Integrand size = 39, antiderivative size = 19 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {(-1+3 x) (3-\log (6-x))}{x^9} \] Output:
(-1+3*x)/x^9*(3-ln(6-x))
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {3}{x^9}+\frac {9}{x^8}+\frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8} \] Input:
Integrate[(-162 + 460*x - 75*x^2 + (54 - 153*x + 24*x^2)*Log[6 - x])/(-6*x ^10 + x^11),x]
Output:
-3/x^9 + 9/x^8 + Log[6 - x]/x^9 - (3*Log[6 - x])/x^8
Time = 0.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2026, 7293, 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.
\(\displaystyle \int \frac {-75 x^2+\left (24 x^2-153 x+54\right ) \log (6-x)+460 x-162}{x^{11}-6 x^{10}} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-75 x^2+\left (24 x^2-153 x+54\right ) \log (6-x)+460 x-162}{(x-6) x^{10}}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 (8 x-3) \log (6-x)}{x^{10}}+\frac {-75 x^2+460 x-162}{(x-6) x^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{x^9}+\frac {\log (6-x)}{x^9}+\frac {9}{x^8}-\frac {3 \log (6-x)}{x^8}\) |
Input:
Int[(-162 + 460*x - 75*x^2 + (54 - 153*x + 24*x^2)*Log[6 - x])/(-6*x^10 + x^11),x]
Output:
-3/x^9 + 9/x^8 + Log[6 - x]/x^9 - (3*Log[6 - x])/x^8
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32
method | result | size |
norman | \(\frac {-3+9 x -3 \ln \left (-x +6\right ) x +\ln \left (-x +6\right )}{x^{9}}\) | \(25\) |
risch | \(-\frac {\left (-1+3 x \right ) \ln \left (-x +6\right )}{x^{9}}+\frac {9 x -3}{x^{9}}\) | \(28\) |
parallelrisch | \(-\frac {36+36 \ln \left (-x +6\right ) x -108 x -12 \ln \left (-x +6\right )}{12 x^{9}}\) | \(28\) |
derivativedivides | \(-\frac {3}{x^{9}}+\frac {9}{x^{8}}-\frac {17 \ln \left (-x +6\right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}+10077696 x -45349632\right )}{10077696 x^{9}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}\) | \(175\) |
default | \(-\frac {3}{x^{9}}+\frac {9}{x^{8}}-\frac {17 \ln \left (-x +6\right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}+10077696 x -45349632\right )}{10077696 x^{9}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}\) | \(175\) |
parts | \(-\frac {17 \ln \left (-x \right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}+\frac {9}{x^{8}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}+10077696 x -45349632\right )}{10077696 x^{9}}-\frac {3}{x^{9}}+\frac {17 \ln \left (x \right )}{10077696}-\frac {17 \ln \left (-6+x \right )}{10077696}\) | \(183\) |
orering | \(-\frac {x \left (153 x^{3}-972 x^{2}+631 x -108\right ) \left (\left (24 x^{2}-153 x +54\right ) \ln \left (-x +6\right )-75 x^{2}+460 x -162\right )}{3 \left (192 x^{3}-1151 x^{2}+783 x -144\right ) \left (x^{11}-6 x^{10}\right )}-\frac {\left (9 x^{2}-6 x +1\right ) x^{2} \left (-6+x \right ) \left (\frac {\left (48 x -153\right ) \ln \left (-x +6\right )-\frac {24 x^{2}-153 x +54}{-x +6}-150 x +460}{x^{11}-6 x^{10}}-\frac {\left (\left (24 x^{2}-153 x +54\right ) \ln \left (-x +6\right )-75 x^{2}+460 x -162\right ) \left (11 x^{10}-60 x^{9}\right )}{\left (x^{11}-6 x^{10}\right )^{2}}\right )}{3 \left (192 x^{3}-1151 x^{2}+783 x -144\right )}\) | \(210\) |
Input:
int(((24*x^2-153*x+54)*ln(-x+6)-75*x^2+460*x-162)/(x^11-6*x^10),x,method=_ RETURNVERBOSE)
Output:
(-3+9*x-3*ln(-x+6)*x+ln(-x+6))/x^9
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (-x + 6\right ) - 9 \, x + 3}{x^{9}} \] Input:
integrate(((24*x^2-153*x+54)*log(6-x)-75*x^2+460*x-162)/(x^11-6*x^10),x, a lgorithm="fricas")
Output:
-((3*x - 1)*log(-x + 6) - 9*x + 3)/x^9
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {\left (1 - 3 x\right ) \log {\left (6 - x \right )}}{x^{9}} - \frac {3 - 9 x}{x^{9}} \] Input:
integrate(((24*x**2-153*x+54)*ln(6-x)-75*x**2+460*x-162)/(x**11-6*x**10),x )
Output:
(1 - 3*x)*log(6 - x)/x**9 - (3 - 9*x)/x**9
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.89 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {210 \, x^{8} + 630 \, x^{7} + 2520 \, x^{6} + 11340 \, x^{5} + 54432 \, x^{4} + 272160 \, x^{3} + 1399680 \, x^{2} + 35 \, {\left (x^{9} - 1119744 \, x + 373248\right )} \log \left (-x + 6\right ) + 124921440 \, x}{13063680 \, x^{9}} - \frac {35 \, x^{8} + 105 \, x^{7} + 420 \, x^{6} + 1890 \, x^{5} + 9072 \, x^{4} + 45360 \, x^{3} + 233280 \, x^{2} + 1224720 \, x + 6531840}{2177280 \, x^{9}} - \frac {1}{373248} \, \log \left (x - 6\right ) \] Input:
integrate(((24*x^2-153*x+54)*log(6-x)-75*x^2+460*x-162)/(x^11-6*x^10),x, a lgorithm="maxima")
Output:
1/13063680*(210*x^8 + 630*x^7 + 2520*x^6 + 11340*x^5 + 54432*x^4 + 272160* x^3 + 1399680*x^2 + 35*(x^9 - 1119744*x + 373248)*log(-x + 6) + 124921440* x)/x^9 - 1/2177280*(35*x^8 + 105*x^7 + 420*x^6 + 1890*x^5 + 9072*x^4 + 453 60*x^3 + 233280*x^2 + 1224720*x + 6531840)/x^9 - 1/373248*log(x - 6)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 7.53 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (-x + 6\right )}{{\left (x - 6\right )}^{9} + 54 \, {\left (x - 6\right )}^{8} + 1296 \, {\left (x - 6\right )}^{7} + 18144 \, {\left (x - 6\right )}^{6} + 163296 \, {\left (x - 6\right )}^{5} + 979776 \, {\left (x - 6\right )}^{4} + 3919104 \, {\left (x - 6\right )}^{3} + 10077696 \, {\left (x - 6\right )}^{2} + 15116544 \, x - 80621568} + \frac {3 \, {\left (3 \, x - 1\right )}}{{\left (x - 6\right )}^{9} + 54 \, {\left (x - 6\right )}^{8} + 1296 \, {\left (x - 6\right )}^{7} + 18144 \, {\left (x - 6\right )}^{6} + 163296 \, {\left (x - 6\right )}^{5} + 979776 \, {\left (x - 6\right )}^{4} + 3919104 \, {\left (x - 6\right )}^{3} + 10077696 \, {\left (x - 6\right )}^{2} + 15116544 \, x - 80621568} \] Input:
integrate(((24*x^2-153*x+54)*log(6-x)-75*x^2+460*x-162)/(x^11-6*x^10),x, a lgorithm="giac")
Output:
-(3*x - 1)*log(-x + 6)/((x - 6)^9 + 54*(x - 6)^8 + 1296*(x - 6)^7 + 18144* (x - 6)^6 + 163296*(x - 6)^5 + 979776*(x - 6)^4 + 3919104*(x - 6)^3 + 1007 7696*(x - 6)^2 + 15116544*x - 80621568) + 3*(3*x - 1)/((x - 6)^9 + 54*(x - 6)^8 + 1296*(x - 6)^7 + 18144*(x - 6)^6 + 163296*(x - 6)^5 + 979776*(x - 6)^4 + 3919104*(x - 6)^3 + 10077696*(x - 6)^2 + 15116544*x - 80621568)
Time = 2.87 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {\left (\ln \left (6-x\right )-3\right )\,\left (3\,x-1\right )}{x^9} \] Input:
int(-(460*x + log(6 - x)*(24*x^2 - 153*x + 54) - 75*x^2 - 162)/(6*x^10 - x ^11),x)
Output:
-((log(6 - x) - 3)*(3*x - 1))/x^9
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.47 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {17 \,\mathrm {log}\left (-x +6\right ) x^{9}-30233088 \,\mathrm {log}\left (-x +6\right ) x +10077696 \,\mathrm {log}\left (-x +6\right )-17 \,\mathrm {log}\left (x -6\right ) x^{9}+90699264 x -30233088}{10077696 x^{9}} \] Input:
int(((24*x^2-153*x+54)*log(6-x)-75*x^2+460*x-162)/(x^11-6*x^10),x)
Output:
(17*log( - x + 6)*x**9 - 30233088*log( - x + 6)*x + 10077696*log( - x + 6) - 17*log(x - 6)*x**9 + 90699264*x - 30233088)/(10077696*x**9)