Optimal. Leaf size=50 \[ -\frac {1}{972} \log \left (4 x^2-6 x+9\right )+\frac {1}{486} \log (2 x+3)+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{162 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1586, 2058, 628, 618, 204} \[ -\frac {1}{972} \log \left (4 x^2-6 x+9\right )+\frac {1}{486} \log (2 x+3)+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{162 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 1586
Rule 2058
Rubi steps
\begin {align*} \int \frac {3-2 x}{729-64 x^6} \, dx &=\int \frac {1}{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5} \, dx\\ &=\int \left (\frac {1}{243 (3+2 x)}+\frac {3-4 x}{486 \left (9-6 x+4 x^2\right )}+\frac {1}{54 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{486} \log (3+2 x)+\frac {1}{486} \int \frac {3-4 x}{9-6 x+4 x^2} \, dx+\frac {1}{54} \int \frac {1}{9+6 x+4 x^2} \, dx\\ &=\frac {1}{486} \log (3+2 x)-\frac {1}{972} \log \left (9-6 x+4 x^2\right )-\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{162 \sqrt {3}}+\frac {1}{486} \log (3+2 x)-\frac {1}{972} \log \left (9-6 x+4 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.00 \[ -\frac {1}{972} \log \left (4 x^2-6 x+9\right )+\frac {1}{486} \log (2 x+3)+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{162 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 0.76 \[ \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{972} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{486} \, \log \left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 39, normalized size = 0.78 \[ \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{972} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{486} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.78 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{486}+\frac {\ln \left (2 x +3\right )}{486}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{972} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.89, size = 38, normalized size = 0.76 \[ \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{972} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{486} \, \log \left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 49, normalized size = 0.98 \[ \frac {\ln \left (x+\frac {3}{2}\right )}{486}-\frac {\ln \left (x^2-\frac {3\,x}{2}+\frac {9}{4}\right )}{972}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{1327104\,\left (\frac {x}{884736}+\frac {1}{884736}\right )}-\frac {\sqrt {3}\,x}{7962624\,\left (\frac {x}{884736}+\frac {1}{884736}\right )}\right )}{486} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 46, normalized size = 0.92 \[ \frac {\log {\left (x + \frac {3}{2} \right )}}{486} - \frac {\log {\left (4 x^{2} - 6 x + 9 \right )}}{972} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{486} \]
Verification of antiderivative is not currently implemented for this CAS.
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