Optimal. Leaf size=146 \[ \frac {x}{236196 \left (4 x^2-6 x+9\right )}-\frac {x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{8503056}+\frac {\log \left (4 x^2+6 x+9\right )}{944784}+\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (2 x+3)}{4251528}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}} \]
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Rubi [A] time = 0.17, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {1586, 2074, 638, 618, 204, 634, 628} \[ \frac {x}{236196 \left (4 x^2-6 x+9\right )}-\frac {x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{8503056}+\frac {\log \left (4 x^2+6 x+9\right )}{944784}+\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (2 x+3)}{4251528}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 638
Rule 1586
Rule 2074
Rubi steps
\begin {align*} \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{(3+2 x) \left (243-162 x+108 x^2-72 x^3+48 x^4-32 x^5\right )^2} \, dx\\ &=\int \left (\frac {1}{354294 (-3+2 x)^2}-\frac {1}{236196 (-3+2 x)}+\frac {1}{2125764 (3+2 x)}+\frac {3-x}{39366 \left (9-6 x+4 x^2\right )^2}+\frac {33-2 x}{2125764 \left (9-6 x+4 x^2\right )}+\frac {x}{39366 \left (9+6 x+4 x^2\right )^2}+\frac {7+6 x}{708588 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}+\frac {\int \frac {33-2 x}{9-6 x+4 x^2} \, dx}{2125764}+\frac {\int \frac {7+6 x}{9+6 x+4 x^2} \, dx}{708588}+\frac {\int \frac {3-x}{\left (9-6 x+4 x^2\right )^2} \, dx}{39366}+\frac {\int \frac {x}{\left (9+6 x+4 x^2\right )^2} \, dx}{39366}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{8503056}+\frac {\int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{944784}-\frac {\int \frac {1}{9+6 x+4 x^2} \, dx}{708588}+\frac {5 \int \frac {1}{9+6 x+4 x^2} \, dx}{1417176}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{236196}+\frac {7 \int \frac {1}{9-6 x+4 x^2} \, dx}{472392}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784}+\frac {\operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{354294}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{708588}-\frac {\operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{118098}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{236196}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 121, normalized size = 0.83 \[ \frac {-\log \left (4 x^2-6 x+9\right )+9 \log \left (4 x^2+6 x+9\right )+\frac {1944 x}{-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243}-18 \log (3-2 x)+2 \log (2 x+3)+18 \sqrt {3} \tan ^{-1}\left (\frac {4 x-3}{3 \sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{8503056} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 257, normalized size = 1.76 \[ \frac {2 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 18 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 9 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x + 3\right ) - 18 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x - 3\right ) - 1944 \, x}{8503056 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 111, normalized size = 0.76 \[ \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} + 6 \, x + 9\right )} {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x - 3\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{472392} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 115, normalized size = 0.79 \[ \frac {x}{944784 x^{2}-1417176 x +2125764}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{4251528}-\frac {\ln \left (2 x -3\right )}{472392}+\frac {\ln \left (2 x +3\right )}{4251528}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{8503056}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{944784}+\frac {-\frac {x}{4}-\frac {3}{4}}{708588 x^{2}+1062882 x +1594323}-\frac {1}{708588 \left (2 x -3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 105, normalized size = 0.72 \[ \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left (2 \, x + 3\right ) - \frac {1}{472392} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.09, size = 121, normalized size = 0.83 \[ \frac {\ln \left (x+\frac {3}{2}\right )}{4251528}-\frac {\ln \left (x-\frac {3}{2}\right )}{472392}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )-\frac {x}{139968\,\left (x^5-\frac {3\,x^4}{2}+\frac {9\,x^3}{4}-\frac {27\,x^2}{8}+\frac {81\,x}{16}-\frac {243}{32}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 124, normalized size = 0.85 \[ - \frac {x}{139968 x^{5} - 209952 x^{4} + 314928 x^{3} - 472392 x^{2} + 708588 x - 1062882} - \frac {\log {\left (x - \frac {3}{2} \right )}}{472392} + \frac {\log {\left (x + \frac {3}{2} \right )}}{4251528} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{8503056} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{4251528} \]
Verification of antiderivative is not currently implemented for this CAS.
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