Optimal. Leaf size=110 \[ -\frac {\log \left (4 x^2-6 x+9\right )}{17496}-\frac {\log \left (4 x^2+6 x+9\right )}{17496}-\frac {1}{2916 (2 x+3)}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (2 x+3)}{17496}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{8748 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{2916 \sqrt {3}} \]
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Rubi [A] time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1586, 2074, 634, 618, 204, 628} \[ -\frac {\log \left (4 x^2-6 x+9\right )}{17496}-\frac {\log \left (4 x^2+6 x+9\right )}{17496}-\frac {1}{2916 (2 x+3)}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (2 x+3)}{17496}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{8748 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{2916 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1586
Rule 2074
Rubi steps
\begin {align*} \int \frac {243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{(3+2 x)^2 \left (243-162 x+108 x^2-72 x^3+48 x^4-32 x^5\right )} \, dx\\ &=\int \left (-\frac {1}{8748 (-3+2 x)}+\frac {1}{1458 (3+2 x)^2}+\frac {5}{8748 (3+2 x)}+\frac {3-2 x}{4374 \left (9-6 x+4 x^2\right )}+\frac {3-2 x}{4374 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{2916 (3+2 x)}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (3+2 x)}{17496}+\frac {\int \frac {3-2 x}{9-6 x+4 x^2} \, dx}{4374}+\frac {\int \frac {3-2 x}{9+6 x+4 x^2} \, dx}{4374}\\ &=-\frac {1}{2916 (3+2 x)}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (3+2 x)}{17496}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{17496}-\frac {\int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{17496}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{2916}+\frac {1}{972} \int \frac {1}{9+6 x+4 x^2} \, dx\\ &=-\frac {1}{2916 (3+2 x)}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (3+2 x)}{17496}-\frac {\log \left (9-6 x+4 x^2\right )}{17496}-\frac {\log \left (9+6 x+4 x^2\right )}{17496}-\frac {\operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{1458}-\frac {1}{486} \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )\\ &=-\frac {1}{2916 (3+2 x)}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{8748 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{2916 \sqrt {3}}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (3+2 x)}{17496}-\frac {\log \left (9-6 x+4 x^2\right )}{17496}-\frac {\log \left (9+6 x+4 x^2\right )}{17496}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 100, normalized size = 0.91 \[ \frac {-3 \log \left (4 x^2-6 x+9\right )-3 \log \left (4 x^2+6 x+9\right )-\frac {18}{2 x+3}-3 \log (3-2 x)+15 \log (2 x+3)+2 \sqrt {3} \tan ^{-1}\left (\frac {4 x-3}{3 \sqrt {3}}\right )+6 \sqrt {3} \tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{52488} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 115, normalized size = 1.05 \[ \frac {6 \, \sqrt {3} {\left (2 \, x + 3\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 2 \, \sqrt {3} {\left (2 \, x + 3\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - 3 \, {\left (2 \, x + 3\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 3 \, {\left (2 \, x + 3\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 15 \, {\left (2 \, x + 3\right )} \log \left (2 \, x + 3\right ) - 3 \, {\left (2 \, x + 3\right )} \log \left (2 \, x - 3\right ) - 18}{52488 \, {\left (2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 86, normalized size = 0.78 \[ \frac {1}{8748} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{26244} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {1}{2916 \, {\left (2 \, x + 3\right )}} - \frac {1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {5}{17496} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{17496} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 85, normalized size = 0.77 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{26244}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{8748}-\frac {\ln \left (2 x -3\right )}{17496}+\frac {5 \ln \left (2 x +3\right )}{17496}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{17496}-\frac {\ln \left (4 x^{2}+6 x +9\right )}{17496}-\frac {1}{2916 \left (2 x +3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 84, normalized size = 0.76 \[ \frac {1}{8748} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{26244} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {1}{2916 \, {\left (2 \, x + 3\right )}} - \frac {1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {5}{17496} \, \log \left (2 \, x + 3\right ) - \frac {1}{17496} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.10, size = 100, normalized size = 0.91 \[ \frac {5\,\ln \left (x+\frac {3}{2}\right )}{17496}-\frac {\ln \left (x-\frac {3}{2}\right )}{17496}-\frac {1}{5832\,\left (x+\frac {3}{2}\right )}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{17496}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{17496}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{52488}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{52488}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 105, normalized size = 0.95 \[ - \frac {\log {\left (x - \frac {3}{2} \right )}}{17496} + \frac {5 \log {\left (x + \frac {3}{2} \right )}}{17496} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{17496} - \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{17496} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{26244} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{8748} - \frac {1}{5832 x + 8748} \]
Verification of antiderivative is not currently implemented for this CAS.
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