Optimal. Leaf size=131 \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]
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Rubi [A] time = 0.147723, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1586, 2074, 638, 618, 204, 634, 628} \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]
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{27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{\left (27-36 x+24 x^2-8 x^3\right )^2 \left (27+36 x+24 x^2+8 x^3\right )} \, dx\\ &=\int \left (\frac{1}{13122 (-3+2 x)^2}-\frac{7}{78732 (-3+2 x)}+\frac{1}{236196 (3+2 x)}+\frac{3+2 x}{4374 \left (9-6 x+4 x^2\right )^2}+\frac{3+17 x}{118098 \left (9-6 x+4 x^2\right )}+\frac{x}{39366 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{\int \frac{3+17 x}{9-6 x+4 x^2} \, dx}{118098}+\frac{\int \frac{x}{9+6 x+4 x^2} \, dx}{39366}+\frac{\int \frac{3+2 x}{\left (9-6 x+4 x^2\right )^2} \, dx}{4374}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{314928}+\frac{17 \int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{944784}-\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{52488}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{13122}+\frac{7 \int \frac{1}{9-6 x+4 x^2} \, dx}{52488}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{17 \log \left (9-6 x+4 x^2\right )}{944784}+\frac{\log \left (9+6 x+4 x^2\right )}{314928}+\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{26244}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{6561}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{26244}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac{11 \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 )}{157464 \sqrt{3}}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{17 \log \left (9-6 x+4 x^2\right )}{944784}+\frac{\log \left (9+6 x+4 x^2\right )}{314928}\\ \end{align*}
Mathematica [A] time = 0.0637757, size = 111, normalized size = 0.85 \[ \frac{\frac{216 x}{-8 x^3+24 x^2-36 x+27}+17 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-42 \log (3-2 x)+2 \log (2 x+3)+22 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{944784} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 102, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{472392}}-{\frac{1}{-78732+52488\,x}}-{\frac{7\,\ln \left ( -3+2\,x \right ) }{157464}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{314928}}-{\frac{\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{118098} \left ({\frac{9\,x}{4}}-{\frac{27}{8}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{17\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{944784}}+{\frac{11\,\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 = 1.3948, size = 128, normalized size = 0.98 \begin{align*} -\frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{11}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{472392} \, \log \left (2 \, x + 3\right ) - \frac{7}{157464} \, \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 = 1.44256, size = 531, normalized size = 4.05 \begin{align*} -\frac{2 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - 22 \, \sqrt{3}{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 3 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 17 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 2 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x + 3\right ) + 42 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \,{\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.46846, size = 119, normalized size = 0.91 \begin{align*} - \frac{x}{34992 x^{3} - 104976 x^{2} + 157464 x - 118098} - \frac{7 \log{\left (x - \frac{3}{2} \right )}}{157464} + \frac{\log{\left (x + \frac{3}{2} \right )}}{472392} + \frac{17 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{314928} + \frac{11 \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 )}}{472392} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04995, size = 134, normalized size = 1.02 \begin{align*} -\frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{11}{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 (2 \, x - 3\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{7}{157464} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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