Optimal. Leaf size=60 \[ -\frac{1}{324} \log \left (4 x^2+6 x+9\right )-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (2 x+3)+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]
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Rubi [A] time = 0.0512375, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1586, 2058, 634, 618, 204, 628} \[ -\frac{1}{324} \log \left (4 x^2+6 x+9\right )-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (2 x+3)+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2058
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{9-6 x+4 x^2}{729-64 x^6} \, dx &=\int \frac{1}{81+54 x-24 x^3-16 x^4} \, dx\\ &=\int \left (-\frac{1}{162 (-3+2 x)}+\frac{1}{54 (3+2 x)}+\frac{3-2 x}{81 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (3+2 x)+\frac{1}{81} \int \frac{3-2 x}{9+6 x+4 x^2} \, dx\\ &=-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (3+2 x)-\frac{1}{324} \int \frac{6+8 x}{9+6 x+4 x^2} \, dx+\frac{1}{18} \int \frac{1}{9+6 x+4 x^2} \, dx\\ &=-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (3+2 x)-\frac{1}{324} \log \left (9+6 x+4 x^2\right )-\frac{1}{9} \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 )}{54 \sqrt{3}}-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (3+2 x)-\frac{1}{324} \log \left (9+6 x+4 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0126314, size = 56, normalized size = 0.93 \[ \frac{1}{324} \left (-\log \left (4 x^2+6 x+9\right )-\log (3-2 x)+3 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 47, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{108}}-{\frac{\ln \left ( -3+2\,x \right ) }{324}}-{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{324}}+{\frac{\sqrt{3}}{162}\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 = 1.37365, size = 62, normalized size = 1.03 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - \frac{1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{108} \, \log \left (2 \, x + 3\right ) - \frac{1}{324} \, \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 = 1.43479, size = 157, normalized size = 2.62 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - \frac{1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{108} \, \log \left (2 \, x + 3\right ) - \frac{1}{324} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.181276, size = 56, normalized size = 0.93 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{324} + \frac{\log{\left (x + \frac{3}{2} \right )}}{108} - \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{324} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{162} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04567, size = 65, normalized size = 1.08 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - \frac{1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{108} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{324} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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