Integrand size = 20, antiderivative size = 142 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}-\frac {5 \log \left (9+6 x+4 x^2\right )}{2834352} \]
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Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1600, 2099, 648, 632, 210, 642, 628} \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\arctan \left (\frac {4 x+3}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}+\frac {4 x+3}{236196 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{944784}-\frac {5 \log \left (4 x^2+6 x+9\right )}{2834352}+\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (2 x+3)}-\frac {\log (3-2 x)}{354294}+\frac {\log (2 x+3)}{118098} \]
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Rule 210
Rule 628
Rule 632
Rule 642
Rule 648
Rule 1600
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (9-6 x+4 x^2\right ) \left (81+54 x-24 x^3-16 x^4\right )^2} \, dx \\ & = \int \left (\frac {1}{236196 (-3+2 x)^2}-\frac {1}{177147 (-3+2 x)}+\frac {1}{78732 (3+2 x)^2}+\frac {1}{59049 (3+2 x)}+\frac {3-2 x}{236196 \left (9-6 x+4 x^2\right )}+\frac {1}{4374 \left (9+6 x+4 x^2\right )^2}+\frac {21-10 x}{708588 \left (9+6 x+4 x^2\right )}\right ) \, dx \\ & = \frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}+\frac {\int \frac {21-10 x}{9+6 x+4 x^2} \, dx}{708588}+\frac {\int \frac {3-2 x}{9-6 x+4 x^2} \, dx}{236196}+\frac {\int \frac {1}{\left (9+6 x+4 x^2\right )^2} \, dx}{4374} \\ & = \frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{944784}-\frac {5 \int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{2834352}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{157464}+\frac {\int \frac {1}{9+6 x+4 x^2} \, dx}{59049}+\frac {19 \int \frac {1}{9+6 x+4 x^2} \, dx}{472392} \\ & = \frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}-\frac {5 \log \left (9+6 x+4 x^2\right )}{2834352}-\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{78732}-\frac {2 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{59049}-\frac {19 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{236196} \\ & = \frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}-\frac {5 \log \left (9+6 x+4 x^2\right )}{2834352} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {648 x}{81+54 x-24 x^3-16 x^4}+2 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )+18 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-8 \log (3-2 x)+24 \log (3+2 x)-3 \log \left (9-6 x+4 x^2\right )-5 \log \left (9+6 x+4 x^2\right )}{2834352} \]
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Time = 1.55 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {x}{69984 \left (x^{4}+\frac {3}{2} x^{3}-\frac {27}{8} x -\frac {81}{16}\right )}-\frac {5 \ln \left (16 x^{2}+24 x +36\right )}{2834352}+\frac {\arctan \left (\frac {\left (4 x +3\right ) \sqrt {3}}{9}\right ) \sqrt {3}}{157464}-\frac {\ln \left (-3+2 x \right )}{354294}+\frac {\ln \left (2 x +3\right )}{118098}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (-\frac {3}{2}+2 x \right ) \sqrt {3}}{9}\right )}{1417176}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{944784}\) | \(94\) |
default | \(-\frac {1}{472392 \left (-3+2 x \right )}-\frac {\ln \left (-3+2 x \right )}{354294}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{1417176}-\frac {1}{157464 \left (2 x +3\right )}+\frac {\ln \left (2 x +3\right )}{118098}-\frac {-3 x -\frac {9}{4}}{708588 \left (x^{2}+\frac {3}{2} x +\frac {9}{4}\right )}-\frac {5 \ln \left (4 x^{2}+6 x +9\right )}{2834352}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{157464}\) | \(111\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{236196}-\frac {i \left (\frac {16 i x^{3}}{27 \left (-\frac {128 x^{6}}{729}+2\right )}+i \operatorname {arctanh}\left (\frac {8 x^{3}}{27}\right )\right )}{236196}+\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{236196}\) | \(270\) |
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Time = 0.32 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.32 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {18 \, \sqrt {3} {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 2 \, \sqrt {3} {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - 5 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 3 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 24 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) - 8 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x - 3\right ) - 648 \, x}{2834352 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=- \frac {x}{69984 x^{4} + 104976 x^{3} - 236196 x - 354294} - \frac {\log {\left (x - \frac {3}{2} \right )}}{354294} + \frac {\log {\left (x + \frac {3}{2} \right )}}{118098} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} - \frac {5 \log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{2834352} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{1417176} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{157464} \]
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Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} - \frac {5}{2834352} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{118098} \, \log \left (2 \, x + 3\right ) - \frac {1}{354294} \, \log \left (2 \, x - 3\right ) \]
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} + 6 \, x + 9\right )} {\left (2 \, x + 3\right )} {\left (2 \, x - 3\right )}} - \frac {5}{2834352} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{118098} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{354294} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
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Time = 11.76 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.77 \[ \int \frac {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {\ln \left (x+\frac {3}{2}\right )}{118098}-\frac {\ln \left (x-\frac {3}{2}\right )}{354294}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right )+\frac {x}{69984\,\left (-x^4-\frac {3\,x^3}{2}+\frac {27\,x}{8}+\frac {81}{16}\right )} \]
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