Integrand size = 8, antiderivative size = 127 \[ \int x^9 \log ^{11}(x) \, dx=-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \log (x)}{15625000}-\frac {6237 x^{10} \log ^2(x)}{3125000}+\frac {2079 x^{10} \log ^3(x)}{312500}-\frac {2079 x^{10} \log ^4(x)}{125000}+\frac {2079 x^{10} \log ^5(x)}{62500}-\frac {693 x^{10} \log ^6(x)}{12500}+\frac {99 x^{10} \log ^7(x)}{1250}-\frac {99 x^{10} \log ^8(x)}{1000}+\frac {11}{100} x^{10} \log ^9(x)-\frac {11}{100} x^{10} \log ^{10}(x)+\frac {1}{10} x^{10} \log ^{11}(x) \]
-6237/156250000*x^10+6237/15625000*x^10*ln(x)-6237/3125000*x^10*ln(x)^2+20 79/312500*x^10*ln(x)^3-2079/125000*x^10*ln(x)^4+2079/62500*x^10*ln(x)^5-69 3/12500*x^10*ln(x)^6+99/1250*x^10*ln(x)^7-99/1000*x^10*ln(x)^8+11/100*x^10 *ln(x)^9-11/100*x^10*ln(x)^10+1/10*x^10*ln(x)^11
Time = 0.00 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int x^9 \log ^{11}(x) \, dx=-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \log (x)}{15625000}-\frac {6237 x^{10} \log ^2(x)}{3125000}+\frac {2079 x^{10} \log ^3(x)}{312500}-\frac {2079 x^{10} \log ^4(x)}{125000}+\frac {2079 x^{10} \log ^5(x)}{62500}-\frac {693 x^{10} \log ^6(x)}{12500}+\frac {99 x^{10} \log ^7(x)}{1250}-\frac {99 x^{10} \log ^8(x)}{1000}+\frac {11}{100} x^{10} \log ^9(x)-\frac {11}{100} x^{10} \log ^{10}(x)+\frac {1}{10} x^{10} \log ^{11}(x) \]
(-6237*x^10)/156250000 + (6237*x^10*Log[x])/15625000 - (6237*x^10*Log[x]^2 )/3125000 + (2079*x^10*Log[x]^3)/312500 - (2079*x^10*Log[x]^4)/125000 + (2 079*x^10*Log[x]^5)/62500 - (693*x^10*Log[x]^6)/12500 + (99*x^10*Log[x]^7)/ 1250 - (99*x^10*Log[x]^8)/1000 + (11*x^10*Log[x]^9)/100 - (11*x^10*Log[x]^ 10)/100 + (x^10*Log[x]^11)/10
Time = 0.56 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {2742, 2742, 2742, 2742, 2742, 2742, 2742, 2742, 2742, 2742, 2741}
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.
| \(\displaystyle \int x^9 \log ^{11}(x) \, dx\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \int x^9 \log ^{10}(x)dx\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {1}{10} x^{10} \log ^{10}(x)-\int x^9 \log ^9(x)dx\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \int x^9 \log ^8(x)dx+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \int x^9 \log ^7(x)dx\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \int x^9 \log ^6(x)dx\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \int x^9 \log ^5(x)dx\right )\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \left (\frac {1}{10} x^{10} \log ^5(x)-\frac {1}{2} \int x^9 \log ^4(x)dx\right )\right )\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \left (\frac {1}{2} \left (\frac {2}{5} \int x^9 \log ^3(x)dx-\frac {1}{10} x^{10} \log ^4(x)\right )+\frac {1}{10} x^{10} \log ^5(x)\right )\right )\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \left (\frac {1}{2} \left (\frac {2}{5} \left (\frac {1}{10} x^{10} \log ^3(x)-\frac {3}{10} \int x^9 \log ^2(x)dx\right )-\frac {1}{10} x^{10} \log ^4(x)\right )+\frac {1}{10} x^{10} \log ^5(x)\right )\right )\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \left (\frac {1}{2} \left (\frac {2}{5} \left (\frac {1}{10} x^{10} \log ^3(x)-\frac {3}{10} \left (\frac {1}{10} x^{10} \log ^2(x)-\frac {1}{5} \int x^9 \log (x)dx\right )\right )-\frac {1}{10} x^{10} \log ^4(x)\right )+\frac {1}{10} x^{10} \log ^5(x)\right )\right )\right )\right )+\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)\right )\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {1}{10} x^{10} \log ^{11}(x)-\frac {11}{10} \left (\frac {1}{10} x^{10} \log ^{10}(x)-\frac {1}{10} x^{10} \log ^9(x)+\frac {9}{10} \left (\frac {1}{10} x^{10} \log ^8(x)-\frac {4}{5} \left (\frac {1}{10} x^{10} \log ^7(x)-\frac {7}{10} \left (\frac {1}{10} x^{10} \log ^6(x)-\frac {3}{5} \left (\frac {1}{10} x^{10} \log ^5(x)+\frac {1}{2} \left (\frac {2}{5} \left (\frac {1}{10} x^{10} \log ^3(x)-\frac {3}{10} \left (\frac {1}{10} x^{10} \log ^2(x)+\frac {1}{5} \left (\frac {x^{10}}{100}-\frac {1}{10} x^{10} \log (x)\right )\right )\right )-\frac {1}{10} x^{10} \log ^4(x)\right )\right )\right )\right )\right )\right )\) |
(x^10*Log[x]^11)/10 - (11*(-1/10*(x^10*Log[x]^9) + (x^10*Log[x]^10)/10 + ( 9*((x^10*Log[x]^8)/10 - (4*((x^10*Log[x]^7)/10 - (7*((x^10*Log[x]^6)/10 - (3*((x^10*Log[x]^5)/10 + (-1/10*(x^10*Log[x]^4) + (2*((x^10*Log[x]^3)/10 - (3*((x^10*Log[x]^2)/10 + (x^10/100 - (x^10*Log[x])/10)/5))/10))/5)/2))/5) )/10))/5))/10))/10
3.1.58.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\) | \(104\) |
| risch | \(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\) | \(104\) |
| parallelrisch | \(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\) | \(104\) |
| parts | \(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\) | \(104\) |
-6237/156250000*x^10+6237/15625000*x^10*ln(x)-6237/3125000*x^10*ln(x)^2+20 79/312500*x^10*ln(x)^3-2079/125000*x^10*ln(x)^4+2079/62500*x^10*ln(x)^5-69 3/12500*x^10*ln(x)^6+99/1250*x^10*ln(x)^7-99/1000*x^10*ln(x)^8+11/100*x^10 *ln(x)^9-11/100*x^10*ln(x)^10+1/10*x^10*ln(x)^11
Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.81 \[ \int x^9 \log ^{11}(x) \, dx=\frac {1}{10} \, x^{10} \log \left (x\right )^{11} - \frac {11}{100} \, x^{10} \log \left (x\right )^{10} + \frac {11}{100} \, x^{10} \log \left (x\right )^{9} - \frac {99}{1000} \, x^{10} \log \left (x\right )^{8} + \frac {99}{1250} \, x^{10} \log \left (x\right )^{7} - \frac {693}{12500} \, x^{10} \log \left (x\right )^{6} + \frac {2079}{62500} \, x^{10} \log \left (x\right )^{5} - \frac {2079}{125000} \, x^{10} \log \left (x\right )^{4} + \frac {2079}{312500} \, x^{10} \log \left (x\right )^{3} - \frac {6237}{3125000} \, x^{10} \log \left (x\right )^{2} + \frac {6237}{15625000} \, x^{10} \log \left (x\right ) - \frac {6237}{156250000} \, x^{10} \]
1/10*x^10*log(x)^11 - 11/100*x^10*log(x)^10 + 11/100*x^10*log(x)^9 - 99/10 00*x^10*log(x)^8 + 99/1250*x^10*log(x)^7 - 693/12500*x^10*log(x)^6 + 2079/ 62500*x^10*log(x)^5 - 2079/125000*x^10*log(x)^4 + 2079/312500*x^10*log(x)^ 3 - 6237/3125000*x^10*log(x)^2 + 6237/15625000*x^10*log(x) - 6237/15625000 0*x^10
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05 \[ \int x^9 \log ^{11}(x) \, dx=\frac {x^{10} \log {\left (x \right )}^{11}}{10} - \frac {11 x^{10} \log {\left (x \right )}^{10}}{100} + \frac {11 x^{10} \log {\left (x \right )}^{9}}{100} - \frac {99 x^{10} \log {\left (x \right )}^{8}}{1000} + \frac {99 x^{10} \log {\left (x \right )}^{7}}{1250} - \frac {693 x^{10} \log {\left (x \right )}^{6}}{12500} + \frac {2079 x^{10} \log {\left (x \right )}^{5}}{62500} - \frac {2079 x^{10} \log {\left (x \right )}^{4}}{125000} + \frac {2079 x^{10} \log {\left (x \right )}^{3}}{312500} - \frac {6237 x^{10} \log {\left (x \right )}^{2}}{3125000} + \frac {6237 x^{10} \log {\left (x \right )}}{15625000} - \frac {6237 x^{10}}{156250000} \]
x**10*log(x)**11/10 - 11*x**10*log(x)**10/100 + 11*x**10*log(x)**9/100 - 9 9*x**10*log(x)**8/1000 + 99*x**10*log(x)**7/1250 - 693*x**10*log(x)**6/125 00 + 2079*x**10*log(x)**5/62500 - 2079*x**10*log(x)**4/125000 + 2079*x**10 *log(x)**3/312500 - 6237*x**10*log(x)**2/3125000 + 6237*x**10*log(x)/15625 000 - 6237*x**10/156250000
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int x^9 \log ^{11}(x) \, dx=\frac {1}{156250000} \, {\left (15625000 \, \log \left (x\right )^{11} - 17187500 \, \log \left (x\right )^{10} + 17187500 \, \log \left (x\right )^{9} - 15468750 \, \log \left (x\right )^{8} + 12375000 \, \log \left (x\right )^{7} - 8662500 \, \log \left (x\right )^{6} + 5197500 \, \log \left (x\right )^{5} - 2598750 \, \log \left (x\right )^{4} + 1039500 \, \log \left (x\right )^{3} - 311850 \, \log \left (x\right )^{2} + 62370 \, \log \left (x\right ) - 6237\right )} x^{10} \]
1/156250000*(15625000*log(x)^11 - 17187500*log(x)^10 + 17187500*log(x)^9 - 15468750*log(x)^8 + 12375000*log(x)^7 - 8662500*log(x)^6 + 5197500*log(x) ^5 - 2598750*log(x)^4 + 1039500*log(x)^3 - 311850*log(x)^2 + 62370*log(x) - 6237)*x^10
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.81 \[ \int x^9 \log ^{11}(x) \, dx=\frac {1}{10} \, x^{10} \log \left (x\right )^{11} - \frac {11}{100} \, x^{10} \log \left (x\right )^{10} + \frac {11}{100} \, x^{10} \log \left (x\right )^{9} - \frac {99}{1000} \, x^{10} \log \left (x\right )^{8} + \frac {99}{1250} \, x^{10} \log \left (x\right )^{7} - \frac {693}{12500} \, x^{10} \log \left (x\right )^{6} + \frac {2079}{62500} \, x^{10} \log \left (x\right )^{5} - \frac {2079}{125000} \, x^{10} \log \left (x\right )^{4} + \frac {2079}{312500} \, x^{10} \log \left (x\right )^{3} - \frac {6237}{3125000} \, x^{10} \log \left (x\right )^{2} + \frac {6237}{15625000} \, x^{10} \log \left (x\right ) - \frac {6237}{156250000} \, x^{10} \]
1/10*x^10*log(x)^11 - 11/100*x^10*log(x)^10 + 11/100*x^10*log(x)^9 - 99/10 00*x^10*log(x)^8 + 99/1250*x^10*log(x)^7 - 693/12500*x^10*log(x)^6 + 2079/ 62500*x^10*log(x)^5 - 2079/125000*x^10*log(x)^4 + 2079/312500*x^10*log(x)^ 3 - 6237/3125000*x^10*log(x)^2 + 6237/15625000*x^10*log(x) - 6237/15625000 0*x^10
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int x^9 \log ^{11}(x) \, dx=\frac {6237\,x^{10}\,\left (\frac {15625000\,{\ln \left (x\right )}^{11}}{6237}-\frac {1562500\,{\ln \left (x\right )}^{10}}{567}+\frac {1562500\,{\ln \left (x\right )}^9}{567}-\frac {156250\,{\ln \left (x\right )}^8}{63}+\frac {125000\,{\ln \left (x\right )}^7}{63}-\frac {12500\,{\ln \left (x\right )}^6}{9}+\frac {2500\,{\ln \left (x\right )}^5}{3}-\frac {1250\,{\ln \left (x\right )}^4}{3}+\frac {500\,{\ln \left (x\right )}^3}{3}-50\,{\ln \left (x\right )}^2+10\,\ln \left (x\right )-1\right )}{156250000} \]