Optimal. Leaf size=142 \[ -\frac{3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac{5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{472392 \sqrt{3}} \]
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Rubi [A] time = 0.144815, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1586, 2074, 614, 618, 204, 634, 628} \[ -\frac{3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac{5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{472392 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2074
Rule 614
Rule 618
Rule 204
Rule 634
Rule 628
Rubi steps
\begin{align*} \int \frac{9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx &=\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}{78732 (-3+2 x)^2}-\frac{1}{59049 (-3+2 x)}+\frac{1}{236196 (3+2 x)^2}+\frac{1}{177147 (3+2 x)}+\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 )}+\frac{3+2 x}{236196 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\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}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{3-4 x}{236196 \left (9-6 x+4 x^2\right )}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\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}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{3-4 x}{236196 \left (9-6 x+4 x^2\right )}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\frac{5 \log \left (9-6 x+4 x^2\right )}{2834352}+\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 )}{78732}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{59049}-\frac{19 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{236196}\\ &=\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (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 )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{472392 \sqrt{3}}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\frac{5 \log \left (9-6 x+4 x^2\right )}{2834352}+\frac{\log \left (9+6 x+4 x^2\right )}{944784}\\ \end{align*}
Mathematica [A] time = 0.0619316, size = 111, normalized size = 0.78 \[ \frac{\frac{648 x}{-16 x^4+24 x^3-54 x+81}+5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-24 \log (3-2 x)+8 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2834352} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 111, normalized size = 0.8 \begin{align*} -{\frac{1}{1417176+944784\,x}}+{\frac{\ln \left ( 3+2\,x \right ) }{354294}}-{\frac{1}{-472392+314928\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{118098}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{1417176}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{708588} \left ( 3\,x-{\frac{9}{4}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{5\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{2834352}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37685, size = 128, normalized size = 0.9 \begin{align*} \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{157464} \, \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{1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{2834352} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{354294} \, \log \left (2 \, x + 3\right ) - \frac{1}{118098} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4611, size = 539, normalized size = 3.8 \begin{align*} \frac{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 ) + 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 ) + 3 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 5 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 8 \,{\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) - 24 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.46865, size = 116, normalized size = 0.82 \begin{align*} - \frac{x}{69984 x^{4} - 104976 x^{3} + 236196 x - 354294} - \frac{\log{\left (x - \frac{3}{2} \right )}}{118098} + \frac{\log{\left (x + \frac{3}{2} \right )}}{354294} + \frac{5 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{2834352} + \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 )}}{157464} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{1417176} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06027, size = 143, normalized size = 1.01 \begin{align*} \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{157464} \, \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{1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{2834352} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{354294} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{118098} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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