Integrand size = 89, antiderivative size = 34 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {x (x+\log ((1-x) (-2+x)))}{9 \left (\frac {5}{9-x}+\frac {x}{4}\right )} \]
Time = 0.54 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.62 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {1}{9} \left (4 x+\frac {80 x}{-20-9 x+x^2}+\frac {80 \log \left (-2+3 x-x^2\right )}{-20-9 x+x^2}+4 \log \left (2-3 x+x^2\right )\right ) \]
Integrate[(720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 + (1440 - 2480*x + 1200*x^2 - 160*x^3)*Log[-2 + 3*x - x^2])/(7200 - 4320*x - 5382* x^2 + 1809*x^3 + 873*x^4 - 189*x^5 + 9*x^6),x]
(4*x + (80*x)/(-20 - 9*x + x^2) + (80*Log[-2 + 3*x - x^2])/(-20 - 9*x + x^ 2) + 4*Log[2 - 3*x + x^2])/9
Leaf count is larger than twice the leaf count of optimal. \(1290\) vs. \(2(34)=68\).
Time = 5.19 (sec) , antiderivative size = 1290, normalized size of antiderivative = 37.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, 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.
| \(\displaystyle \int \frac {4 x^6-76 x^5+152 x^4+1748 x^3-3444 x^2+\left (-160 x^3+1200 x^2-2480 x+1440\right ) \log \left (-x^2+3 x-2\right )+720 x}{9 x^6-189 x^5+873 x^4+1809 x^3-5382 x^2-4320 x+7200} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 x^6-76 x^5+152 x^4+1748 x^3-3444 x^2+\left (-160 x^3+1200 x^2-2480 x+1440\right ) \log \left (-x^2+3 x-2\right )+720 x}{10404 (x-2)}-\frac {4 x^6-76 x^5+152 x^4+1748 x^3-3444 x^2+\left (-160 x^3+1200 x^2-2480 x+1440\right ) \log \left (-x^2+3 x-2\right )+720 x}{7056 (x-1)}+\frac {(93 x-940) \left (4 x^6-76 x^5+152 x^4+1748 x^3-3444 x^2+\left (-160 x^3+1200 x^2-2480 x+1440\right ) \log \left (-x^2+3 x-2\right )+720 x\right )}{2039184 \left (x^2-9 x-20\right )}+\frac {(38-3 x) \left (4 x^6-76 x^5+152 x^4+1748 x^3-3444 x^2+\left (-160 x^3+1200 x^2-2480 x+1440\right ) \log \left (-x^2+3 x-2\right )+720 x\right )}{4284 \left (x^2-9 x-20\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(140-3 x) x^5}{24633 \left (-x^2+9 x+20\right )}-\frac {19 (140-3 x) x^4}{24633 \left (-x^2+9 x+20\right )}-\frac {x^4}{8211}+\frac {38 (140-3 x) x^3}{24633 \left (-x^2+9 x+20\right )}+\frac {10 x^3}{1449}+\frac {19 (140-3 x) x^2}{1071 \left (-x^2+9 x+20\right )}-\frac {1304 x^2}{24633}-\frac {41 (140-3 x) x}{1173 \left (-x^2+9 x+20\right )}+\frac {2207 x}{8211}-\frac {5 \left (2369+399 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{651406}+\frac {41 \left (11109-341 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{377706}-\frac {152 \left (3703-387 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{172431}-\frac {41 \left (21459-611 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{1116696}-\frac {95 \left (40733-2957 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{1321971}+\frac {19 \left (145751-13479 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{1019592}+\frac {19 \left (1740939-133531 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{11725308}+\frac {19 \left (2284751-184479 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{3965913}-\frac {19 \left (18583471-1471359 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{23450616}-\frac {\left (31427361-2457169 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{7931826}+\frac {\left (202070019-15912851 \sqrt {161}\right ) \log \left (-2 x-\sqrt {161}+9\right )}{23450616}+\frac {1320 \log \left (-2 x-\sqrt {161}+9\right )}{2737 \sqrt {161}}+\frac {\left (202070019+15912851 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{23450616}-\frac {\left (31427361+2457169 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{7931826}-\frac {19 \left (18583471+1471359 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{23450616}+\frac {19 \left (2284751+184479 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{3965913}+\frac {19 \left (1740939+133531 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{11725308}+\frac {19 \left (145751+13479 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{1019592}-\frac {95 \left (40733+2957 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{1321971}-\frac {41 \left (21459+611 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{1116696}-\frac {152 \left (3703+387 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{172431}+\frac {41 \left (11109+341 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{377706}-\frac {5 \left (2369-399 \sqrt {161}\right ) \log \left (-2 x+\sqrt {161}+9\right )}{651406}-\frac {1320 \log \left (-2 x+\sqrt {161}+9\right )}{2737 \sqrt {161}}+\frac {160 \left (9+\sqrt {161}\right ) \log (1-x)}{1449 \left (7+\sqrt {161}\right )}-\frac {160 \log (1-x)}{161 \left (7+\sqrt {161}\right )}+\frac {160 \left (9-\sqrt {161}\right ) \log (1-x)}{1449 \left (7-\sqrt {161}\right )}-\frac {160 \log (1-x)}{161 \left (7-\sqrt {161}\right )}+\frac {8}{63} \log (1-x)+\frac {160 \left (9+\sqrt {161}\right ) \log (2-x)}{1449 \left (5+\sqrt {161}\right )}-\frac {160 \log (2-x)}{161 \left (5+\sqrt {161}\right )}+\frac {160 \left (9-\sqrt {161}\right ) \log (2-x)}{1449 \left (5-\sqrt {161}\right )}-\frac {160 \log (2-x)}{161 \left (5-\sqrt {161}\right )}+\frac {28}{153} \log (2-x)+\frac {80 \left (6-\sqrt {161}\right ) \log \left (\left (49-3 \sqrt {161}\right ) x-2 \left (231-19 \sqrt {161}\right )\right )}{161 \left (49-3 \sqrt {161}\right )}-\frac {400 \left (43-3 \sqrt {161}\right ) \log \left (\left (49-3 \sqrt {161}\right ) x-2 \left (231-19 \sqrt {161}\right )\right )}{1449 \left (49-3 \sqrt {161}\right )}-\frac {400 \left (43+3 \sqrt {161}\right ) \log \left (2 \left (231+19 \sqrt {161}\right )-\left (49+3 \sqrt {161}\right ) x\right )}{1449 \left (49+3 \sqrt {161}\right )}+\frac {80 \left (6+\sqrt {161}\right ) \log \left (2 \left (231+19 \sqrt {161}\right )-\left (49+3 \sqrt {161}\right ) x\right )}{161 \left (49+3 \sqrt {161}\right )}-\frac {160 \left (9-\sqrt {161}\right ) \log \left (-x^2+3 x-2\right )}{1449 \left (-2 x-\sqrt {161}+9\right )}+\frac {160 \log \left (-x^2+3 x-2\right )}{161 \left (-2 x-\sqrt {161}+9\right )}-\frac {160 \left (9+\sqrt {161}\right ) \log \left (-x^2+3 x-2\right )}{1449 \left (-2 x+\sqrt {161}+9\right )}+\frac {160 \log \left (-x^2+3 x-2\right )}{161 \left (-2 x+\sqrt {161}+9\right )}+\frac {20 (140-3 x)}{2737 \left (-x^2+9 x+20\right )}\) |
Int[(720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 + (1440 - 2480 *x + 1200*x^2 - 160*x^3)*Log[-2 + 3*x - x^2])/(7200 - 4320*x - 5382*x^2 + 1809*x^3 + 873*x^4 - 189*x^5 + 9*x^6),x]
(2207*x)/8211 - (1304*x^2)/24633 + (10*x^3)/1449 - x^4/8211 + (20*(140 - 3 *x))/(2737*(20 + 9*x - x^2)) - (41*(140 - 3*x)*x)/(1173*(20 + 9*x - x^2)) + (19*(140 - 3*x)*x^2)/(1071*(20 + 9*x - x^2)) + (38*(140 - 3*x)*x^3)/(246 33*(20 + 9*x - x^2)) - (19*(140 - 3*x)*x^4)/(24633*(20 + 9*x - x^2)) + ((1 40 - 3*x)*x^5)/(24633*(20 + 9*x - x^2)) + (1320*Log[9 - Sqrt[161] - 2*x])/ (2737*Sqrt[161]) + ((202070019 - 15912851*Sqrt[161])*Log[9 - Sqrt[161] - 2 *x])/23450616 - ((31427361 - 2457169*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/ 7931826 - (19*(18583471 - 1471359*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/234 50616 + (19*(2284751 - 184479*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/3965913 + (19*(1740939 - 133531*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/11725308 + ( 19*(145751 - 13479*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/1019592 - (95*(407 33 - 2957*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/1321971 - (41*(21459 - 611* Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/1116696 - (152*(3703 - 387*Sqrt[161]) *Log[9 - Sqrt[161] - 2*x])/172431 + (41*(11109 - 341*Sqrt[161])*Log[9 - Sq rt[161] - 2*x])/377706 - (5*(2369 + 399*Sqrt[161])*Log[9 - Sqrt[161] - 2*x ])/651406 - (1320*Log[9 + Sqrt[161] - 2*x])/(2737*Sqrt[161]) - (5*(2369 - 399*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/651406 + (41*(11109 + 341*Sqrt[16 1])*Log[9 + Sqrt[161] - 2*x])/377706 - (152*(3703 + 387*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/172431 - (41*(21459 + 611*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/1116696 - (95*(40733 + 2957*Sqrt[161])*Log[9 + Sqrt[161] - 2*x]...
3.9.68.3.1 Defintions of rubi rules used
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]
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53
| method | result | size |
| norman | \(\frac {-36 x -4 \ln \left (-x^{2}+3 x -2\right ) x +\frac {4 \ln \left (-x^{2}+3 x -2\right ) x^{2}}{9}+\frac {4 x^{3}}{9}-80}{x^{2}-9 x -20}\) | \(52\) |
| parallelrisch | \(\frac {-1200+4 x^{3}+4 \ln \left (-x^{2}+3 x -2\right ) x^{2}+24 x^{2}-36 \ln \left (-x^{2}+3 x -2\right ) x -540 x}{9 x^{2}-81 x -180}\) | \(58\) |
| risch | \(\frac {80 \ln \left (-x^{2}+3 x -2\right )}{9 \left (x^{2}-9 x -20\right )}+\frac {\frac {4 \ln \left (x^{2}-3 x +2\right ) x^{2}}{9}+\frac {4 x^{3}}{9}-4 \ln \left (x^{2}-3 x +2\right ) x -4 x^{2}-\frac {80 \ln \left (x^{2}-3 x +2\right )}{9}}{x^{2}-9 x -20}\) | \(82\) |
int(((-160*x^3+1200*x^2-2480*x+1440)*ln(-x^2+3*x-2)+4*x^6-76*x^5+152*x^4+1 748*x^3-3444*x^2+720*x)/(9*x^6-189*x^5+873*x^4+1809*x^3-5382*x^2-4320*x+72 00),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {4 \, {\left (x^{3} - 9 \, x^{2} + {\left (x^{2} - 9 \, x\right )} \log \left (-x^{2} + 3 \, x - 2\right )\right )}}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} \]
integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+15 2*x^4+1748*x^3-3444*x^2+720*x)/(9*x^6-189*x^5+873*x^4+1809*x^3-5382*x^2-43 20*x+7200),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {4 x}{9} + \frac {80 x}{9 x^{2} - 81 x - 180} + \frac {4 \log {\left (x^{2} - 3 x + 2 \right )}}{9} + \frac {80 \log {\left (- x^{2} + 3 x - 2 \right )}}{9 x^{2} - 81 x - 180} \]
integrate(((-160*x**3+1200*x**2-2480*x+1440)*ln(-x**2+3*x-2)+4*x**6-76*x** 5+152*x**4+1748*x**3-3444*x**2+720*x)/(9*x**6-189*x**5+873*x**4+1809*x**3- 5382*x**2-4320*x+7200),x)
4*x/9 + 80*x/(9*x**2 - 81*x - 180) + 4*log(x**2 - 3*x + 2)/9 + 80*log(-x** 2 + 3*x - 2)/(9*x**2 - 81*x - 180)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (25) = 50\).
Time = 0.35 (sec) , antiderivative size = 161, normalized size of antiderivative = 4.74 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {4}{9} \, x + \frac {20 \, {\left (17 \, {\left (x^{2} - 9 \, x + 8\right )} \log \left (x - 1\right ) + 14 \, {\left (x^{2} - 9 \, x + 14\right )} \log \left (-x + 2\right )\right )}}{1071 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {1270433 \, x + 2339940}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {19 \, {\left (116997 \, x + 217460\right )}}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {38 \, {\left (10873 \, x + 19140\right )}}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {19 \, {\left (957 \, x + 2260\right )}}{1071 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {41 \, {\left (113 \, x - 60\right )}}{1173 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {20 \, {\left (3 \, x - 140\right )}}{2737 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {8}{63} \, \log \left (x - 1\right ) + \frac {28}{153} \, \log \left (x - 2\right ) \]
integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+15 2*x^4+1748*x^3-3444*x^2+720*x)/(9*x^6-189*x^5+873*x^4+1809*x^3-5382*x^2-43 20*x+7200),x, algorithm=\
4/9*x + 20/1071*(17*(x^2 - 9*x + 8)*log(x - 1) + 14*(x^2 - 9*x + 14)*log(- x + 2))/(x^2 - 9*x - 20) - 1/24633*(1270433*x + 2339940)/(x^2 - 9*x - 20) + 19/24633*(116997*x + 217460)/(x^2 - 9*x - 20) - 38/24633*(10873*x + 1914 0)/(x^2 - 9*x - 20) - 19/1071*(957*x + 2260)/(x^2 - 9*x - 20) + 41/1173*(1 13*x - 60)/(x^2 - 9*x - 20) + 20/2737*(3*x - 140)/(x^2 - 9*x - 20) + 8/63* log(x - 1) + 28/153*log(x - 2)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {4}{9} \, x + \frac {80 \, x}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {80 \, \log \left (-x^{2} + 3 \, x - 2\right )}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {4}{9} \, \log \left (x^{2} - 3 \, x + 2\right ) \]
integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+15 2*x^4+1748*x^3-3444*x^2+720*x)/(9*x^6-189*x^5+873*x^4+1809*x^3-5382*x^2-43 20*x+7200),x, algorithm=\
4/9*x + 80/9*x/(x^2 - 9*x - 20) + 80/9*log(-x^2 + 3*x - 2)/(x^2 - 9*x - 20 ) + 4/9*log(x^2 - 3*x + 2)
Time = 9.38 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.62 \[ \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx=\frac {4\,x}{9}+\frac {4\,\ln \left (x^2-3\,x+2\right )}{9}-\frac {80\,\ln \left (-x^2+3\,x-2\right )}{9\,\left (-x^2+9\,x+20\right )}-\frac {80\,x}{9\,\left (-x^2+9\,x+20\right )} \]
int((720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 - log(3*x - x^ 2 - 2)*(2480*x - 1200*x^2 + 160*x^3 - 1440))/(1809*x^3 - 5382*x^2 - 4320*x + 873*x^4 - 189*x^5 + 9*x^6 + 7200),x)