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 = 0.172511, antiderivative size = 146, normalized size of antiderivative = 1., 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 1586
Rule 2074
Rule 638
Rule 618
Rule 204
Rule 634
Rule 628
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*}
Mathematica [A] time = 0.0817094, size = 121, normalized size = 0.83 \[ \frac{\frac{1944 x}{-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243}-\log \left (4 x^2-6 x+9\right )+9 \log \left (4 x^2+6 x+9\right )-18 \log (3-2 x)+2 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8503056} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 115, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{4251528}}-{\frac{1}{-2125764+1417176\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{472392}}+{\frac{1}{708588} \left ( -{\frac{x}{4}}-{\frac{3}{4}} \right ) \left ({x}^{2}+{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{4251528}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{x}{944784} \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{8503056}}+{\frac{\sqrt{3}}{472392}\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 = 1.38129, size = 142, normalized size = 0.97 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50641, size = 736, normalized size = 5.04 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.501011, size = 124, normalized size = 0.85 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06199, size = 150, normalized size = 1.03 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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