x**8 + 2*log(x)**2*x**7 - 9*log(x)**2*x**6 - 6*log(x)**2*x* *5 + 29*log(x)**2*x**4 - 20*log(x)**2*x**3 - 4*log(x)*x**7 - 8*log(x)*x**6 + 36*log(x)*x**5 + 24*log(x)*x**4 - 116*log(x)*x**3 + 80*log(x)*x**2 - 16 *log(x)*x + 4*x**6 + 8*x**5 - 36*x**4 - 24*x**3 + 116*x**2 - 80*x + 16))/( 25*log(x)**2*x**2)